Research ArticleA Generalization Belief Propagation Decoding Algorithm forPolar Codes Based on Particle Swarm Optimization

Yingxian Zhang1 Aijun Liu1 Xiaofei Pan1 Shi He2 and Chao Gong1

1 Key Laboratory of Military Satellite Communications College of Communications EngineeringPLA University of Science and Technology Nanjing 210007 China

2Unit 75706 PLA Guangzhou 510000 China

Correspondence should be addressed to Yingxian Zhang jenusxyzgmailcom

Received 31 December 2013 Accepted 21 April 2014 Published 13 May 2014

Academic Editor Balaji Raghavan

Copyright copy 2014 Yingxian Zhang et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a generalization belief propagation (BP) decoding algorithm based on particle swarm optimization (PSO) to improvethe performance of the polar codes Through the analysis of the existing BP decoding algorithm we first introduce a probabilitymodifying factor to each node of the BP decoder so as to enhance the error correcting capacity of the decoding Then wegeneralize the BP decoding algorithm based on thesemodifying factors and drive the probability update equations for the proposeddecoding Based on the new probability update equations we show the intrinsic relationship of the existing decoding algorithmsFinally in order to achieve the best performance we formulate an optimization problem to find the optimal probability modifyingfactors for the proposed decoding algorithm Furthermore a method based on the modified PSO algorithm is also introduced tosolve that optimization problem Numerical results show that the proposed generalization BP decoding algorithm achieves betterperformance than that of the existing BP decoding which suggests the effectiveness of the proposed decoding algorithm

1 Introduction

Since the ability of achieving Shannon capacity and theirlow encoding and decoding complexity the polar codeshave received much attention in the research field of error-correcting coding recently [1ndash17] While compared to someexisting coding schemes such as low-density parity-check(LDPC) and turbo codes the performance of the polar codesin the finite length regime is disappointing [2 3] Hencemotivated by this observation researchers have proposedmany decoding algorithms to improve the performance of thepolar codes [4ndash17]

Based on the successive-cancelation (SC) decoding algo-rithm proposed by Arıkan [1] authors of [4 5] had intro-duced a list successive-cancelation (SCL) decoding algorithmwith consideration of 119871 SC decoding paths where the resultsshowed that performance of SCL was very close to that ofmaximum-likelihood (ML) decoding Then to decrease thetime complexity of the SCL another decoding algorithm

derived from SC called stack successive-cancelation (SCS)was proposed in [6] Furthermore it was proven in [7]that with cyclic redundancy check (CRC) aided SCL evenoutperformed more than some turbo codes However due tothe serial processing nature of the SC all the algorithms in [4ndash7] would suffer a low decoding throughput and high latencyTherefore some improved versions of SC were further pro-posedwith the explicit aim to increase throughput and reducethe latency without sacrificing error-rate performance suchas simplified successive-cancellation (SSC) [8] maximum-likelihood SSC (ML-SSC) [9] and repetition single paritycheck ML-SSC (RSM-SSC) [10 11] Besides from those SCbased algorithms researchers had also investigated someother algorithms more in parallel one typical representationof which was the belief propagation (BP) [12] decodingalgorithm With the factor graph representation of polarcodes [13] authors in [14 15] showed that BP polar decodinghas particular advantages with respect to the decodingthroughput while the performancewas better than that of the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 606913 10 pageshttpdxdoiorg1011552014606913

2 Mathematical Problems in Engineering

SC and some improved SC decoding What is more with theminimal stopping set optimized results of [16 17] had shownthat the error floor performance of polar codes was superiorto that of LDPC codes

Indeed all the decoding algorithms in [4ndash17] can improvethe performance of polar codes to a certain degree Howeveras the capacity achieving coding scheme the results of thosealgorithms is still limited Hence in this paper we pro-pose a generalization belief propagation (BP) decoding algo-rithm based on particle swarm optimization (PSO) toimprove the performance of the polar codes with the finitelength Based on the analysis results of the existing BP decod-ing algorithmwe first show that the error-correcting capacityof the BP decoding algorithm is important for perform-ance of the decoding Motivated by that observation weintroduce a probability modifying factor to each node ofthe BP decoder to enhance the reliability of the probabilityupdated Then we generalize the BP decoding algorithm onthe basis of thesemodifying factors and further drive the pro-bability update equations for the proposed generalization BPdecoding Finally in order to achieve the best performancewe formulate an optimization problem to find the optimalprobabilitymodifying factors for the proposed generalizationBP decoding Furthermore a method based on the modifiedPSO algorithm is also introduced to solve the optimizationproblem The main contributions of this paper can be sum-marized as follows

(i) A generalization belief propagation (BP) decodingalgorithm based on probability modifying factor isintroduced Furthermore as to improve the perfor-mance of the proposed decoding a BP decodingalgorithm optimization problem is formulated

(ii) A method based on the modified PSO algorithm isintroduced to solve the optimization problem

Thefinding of this paper suggests thatwith the probabilitymodifying factor the error-correcting capacity of the BPdecoding can be enhanced and the performance of the polarcodes can be improved correspondingly which is finallyproven by the simulation results

The remainder of this paper is organized as follows InSection 2 we explain some notations and introduce somebackground on the polar codes And in Section 3 the gener-alization belief propagation (BP) decoding algorithm basedon the probability modifying factors is described in detail InSection 4 we formulate and solve the optimization problemof finding the probability modifying factors And Section 5provides the simulation results for the complexity and biterror performance of the proposed decoding Finally wemake some conclusions in Section 6

2 Preliminary

21 Notations In this paper the blackboard bold letterssuch as X denote the sets and |X| denotes the num-ber of elements in X The notation 119906

119873minus1

0denotes an 119873-

dimensional vector (1199060 1199061 119906

119873minus1) and 119906

119895

119894indicates a

subvector (119906119894 119906119894+1

119906119895minus1

119906119895) of 119906119873minus1

0 0 le 119894 119895 le 119873 minus 1

When 119894 gt 119895 119906119895119894is an empty vector Further given a vector set

U vector 119894is the 119894th element of U

The matrices in this paper are denoted by bold lettersThe subscript of a matrix indicates its size for exampleA119873times119872

represents an119873times119872matrixA Specifically the squarematrices are written as A

119873 size of which is 119873 times 119873 and Aminus1

119873

is the inverse of A119873 Furthermore the Kronecker product

of two matrices A and B is written as A otimes B and the 119899thKronecker power of A is Aotimes119899

During the procedure of the encoding and decoding wedenote the intermediate node by V

119868(119894 119895) or V

119874(119894 119895) 0 le 119894 le

119899 0 le 119895 le 119873 minus 1 where119873 = 2119899 is the code length Besides

we also indicate the probability messages of the intermediatenode V

119868(119894 119895) as 119901

119868(119894119895) where the probability of V

119868(119894 119895) being

equal to 0 or 1 is119901119868(119894119895)

(0) or119901119868(119894119895)

(1) Similarly the probabilitymessages of V

119874(119894 119895) are 119901

119874(119894119895)(0) and 119901

119874(119894119895)(1)

Throughout this paper ldquooplusrdquo denotes the modulo-twosum and ldquosum119872

119894=0oplus119909119894rdquo means ldquo119909

0oplus 1199091oplus oplus119909

119872rdquo While the

operators ldquootimesrdquo and ldquo⊙rdquo are defined as 119901119909otimes 119901119910= 119901(119909oplus119910)

and119901119909⊙ 119901119910= 119901119909lowast 119901119910 respectively 119909 119910 isin 0 1

22 Background of Polar Codes A polar coding scheme canbe uniquely defined by three parameters block-length 119873 =

2119899 code rate 119877 = 119870119873 and an information set I sub N =

0 1 119873 minus 1 where119870 = |I| With these three parametersa source binary vector 119906119873minus1

0consisting of 119870 information bits

and 119873 minus 119870 frozen bits can be mapped a codeword 119909119873minus1

0by

a linear matric G119873

= B119873Fotimes1198992 where F

2= [1 0

1 1] B119873

is abit-reversal permutation matrix defined in [1] and 119909

119873minus1

0=

119906119873minus1

0G119873

In practice the construction procedure of a polar codecan be divided into 119899 = log

2119873 stages as shown in Figure

1(a) where the circle nodes in the leftmost column are theinput nodes of encoder values of which are equal to binarysource vectorthat is V

119868(0 119894) = 119906

119894 and the circle nodes in

the rightmost column are the output nodes of encoderV119874(119899 minus 1 119894) = 119909

119894 It is also noticed that each encoding stage

has 1198732 so-called processing units (PU) with a generationmatrix of F

2 and each PU has two input and two output

variable nodes as shown in Figure 1(b) During the processof polar encoding based on F

2 we have

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1)

(1)

where V119868(119894 2119895) and V

119868(119894 2119895 + 1) are the input nodes of the 119895th

PU in the 119894th stage 0 le 119894 le 119899 minus 1 0 le 119895 le 1198732 minus 1 andsimilarly V

119874(119894 2119895) and V

119874(119894 2119895+1) are the output nodes After

the procedure of the polar encoding all the bits in the code-word 119909

119873minus1

0are passed to the119873-channels which consisted of

119873 independent channels of119882 with a transition probability of119882(119910119894| 119909119894) where119910

119894is 119894th element of the received vector119910119873minus1

0

Then the decoder of the receiver will output the estimatedcodeword 119909

119873minus1

0and the estimated source binary vector 119873minus1

0

with different decoding algorithms [4ndash17]

Mathematical Problems in Engineering 3

Stage 0u0

I(0 0)

u1

I(0 1)

u2

I(0 2)

u3

I(0 3)

u4

I(0 4)

u5

I(0 5)

u6

I(0 6)

u7

I(0 7)

O(0 0)

O(0 1)

O(0 2)

O(0 3)

O(0 4)

O(0 5)

O(0 6)

O(0 7)

Stage 1

I(1 0)

I(1 1)

I(1 2)

I(1 3)

I(1 4)

I(1 5)

I(1 6)

I(1 7)

O(1 0)

O(1 1)

O(1 2)

O(1 3)

O(1 4)

O(1 5)

O(1 6)

O(1 7)

Stage 2

I(2 0)

I(2 1)

I(2 2)

I(2 3)

I(2 4)

I(2 5)

I(2 6)

I(2 7)

O(2 0)

O(2 1)

O(2 2)

O(2 3)

O(2 4)

O(2 5)

O(2 6)

O(2 7)

x0

x1

x2

x3

x4

x5

x6

x7

W

W

W

W

W

W

W

W

y0

y1

y2

y3

y4

y5

y6

y7

= = =

= = =

+ + +

+ + +

= = =

+ + +

= = =

+ + +

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)=

+

(b)

Figure 1 (a) Construction procedure of polar codes with119873 = 8 (b) Diagram of PU in code construction

3 BP Decoding for Polar Codes

In this section we will generalize the BP decoding algorithmfor polar codes with the analysis of the existing polar BPdecoding

31 Existing BP Decoding The basic BP process unit (BP-PU) of polar codes is shown in Figure 2 where 119901119905

119868119871(1198942119895)and

119901119905

119868119871(1198942119895+1)are the right-to-left probability messages passed to

V119868(119894 2119895) and V

119868(119894 2119895 + 1) 119905 is the iteration number 119901119905minus1

119868119877(1198942119895)

and119901119905minus1119868119877(1198942119895+1)

are the left-to-right probabilitymessages passedfrom the nodes 119901119905minus1

119874119871(1198942119895)and 119901

119905minus1

119874119871(1198942119895+1)are the right-to-left

probability messages passed from V119874(119894 2119895) and V

119874(119894 2119895 + 1)

and 119901119905

119874119877(1198942119895)and 119901

119905

119874119877(1198942119895+1)are the left-to-right probability

messages passed to the nodes According to the formula oftotal probability there has

119901119905

119868119871(119894119895)(0) + 119901

119905

119868119871(119894119895)(1) = 1

119901119905

119868119877(119894119895)(0) + 119901

119905

119868119877(119894119895)(1) = 1

119901119905

119874119871(119894119895)(0) + 119901

119905

119874119871(119894119895)(1) = 1

119901119905

119874119877(119894119895)(0) + 119901

119905

119874119877(119894119895)(1) = 1

(2)

Furthermore based on the transition probability of thechannel119882 that is119882(119910

119894| 119909119894) and the received vector 119910119873minus1

0

we have

1199010

119874119871(119899minus1119894)(0) = 119882(119910

119894| 119909119894= 0)

119875 (119909119894= 0)

119875 (119910119894)

1199010

119874119871(119899minus1119894)(1) = 119882(119910

119894| 119909119894= 1)

119875 (119909119894= 1)

119875 (119910119894)

(3)

where 119875(119909119894) and 119875(119910

119894) are the probability distribution func-

tions of coded bit 119909119894and received signal sample 119910

119894 In the BP

4 Mathematical Problems in Engineering

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i2j)

ptIL(i2j+1)

ptOR(i2j)

ptOR(i2j+1)

=

+

ptminus1OL(i2j)

ptminus1OL(i2j+1)

ptminus1IR(i2j)

ptminus1IR(i2j+1)

Figure 2 The BP decoding algorithm for polar codes is basedon the factor graph representation of the codes [13] In the algo-rithm each node V

119868(119894 119895) or V

119874(119894 119895) is associated with two types of

probability messages left-to-right probability messages and right-to-left probability messages Both of messages are propagated andupdated iteratively between adjacent nodes in the procedure of BPAnd the messages updated schedule is the same as that of [14]that is messages will firstly propagate from the rightmost nodes tothe leftmost nodes After arriving at the leftmost nodes the coursewill be reversed and messages propagation will move toward therightmost nodes This procedure makes one round iteration of BPDiagram of BP-PU

decoding algorithm [13ndash17] the messages updated equationsare given by

119901119905

119868119871(1198942119895)= 119901119905minus1

119874119871(1198942119895)otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 119901119905minus1

119868119877(1198942119895+1))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119874119871(1198942119895+1)

119901119905

119874119877(1198942119895)= 119901119905minus1

119868119877(1198942119895)otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 119901119905minus1

119874119871(1198942119895+1))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119868119877(1198942119895+1)

(4)

Based on (4) when the decoding reaches maximumiteration number (mIter) BP decoder output the decodedvector u = (

0

119873minus1) each elements of which depended

on

119894=

0 119901mIter119868119871(0119894)

(0) gt 119901mIter119868119871(0119894)

(1)

1 otherwise(5)

32 Analysis of BP Decoder It is noticed from [13ndash17] basedon the BP decoding the performance of polar codes canbe improved to a certain degree However as the capacityachieving coding scheme the results of those algorithms isstill disappointingHence we cannot helpwonderingwhy theperformance of the polar codes with finite length is inferiorto that of LDPC codes [16 17] and how we can improve it Toanswer the questions we need to make a further analysis ofthe existing BP decoder

In fact it is noticed from the factor graph of polar codesin [13] that the degree of the check or variable nodes in thepolar BP decoder is 2 or 3 While the average degree of theLDPC codes is usually greater than 3 [12 18] which meansthat the error-correcting capacity of the polar BP decodingwill be weaken Hence the performance of the polar codesis inferior to that of LDPC codes with the same length [18]As shown in Figure 3 two Tanner graphes with different

degree distributions are given We assume that there existsan error variable node V

1in a certain iteration Compared

to the check nodes 1198880and 1198881in Figure 3(a) the influence

of error probability messages from V1to 1198880and 1198881in Figure

3(a) is weak because with a greater degree number the pro-portion of the error probability messages in the input prob-ability messages of 119888

0and 1198881will be reduced What is more

the convergence of BP decoding with a greater average degreewill be quicker which results in less computational iterationnumber as shown in [12ndash18] Therefore to improve theperformance of polar BP decoding it is important to enhancethe error-correcting capacity of the algorithm

33 Generalization of BP Decoding Motivated by aforemen-tioned observation in this subsection we will introduce ageneralization framework of BP decoder for polar codes Inorder to improve the error-correcting capacity of polar BPdecoding twelve probability messages modifying factors areintroduced to the nodes of the BP-PU shown in Figure 2When the probability messages of a node of the BP-PU isupdated three probability messages modifying factors areadded to other three nodes respectively

As illustrated in Figure 4 1205741199051198710(119894119895)

120574119905

1198715(119894119895)and 120574

119905

1198770(119894119895)

120574119905

1198775(119894119895)are the probability messages modifying factors of

the 119895th BP-PU in 119894th stage In this case each equation in(4) will be correspondingly added with three variables forexample the first equation of (4) is added with 120574119905

1198710(119894119895) 1205741199051198711(119894119895)

and 120574

119905

1198712(119894119895) Besides based on the new construction of BP-

PU we further define the result of each nodesrsquo probabilitymessages calculation as a function with respect to these threevariables and the probability messages of other three nodeshence we can get the new messages update equations as (6)Consider

119901119905

119868119871(1198942119895)= 1198910

(119894119895)(119901119905minus1

119874119871(1198942119895) 119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119868119877(1198942119895+1)

120574119905

1198710(119894119895) 120574119905

1198711(119894119895) 120574119905

1198712(119894119895))

119901119905

119868119871(1198942119895+1)= 1198911

(119894119895)(119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119874119871(1198942119895) 119901119905minus1

119868119877(1198942119895)

120574119905

1198713(119894119895) 120574119905

1198714(119894119895) 120574119905

1198715(119894119895))

119901119905

119874119877(1198942119895)= 1198912

(119894119895)(119901119905minus1

119868119877(1198942119895) 119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119874119871(1198942119895+1)

120574119905

1198770(119894119895) 120574119905

1198771(119894119895) 120574119905

1198772(119894119895))

119901119905

119874119877(1198942119895+1)= 1198913

(119894119895)(119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119868119877(1198942119895) 119901119905minus1

119874119871(1198942119895)

ℎ 120574119905

1198773(119894119895) 120574119905

1198774(119894119895) 120574119905

1198775(119894119895))

(6)

In (6) 1198910(119894119895)

(1205935

0) 1198911(119894119895)

(1205935

0) 1198912(119894119895)

(1205935

0) and 119891

3

(119894119895)(1205935

0) are the

four probability messages update functions of the 119895th BP-PU in 119894th stage and 120593

0 120593

5are six probability messages

variables Based on (6) we can easily get a new expression of

Mathematical Problems in Engineering 5

0 1 2 3

c0 c1 c2

(a)

0 1 2 3

c0 c1 c2

(b)

Figure 3 (a) Tanner graph with average degree less than 3 (b) Tanner graph with average degree more than 3

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL0(ij)

120574tL1(ij)120574tL2(ij)

=

+

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOR(i 2j)

ptOL(i 2j + 1)

120574tR0(ij)

120574tR1(ij) 120574tR2(ij)

=

+

(b)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIL (i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL3(ij)

120574tL4(ij)120574tL5(ij)

=

+

(c)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOR(i 2j + 1)

120574tR3(ij)

120574tR4(ij) 120574tR5(ij)

=

+

(d)

Figure 4 Diagrams of the new BP-PU

(4) defined as the functions of 1198910(119894119895)

(1205935

0) 1198911

(119894119895)(1205935

0) 1198912

(119894119895)(1205935

0)

and 1198913

(119894119895)(1205935

0) that is

1198910

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198913

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

(7)

That is to say the existing BP decoding algorithm is aspecial case of the generalization BP decoding proposed inthis work Furthermore with the four probability messagesupdate functions of

1198910

(119894119895)(1205935

0) = 1205930otimes 1205931

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes 1205931

1198913

(119894119895)(1205935

0) = 1205930

(8)

we can also get the probability messages update equationsof the SC based decoding algorithms as shown in [1 4ndash11] Similarly it is concluded that the SC based decodingalgorithms are also the special cases of our generalization BPdecoding which has shown the intrinsic relationship of theSC based decoding and BP decoding algorithms

Therefore we can conclude from (6)ndash(8) that in order toimprove the performance of the polar codes it is needed tofind the optimal functions of1198910

(119894119895)(1205935

0)1198911(119894119895)

(1205935

0)1198912(119894119895)

(1205935

0) and

1198913

(119894119895)(1205935

0) and the optimal parameters of 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) which will be discussed in the next section

6 Mathematical Problems in Engineering

4 BP Decoding Algorithm Optimization

Motivated by the aforementioned conclusion we now con-sider how to optimize the generalization BP decoding algo-rithm so as to improve the performance of the polar codes

41 Messages Update Functions It is noticed from (4) and(6)ndash(8) that in order to enhance the reliability of messagespropagation it is important to determine the appropriatemessages update functions for the generalization BP decod-ing algorithm However in general it is difficult to find theoptimal messages update functions therefore in this workthe messages update functions are introduced by the analysisof the construction of polar encoding And based on (1) wehave

V119868(119894 2119895) = V

119874(119894 2119895) oplus V

119874(119894 2119895 + 1)

= V119874(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119868(119894 2119895 + 1) = V

119874(119894 2119895 + 1) = V

119874(119894 2119895) oplus V

119868(119894 2119895)

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

= V119868(119894 2119895) oplus V

119874(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1) = V

119868(119894 2119895) oplus V

119874(119894 2119895)

(9)

Therefore based on the error coding theory and (9) wecan get the messages update functions as

1198910

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198911

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

1198912

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198913

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

(10)

That is to say with the probability messages modifyingfactors the new probability messages update equations willbe written as

119901119905

119868119871(1198942119895)= ((119901

119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895)) otimes (119901

119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198711(119894119895)))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895))

otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198712(119894119895)))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198713(119894119895))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198714(119894119895))

otimes (119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198715(119894119895)))

119901119905

119874119877(1198942119895)= ((119901

119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895)) otimes (119901

119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198771(119894119895)))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895))

otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198772(119894119895)))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198773(119894119895))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198774(119894119895))

otimes (119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198775(119894119895)))

(11)

Next our task is to determine the optimal probabilitymessages modifying factors that is 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) so as to achieve the best performance

42 Probability Modifying Factors Optimization In practicedue to the input of frozen bits [1] there exists some so-called frozen nodes in the decoder and values of these nodesare determined and independent of the decoding algorithmHence if the decoding is correct the probabilitymessages of afrozen node V

119868(119894 119895) or V

119874(119894 119895) (the determined value of frozen

nodes is 0) must satisfy the condition of

119901119905

119868119871(119894119895)(0) gt 119901

119905

119868119871(119894119895)(1)

119901119905

119868119877(119894119895)(0) gt 119901

119905

119868119877(119894119895)(1)

or

119901119905

119874119871(119894119895)(0) gt 119901

119905

119874119871(119894119895)(1)

119901119905

119874119877(119894119895)(0) gt 119901

119905

119874119877(119894119895)(1)

(12)

While in practice due to the noise of received signalthere may exist some frozen nodes unsatisfied the reliabilitycondition Therefore in order to improve the accuracy ofdecoding it is needed to find the optimal probability mes-sages modifying factors such that all the frozen nodes couldalways satisfy the conditions of (12) during the procedure ofBP decoding

To achieve above goal we introduce a pair parametersto each node in the decoder which are called left and rightreliability degrees And for a node V

119868(119894 119895) (similar to V

119874(119894 119895))

the left and right reliability degrees are denoted as 120577119905119868119871(119894119895)

and120577119905

119868119877(119894119895) respectively Where if V

119868(119894 119895) is a frozen node we can

get 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

(13)

Mathematical Problems in Engineering 7

otherwise the values of 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)will be given by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

119901119868119871(119894119895) (0)

119901119868119871(119894119895) (1)

gt 1

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(1)

119901119905

119868119871(119894119895)(0)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(1)

119901119905

119868119877(119894119895)(0)

otherwise

(14)

where 120577119905119868119871(119894119895)

gt 1 and 120577119905

119868119877(119894119895)gt 1

The reliability degrees indicate the reliability of the nodersquosdecision value and the larger reliability degree the higherreliability of that value Specially when there is no noise inthe channel the reliability degrees of a node will approachto infinity Based on previous observations the problem ofgetting the optimal probability messages modifying factorscan be formulated as an optimization problem to maximizethe reliability degrees of all the node in decoder while thetarget function is written as

arg max0le120574119905

1198710(119894119896)120574119905

1198715(119894119896)le1

0le120574119905

1198770(119894119896)120574119905

1198775(119894119896)le1

log2119873

sum

119894=0

119873

sum

119895=0

(120577119905

119868119871(119894119895)+ 120577119905

119868119877(119894119895))

st 120574119905

1198710(119894119896)(0) + 120574

119905

1198710(119894119896)(1) = 1

120574119905

1198715(119894119896)(0) + 120574

119905

1198715(119894119896)(1) = 1

120574119905

1198770(119894119896)(0) + 120574

119905

1198770(119894119896)(1) = 1

120574119905

1198775(119894119896)(0) + 120574

119905

1198775(119894119896)(1) = 1

(15)

where 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)are decided by (11) and (13)-(14) and

119896 is the integral value of 1198952 that is 119896 = lfloor1198952rfloor

43 Problem Solving Based on Modified PSO AlgorithmBased on (15) we have formulated the problem to find theoptimal probabilitymodifying factors and in this subsectionwe will consider the solution of the problem so as to get theoptimized BP decoding algorithm

Among the different global optimizers particle swarmoptimization (PSO) has demonstrated its usefulness as anoptimizer capable of finding global optimum The idea ofPSO is based on simulation of simplified social models suchas birds flocking and fish schooling Like other stochasticevolutionary algorithms PSO is independent of mathemat-ical characteristics of objective problems However unlikethese algorithms with PSO each particle has its own unique

memory to ldquorememberrdquo its own best solution Thus eachparticle has its own ldquoideardquo of whether it has found theoptimum However in other algorithms previous knowledgeof the problemwill be destroyed once the population changesPSO has attracted increasing attention because of its simpleconcept easy implementation and quick These advantagesallow it to be applied successfully in a variety of fields mainlyfor unconstrained continuous optimization problems [19ndash26] Therefore in this work we explore the PSO algorithmwith certain modification based on particle fitness evaluationto solve the problem (15)

To present our modified PSO algorithm we first intro-duce some preliminary definitions and concepts used in theoperation of the PSOalgorithmwhich is described as follows

(1) ParametersDefinitionsWe consider a swarmof119871particlesz0 z

119871minus1 where the position of z

119894= (1199111198940 1199111198941 119911

119894(119879minus1))

represents a possible solution point of those probabilitymodifying factors in the problem (15) 0 le 119894 le 119871 minus 1 and119879 is the number of the probability modifying factors Eachelement of z

119894is the value of a probability modifying factor

that is 0 le 119911119894119895le 1 0 le 119895 le 119879 minus 1 Based on the work of [19]

the position of particle z119894can be updated by

z119905+1119894

= z119905119894+ w119905+1119894

(16)

In (16) the pseudovelocity w119905+1119894

is calculated as follows

w119905+1119894

= 120596119905w119905119894+ 11988811199031(p119894minus z119905119894) + 11988821199032(p119892minus z119905119894) (17)

where120596119905 is particle inertia 119905 indicates the iteration number p119894

represents the best ever position of particle 119894 at iteration 119905 andp119892represents the global best position in the swarmat iteration

119905 1199031and 1199032represent uniform random numbers between 0

and 1 It is proven in [19] that when 1198881= 1198882= 2 the products

11988811199031or 11988821199032 have a mean of 1

(2) Fitness Evaluation In general with the parameter defini-tions aforementioned we can get the global optimal solutionof a optimization problem However in problem (15) thesituation will be different because during the procedureof the BP decoding for polar codes a nodersquos probabilitymessages calculated on the basis of the probability modifyingfactors must satisfy the conditions of (12) or (14) In orderto check these conditions during the decoding we furtherdefine two functions as

1198921(z119905119894) =

1003816100381610038161003816V119890

119865

10038161003816100381610038161003816100381610038161003816V119865

1003816100381610038161003816

1198922(z119905119894) =

1003816100381610038161003816V119890

119868

10038161003816100381610038161003816100381610038161003816V119868

1003816100381610038161003816

(18)

where V119865is the frozen nodes set of the decoder and V119890

119865is the

set of frozen nodeswhose probabilitymessages based on z119905119894do

not satisfy the condition of (12) Similarly V119868is the nonfrozen

nodes set of the decoder and V119890119868is the set of nonfrozen nodes

whose left and right reliability degrees based on z119905119894do not

satisfy (14) Furthermore we have 1198921(z119905119894) 1198922(z119905119894) isin [0 1]

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

SC and some improved SC decoding What is more with theminimal stopping set optimized results of [16 17] had shownthat the error floor performance of polar codes was superiorto that of LDPC codes

Indeed all the decoding algorithms in [4ndash17] can improvethe performance of polar codes to a certain degree Howeveras the capacity achieving coding scheme the results of thosealgorithms is still limited Hence in this paper we pro-pose a generalization belief propagation (BP) decoding algo-rithm based on particle swarm optimization (PSO) toimprove the performance of the polar codes with the finitelength Based on the analysis results of the existing BP decod-ing algorithmwe first show that the error-correcting capacityof the BP decoding algorithm is important for perform-ance of the decoding Motivated by that observation weintroduce a probability modifying factor to each node ofthe BP decoder to enhance the reliability of the probabilityupdated Then we generalize the BP decoding algorithm onthe basis of thesemodifying factors and further drive the pro-bability update equations for the proposed generalization BPdecoding Finally in order to achieve the best performancewe formulate an optimization problem to find the optimalprobabilitymodifying factors for the proposed generalizationBP decoding Furthermore a method based on the modifiedPSO algorithm is also introduced to solve the optimizationproblem The main contributions of this paper can be sum-marized as follows

(i) A generalization belief propagation (BP) decodingalgorithm based on probability modifying factor isintroduced Furthermore as to improve the perfor-mance of the proposed decoding a BP decodingalgorithm optimization problem is formulated

(ii) A method based on the modified PSO algorithm isintroduced to solve the optimization problem

Thefinding of this paper suggests thatwith the probabilitymodifying factor the error-correcting capacity of the BPdecoding can be enhanced and the performance of the polarcodes can be improved correspondingly which is finallyproven by the simulation results

The remainder of this paper is organized as follows InSection 2 we explain some notations and introduce somebackground on the polar codes And in Section 3 the gener-alization belief propagation (BP) decoding algorithm basedon the probability modifying factors is described in detail InSection 4 we formulate and solve the optimization problemof finding the probability modifying factors And Section 5provides the simulation results for the complexity and biterror performance of the proposed decoding Finally wemake some conclusions in Section 6

2 Preliminary

21 Notations In this paper the blackboard bold letterssuch as X denote the sets and |X| denotes the num-ber of elements in X The notation 119906

119873minus1

0denotes an 119873-

dimensional vector (1199060 1199061 119906

119873minus1) and 119906

119895

119894indicates a

subvector (119906119894 119906119894+1

119906119895minus1

119906119895) of 119906119873minus1

0 0 le 119894 119895 le 119873 minus 1

When 119894 gt 119895 119906119895119894is an empty vector Further given a vector set

U vector 119894is the 119894th element of U

The matrices in this paper are denoted by bold lettersThe subscript of a matrix indicates its size for exampleA119873times119872

represents an119873times119872matrixA Specifically the squarematrices are written as A

119873 size of which is 119873 times 119873 and Aminus1

119873

is the inverse of A119873 Furthermore the Kronecker product

of two matrices A and B is written as A otimes B and the 119899thKronecker power of A is Aotimes119899

During the procedure of the encoding and decoding wedenote the intermediate node by V

119868(119894 119895) or V

119874(119894 119895) 0 le 119894 le

119899 0 le 119895 le 119873 minus 1 where119873 = 2119899 is the code length Besides

we also indicate the probability messages of the intermediatenode V

119868(119894 119895) as 119901

119868(119894119895) where the probability of V

119868(119894 119895) being

equal to 0 or 1 is119901119868(119894119895)

(0) or119901119868(119894119895)

(1) Similarly the probabilitymessages of V

119874(119894 119895) are 119901

119874(119894119895)(0) and 119901

119874(119894119895)(1)

Throughout this paper ldquooplusrdquo denotes the modulo-twosum and ldquosum119872

119894=0oplus119909119894rdquo means ldquo119909

0oplus 1199091oplus oplus119909

119872rdquo While the

operators ldquootimesrdquo and ldquo⊙rdquo are defined as 119901119909otimes 119901119910= 119901(119909oplus119910)

and119901119909⊙ 119901119910= 119901119909lowast 119901119910 respectively 119909 119910 isin 0 1

22 Background of Polar Codes A polar coding scheme canbe uniquely defined by three parameters block-length 119873 =

2119899 code rate 119877 = 119870119873 and an information set I sub N =

0 1 119873 minus 1 where119870 = |I| With these three parametersa source binary vector 119906119873minus1

0consisting of 119870 information bits

and 119873 minus 119870 frozen bits can be mapped a codeword 119909119873minus1

0by

a linear matric G119873

= B119873Fotimes1198992 where F

2= [1 0

1 1] B119873

is abit-reversal permutation matrix defined in [1] and 119909

119873minus1

0=

119906119873minus1

0G119873

In practice the construction procedure of a polar codecan be divided into 119899 = log

2119873 stages as shown in Figure

1(a) where the circle nodes in the leftmost column are theinput nodes of encoder values of which are equal to binarysource vectorthat is V

119868(0 119894) = 119906

119894 and the circle nodes in

the rightmost column are the output nodes of encoderV119874(119899 minus 1 119894) = 119909

119894 It is also noticed that each encoding stage

has 1198732 so-called processing units (PU) with a generationmatrix of F

2 and each PU has two input and two output

variable nodes as shown in Figure 1(b) During the processof polar encoding based on F

2 we have

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1)

(1)

where V119868(119894 2119895) and V

119868(119894 2119895 + 1) are the input nodes of the 119895th

PU in the 119894th stage 0 le 119894 le 119899 minus 1 0 le 119895 le 1198732 minus 1 andsimilarly V

119874(119894 2119895) and V

119874(119894 2119895+1) are the output nodes After

the procedure of the polar encoding all the bits in the code-word 119909

119873minus1

0are passed to the119873-channels which consisted of

119873 independent channels of119882 with a transition probability of119882(119910119894| 119909119894) where119910

119894is 119894th element of the received vector119910119873minus1

0

Then the decoder of the receiver will output the estimatedcodeword 119909

119873minus1

0and the estimated source binary vector 119873minus1

0

with different decoding algorithms [4ndash17]

Mathematical Problems in Engineering 3

Stage 0u0

I(0 0)

u1

I(0 1)

u2

I(0 2)

u3

I(0 3)

u4

I(0 4)

u5

I(0 5)

u6

I(0 6)

u7

I(0 7)

O(0 0)

O(0 1)

O(0 2)

O(0 3)

O(0 4)

O(0 5)

O(0 6)

O(0 7)

Stage 1

I(1 0)

I(1 1)

I(1 2)

I(1 3)

I(1 4)

I(1 5)

I(1 6)

I(1 7)

O(1 0)

O(1 1)

O(1 2)

O(1 3)

O(1 4)

O(1 5)

O(1 6)

O(1 7)

Stage 2

I(2 0)

I(2 1)

I(2 2)

I(2 3)

I(2 4)

I(2 5)

I(2 6)

I(2 7)

O(2 0)

O(2 1)

O(2 2)

O(2 3)

O(2 4)

O(2 5)

O(2 6)

O(2 7)

x0

x1

x2

x3

x4

x5

x6

x7

W

W

W

W

W

W

W

W

y0

y1

y2

y3

y4

y5

y6

y7

= = =

= = =

+ + +

+ + +

= = =

+ + +

= = =

+ + +

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)=

+

(b)

Figure 1 (a) Construction procedure of polar codes with119873 = 8 (b) Diagram of PU in code construction

3 BP Decoding for Polar Codes

In this section we will generalize the BP decoding algorithmfor polar codes with the analysis of the existing polar BPdecoding

31 Existing BP Decoding The basic BP process unit (BP-PU) of polar codes is shown in Figure 2 where 119901119905

119868119871(1198942119895)and

119901119905

119868119871(1198942119895+1)are the right-to-left probability messages passed to

V119868(119894 2119895) and V

119868(119894 2119895 + 1) 119905 is the iteration number 119901119905minus1

119868119877(1198942119895)

and119901119905minus1119868119877(1198942119895+1)

are the left-to-right probabilitymessages passedfrom the nodes 119901119905minus1

119874119871(1198942119895)and 119901

119905minus1

119874119871(1198942119895+1)are the right-to-left

probability messages passed from V119874(119894 2119895) and V

119874(119894 2119895 + 1)

and 119901119905

119874119877(1198942119895)and 119901

119905

119874119877(1198942119895+1)are the left-to-right probability

messages passed to the nodes According to the formula oftotal probability there has

119901119905

119868119871(119894119895)(0) + 119901

119905

119868119871(119894119895)(1) = 1

119901119905

119868119877(119894119895)(0) + 119901

119905

119868119877(119894119895)(1) = 1

119901119905

119874119871(119894119895)(0) + 119901

119905

119874119871(119894119895)(1) = 1

119901119905

119874119877(119894119895)(0) + 119901

119905

119874119877(119894119895)(1) = 1

(2)

Furthermore based on the transition probability of thechannel119882 that is119882(119910

119894| 119909119894) and the received vector 119910119873minus1

0

we have

1199010

119874119871(119899minus1119894)(0) = 119882(119910

119894| 119909119894= 0)

119875 (119909119894= 0)

119875 (119910119894)

1199010

119874119871(119899minus1119894)(1) = 119882(119910

119894| 119909119894= 1)

119875 (119909119894= 1)

119875 (119910119894)

(3)

where 119875(119909119894) and 119875(119910

119894) are the probability distribution func-

tions of coded bit 119909119894and received signal sample 119910

119894 In the BP

4 Mathematical Problems in Engineering

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i2j)

ptIL(i2j+1)

ptOR(i2j)

ptOR(i2j+1)

=

+

ptminus1OL(i2j)

ptminus1OL(i2j+1)

ptminus1IR(i2j)

ptminus1IR(i2j+1)

Figure 2 The BP decoding algorithm for polar codes is basedon the factor graph representation of the codes [13] In the algo-rithm each node V

119868(119894 119895) or V

119874(119894 119895) is associated with two types of

probability messages left-to-right probability messages and right-to-left probability messages Both of messages are propagated andupdated iteratively between adjacent nodes in the procedure of BPAnd the messages updated schedule is the same as that of [14]that is messages will firstly propagate from the rightmost nodes tothe leftmost nodes After arriving at the leftmost nodes the coursewill be reversed and messages propagation will move toward therightmost nodes This procedure makes one round iteration of BPDiagram of BP-PU

decoding algorithm [13ndash17] the messages updated equationsare given by

119901119905

119868119871(1198942119895)= 119901119905minus1

119874119871(1198942119895)otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 119901119905minus1

119868119877(1198942119895+1))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119874119871(1198942119895+1)

119901119905

119874119877(1198942119895)= 119901119905minus1

119868119877(1198942119895)otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 119901119905minus1

119874119871(1198942119895+1))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119868119877(1198942119895+1)

(4)

Based on (4) when the decoding reaches maximumiteration number (mIter) BP decoder output the decodedvector u = (

0

119873minus1) each elements of which depended

on

119894=

0 119901mIter119868119871(0119894)

(0) gt 119901mIter119868119871(0119894)

(1)

1 otherwise(5)

32 Analysis of BP Decoder It is noticed from [13ndash17] basedon the BP decoding the performance of polar codes canbe improved to a certain degree However as the capacityachieving coding scheme the results of those algorithms isstill disappointingHence we cannot helpwonderingwhy theperformance of the polar codes with finite length is inferiorto that of LDPC codes [16 17] and how we can improve it Toanswer the questions we need to make a further analysis ofthe existing BP decoder

In fact it is noticed from the factor graph of polar codesin [13] that the degree of the check or variable nodes in thepolar BP decoder is 2 or 3 While the average degree of theLDPC codes is usually greater than 3 [12 18] which meansthat the error-correcting capacity of the polar BP decodingwill be weaken Hence the performance of the polar codesis inferior to that of LDPC codes with the same length [18]As shown in Figure 3 two Tanner graphes with different

degree distributions are given We assume that there existsan error variable node V

1in a certain iteration Compared

to the check nodes 1198880and 1198881in Figure 3(a) the influence

of error probability messages from V1to 1198880and 1198881in Figure

3(a) is weak because with a greater degree number the pro-portion of the error probability messages in the input prob-ability messages of 119888

0and 1198881will be reduced What is more

the convergence of BP decoding with a greater average degreewill be quicker which results in less computational iterationnumber as shown in [12ndash18] Therefore to improve theperformance of polar BP decoding it is important to enhancethe error-correcting capacity of the algorithm

33 Generalization of BP Decoding Motivated by aforemen-tioned observation in this subsection we will introduce ageneralization framework of BP decoder for polar codes Inorder to improve the error-correcting capacity of polar BPdecoding twelve probability messages modifying factors areintroduced to the nodes of the BP-PU shown in Figure 2When the probability messages of a node of the BP-PU isupdated three probability messages modifying factors areadded to other three nodes respectively

As illustrated in Figure 4 1205741199051198710(119894119895)

120574119905

1198715(119894119895)and 120574

119905

1198770(119894119895)

120574119905

1198775(119894119895)are the probability messages modifying factors of

the 119895th BP-PU in 119894th stage In this case each equation in(4) will be correspondingly added with three variables forexample the first equation of (4) is added with 120574119905

1198710(119894119895) 1205741199051198711(119894119895)

and 120574

119905

1198712(119894119895) Besides based on the new construction of BP-

PU we further define the result of each nodesrsquo probabilitymessages calculation as a function with respect to these threevariables and the probability messages of other three nodeshence we can get the new messages update equations as (6)Consider

119901119905

119868119871(1198942119895)= 1198910

(119894119895)(119901119905minus1

119874119871(1198942119895) 119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119868119877(1198942119895+1)

120574119905

1198710(119894119895) 120574119905

1198711(119894119895) 120574119905

1198712(119894119895))

119901119905

119868119871(1198942119895+1)= 1198911

(119894119895)(119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119874119871(1198942119895) 119901119905minus1

119868119877(1198942119895)

120574119905

1198713(119894119895) 120574119905

1198714(119894119895) 120574119905

1198715(119894119895))

119901119905

119874119877(1198942119895)= 1198912

(119894119895)(119901119905minus1

119868119877(1198942119895) 119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119874119871(1198942119895+1)

120574119905

1198770(119894119895) 120574119905

1198771(119894119895) 120574119905

1198772(119894119895))

119901119905

119874119877(1198942119895+1)= 1198913

(119894119895)(119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119868119877(1198942119895) 119901119905minus1

119874119871(1198942119895)

ℎ 120574119905

1198773(119894119895) 120574119905

1198774(119894119895) 120574119905

1198775(119894119895))

(6)

In (6) 1198910(119894119895)

(1205935

0) 1198911(119894119895)

(1205935

0) 1198912(119894119895)

(1205935

0) and 119891

3

(119894119895)(1205935

0) are the

four probability messages update functions of the 119895th BP-PU in 119894th stage and 120593

0 120593

5are six probability messages

variables Based on (6) we can easily get a new expression of

Mathematical Problems in Engineering 5

0 1 2 3

c0 c1 c2

(a)

0 1 2 3

c0 c1 c2

(b)

Figure 3 (a) Tanner graph with average degree less than 3 (b) Tanner graph with average degree more than 3

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL0(ij)

120574tL1(ij)120574tL2(ij)

=

+

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOR(i 2j)

ptOL(i 2j + 1)

120574tR0(ij)

120574tR1(ij) 120574tR2(ij)

=

+

(b)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIL (i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL3(ij)

120574tL4(ij)120574tL5(ij)

=

+

(c)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOR(i 2j + 1)

120574tR3(ij)

120574tR4(ij) 120574tR5(ij)

=

+

(d)

Figure 4 Diagrams of the new BP-PU

(4) defined as the functions of 1198910(119894119895)

(1205935

0) 1198911

(119894119895)(1205935

0) 1198912

(119894119895)(1205935

0)

and 1198913

(119894119895)(1205935

0) that is

1198910

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198913

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

(7)

That is to say the existing BP decoding algorithm is aspecial case of the generalization BP decoding proposed inthis work Furthermore with the four probability messagesupdate functions of

1198910

(119894119895)(1205935

0) = 1205930otimes 1205931

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes 1205931

1198913

(119894119895)(1205935

0) = 1205930

(8)

we can also get the probability messages update equationsof the SC based decoding algorithms as shown in [1 4ndash11] Similarly it is concluded that the SC based decodingalgorithms are also the special cases of our generalization BPdecoding which has shown the intrinsic relationship of theSC based decoding and BP decoding algorithms

Therefore we can conclude from (6)ndash(8) that in order toimprove the performance of the polar codes it is needed tofind the optimal functions of1198910

(119894119895)(1205935

0)1198911(119894119895)

(1205935

0)1198912(119894119895)

(1205935

0) and

1198913

(119894119895)(1205935

0) and the optimal parameters of 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) which will be discussed in the next section

6 Mathematical Problems in Engineering

4 BP Decoding Algorithm Optimization

Motivated by the aforementioned conclusion we now con-sider how to optimize the generalization BP decoding algo-rithm so as to improve the performance of the polar codes

41 Messages Update Functions It is noticed from (4) and(6)ndash(8) that in order to enhance the reliability of messagespropagation it is important to determine the appropriatemessages update functions for the generalization BP decod-ing algorithm However in general it is difficult to find theoptimal messages update functions therefore in this workthe messages update functions are introduced by the analysisof the construction of polar encoding And based on (1) wehave

V119868(119894 2119895) = V

119874(119894 2119895) oplus V

119874(119894 2119895 + 1)

= V119874(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119868(119894 2119895 + 1) = V

119874(119894 2119895 + 1) = V

119874(119894 2119895) oplus V

119868(119894 2119895)

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

= V119868(119894 2119895) oplus V

119874(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1) = V

119868(119894 2119895) oplus V

119874(119894 2119895)

(9)

Therefore based on the error coding theory and (9) wecan get the messages update functions as

1198910

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198911

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

1198912

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198913

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

(10)

That is to say with the probability messages modifyingfactors the new probability messages update equations willbe written as

119901119905

119868119871(1198942119895)= ((119901

119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895)) otimes (119901

119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198711(119894119895)))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895))

otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198712(119894119895)))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198713(119894119895))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198714(119894119895))

otimes (119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198715(119894119895)))

119901119905

119874119877(1198942119895)= ((119901

119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895)) otimes (119901

119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198771(119894119895)))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895))

otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198772(119894119895)))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198773(119894119895))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198774(119894119895))

otimes (119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198775(119894119895)))

(11)

Next our task is to determine the optimal probabilitymessages modifying factors that is 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) so as to achieve the best performance

42 Probability Modifying Factors Optimization In practicedue to the input of frozen bits [1] there exists some so-called frozen nodes in the decoder and values of these nodesare determined and independent of the decoding algorithmHence if the decoding is correct the probabilitymessages of afrozen node V

119868(119894 119895) or V

119874(119894 119895) (the determined value of frozen

nodes is 0) must satisfy the condition of

119901119905

119868119871(119894119895)(0) gt 119901

119905

119868119871(119894119895)(1)

119901119905

119868119877(119894119895)(0) gt 119901

119905

119868119877(119894119895)(1)

or

119901119905

119874119871(119894119895)(0) gt 119901

119905

119874119871(119894119895)(1)

119901119905

119874119877(119894119895)(0) gt 119901

119905

119874119877(119894119895)(1)

(12)

While in practice due to the noise of received signalthere may exist some frozen nodes unsatisfied the reliabilitycondition Therefore in order to improve the accuracy ofdecoding it is needed to find the optimal probability mes-sages modifying factors such that all the frozen nodes couldalways satisfy the conditions of (12) during the procedure ofBP decoding

To achieve above goal we introduce a pair parametersto each node in the decoder which are called left and rightreliability degrees And for a node V

119868(119894 119895) (similar to V

119874(119894 119895))

the left and right reliability degrees are denoted as 120577119905119868119871(119894119895)

and120577119905

119868119877(119894119895) respectively Where if V

119868(119894 119895) is a frozen node we can

get 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

(13)

Mathematical Problems in Engineering 7

otherwise the values of 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)will be given by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

119901119868119871(119894119895) (0)

119901119868119871(119894119895) (1)

gt 1

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(1)

119901119905

119868119871(119894119895)(0)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(1)

119901119905

119868119877(119894119895)(0)

otherwise

(14)

where 120577119905119868119871(119894119895)

gt 1 and 120577119905

119868119877(119894119895)gt 1

The reliability degrees indicate the reliability of the nodersquosdecision value and the larger reliability degree the higherreliability of that value Specially when there is no noise inthe channel the reliability degrees of a node will approachto infinity Based on previous observations the problem ofgetting the optimal probability messages modifying factorscan be formulated as an optimization problem to maximizethe reliability degrees of all the node in decoder while thetarget function is written as

arg max0le120574119905

1198710(119894119896)120574119905

1198715(119894119896)le1

0le120574119905

1198770(119894119896)120574119905

1198775(119894119896)le1

log2119873

sum

119894=0

119873

sum

119895=0

(120577119905

119868119871(119894119895)+ 120577119905

119868119877(119894119895))

st 120574119905

1198710(119894119896)(0) + 120574

119905

1198710(119894119896)(1) = 1

120574119905

1198715(119894119896)(0) + 120574

119905

1198715(119894119896)(1) = 1

120574119905

1198770(119894119896)(0) + 120574

119905

1198770(119894119896)(1) = 1

120574119905

1198775(119894119896)(0) + 120574

119905

1198775(119894119896)(1) = 1

(15)

where 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)are decided by (11) and (13)-(14) and

119896 is the integral value of 1198952 that is 119896 = lfloor1198952rfloor

43 Problem Solving Based on Modified PSO AlgorithmBased on (15) we have formulated the problem to find theoptimal probabilitymodifying factors and in this subsectionwe will consider the solution of the problem so as to get theoptimized BP decoding algorithm

Among the different global optimizers particle swarmoptimization (PSO) has demonstrated its usefulness as anoptimizer capable of finding global optimum The idea ofPSO is based on simulation of simplified social models suchas birds flocking and fish schooling Like other stochasticevolutionary algorithms PSO is independent of mathemat-ical characteristics of objective problems However unlikethese algorithms with PSO each particle has its own unique

memory to ldquorememberrdquo its own best solution Thus eachparticle has its own ldquoideardquo of whether it has found theoptimum However in other algorithms previous knowledgeof the problemwill be destroyed once the population changesPSO has attracted increasing attention because of its simpleconcept easy implementation and quick These advantagesallow it to be applied successfully in a variety of fields mainlyfor unconstrained continuous optimization problems [19ndash26] Therefore in this work we explore the PSO algorithmwith certain modification based on particle fitness evaluationto solve the problem (15)

To present our modified PSO algorithm we first intro-duce some preliminary definitions and concepts used in theoperation of the PSOalgorithmwhich is described as follows

(1) ParametersDefinitionsWe consider a swarmof119871particlesz0 z

119871minus1 where the position of z

119894= (1199111198940 1199111198941 119911

119894(119879minus1))

represents a possible solution point of those probabilitymodifying factors in the problem (15) 0 le 119894 le 119871 minus 1 and119879 is the number of the probability modifying factors Eachelement of z

119894is the value of a probability modifying factor

that is 0 le 119911119894119895le 1 0 le 119895 le 119879 minus 1 Based on the work of [19]

the position of particle z119894can be updated by

z119905+1119894

= z119905119894+ w119905+1119894

(16)

In (16) the pseudovelocity w119905+1119894

is calculated as follows

w119905+1119894

= 120596119905w119905119894+ 11988811199031(p119894minus z119905119894) + 11988821199032(p119892minus z119905119894) (17)

where120596119905 is particle inertia 119905 indicates the iteration number p119894

represents the best ever position of particle 119894 at iteration 119905 andp119892represents the global best position in the swarmat iteration

119905 1199031and 1199032represent uniform random numbers between 0

and 1 It is proven in [19] that when 1198881= 1198882= 2 the products

11988811199031or 11988821199032 have a mean of 1

(2) Fitness Evaluation In general with the parameter defini-tions aforementioned we can get the global optimal solutionof a optimization problem However in problem (15) thesituation will be different because during the procedureof the BP decoding for polar codes a nodersquos probabilitymessages calculated on the basis of the probability modifyingfactors must satisfy the conditions of (12) or (14) In orderto check these conditions during the decoding we furtherdefine two functions as

1198921(z119905119894) =

1003816100381610038161003816V119890

119865

10038161003816100381610038161003816100381610038161003816V119865

1003816100381610038161003816

1198922(z119905119894) =

1003816100381610038161003816V119890

119868

10038161003816100381610038161003816100381610038161003816V119868

1003816100381610038161003816

(18)

where V119865is the frozen nodes set of the decoder and V119890

119865is the

set of frozen nodeswhose probabilitymessages based on z119905119894do

not satisfy the condition of (12) Similarly V119868is the nonfrozen

nodes set of the decoder and V119890119868is the set of nonfrozen nodes

whose left and right reliability degrees based on z119905119894do not

satisfy (14) Furthermore we have 1198921(z119905119894) 1198922(z119905119894) isin [0 1]

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Stage 0u0

I(0 0)

u1

I(0 1)

u2

I(0 2)

u3

I(0 3)

u4

I(0 4)

u5

I(0 5)

u6

I(0 6)

u7

I(0 7)

O(0 0)

O(0 1)

O(0 2)

O(0 3)

O(0 4)

O(0 5)

O(0 6)

O(0 7)

Stage 1

I(1 0)

I(1 1)

I(1 2)

I(1 3)

I(1 4)

I(1 5)

I(1 6)

I(1 7)

O(1 0)

O(1 1)

O(1 2)

O(1 3)

O(1 4)

O(1 5)

O(1 6)

O(1 7)

Stage 2

I(2 0)

I(2 1)

I(2 2)

I(2 3)

I(2 4)

I(2 5)

I(2 6)

I(2 7)

O(2 0)

O(2 1)

O(2 2)

O(2 3)

O(2 4)

O(2 5)

O(2 6)

O(2 7)

x0

x1

x2

x3

x4

x5

x6

x7

W

W

W

W

W

W

W

W

y0

y1

y2

y3

y4

y5

y6

y7

= = =

= = =

+ + +

+ + +

= = =

+ + +

= = =

+ + +

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)=

+

(b)

Figure 1 (a) Construction procedure of polar codes with119873 = 8 (b) Diagram of PU in code construction

3 BP Decoding for Polar Codes

In this section we will generalize the BP decoding algorithmfor polar codes with the analysis of the existing polar BPdecoding

31 Existing BP Decoding The basic BP process unit (BP-PU) of polar codes is shown in Figure 2 where 119901119905

119868119871(1198942119895)and

119901119905

119868119871(1198942119895+1)are the right-to-left probability messages passed to

V119868(119894 2119895) and V

119868(119894 2119895 + 1) 119905 is the iteration number 119901119905minus1

119868119877(1198942119895)

and119901119905minus1119868119877(1198942119895+1)

are the left-to-right probabilitymessages passedfrom the nodes 119901119905minus1

119874119871(1198942119895)and 119901

119905minus1

119874119871(1198942119895+1)are the right-to-left

probability messages passed from V119874(119894 2119895) and V

119874(119894 2119895 + 1)

and 119901119905

119874119877(1198942119895)and 119901

119905

119874119877(1198942119895+1)are the left-to-right probability

messages passed to the nodes According to the formula oftotal probability there has

119901119905

119868119871(119894119895)(0) + 119901

119905

119868119871(119894119895)(1) = 1

119901119905

119868119877(119894119895)(0) + 119901

119905

119868119877(119894119895)(1) = 1

119901119905

119874119871(119894119895)(0) + 119901

119905

119874119871(119894119895)(1) = 1

119901119905

119874119877(119894119895)(0) + 119901

119905

119874119877(119894119895)(1) = 1

(2)

Furthermore based on the transition probability of thechannel119882 that is119882(119910

119894| 119909119894) and the received vector 119910119873minus1

0

we have

1199010

119874119871(119899minus1119894)(0) = 119882(119910

119894| 119909119894= 0)

119875 (119909119894= 0)

119875 (119910119894)

1199010

119874119871(119899minus1119894)(1) = 119882(119910

119894| 119909119894= 1)

119875 (119909119894= 1)

119875 (119910119894)

(3)

where 119875(119909119894) and 119875(119910

119894) are the probability distribution func-

tions of coded bit 119909119894and received signal sample 119910

119894 In the BP

4 Mathematical Problems in Engineering

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i2j)

ptIL(i2j+1)

ptOR(i2j)

ptOR(i2j+1)

=

+

ptminus1OL(i2j)

ptminus1OL(i2j+1)

ptminus1IR(i2j)

ptminus1IR(i2j+1)

Figure 2 The BP decoding algorithm for polar codes is basedon the factor graph representation of the codes [13] In the algo-rithm each node V

119868(119894 119895) or V

119874(119894 119895) is associated with two types of

probability messages left-to-right probability messages and right-to-left probability messages Both of messages are propagated andupdated iteratively between adjacent nodes in the procedure of BPAnd the messages updated schedule is the same as that of [14]that is messages will firstly propagate from the rightmost nodes tothe leftmost nodes After arriving at the leftmost nodes the coursewill be reversed and messages propagation will move toward therightmost nodes This procedure makes one round iteration of BPDiagram of BP-PU

decoding algorithm [13ndash17] the messages updated equationsare given by

119901119905

119868119871(1198942119895)= 119901119905minus1

119874119871(1198942119895)otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 119901119905minus1

119868119877(1198942119895+1))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119874119871(1198942119895+1)

119901119905

119874119877(1198942119895)= 119901119905minus1

119868119877(1198942119895)otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 119901119905minus1

119874119871(1198942119895+1))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119868119877(1198942119895+1)

(4)

Based on (4) when the decoding reaches maximumiteration number (mIter) BP decoder output the decodedvector u = (

0

119873minus1) each elements of which depended

on

119894=

0 119901mIter119868119871(0119894)

(0) gt 119901mIter119868119871(0119894)

(1)

1 otherwise(5)

32 Analysis of BP Decoder It is noticed from [13ndash17] basedon the BP decoding the performance of polar codes canbe improved to a certain degree However as the capacityachieving coding scheme the results of those algorithms isstill disappointingHence we cannot helpwonderingwhy theperformance of the polar codes with finite length is inferiorto that of LDPC codes [16 17] and how we can improve it Toanswer the questions we need to make a further analysis ofthe existing BP decoder

In fact it is noticed from the factor graph of polar codesin [13] that the degree of the check or variable nodes in thepolar BP decoder is 2 or 3 While the average degree of theLDPC codes is usually greater than 3 [12 18] which meansthat the error-correcting capacity of the polar BP decodingwill be weaken Hence the performance of the polar codesis inferior to that of LDPC codes with the same length [18]As shown in Figure 3 two Tanner graphes with different

degree distributions are given We assume that there existsan error variable node V

1in a certain iteration Compared

to the check nodes 1198880and 1198881in Figure 3(a) the influence

of error probability messages from V1to 1198880and 1198881in Figure

3(a) is weak because with a greater degree number the pro-portion of the error probability messages in the input prob-ability messages of 119888

0and 1198881will be reduced What is more

the convergence of BP decoding with a greater average degreewill be quicker which results in less computational iterationnumber as shown in [12ndash18] Therefore to improve theperformance of polar BP decoding it is important to enhancethe error-correcting capacity of the algorithm

33 Generalization of BP Decoding Motivated by aforemen-tioned observation in this subsection we will introduce ageneralization framework of BP decoder for polar codes Inorder to improve the error-correcting capacity of polar BPdecoding twelve probability messages modifying factors areintroduced to the nodes of the BP-PU shown in Figure 2When the probability messages of a node of the BP-PU isupdated three probability messages modifying factors areadded to other three nodes respectively

As illustrated in Figure 4 1205741199051198710(119894119895)

120574119905

1198715(119894119895)and 120574

119905

1198770(119894119895)

120574119905

1198775(119894119895)are the probability messages modifying factors of

the 119895th BP-PU in 119894th stage In this case each equation in(4) will be correspondingly added with three variables forexample the first equation of (4) is added with 120574119905

1198710(119894119895) 1205741199051198711(119894119895)

and 120574

119905

1198712(119894119895) Besides based on the new construction of BP-

PU we further define the result of each nodesrsquo probabilitymessages calculation as a function with respect to these threevariables and the probability messages of other three nodeshence we can get the new messages update equations as (6)Consider

119901119905

119868119871(1198942119895)= 1198910

(119894119895)(119901119905minus1

119874119871(1198942119895) 119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119868119877(1198942119895+1)

120574119905

1198710(119894119895) 120574119905

1198711(119894119895) 120574119905

1198712(119894119895))

119901119905

119868119871(1198942119895+1)= 1198911

(119894119895)(119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119874119871(1198942119895) 119901119905minus1

119868119877(1198942119895)

120574119905

1198713(119894119895) 120574119905

1198714(119894119895) 120574119905

1198715(119894119895))

119901119905

119874119877(1198942119895)= 1198912

(119894119895)(119901119905minus1

119868119877(1198942119895) 119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119874119871(1198942119895+1)

120574119905

1198770(119894119895) 120574119905

1198771(119894119895) 120574119905

1198772(119894119895))

119901119905

119874119877(1198942119895+1)= 1198913

(119894119895)(119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119868119877(1198942119895) 119901119905minus1

119874119871(1198942119895)

ℎ 120574119905

1198773(119894119895) 120574119905

1198774(119894119895) 120574119905

1198775(119894119895))

(6)

In (6) 1198910(119894119895)

(1205935

0) 1198911(119894119895)

(1205935

0) 1198912(119894119895)

(1205935

0) and 119891

3

(119894119895)(1205935

0) are the

four probability messages update functions of the 119895th BP-PU in 119894th stage and 120593

0 120593

5are six probability messages

variables Based on (6) we can easily get a new expression of

Mathematical Problems in Engineering 5

0 1 2 3

c0 c1 c2

(a)

0 1 2 3

c0 c1 c2

(b)

Figure 3 (a) Tanner graph with average degree less than 3 (b) Tanner graph with average degree more than 3

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL0(ij)

120574tL1(ij)120574tL2(ij)

=

+

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOR(i 2j)

ptOL(i 2j + 1)

120574tR0(ij)

120574tR1(ij) 120574tR2(ij)

=

+

(b)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIL (i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL3(ij)

120574tL4(ij)120574tL5(ij)

=

+

(c)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOR(i 2j + 1)

120574tR3(ij)

120574tR4(ij) 120574tR5(ij)

=

+

(d)

Figure 4 Diagrams of the new BP-PU

(4) defined as the functions of 1198910(119894119895)

(1205935

0) 1198911

(119894119895)(1205935

0) 1198912

(119894119895)(1205935

0)

and 1198913

(119894119895)(1205935

0) that is

1198910

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198913

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

(7)

That is to say the existing BP decoding algorithm is aspecial case of the generalization BP decoding proposed inthis work Furthermore with the four probability messagesupdate functions of

1198910

(119894119895)(1205935

0) = 1205930otimes 1205931

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes 1205931

1198913

(119894119895)(1205935

0) = 1205930

(8)

we can also get the probability messages update equationsof the SC based decoding algorithms as shown in [1 4ndash11] Similarly it is concluded that the SC based decodingalgorithms are also the special cases of our generalization BPdecoding which has shown the intrinsic relationship of theSC based decoding and BP decoding algorithms

Therefore we can conclude from (6)ndash(8) that in order toimprove the performance of the polar codes it is needed tofind the optimal functions of1198910

(119894119895)(1205935

0)1198911(119894119895)

(1205935

0)1198912(119894119895)

(1205935

0) and

1198913

(119894119895)(1205935

0) and the optimal parameters of 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) which will be discussed in the next section

6 Mathematical Problems in Engineering

4 BP Decoding Algorithm Optimization

Motivated by the aforementioned conclusion we now con-sider how to optimize the generalization BP decoding algo-rithm so as to improve the performance of the polar codes

41 Messages Update Functions It is noticed from (4) and(6)ndash(8) that in order to enhance the reliability of messagespropagation it is important to determine the appropriatemessages update functions for the generalization BP decod-ing algorithm However in general it is difficult to find theoptimal messages update functions therefore in this workthe messages update functions are introduced by the analysisof the construction of polar encoding And based on (1) wehave

V119868(119894 2119895) = V

119874(119894 2119895) oplus V

119874(119894 2119895 + 1)

= V119874(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119868(119894 2119895 + 1) = V

119874(119894 2119895 + 1) = V

119874(119894 2119895) oplus V

119868(119894 2119895)

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

= V119868(119894 2119895) oplus V

119874(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1) = V

119868(119894 2119895) oplus V

119874(119894 2119895)

(9)

Therefore based on the error coding theory and (9) wecan get the messages update functions as

1198910

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198911

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

1198912

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198913

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

(10)

That is to say with the probability messages modifyingfactors the new probability messages update equations willbe written as

119901119905

119868119871(1198942119895)= ((119901

119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895)) otimes (119901

119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198711(119894119895)))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895))

otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198712(119894119895)))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198713(119894119895))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198714(119894119895))

otimes (119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198715(119894119895)))

119901119905

119874119877(1198942119895)= ((119901

119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895)) otimes (119901

119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198771(119894119895)))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895))

otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198772(119894119895)))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198773(119894119895))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198774(119894119895))

otimes (119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198775(119894119895)))

(11)

Next our task is to determine the optimal probabilitymessages modifying factors that is 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) so as to achieve the best performance

42 Probability Modifying Factors Optimization In practicedue to the input of frozen bits [1] there exists some so-called frozen nodes in the decoder and values of these nodesare determined and independent of the decoding algorithmHence if the decoding is correct the probabilitymessages of afrozen node V

119868(119894 119895) or V

119874(119894 119895) (the determined value of frozen

nodes is 0) must satisfy the condition of

119901119905

119868119871(119894119895)(0) gt 119901

119905

119868119871(119894119895)(1)

119901119905

119868119877(119894119895)(0) gt 119901

119905

119868119877(119894119895)(1)

or

119901119905

119874119871(119894119895)(0) gt 119901

119905

119874119871(119894119895)(1)

119901119905

119874119877(119894119895)(0) gt 119901

119905

119874119877(119894119895)(1)

(12)

While in practice due to the noise of received signalthere may exist some frozen nodes unsatisfied the reliabilitycondition Therefore in order to improve the accuracy ofdecoding it is needed to find the optimal probability mes-sages modifying factors such that all the frozen nodes couldalways satisfy the conditions of (12) during the procedure ofBP decoding

To achieve above goal we introduce a pair parametersto each node in the decoder which are called left and rightreliability degrees And for a node V

119868(119894 119895) (similar to V

119874(119894 119895))

the left and right reliability degrees are denoted as 120577119905119868119871(119894119895)

and120577119905

119868119877(119894119895) respectively Where if V

119868(119894 119895) is a frozen node we can

get 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

(13)

Mathematical Problems in Engineering 7

otherwise the values of 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)will be given by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

119901119868119871(119894119895) (0)

119901119868119871(119894119895) (1)

gt 1

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(1)

119901119905

119868119871(119894119895)(0)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(1)

119901119905

119868119877(119894119895)(0)

otherwise

(14)

where 120577119905119868119871(119894119895)

gt 1 and 120577119905

119868119877(119894119895)gt 1

The reliability degrees indicate the reliability of the nodersquosdecision value and the larger reliability degree the higherreliability of that value Specially when there is no noise inthe channel the reliability degrees of a node will approachto infinity Based on previous observations the problem ofgetting the optimal probability messages modifying factorscan be formulated as an optimization problem to maximizethe reliability degrees of all the node in decoder while thetarget function is written as

arg max0le120574119905

1198710(119894119896)120574119905

1198715(119894119896)le1

0le120574119905

1198770(119894119896)120574119905

1198775(119894119896)le1

log2119873

sum

119894=0

119873

sum

119895=0

(120577119905

119868119871(119894119895)+ 120577119905

119868119877(119894119895))

st 120574119905

1198710(119894119896)(0) + 120574

119905

1198710(119894119896)(1) = 1

120574119905

1198715(119894119896)(0) + 120574

119905

1198715(119894119896)(1) = 1

120574119905

1198770(119894119896)(0) + 120574

119905

1198770(119894119896)(1) = 1

120574119905

1198775(119894119896)(0) + 120574

119905

1198775(119894119896)(1) = 1

(15)

where 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)are decided by (11) and (13)-(14) and

119896 is the integral value of 1198952 that is 119896 = lfloor1198952rfloor

43 Problem Solving Based on Modified PSO AlgorithmBased on (15) we have formulated the problem to find theoptimal probabilitymodifying factors and in this subsectionwe will consider the solution of the problem so as to get theoptimized BP decoding algorithm

Among the different global optimizers particle swarmoptimization (PSO) has demonstrated its usefulness as anoptimizer capable of finding global optimum The idea ofPSO is based on simulation of simplified social models suchas birds flocking and fish schooling Like other stochasticevolutionary algorithms PSO is independent of mathemat-ical characteristics of objective problems However unlikethese algorithms with PSO each particle has its own unique

memory to ldquorememberrdquo its own best solution Thus eachparticle has its own ldquoideardquo of whether it has found theoptimum However in other algorithms previous knowledgeof the problemwill be destroyed once the population changesPSO has attracted increasing attention because of its simpleconcept easy implementation and quick These advantagesallow it to be applied successfully in a variety of fields mainlyfor unconstrained continuous optimization problems [19ndash26] Therefore in this work we explore the PSO algorithmwith certain modification based on particle fitness evaluationto solve the problem (15)

To present our modified PSO algorithm we first intro-duce some preliminary definitions and concepts used in theoperation of the PSOalgorithmwhich is described as follows

(1) ParametersDefinitionsWe consider a swarmof119871particlesz0 z

119871minus1 where the position of z

119894= (1199111198940 1199111198941 119911

119894(119879minus1))

represents a possible solution point of those probabilitymodifying factors in the problem (15) 0 le 119894 le 119871 minus 1 and119879 is the number of the probability modifying factors Eachelement of z

119894is the value of a probability modifying factor

that is 0 le 119911119894119895le 1 0 le 119895 le 119879 minus 1 Based on the work of [19]

the position of particle z119894can be updated by

z119905+1119894

= z119905119894+ w119905+1119894

(16)

In (16) the pseudovelocity w119905+1119894

is calculated as follows

w119905+1119894

= 120596119905w119905119894+ 11988811199031(p119894minus z119905119894) + 11988821199032(p119892minus z119905119894) (17)

where120596119905 is particle inertia 119905 indicates the iteration number p119894

represents the best ever position of particle 119894 at iteration 119905 andp119892represents the global best position in the swarmat iteration

119905 1199031and 1199032represent uniform random numbers between 0

and 1 It is proven in [19] that when 1198881= 1198882= 2 the products

11988811199031or 11988821199032 have a mean of 1

(2) Fitness Evaluation In general with the parameter defini-tions aforementioned we can get the global optimal solutionof a optimization problem However in problem (15) thesituation will be different because during the procedureof the BP decoding for polar codes a nodersquos probabilitymessages calculated on the basis of the probability modifyingfactors must satisfy the conditions of (12) or (14) In orderto check these conditions during the decoding we furtherdefine two functions as

1198921(z119905119894) =

1003816100381610038161003816V119890

119865

10038161003816100381610038161003816100381610038161003816V119865

1003816100381610038161003816

1198922(z119905119894) =

1003816100381610038161003816V119890

119868

10038161003816100381610038161003816100381610038161003816V119868

1003816100381610038161003816

(18)

where V119865is the frozen nodes set of the decoder and V119890

119865is the

set of frozen nodeswhose probabilitymessages based on z119905119894do

not satisfy the condition of (12) Similarly V119868is the nonfrozen

nodes set of the decoder and V119890119868is the set of nonfrozen nodes

whose left and right reliability degrees based on z119905119894do not

satisfy (14) Furthermore we have 1198921(z119905119894) 1198922(z119905119894) isin [0 1]

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i2j)

ptIL(i2j+1)

ptOR(i2j)

ptOR(i2j+1)

=

+

ptminus1OL(i2j)

ptminus1OL(i2j+1)

ptminus1IR(i2j)

ptminus1IR(i2j+1)

Figure 2 The BP decoding algorithm for polar codes is basedon the factor graph representation of the codes [13] In the algo-rithm each node V

119868(119894 119895) or V

119874(119894 119895) is associated with two types of

probability messages left-to-right probability messages and right-to-left probability messages Both of messages are propagated andupdated iteratively between adjacent nodes in the procedure of BPAnd the messages updated schedule is the same as that of [14]that is messages will firstly propagate from the rightmost nodes tothe leftmost nodes After arriving at the leftmost nodes the coursewill be reversed and messages propagation will move toward therightmost nodes This procedure makes one round iteration of BPDiagram of BP-PU

decoding algorithm [13ndash17] the messages updated equationsare given by

119901119905

119868119871(1198942119895)= 119901119905minus1

119874119871(1198942119895)otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 119901119905minus1

119868119877(1198942119895+1))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119874119871(1198942119895+1)

119901119905

119874119877(1198942119895)= 119901119905minus1

119868119877(1198942119895)otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 119901119905minus1

119874119871(1198942119895+1))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895)otimes 119901119905minus1

119874119871(1198942119895)) ⊙ 119901119905minus1

119868119877(1198942119895+1)

(4)

Based on (4) when the decoding reaches maximumiteration number (mIter) BP decoder output the decodedvector u = (

0

119873minus1) each elements of which depended

on

119894=

0 119901mIter119868119871(0119894)

(0) gt 119901mIter119868119871(0119894)

(1)

1 otherwise(5)

32 Analysis of BP Decoder It is noticed from [13ndash17] basedon the BP decoding the performance of polar codes canbe improved to a certain degree However as the capacityachieving coding scheme the results of those algorithms isstill disappointingHence we cannot helpwonderingwhy theperformance of the polar codes with finite length is inferiorto that of LDPC codes [16 17] and how we can improve it Toanswer the questions we need to make a further analysis ofthe existing BP decoder

In fact it is noticed from the factor graph of polar codesin [13] that the degree of the check or variable nodes in thepolar BP decoder is 2 or 3 While the average degree of theLDPC codes is usually greater than 3 [12 18] which meansthat the error-correcting capacity of the polar BP decodingwill be weaken Hence the performance of the polar codesis inferior to that of LDPC codes with the same length [18]As shown in Figure 3 two Tanner graphes with different

degree distributions are given We assume that there existsan error variable node V

1in a certain iteration Compared

to the check nodes 1198880and 1198881in Figure 3(a) the influence

of error probability messages from V1to 1198880and 1198881in Figure

3(a) is weak because with a greater degree number the pro-portion of the error probability messages in the input prob-ability messages of 119888

0and 1198881will be reduced What is more

the convergence of BP decoding with a greater average degreewill be quicker which results in less computational iterationnumber as shown in [12ndash18] Therefore to improve theperformance of polar BP decoding it is important to enhancethe error-correcting capacity of the algorithm

33 Generalization of BP Decoding Motivated by aforemen-tioned observation in this subsection we will introduce ageneralization framework of BP decoder for polar codes Inorder to improve the error-correcting capacity of polar BPdecoding twelve probability messages modifying factors areintroduced to the nodes of the BP-PU shown in Figure 2When the probability messages of a node of the BP-PU isupdated three probability messages modifying factors areadded to other three nodes respectively

As illustrated in Figure 4 1205741199051198710(119894119895)

120574119905

1198715(119894119895)and 120574

119905

1198770(119894119895)

120574119905

1198775(119894119895)are the probability messages modifying factors of

the 119895th BP-PU in 119894th stage In this case each equation in(4) will be correspondingly added with three variables forexample the first equation of (4) is added with 120574119905

1198710(119894119895) 1205741199051198711(119894119895)

and 120574

119905

1198712(119894119895) Besides based on the new construction of BP-

PU we further define the result of each nodesrsquo probabilitymessages calculation as a function with respect to these threevariables and the probability messages of other three nodeshence we can get the new messages update equations as (6)Consider

119901119905

119868119871(1198942119895)= 1198910

(119894119895)(119901119905minus1

119874119871(1198942119895) 119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119868119877(1198942119895+1)

120574119905

1198710(119894119895) 120574119905

1198711(119894119895) 120574119905

1198712(119894119895))

119901119905

119868119871(1198942119895+1)= 1198911

(119894119895)(119901119905minus1

119874119871(1198942119895+1) 119901119905minus1

119874119871(1198942119895) 119901119905minus1

119868119877(1198942119895)

120574119905

1198713(119894119895) 120574119905

1198714(119894119895) 120574119905

1198715(119894119895))

119901119905

119874119877(1198942119895)= 1198912

(119894119895)(119901119905minus1

119868119877(1198942119895) 119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119874119871(1198942119895+1)

120574119905

1198770(119894119895) 120574119905

1198771(119894119895) 120574119905

1198772(119894119895))

119901119905

119874119877(1198942119895+1)= 1198913

(119894119895)(119901119905minus1

119868119877(1198942119895+1) 119901119905minus1

119868119877(1198942119895) 119901119905minus1

119874119871(1198942119895)

ℎ 120574119905

1198773(119894119895) 120574119905

1198774(119894119895) 120574119905

1198775(119894119895))

(6)

In (6) 1198910(119894119895)

(1205935

0) 1198911(119894119895)

(1205935

0) 1198912(119894119895)

(1205935

0) and 119891

3

(119894119895)(1205935

0) are the

four probability messages update functions of the 119895th BP-PU in 119894th stage and 120593

0 120593

5are six probability messages

variables Based on (6) we can easily get a new expression of

Mathematical Problems in Engineering 5

0 1 2 3

c0 c1 c2

(a)

0 1 2 3

c0 c1 c2

(b)

Figure 3 (a) Tanner graph with average degree less than 3 (b) Tanner graph with average degree more than 3

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL0(ij)

120574tL1(ij)120574tL2(ij)

=

+

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOR(i 2j)

ptOL(i 2j + 1)

120574tR0(ij)

120574tR1(ij) 120574tR2(ij)

=

+

(b)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIL (i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL3(ij)

120574tL4(ij)120574tL5(ij)

=

+

(c)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOR(i 2j + 1)

120574tR3(ij)

120574tR4(ij) 120574tR5(ij)

=

+

(d)

Figure 4 Diagrams of the new BP-PU

(4) defined as the functions of 1198910(119894119895)

(1205935

0) 1198911

(119894119895)(1205935

0) 1198912

(119894119895)(1205935

0)

and 1198913

(119894119895)(1205935

0) that is

1198910

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198913

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

(7)

That is to say the existing BP decoding algorithm is aspecial case of the generalization BP decoding proposed inthis work Furthermore with the four probability messagesupdate functions of

1198910

(119894119895)(1205935

0) = 1205930otimes 1205931

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes 1205931

1198913

(119894119895)(1205935

0) = 1205930

(8)

we can also get the probability messages update equationsof the SC based decoding algorithms as shown in [1 4ndash11] Similarly it is concluded that the SC based decodingalgorithms are also the special cases of our generalization BPdecoding which has shown the intrinsic relationship of theSC based decoding and BP decoding algorithms

Therefore we can conclude from (6)ndash(8) that in order toimprove the performance of the polar codes it is needed tofind the optimal functions of1198910

(119894119895)(1205935

0)1198911(119894119895)

(1205935

0)1198912(119894119895)

(1205935

0) and

1198913

(119894119895)(1205935

0) and the optimal parameters of 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) which will be discussed in the next section

6 Mathematical Problems in Engineering

4 BP Decoding Algorithm Optimization

Motivated by the aforementioned conclusion we now con-sider how to optimize the generalization BP decoding algo-rithm so as to improve the performance of the polar codes

41 Messages Update Functions It is noticed from (4) and(6)ndash(8) that in order to enhance the reliability of messagespropagation it is important to determine the appropriatemessages update functions for the generalization BP decod-ing algorithm However in general it is difficult to find theoptimal messages update functions therefore in this workthe messages update functions are introduced by the analysisof the construction of polar encoding And based on (1) wehave

V119868(119894 2119895) = V

119874(119894 2119895) oplus V

119874(119894 2119895 + 1)

= V119874(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119868(119894 2119895 + 1) = V

119874(119894 2119895 + 1) = V

119874(119894 2119895) oplus V

119868(119894 2119895)

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

= V119868(119894 2119895) oplus V

119874(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1) = V

119868(119894 2119895) oplus V

119874(119894 2119895)

(9)

Therefore based on the error coding theory and (9) wecan get the messages update functions as

1198910

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198911

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

1198912

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198913

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

(10)

That is to say with the probability messages modifyingfactors the new probability messages update equations willbe written as

119901119905

119868119871(1198942119895)= ((119901

119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895)) otimes (119901

119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198711(119894119895)))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895))

otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198712(119894119895)))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198713(119894119895))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198714(119894119895))

otimes (119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198715(119894119895)))

119901119905

119874119877(1198942119895)= ((119901

119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895)) otimes (119901

119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198771(119894119895)))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895))

otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198772(119894119895)))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198773(119894119895))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198774(119894119895))

otimes (119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198775(119894119895)))

(11)

Next our task is to determine the optimal probabilitymessages modifying factors that is 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) so as to achieve the best performance

42 Probability Modifying Factors Optimization In practicedue to the input of frozen bits [1] there exists some so-called frozen nodes in the decoder and values of these nodesare determined and independent of the decoding algorithmHence if the decoding is correct the probabilitymessages of afrozen node V

119868(119894 119895) or V

119874(119894 119895) (the determined value of frozen

nodes is 0) must satisfy the condition of

119901119905

119868119871(119894119895)(0) gt 119901

119905

119868119871(119894119895)(1)

119901119905

119868119877(119894119895)(0) gt 119901

119905

119868119877(119894119895)(1)

or

119901119905

119874119871(119894119895)(0) gt 119901

119905

119874119871(119894119895)(1)

119901119905

119874119877(119894119895)(0) gt 119901

119905

119874119877(119894119895)(1)

(12)

While in practice due to the noise of received signalthere may exist some frozen nodes unsatisfied the reliabilitycondition Therefore in order to improve the accuracy ofdecoding it is needed to find the optimal probability mes-sages modifying factors such that all the frozen nodes couldalways satisfy the conditions of (12) during the procedure ofBP decoding

To achieve above goal we introduce a pair parametersto each node in the decoder which are called left and rightreliability degrees And for a node V

119868(119894 119895) (similar to V

119874(119894 119895))

the left and right reliability degrees are denoted as 120577119905119868119871(119894119895)

and120577119905

119868119877(119894119895) respectively Where if V

119868(119894 119895) is a frozen node we can

get 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

(13)

Mathematical Problems in Engineering 7

otherwise the values of 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)will be given by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

119901119868119871(119894119895) (0)

119901119868119871(119894119895) (1)

gt 1

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(1)

119901119905

119868119871(119894119895)(0)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(1)

119901119905

119868119877(119894119895)(0)

otherwise

(14)

where 120577119905119868119871(119894119895)

gt 1 and 120577119905

119868119877(119894119895)gt 1

The reliability degrees indicate the reliability of the nodersquosdecision value and the larger reliability degree the higherreliability of that value Specially when there is no noise inthe channel the reliability degrees of a node will approachto infinity Based on previous observations the problem ofgetting the optimal probability messages modifying factorscan be formulated as an optimization problem to maximizethe reliability degrees of all the node in decoder while thetarget function is written as

arg max0le120574119905

1198710(119894119896)120574119905

1198715(119894119896)le1

0le120574119905

1198770(119894119896)120574119905

1198775(119894119896)le1

log2119873

sum

119894=0

119873

sum

119895=0

(120577119905

119868119871(119894119895)+ 120577119905

119868119877(119894119895))

st 120574119905

1198710(119894119896)(0) + 120574

119905

1198710(119894119896)(1) = 1

120574119905

1198715(119894119896)(0) + 120574

119905

1198715(119894119896)(1) = 1

120574119905

1198770(119894119896)(0) + 120574

119905

1198770(119894119896)(1) = 1

120574119905

1198775(119894119896)(0) + 120574

119905

1198775(119894119896)(1) = 1

(15)

where 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)are decided by (11) and (13)-(14) and

119896 is the integral value of 1198952 that is 119896 = lfloor1198952rfloor

43 Problem Solving Based on Modified PSO AlgorithmBased on (15) we have formulated the problem to find theoptimal probabilitymodifying factors and in this subsectionwe will consider the solution of the problem so as to get theoptimized BP decoding algorithm

Among the different global optimizers particle swarmoptimization (PSO) has demonstrated its usefulness as anoptimizer capable of finding global optimum The idea ofPSO is based on simulation of simplified social models suchas birds flocking and fish schooling Like other stochasticevolutionary algorithms PSO is independent of mathemat-ical characteristics of objective problems However unlikethese algorithms with PSO each particle has its own unique

memory to ldquorememberrdquo its own best solution Thus eachparticle has its own ldquoideardquo of whether it has found theoptimum However in other algorithms previous knowledgeof the problemwill be destroyed once the population changesPSO has attracted increasing attention because of its simpleconcept easy implementation and quick These advantagesallow it to be applied successfully in a variety of fields mainlyfor unconstrained continuous optimization problems [19ndash26] Therefore in this work we explore the PSO algorithmwith certain modification based on particle fitness evaluationto solve the problem (15)

To present our modified PSO algorithm we first intro-duce some preliminary definitions and concepts used in theoperation of the PSOalgorithmwhich is described as follows

(1) ParametersDefinitionsWe consider a swarmof119871particlesz0 z

119871minus1 where the position of z

119894= (1199111198940 1199111198941 119911

119894(119879minus1))

represents a possible solution point of those probabilitymodifying factors in the problem (15) 0 le 119894 le 119871 minus 1 and119879 is the number of the probability modifying factors Eachelement of z

119894is the value of a probability modifying factor

that is 0 le 119911119894119895le 1 0 le 119895 le 119879 minus 1 Based on the work of [19]

the position of particle z119894can be updated by

z119905+1119894

= z119905119894+ w119905+1119894

(16)

In (16) the pseudovelocity w119905+1119894

is calculated as follows

w119905+1119894

= 120596119905w119905119894+ 11988811199031(p119894minus z119905119894) + 11988821199032(p119892minus z119905119894) (17)

where120596119905 is particle inertia 119905 indicates the iteration number p119894

represents the best ever position of particle 119894 at iteration 119905 andp119892represents the global best position in the swarmat iteration

119905 1199031and 1199032represent uniform random numbers between 0

and 1 It is proven in [19] that when 1198881= 1198882= 2 the products

11988811199031or 11988821199032 have a mean of 1

(2) Fitness Evaluation In general with the parameter defini-tions aforementioned we can get the global optimal solutionof a optimization problem However in problem (15) thesituation will be different because during the procedureof the BP decoding for polar codes a nodersquos probabilitymessages calculated on the basis of the probability modifyingfactors must satisfy the conditions of (12) or (14) In orderto check these conditions during the decoding we furtherdefine two functions as

1198921(z119905119894) =

1003816100381610038161003816V119890

119865

10038161003816100381610038161003816100381610038161003816V119865

1003816100381610038161003816

1198922(z119905119894) =

1003816100381610038161003816V119890

119868

10038161003816100381610038161003816100381610038161003816V119868

1003816100381610038161003816

(18)

where V119865is the frozen nodes set of the decoder and V119890

119865is the

set of frozen nodeswhose probabilitymessages based on z119905119894do

not satisfy the condition of (12) Similarly V119868is the nonfrozen

nodes set of the decoder and V119890119868is the set of nonfrozen nodes

whose left and right reliability degrees based on z119905119894do not

satisfy (14) Furthermore we have 1198921(z119905119894) 1198922(z119905119894) isin [0 1]

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 1 2 3

c0 c1 c2

(a)

0 1 2 3

c0 c1 c2

(b)

Figure 3 (a) Tanner graph with average degree less than 3 (b) Tanner graph with average degree more than 3

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIL(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL0(ij)

120574tL1(ij)120574tL2(ij)

=

+

(a)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOR(i 2j)

ptOL(i 2j + 1)

120574tR0(ij)

120574tR1(ij) 120574tR2(ij)

=

+

(b)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIL (i 2j + 1)

ptOL(i 2j)

ptOL(i 2j + 1)

120574tL3(ij)

120574tL4(ij)120574tL5(ij)

=

+

(c)

I(i 2j)

I(i 2j + 1)

O(i 2j)

O(i 2j + 1)

ptIR(i 2j)

ptIR(i 2j + 1)

ptOL(i 2j)

ptOR(i 2j + 1)

120574tR3(ij)

120574tR4(ij) 120574tR5(ij)

=

+

(d)

Figure 4 Diagrams of the new BP-PU

(4) defined as the functions of 1198910(119894119895)

(1205935

0) 1198911

(119894119895)(1205935

0) 1198912

(119894119895)(1205935

0)

and 1198913

(119894119895)(1205935

0) that is

1198910

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes (1205931⊙ 1205932)

1198913

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

(7)

That is to say the existing BP decoding algorithm is aspecial case of the generalization BP decoding proposed inthis work Furthermore with the four probability messagesupdate functions of

1198910

(119894119895)(1205935

0) = 1205930otimes 1205931

1198911

(119894119895)(1205935

0) = 1205930⊙ (1205931otimes 1205932)

1198912

(119894119895)(1205935

0) = 1205930otimes 1205931

1198913

(119894119895)(1205935

0) = 1205930

(8)

we can also get the probability messages update equationsof the SC based decoding algorithms as shown in [1 4ndash11] Similarly it is concluded that the SC based decodingalgorithms are also the special cases of our generalization BPdecoding which has shown the intrinsic relationship of theSC based decoding and BP decoding algorithms

Therefore we can conclude from (6)ndash(8) that in order toimprove the performance of the polar codes it is needed tofind the optimal functions of1198910

(119894119895)(1205935

0)1198911(119894119895)

(1205935

0)1198912(119894119895)

(1205935

0) and

1198913

(119894119895)(1205935

0) and the optimal parameters of 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) which will be discussed in the next section

6 Mathematical Problems in Engineering

4 BP Decoding Algorithm Optimization

Motivated by the aforementioned conclusion we now con-sider how to optimize the generalization BP decoding algo-rithm so as to improve the performance of the polar codes

41 Messages Update Functions It is noticed from (4) and(6)ndash(8) that in order to enhance the reliability of messagespropagation it is important to determine the appropriatemessages update functions for the generalization BP decod-ing algorithm However in general it is difficult to find theoptimal messages update functions therefore in this workthe messages update functions are introduced by the analysisof the construction of polar encoding And based on (1) wehave

V119868(119894 2119895) = V

119874(119894 2119895) oplus V

119874(119894 2119895 + 1)

= V119874(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119868(119894 2119895 + 1) = V

119874(119894 2119895 + 1) = V

119874(119894 2119895) oplus V

119868(119894 2119895)

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

= V119868(119894 2119895) oplus V

119874(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1) = V

119868(119894 2119895) oplus V

119874(119894 2119895)

(9)

Therefore based on the error coding theory and (9) wecan get the messages update functions as

1198910

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198911

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

1198912

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198913

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

(10)

That is to say with the probability messages modifyingfactors the new probability messages update equations willbe written as

119901119905

119868119871(1198942119895)= ((119901

119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895)) otimes (119901

119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198711(119894119895)))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895))

otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198712(119894119895)))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198713(119894119895))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198714(119894119895))

otimes (119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198715(119894119895)))

119901119905

119874119877(1198942119895)= ((119901

119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895)) otimes (119901

119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198771(119894119895)))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895))

otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198772(119894119895)))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198773(119894119895))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198774(119894119895))

otimes (119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198775(119894119895)))

(11)

Next our task is to determine the optimal probabilitymessages modifying factors that is 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) so as to achieve the best performance

42 Probability Modifying Factors Optimization In practicedue to the input of frozen bits [1] there exists some so-called frozen nodes in the decoder and values of these nodesare determined and independent of the decoding algorithmHence if the decoding is correct the probabilitymessages of afrozen node V

119868(119894 119895) or V

119874(119894 119895) (the determined value of frozen

nodes is 0) must satisfy the condition of

119901119905

119868119871(119894119895)(0) gt 119901

119905

119868119871(119894119895)(1)

119901119905

119868119877(119894119895)(0) gt 119901

119905

119868119877(119894119895)(1)

or

119901119905

119874119871(119894119895)(0) gt 119901

119905

119874119871(119894119895)(1)

119901119905

119874119877(119894119895)(0) gt 119901

119905

119874119877(119894119895)(1)

(12)

While in practice due to the noise of received signalthere may exist some frozen nodes unsatisfied the reliabilitycondition Therefore in order to improve the accuracy ofdecoding it is needed to find the optimal probability mes-sages modifying factors such that all the frozen nodes couldalways satisfy the conditions of (12) during the procedure ofBP decoding

To achieve above goal we introduce a pair parametersto each node in the decoder which are called left and rightreliability degrees And for a node V

119868(119894 119895) (similar to V

119874(119894 119895))

the left and right reliability degrees are denoted as 120577119905119868119871(119894119895)

and120577119905

119868119877(119894119895) respectively Where if V

119868(119894 119895) is a frozen node we can

get 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

(13)

Mathematical Problems in Engineering 7

otherwise the values of 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)will be given by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

119901119868119871(119894119895) (0)

119901119868119871(119894119895) (1)

gt 1

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(1)

119901119905

119868119871(119894119895)(0)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(1)

119901119905

119868119877(119894119895)(0)

otherwise

(14)

where 120577119905119868119871(119894119895)

gt 1 and 120577119905

119868119877(119894119895)gt 1

The reliability degrees indicate the reliability of the nodersquosdecision value and the larger reliability degree the higherreliability of that value Specially when there is no noise inthe channel the reliability degrees of a node will approachto infinity Based on previous observations the problem ofgetting the optimal probability messages modifying factorscan be formulated as an optimization problem to maximizethe reliability degrees of all the node in decoder while thetarget function is written as

arg max0le120574119905

1198710(119894119896)120574119905

1198715(119894119896)le1

0le120574119905

1198770(119894119896)120574119905

1198775(119894119896)le1

log2119873

sum

119894=0

119873

sum

119895=0

(120577119905

119868119871(119894119895)+ 120577119905

119868119877(119894119895))

st 120574119905

1198710(119894119896)(0) + 120574

119905

1198710(119894119896)(1) = 1

120574119905

1198715(119894119896)(0) + 120574

119905

1198715(119894119896)(1) = 1

120574119905

1198770(119894119896)(0) + 120574

119905

1198770(119894119896)(1) = 1

120574119905

1198775(119894119896)(0) + 120574

119905

1198775(119894119896)(1) = 1

(15)

where 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)are decided by (11) and (13)-(14) and

119896 is the integral value of 1198952 that is 119896 = lfloor1198952rfloor

43 Problem Solving Based on Modified PSO AlgorithmBased on (15) we have formulated the problem to find theoptimal probabilitymodifying factors and in this subsectionwe will consider the solution of the problem so as to get theoptimized BP decoding algorithm

Among the different global optimizers particle swarmoptimization (PSO) has demonstrated its usefulness as anoptimizer capable of finding global optimum The idea ofPSO is based on simulation of simplified social models suchas birds flocking and fish schooling Like other stochasticevolutionary algorithms PSO is independent of mathemat-ical characteristics of objective problems However unlikethese algorithms with PSO each particle has its own unique

memory to ldquorememberrdquo its own best solution Thus eachparticle has its own ldquoideardquo of whether it has found theoptimum However in other algorithms previous knowledgeof the problemwill be destroyed once the population changesPSO has attracted increasing attention because of its simpleconcept easy implementation and quick These advantagesallow it to be applied successfully in a variety of fields mainlyfor unconstrained continuous optimization problems [19ndash26] Therefore in this work we explore the PSO algorithmwith certain modification based on particle fitness evaluationto solve the problem (15)

To present our modified PSO algorithm we first intro-duce some preliminary definitions and concepts used in theoperation of the PSOalgorithmwhich is described as follows

(1) ParametersDefinitionsWe consider a swarmof119871particlesz0 z

119871minus1 where the position of z

119894= (1199111198940 1199111198941 119911

119894(119879minus1))

represents a possible solution point of those probabilitymodifying factors in the problem (15) 0 le 119894 le 119871 minus 1 and119879 is the number of the probability modifying factors Eachelement of z

119894is the value of a probability modifying factor

that is 0 le 119911119894119895le 1 0 le 119895 le 119879 minus 1 Based on the work of [19]

the position of particle z119894can be updated by

z119905+1119894

= z119905119894+ w119905+1119894

(16)

In (16) the pseudovelocity w119905+1119894

is calculated as follows

w119905+1119894

= 120596119905w119905119894+ 11988811199031(p119894minus z119905119894) + 11988821199032(p119892minus z119905119894) (17)

where120596119905 is particle inertia 119905 indicates the iteration number p119894

represents the best ever position of particle 119894 at iteration 119905 andp119892represents the global best position in the swarmat iteration

119905 1199031and 1199032represent uniform random numbers between 0

and 1 It is proven in [19] that when 1198881= 1198882= 2 the products

11988811199031or 11988821199032 have a mean of 1

(2) Fitness Evaluation In general with the parameter defini-tions aforementioned we can get the global optimal solutionof a optimization problem However in problem (15) thesituation will be different because during the procedureof the BP decoding for polar codes a nodersquos probabilitymessages calculated on the basis of the probability modifyingfactors must satisfy the conditions of (12) or (14) In orderto check these conditions during the decoding we furtherdefine two functions as

1198921(z119905119894) =

1003816100381610038161003816V119890

119865

10038161003816100381610038161003816100381610038161003816V119865

1003816100381610038161003816

1198922(z119905119894) =

1003816100381610038161003816V119890

119868

10038161003816100381610038161003816100381610038161003816V119868

1003816100381610038161003816

(18)

where V119865is the frozen nodes set of the decoder and V119890

119865is the

set of frozen nodeswhose probabilitymessages based on z119905119894do

not satisfy the condition of (12) Similarly V119868is the nonfrozen

nodes set of the decoder and V119890119868is the set of nonfrozen nodes

whose left and right reliability degrees based on z119905119894do not

satisfy (14) Furthermore we have 1198921(z119905119894) 1198922(z119905119894) isin [0 1]

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

4 BP Decoding Algorithm Optimization

Motivated by the aforementioned conclusion we now con-sider how to optimize the generalization BP decoding algo-rithm so as to improve the performance of the polar codes

41 Messages Update Functions It is noticed from (4) and(6)ndash(8) that in order to enhance the reliability of messagespropagation it is important to determine the appropriatemessages update functions for the generalization BP decod-ing algorithm However in general it is difficult to find theoptimal messages update functions therefore in this workthe messages update functions are introduced by the analysisof the construction of polar encoding And based on (1) wehave

V119868(119894 2119895) = V

119874(119894 2119895) oplus V

119874(119894 2119895 + 1)

= V119874(119894 2119895) oplus V

119868(119894 2119895 + 1)

V119868(119894 2119895 + 1) = V

119874(119894 2119895 + 1) = V

119874(119894 2119895) oplus V

119868(119894 2119895)

V119874(119894 2119895) = V

119868(119894 2119895) oplus V

119868(119894 2119895 + 1)

= V119868(119894 2119895) oplus V

119874(119894 2119895 + 1)

V119874(119894 2119895 + 1) = V

119868(119894 2119895 + 1) = V

119868(119894 2119895) oplus V

119874(119894 2119895)

(9)

Therefore based on the error coding theory and (9) wecan get the messages update functions as

1198910

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198911

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

1198912

(119894119895)(1205935

0) = ((120593

0⊙ 1205933) otimes (120593

1⊙ 1205934))

⊙ ((1205930⊙ 1205933) otimes (120593

2⊙ 1205935))

1198913

(119894119895)(1205935

0) = (120593

0⊙ 1205933) ⊙ ((120593

1⊙ 1205934) otimes (120593

2⊙ 1205935))

(10)

That is to say with the probability messages modifyingfactors the new probability messages update equations willbe written as

119901119905

119868119871(1198942119895)= ((119901

119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895)) otimes (119901

119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198711(119894119895)))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198710(119894119895))

otimes (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198712(119894119895)))

119901119905

119868119871(1198942119895+1)= (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198713(119894119895))

⊙ ((119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198714(119894119895))

otimes (119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198715(119894119895)))

119901119905

119874119877(1198942119895)= ((119901

119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895)) otimes (119901

119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198771(119894119895)))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198770(119894119895))

otimes (119901119905minus1

119874119871(1198942119895+1)⊙ 120574119905

1198772(119894119895)))

119901119905

119874119877(1198942119895+1)= (119901119905minus1

119868119877(1198942119895+1)⊙ 120574119905

1198773(119894119895))

⊙ ((119901119905minus1

119868119877(1198942119895)⊙ 120574119905

1198774(119894119895))

otimes (119901119905minus1

119874119871(1198942119895)⊙ 120574119905

1198775(119894119895)))

(11)

Next our task is to determine the optimal probabilitymessages modifying factors that is 120574119905

1198710(119894119895) 120574

119905

1198715(119894119895)and

120574119905

1198770(119894119895) 120574

119905

1198775(119894119895) so as to achieve the best performance

42 Probability Modifying Factors Optimization In practicedue to the input of frozen bits [1] there exists some so-called frozen nodes in the decoder and values of these nodesare determined and independent of the decoding algorithmHence if the decoding is correct the probabilitymessages of afrozen node V

119868(119894 119895) or V

119874(119894 119895) (the determined value of frozen

nodes is 0) must satisfy the condition of

119901119905

119868119871(119894119895)(0) gt 119901

119905

119868119871(119894119895)(1)

119901119905

119868119877(119894119895)(0) gt 119901

119905

119868119877(119894119895)(1)

or

119901119905

119874119871(119894119895)(0) gt 119901

119905

119874119871(119894119895)(1)

119901119905

119874119877(119894119895)(0) gt 119901

119905

119874119877(119894119895)(1)

(12)

While in practice due to the noise of received signalthere may exist some frozen nodes unsatisfied the reliabilitycondition Therefore in order to improve the accuracy ofdecoding it is needed to find the optimal probability mes-sages modifying factors such that all the frozen nodes couldalways satisfy the conditions of (12) during the procedure ofBP decoding

To achieve above goal we introduce a pair parametersto each node in the decoder which are called left and rightreliability degrees And for a node V

119868(119894 119895) (similar to V

119874(119894 119895))

the left and right reliability degrees are denoted as 120577119905119868119871(119894119895)

and120577119905

119868119877(119894119895) respectively Where if V

119868(119894 119895) is a frozen node we can

get 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

(13)

Mathematical Problems in Engineering 7

otherwise the values of 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)will be given by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

119901119868119871(119894119895) (0)

119901119868119871(119894119895) (1)

gt 1

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(1)

119901119905

119868119871(119894119895)(0)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(1)

119901119905

119868119877(119894119895)(0)

otherwise

(14)

where 120577119905119868119871(119894119895)

gt 1 and 120577119905

119868119877(119894119895)gt 1

The reliability degrees indicate the reliability of the nodersquosdecision value and the larger reliability degree the higherreliability of that value Specially when there is no noise inthe channel the reliability degrees of a node will approachto infinity Based on previous observations the problem ofgetting the optimal probability messages modifying factorscan be formulated as an optimization problem to maximizethe reliability degrees of all the node in decoder while thetarget function is written as

arg max0le120574119905

1198710(119894119896)120574119905

1198715(119894119896)le1

0le120574119905

1198770(119894119896)120574119905

1198775(119894119896)le1

log2119873

sum

119894=0

119873

sum

119895=0

(120577119905

119868119871(119894119895)+ 120577119905

119868119877(119894119895))

st 120574119905

1198710(119894119896)(0) + 120574

119905

1198710(119894119896)(1) = 1

120574119905

1198715(119894119896)(0) + 120574

119905

1198715(119894119896)(1) = 1

120574119905

1198770(119894119896)(0) + 120574

119905

1198770(119894119896)(1) = 1

120574119905

1198775(119894119896)(0) + 120574

119905

1198775(119894119896)(1) = 1

(15)

where 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)are decided by (11) and (13)-(14) and

119896 is the integral value of 1198952 that is 119896 = lfloor1198952rfloor

43 Problem Solving Based on Modified PSO AlgorithmBased on (15) we have formulated the problem to find theoptimal probabilitymodifying factors and in this subsectionwe will consider the solution of the problem so as to get theoptimized BP decoding algorithm

Among the different global optimizers particle swarmoptimization (PSO) has demonstrated its usefulness as anoptimizer capable of finding global optimum The idea ofPSO is based on simulation of simplified social models suchas birds flocking and fish schooling Like other stochasticevolutionary algorithms PSO is independent of mathemat-ical characteristics of objective problems However unlikethese algorithms with PSO each particle has its own unique

memory to ldquorememberrdquo its own best solution Thus eachparticle has its own ldquoideardquo of whether it has found theoptimum However in other algorithms previous knowledgeof the problemwill be destroyed once the population changesPSO has attracted increasing attention because of its simpleconcept easy implementation and quick These advantagesallow it to be applied successfully in a variety of fields mainlyfor unconstrained continuous optimization problems [19ndash26] Therefore in this work we explore the PSO algorithmwith certain modification based on particle fitness evaluationto solve the problem (15)

To present our modified PSO algorithm we first intro-duce some preliminary definitions and concepts used in theoperation of the PSOalgorithmwhich is described as follows

(1) ParametersDefinitionsWe consider a swarmof119871particlesz0 z

119871minus1 where the position of z

119894= (1199111198940 1199111198941 119911

119894(119879minus1))

represents a possible solution point of those probabilitymodifying factors in the problem (15) 0 le 119894 le 119871 minus 1 and119879 is the number of the probability modifying factors Eachelement of z

119894is the value of a probability modifying factor

that is 0 le 119911119894119895le 1 0 le 119895 le 119879 minus 1 Based on the work of [19]

the position of particle z119894can be updated by

z119905+1119894

= z119905119894+ w119905+1119894

(16)

In (16) the pseudovelocity w119905+1119894

is calculated as follows

w119905+1119894

= 120596119905w119905119894+ 11988811199031(p119894minus z119905119894) + 11988821199032(p119892minus z119905119894) (17)

where120596119905 is particle inertia 119905 indicates the iteration number p119894

represents the best ever position of particle 119894 at iteration 119905 andp119892represents the global best position in the swarmat iteration

119905 1199031and 1199032represent uniform random numbers between 0

and 1 It is proven in [19] that when 1198881= 1198882= 2 the products

11988811199031or 11988821199032 have a mean of 1

(2) Fitness Evaluation In general with the parameter defini-tions aforementioned we can get the global optimal solutionof a optimization problem However in problem (15) thesituation will be different because during the procedureof the BP decoding for polar codes a nodersquos probabilitymessages calculated on the basis of the probability modifyingfactors must satisfy the conditions of (12) or (14) In orderto check these conditions during the decoding we furtherdefine two functions as

1198921(z119905119894) =

1003816100381610038161003816V119890

119865

10038161003816100381610038161003816100381610038161003816V119865

1003816100381610038161003816

1198922(z119905119894) =

1003816100381610038161003816V119890

119868

10038161003816100381610038161003816100381610038161003816V119868

1003816100381610038161003816

(18)

where V119865is the frozen nodes set of the decoder and V119890

119865is the

set of frozen nodeswhose probabilitymessages based on z119905119894do

not satisfy the condition of (12) Similarly V119868is the nonfrozen

nodes set of the decoder and V119890119868is the set of nonfrozen nodes

whose left and right reliability degrees based on z119905119894do not

satisfy (14) Furthermore we have 1198921(z119905119894) 1198922(z119905119894) isin [0 1]

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

otherwise the values of 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)will be given by

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(0)

119901119905

119868119871(119894119895)(1)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(0)

119901119905

119868119877(119894119895)(1)

119901119868119871(119894119895) (0)

119901119868119871(119894119895) (1)

gt 1

120577119905

119868119871(119894119895)=

119901119905

119868119871(119894119895)(1)

119901119905

119868119871(119894119895)(0)

120577119905

119868119877(119894119895)=

119901119905

119868119877(119894119895)(1)

119901119905

119868119877(119894119895)(0)

otherwise

(14)

where 120577119905119868119871(119894119895)

gt 1 and 120577119905

119868119877(119894119895)gt 1

The reliability degrees indicate the reliability of the nodersquosdecision value and the larger reliability degree the higherreliability of that value Specially when there is no noise inthe channel the reliability degrees of a node will approachto infinity Based on previous observations the problem ofgetting the optimal probability messages modifying factorscan be formulated as an optimization problem to maximizethe reliability degrees of all the node in decoder while thetarget function is written as

arg max0le120574119905

1198710(119894119896)120574119905

1198715(119894119896)le1

0le120574119905

1198770(119894119896)120574119905

1198775(119894119896)le1

log2119873

sum

119894=0

119873

sum

119895=0

(120577119905

119868119871(119894119895)+ 120577119905

119868119877(119894119895))

st 120574119905

1198710(119894119896)(0) + 120574

119905

1198710(119894119896)(1) = 1

120574119905

1198715(119894119896)(0) + 120574

119905

1198715(119894119896)(1) = 1

120574119905

1198770(119894119896)(0) + 120574

119905

1198770(119894119896)(1) = 1

120574119905

1198775(119894119896)(0) + 120574

119905

1198775(119894119896)(1) = 1

(15)

where 120577119905119868119871(119894119895)

and 120577119905

119868119877(119894119895)are decided by (11) and (13)-(14) and

119896 is the integral value of 1198952 that is 119896 = lfloor1198952rfloor

43 Problem Solving Based on Modified PSO AlgorithmBased on (15) we have formulated the problem to find theoptimal probabilitymodifying factors and in this subsectionwe will consider the solution of the problem so as to get theoptimized BP decoding algorithm

Among the different global optimizers particle swarmoptimization (PSO) has demonstrated its usefulness as anoptimizer capable of finding global optimum The idea ofPSO is based on simulation of simplified social models suchas birds flocking and fish schooling Like other stochasticevolutionary algorithms PSO is independent of mathemat-ical characteristics of objective problems However unlikethese algorithms with PSO each particle has its own unique

memory to ldquorememberrdquo its own best solution Thus eachparticle has its own ldquoideardquo of whether it has found theoptimum However in other algorithms previous knowledgeof the problemwill be destroyed once the population changesPSO has attracted increasing attention because of its simpleconcept easy implementation and quick These advantagesallow it to be applied successfully in a variety of fields mainlyfor unconstrained continuous optimization problems [19ndash26] Therefore in this work we explore the PSO algorithmwith certain modification based on particle fitness evaluationto solve the problem (15)

To present our modified PSO algorithm we first intro-duce some preliminary definitions and concepts used in theoperation of the PSOalgorithmwhich is described as follows

(1) ParametersDefinitionsWe consider a swarmof119871particlesz0 z

119871minus1 where the position of z

119894= (1199111198940 1199111198941 119911

119894(119879minus1))

represents a possible solution point of those probabilitymodifying factors in the problem (15) 0 le 119894 le 119871 minus 1 and119879 is the number of the probability modifying factors Eachelement of z

119894is the value of a probability modifying factor

that is 0 le 119911119894119895le 1 0 le 119895 le 119879 minus 1 Based on the work of [19]

the position of particle z119894can be updated by

z119905+1119894

= z119905119894+ w119905+1119894

(16)

In (16) the pseudovelocity w119905+1119894

is calculated as follows

w119905+1119894

= 120596119905w119905119894+ 11988811199031(p119894minus z119905119894) + 11988821199032(p119892minus z119905119894) (17)

where120596119905 is particle inertia 119905 indicates the iteration number p119894

represents the best ever position of particle 119894 at iteration 119905 andp119892represents the global best position in the swarmat iteration

119905 1199031and 1199032represent uniform random numbers between 0

and 1 It is proven in [19] that when 1198881= 1198882= 2 the products

11988811199031or 11988821199032 have a mean of 1

(2) Fitness Evaluation In general with the parameter defini-tions aforementioned we can get the global optimal solutionof a optimization problem However in problem (15) thesituation will be different because during the procedureof the BP decoding for polar codes a nodersquos probabilitymessages calculated on the basis of the probability modifyingfactors must satisfy the conditions of (12) or (14) In orderto check these conditions during the decoding we furtherdefine two functions as

1198921(z119905119894) =

1003816100381610038161003816V119890

119865

10038161003816100381610038161003816100381610038161003816V119865

1003816100381610038161003816

1198922(z119905119894) =

1003816100381610038161003816V119890

119868

10038161003816100381610038161003816100381610038161003816V119868

1003816100381610038161003816

(18)

where V119865is the frozen nodes set of the decoder and V119890

119865is the

set of frozen nodeswhose probabilitymessages based on z119905119894do

not satisfy the condition of (12) Similarly V119868is the nonfrozen

nodes set of the decoder and V119890119868is the set of nonfrozen nodes

whose left and right reliability degrees based on z119905119894do not

satisfy (14) Furthermore we have 1198921(z119905119894) 1198922(z119905119894) isin [0 1]

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

InputThe stopping parameters 120598

1and 1205982

The frozen nodes set V119865

The probability messages of the input nodes of the decoder 119882 (1199100| 1199090) 119882 (119910

119873minus1| 119909119873minus1

)Output

The decoded source binary vector 119873minus10

(1) Initialize the parameters used in the PSO algorithm including particles initial set z0

0 z0

119871minus1

the pseudovelocities w00 w0

119871minus1 particle inertia 1205960 maximum iteration number and 119888

1 1198882

(2) while (Stopping criterion is not satisfied) do(3) Update particle pseudovelocity vector w119905+1

119894using (17)

(4) Update particle position vector z119905+1119894

using (16)(5) Calculate the probability messages of each nodes in the decoder using (11)(6) if 119892

1(z119905+1119894

) gt 1205981and 119892

2(z119905+1119894

) gt 1205982then

(7) Calculate the fitness value ℎ119905+1119894

using (19)(8) if ℎ119905+1

119894ge ℎ119887119890119904119905

119894then

(9) ℎ119887119890119904119905

119894= ℎ119905+1

119894 p119894= z119905+1119894

(10) end if(11) if ℎ119905+1

119894ge ℎ119887119890119904119905

119892then

(12) ℎ119887119890119904119905

119892= ℎ119905+1

119894 p119892= z119905+1119894

(13) end if(14) end if(15) end while(16) Output the decoded source binary vector by hard decision of the output nodes of the decoder(17) return C

Algorithm 1 BP decoding algorithm based on modified PSO optimization

Based on (18) we can evaluate the fitness of a particlersquos currentposition so as to accelerate the finding of the the optimalsolution After the checking of conditions of (12) and (14) wefurther get the fitness value ℎ119905

119894of a particle by

ℎ119905

119894(z119905119894) =

log2119873

sum

119896=0

119873

sum

119895=0

(120577119905

119868119871(119896119895)+ 120577119905

119868119877(119896119895)) (19)

As the summarization of previous description we nowprovide the whole procedure of the proposed BP decodingalgorithm based on PSO optimization with the form ofpseudocode as shown in Algorithm 1

It is noticed form Algorithm 1 that the steps 2ndash15 are thekey of the algorithm The main consideration of these stepsis based on the following two observations On the one handif there is no noise in the channel the BP decoding of thepolar codes will be errorless which means that all the nodesin the decoder will satisfy the condition of (12) that is 119892

1(z119905119894)

and 1198921(z119905119894) are both equal to 0 However in practice this

case is impossible to happen due to the noise of channelhence we introduce two stopping parameters 120598

1and 1205982for

1198921(z119905119894) and 119892

1(z119905119894) Since the values of the frozen nodes are

known to the decoder we set 1205982to 0 To ensure that the

correction of the decoding 1205981should be as small as possible

such that when a particle z119905119894is found to make 119892

1(z119905+1119894

) le 1205981

and 1198922(z119905+1119894

) le 1205982both true which indicates that stopping

criterion is satisfied the loop of (2)ndash(15) can be stopped Onthe other hand when there exist multiparticles that make thestopping criterion satisfied we can choose the one which can

generate maximum fitness value to be our final solution asshown in steps (11)ndash(13)

5 Numerical Results

In this section Monte Carlo simulation is provided to showthe performance of the proposed decoding algorithm Inthe simulation the BPSK modulation and the additive whiteGaussian noise (AWGN) channel are assumed The codelength is 119873 = 2

3= 8 code rate 119877 is 05 and the index of

the information bits is the same as [1]As it is shown in Figure 5 the proposed decoding algo-

rithm based on PSO achieves better bit error rate (BERdefined as as the ratio of error decoding bits to the totaldecoding bits) performance than that of the existing BPdecoding Particularly in the low region of signal to noiseratio (SNR) that is 119864

1198871198730(119864119887is the average power of one

bit and 1198730is the average value of the noise power spectrum

density) the proposed algorithm provides a higher SNRadvantage for example when the BER is 10minus3 the proposeddecoding algorithm provides SNR advantages of 09 dB andwhen the BER is 10minus4 the SNR advantages are 06 dB Hencewe can conclude that with the proposed generalization BPdecoding and PSO performance of the belief propagationdecoding algorithm for polar codes could be improved

Additionally one important thing to note is that sincethe proposed algorithm is implemented depending on theprobability messages modifying factors while values of thesefactors are optimized by the PSO algorithm Hence the timecomplexity of the existing BP decoding is lower than that of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Existing BP decodingProposed BP decoding based on PSO

EbN0 (dB)0 1 2 3 4 5 6 7 8

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

BER

Figure 5 Comparison of BER between the proposed and existingBP decoding algorithms for polar codes with (4 8) The maximumiteration number is set to 60

the proposed decoding which is tightly depending on theconvergence time of the PSO algorithm And the work for thetime complexity optimization of the proposed algorithm willbe conducted in our future research

6 Conclusion

In this work we proposed a generalization belief propagation(BP) decoding algorithm based on particle swarm optimiza-tion (PSO) to improve the performance of the polar codesThrough the analysis of the existing BP decoding algorithmfor polar codes we first introduced a probability modifyingfactor to each node of the BP decoder so as to enhance theerror-correcting capacity of the BP decoding Then we gen-eralized the BP decoding algorithmbased on thesemodifyingfactors and drive the probability update equations for theproposed generalization BP decoding Based on these newprobability update equations we further proved the intrinsicrelationship of the existing decoding algorithms Finally inorder to achieve the best performance we formulated anoptimization problem to find the optimal probability mod-ifying factors for the proposed generalization BP decodingFurthermore a method based on the modified PSO algo-rithmwas also introduced to solve the optimization problemNumerical results showed that the proposed generalizationBP decoding algorithm achieved better performance thanthat of the existing decoding which suggested the effective-ness of the proposed decoding algorithm

Conflict of Interests

The authors declare that they do not have any commercialor associative interests that represent a conflict of interests inconnection with the work submitted

References

[1] E Arıkan ldquoChannel polarization a method for constructingcapacity-achieving codes for symmetric binary-input memory-less channelsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 55 no 7 pp 3051ndash30732009

[2] EArikan andE Telatar ldquoOn the rate of channel polarizationrdquo inProceedings of the IEEE International Symposiumon InformationTheory (ISIT rsquo09) pp 1493ndash1495 Seoul South Korea July 2009

[3] S B Korada E Sasoglu and R Urbanke ldquoPolar codes charac-terization of exponent bounds and constructionsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 56 no 12 pp 6253ndash6264 2010

[4] I Tal andAVardy ldquoList decoding of polar codesrdquo inProceedingsof the IEEE International Symposium on Information Theory(ISIT rsquo11) pp 1ndash5 St Petersburg Russia August 2011

[5] I Tal and A Vardy ldquoList decoding of polar codesrdquo httparxivorgabs12060050

[6] K Chen K Niu and J R Lin ldquoImproved successive cancella-tion decoding of polar codesrdquo IEEE Transactions on Communi-cations vol 61 no 8 pp 3100ndash3107 2013

[7] KNiu andKChen ldquoCRC-aided decoding of polar codesrdquo IEEECommunications Letters vol 16 no 10 pp 1668ndash1671 2012

[8] A Alamdar-Yazdi and F R Kschischang ldquoA simplifiedsuccessive-cancellation decoder for polar codesrdquo IEEE Commu-nications Letters vol 15 no 12 pp 1378ndash1380 2011

[9] G Sarkis and W J Gross ldquoIncreasing the throughput of polardecodersrdquo IEEE Communications Letters vol 17 no 4 pp 725ndash728 2013

[10] G Sarkis P Giard A Vardy C Thibeault and W J GrossldquoFast polar decoders algorithm and implementationrdquo httparxivorgabs13066311

[11] P Giard G Sarkis CThibeault andW J Gross ldquoA fast softwarepolar decoderrdquo httparxivorgabs13066311

[12] R G Gallager ldquoLow-density parity-check codesrdquo IRE Transac-tions on Information Theory vol IT-8 pp 21ndash28 1962

[13] E Arikan ldquoA performance comparison of polar codes and reed-muller codesrdquo IEEE Communications Letters vol 12 no 6 pp447ndash449 2008

[14] N Hussami S B Korada and R Urbanke ldquoPerformance ofpolar codes for channel and source codingrdquo in Proceedings ofthe IEEE International Symposium on Information Theory (ISITrsquo09) pp 1488ndash1492 Seoul South Korea July 2009

[15] E Arikan ldquoPolar codes a pipelined implementationrdquo in Pro-ceedings of the 4th International Symposium on BroadbandCommunication (ISBC rsquo10) pp 11ndash14 July 2010

[16] A Eslami and H Pishro-Nik ldquoOn bit error rate performanceof polar codes in finite regimerdquo in Proceedings of the 48thAnnual Allerton Conference on Communication Control andComputing Allerton (AACCCC rsquo10) pp 188ndash194 Allerton IllUSA October 2010

[17] A Eslami and H Pishro-Nik ldquoOn finite-length performance ofpolar codes stopping sets error floor and concatenateddesignrdquohttparxivorgabs12112187

[18] D Divsalar and C Jones ldquoProtograph LDPC codes withnode degrees at least 3rdquo in Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM rsquo06) pp 1ndash5 SanFrancisco Calif USA November 2006

[19] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[20] C-L Sun J-C Zeng and J-S Pan ldquoAn improved vectorparticle swarmoptimization for constrained optimization prob-lemsrdquo Information Sciences vol 181 no 6 pp 1153ndash1163 2011

[21] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[22] X Hu and R C Eberhart ldquoSolving constrained nonlinearoptimization problems with particle swarm optimizationrdquo inProceedings of the 6th World Multiconference on SystemicsCybernetics and Informatics 2002

[23] F Y H Ahmed S M Shamsuddin and S Z M HashimldquoImproved SpikeProp for using particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID257085 13 pages 2013

[24] A Y Alanis E Rangel J Rivera N Arana-Daniel and CLopez-Franco ldquoParticle swarm based approach of a real-timediscrete neural identifier for linear induction motorsrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 715094 9pages 2013

[25] A M Arasomwan and A O Adewumi ldquoAn investigation intothe performance of particle swarm optimization with variouschaotic mapsrdquoMathematical Problems in Engineering vol 2014Article ID 178959 17 pages 2014

[26] N G Hedeshi and M S Abadeh ldquoCoronary artery diseasedetection using a fuzzy-boosting PSO approachrdquoMathematicalProblems in Engineering vol 2014 Article ID 783734 12 pages2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of