Study on the MAP-Based Decoding Algorithm of Turbo Codes over the Asymmetric Channel

Xingcheng Liu, Jingying Zhou Department of Electronic and Communications Engineering

Sun Yat-sen University Guangzhou, Guangdong, 510275, China

isslxc@mail.sysu.edu.cn

Abstract—The asymmetric Z-channel is a discrete memoryless channel with one of the input symbols transmitted with noise. In this paper, based on the maximum a posterior probability (MAP) decoding algorithm, a novel decoding algorithm for the asymmetric Z-channel is proposed and the performance of turbo codes with the proposed algorithm is simulated. The simulation results show that the proposed turbo decoding algorithm can achieve excellent bit error rate performance over the asymmetric Z-channel.

Keywords-Z-Channel; Turbo Codes; MAP Algorithm; Decoding Algorithm; Bit Error Rate (BER)

I. INTRODUCTION Since the invention of turbo codes [1] in 1993, turbo codes

have obtained much attention due to their near-Shannon limit performance. Therefore, turbo codes have been widely studied for many actual communication systems, such as mobile satellite communication, digital audio and video broadcasting, deep-space communication, UMTS and cdma2000 systems and so on [2-4]. Besides, turbo codes have been applied in the field of information security, such as the video and image encryption and digital watermark technology. Meanwhile, some inherent concepts of turbo codes have been employed in the field of joint source-channel coding [5].

Decoding performance is varied for different transmission channels with different coding methods. Intuitively, a proper decoding algorithm can be chosen to adapt to corresponding channel. Although turbo codes have been widely studied over an additive white Gaussian noise (AWGN) channel and fading channels [2-7], very few studies have been done over the asymmetric channels. This paper aims at studying the turbo decoding algorithm over the asymmetric channels. Asymmetric errors are typical in optical communications because, upon transmission, photons may fade or decay, but new photons cannot be generated. Also, the most likely faults that affect address decoders and word lines in very large scale integration (VLSI) memories and stuck-faults in a serial bus will cause asymmetric errors [8]. The commonly used channel model for this type of errors is the asymmetric Z-channel [9] [10]. Experiments show that the iterative decoding of turbo codes can be employed to the Z-channel.

This paper is organized as follows. In Section Ⅱ, the structures of the turbo encoder are introduced, followed by the introduction of the turbo decoder and MAP algorithm. In Section Ⅳ, the asymmetric Z-channel model is introduced, and the turbo MAP decoding algorithm for the Z-channel is developed. The simulation results are presented in Section Ⅴ. Finally, the conclusion is drawn.

II. TURBO ENCODER Typical turbo codes are parallel concatenated convolutional

codes (PCCC), in which two identical recursive systematic convolutional (RSC) encoders are joined through an interleaver. The encoded parity bit sequence can be punctured through a puncturing matrix and codewords sequence with various code rates can be obtained.

The turbo encoder is shown in Figures 1 and 2. The input information bit sequence {dk} inputs to the encoder RSC1, resulting in a parity bit sequence {v1k}. Meanwhile, the sequence {dk} passes through the interleaver, which produces the position-permuted output sequence {d2k}. The sequence {d2k} passes through the encoder RSC2 and produces a parity bit sequence {v2k}. The sequence {dk} is directly transmitted to the multiplexer to produce the systematic output sequence {v0k}. In order to obtain higher code rates, the parity bit sequences {v1k} and {v2k} must be punctured periodically and forms a new parity sequence {vk}. At last, the sequences {v0k} and {vk} are passed through a multiplexer and a turbo codes sequence {v0k, vk} is generated.

Figure 1. A turbo encoder with vk puncturing

In Figure 2, a switch in the encoder is in position “A” for the first N clock cycles and in position “B” for m additional cycles, where m is the memory length of the encoder. After the

This work was supported by the National Natural Science Foundation of China (No. 60673086, 60711140419), the Science and Technology Plan of Guangdong Province of China (No. 2006B50101003), and the Science and Technology Plan of Guangzhou City of China (2007Z3-D0071).

Figure 2. (37, 21) RSC trellis termination

m tail bits are transmitted, the encoder RSC1 will come to the all-zero state. The input to the encoder RSC2 will be the interleaved sequence of the input to the encoder RSC1. Usually, only the first component encoder is forced to return to the all-zero state [11].

III. TURBO DECODER A key development in turbo codes is the iterative decoding

algorithm. By exchanging extrinsic information between the component decoders, turbo codes significantly improve the decoding performance. As shown in figure 3, the iterative decoder consists of two soft-input/soft-output decoders DEC1 and DEC2, which are serially concatenated via an interleaver. The essence of iterative decoding is that each decoder processes its own data, and passes the extrinsic information to the other. The process of exchanging extrinsic information back and forth between the decoders continues until iteration numbers is reached or other stopping conditions are met [13].

As shown in figure 3, the component decoder DEC1 is associated with the encoder RSC1 and produces soft output, in which extrinsic information exists. The extrinsic information

1 ( )e kdΛ of DEC1 is interleaved and passed to the component decoder DEC2, where it is used as a priori probability. The decoder DEC2 is associated with the encoder RSC2 and produces a likelihood ratio for each input information bit. The extrinsic information 2 ( )e kdΛ of DEC2 is deinterleaved and passed back to the decoder DEC1 as a priori probability. Then the second cycle of iterative decoding can begin. In this way, each decoder takes advantage of the extrinsic information from the other one as suggestions to improving the decoding results. After a number of decoding iterations, the extrinsic information becomes more and more stable and the likelihood ratio of the probabilities is close to that in the maximum-likelihood decoding. Finally, the decoder can make a hard decision and get an estimated sequence by comparing the likelihood ratio to a threshold.

1 ( )e kdΛ

2 ( )e kdΛ

$kd

1 ( )e kdΛ

2 ( )e kdΛ

2 ( )kdΛ

Figure 3. Iterative decoder based on the MAP algorithm

Now let us consider the MAP decoding algorithm for the

turbo encoder shown in figure 1. For a given input information sequence with N bits, 1 1 2{ }, , , , ,N

k Nd d d d d= L L , the output

codeword sequence of the encoder, 1 1 2{ , , , , ,N

kC C C C= L L }NC , is input to an asymmetric Z-channel. After transmitted

over the channel and contaminated by noises, an output sequence 1 1 2{ , , , , , }N

k NR R R R R= L L is obtained. Here,

0( , )k k kC v v= , ( , )k k kR x y= , k is the time index, xk is the received information corresponding to 0kv , yk is the received parity information and sent to decoder DEC1 as y1k

corresponding to 1kv , and to decoder DEC2 as y2k

corresponding to 2kv respectively, as shown in Figure 3. The log likelihood ratio (LLR) ( )kdΛ corresponding to bit dk can be written as [1]

1

1

Pr{ 1| }( ) log

Pr{ 0 | }

Nk

k Nk

d Rd

d R=

Λ ==

. (1)

Finally, the decoder can make a hard decision by comparing ( )kdΛ to a threshold defined by

$ 10

kd ⎧= ⎨⎩

( ) 0( ) 0

k

k

dd

Λ ≥

Λ <. (2)

In order to compute the LLR, ( )kdΛ , we denote the encoder state at time k by kS m= and at time 1k − by 1 'kS m− = . The probability functions,

( )k

i mα , ( )k mβ and ( , ', )i kR m mγ ,can be defined as[1]:

11

1

Pr{ , , }( ) Pr{ , / }Pr{ }k

ki kk k

k kk

d i S m Rm d i S m RR

α = == ⋅ = =

1

1

1

' 01 1

' 0 0

( , ', ) ( '),

( , ', ) ( ')

k

k

ji k

m j

ji k

m m i j

R m m m

R m m m

γ α

γ α

−

−

=

= =

=∑∑

∑∑∑∑ (3)

For k = 0, we have the boundary conditions

0 0(0) 1, ( ) 0, 0, 0,1.i i m m iα α= = ∀ ≠ =

1

1 11

1 1' 0

1 1

1' 0 0

Pr{ | }( )Pr{ | }

( , , ') ( '),

( , ', ) ( ')k

Nk k

k N kk

i k km i

ji k

m m i j

R S mmR R

R m m m

R m m m

β

γ β

γ α

+

+

+ +=

+= =

==

=∑∑

∑∑∑∑

(4)

If the encoder terminates in state zero, the boundary conditions are (0) 1Nβ = , ( ) 0, 0.N m mβ = ∀ ≠ else ( ) 1, .N m mβ = ∀

1

1

1 1

( , ', ) Pr{ , , | '}Pr{ | , , '}Pr{ | , '} Pr{ | '}Pr{ | } Pr{ }, 0,1

i k k k k k

k k k k

k k k k k

k k k

R m m d i R S m S mR d i S m S md i S m S m S m S mR C d i i

γ −

−

− −

= = = == = = = ⋅

= = = ⋅ = == ⋅ = =

(5)

Using equations (1) – (5), we obtain the LLR of bit dk [1]

1

1

1

1' 0

1

0' 0

( ') ( , ', ) ( )( ) log

( ') ( , ', ) ( )

k

k

jk k

m m jk

jk k

m m j

m R m m md

m R m m m

α γ β

α γ β

−

−

=

=

Λ =∑∑∑

∑∑∑. (6)

Equations (1) ~ (6) form the turbo MAP decoding algorithm.

IV. DECODING ALGORITHM ON Z-CHANNEL The traditional decoding algorithm for Turbo codes on the

AWGN has to be modified to adapt to the Z-channel. The below is discussed this algorithm.

A. Decoding Algorithm for the Z-Channel Transmission

Figure 4. The Z-channel

The Z-channel [9] [10], shown in figure 4, is a discrete

memoryless channel with two input symbols. One symbol, x1, is transmitted with noise and received as either y1 or y2, while the other input symbol x2 is transmitted without noise and received as y2. The channel matrix is given by the following

( ) ( )( ) ( )

( ) ( )( ) ( )

1 1 2 1

1 2 2 2

| | 0 | 0 1 | 0 1-0 1| | 0 |1 1 |1

P y x P y x P P p pP y x P y x P P

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦,

(7) where p is the Z-channel transfer probability. Based on the probability theory, we can develop the conditional probability.

{ }

2

2

(1 ) , for 00 00(1 ),for 00 01; or 00 10

(1 ),for 01; or 10

Pr | ,for 00 11,for 01 11; or 10 11

1,for 11 11 0, otherwise

k k

k k k k

k k k k

k k k k

k k k k

k k

p C Rp p C R C R

p C R C R

R C p C Rp C R C R

C R

⎧ − = =⎪ − = = = =⎪⎪ − = = = =⎪⎪= = =⎨⎪ = = = =⎪

= =⎪⎪⎪⎩

，

， ，

，

， ，

，

(8)

Since the code is systematic, xk is independent of the trellis and state m. So we have [1],

1

1

1

Pr{ | , , '}Pr{ | , , '}Pr{ | , , '}

k k k k

k k k k

k k k k

R d i S m S mx d i S m S my d i S m S m

−

−

−

= = =

= = = =⋅ = = =

1Pr{ | } Pr{ | , , '}k k k k k kx d i y d i S m S m−= = ⋅ = = =

Pr{ | } ( , ', ) ,k k i kx d i y m mγ= = ⋅ (9)

Thus, the probability function ( , ', )i kR m mγ can be rewritten as [1] ,

( , ', ) Pr{ | } ( , ', ) Pr{ }i k k k i k kR m m x d i y m m d iγ γ= = ⋅ ⋅ = . (10)

( )kdΛ could be further decomposed into

1

1 1' 0

1

1 1' 0

Pr{ | 1} Pr{ 1}( ) logPr{ | 0} Pr{ 0}

( , ', ) ( ') ( )

( , ', ) ( ') ( )

Pr{ 1} Pr{ | 1}log log ( ).Pr{ 0} Pr{ | 0}

k k kk

k k k

jk k k

m m j

jk k k

m m j

k k ke k

k k k

x d ddx d d

y m m m m

y m m m m

d x d dd x d

γ α β

γ α β

−=

−=

= ⋅ = ⋅→Λ =

= ⋅ = ⋅→

= == + + Λ

= =

∑∑∑

∑∑∑

(11)

where Pr{ 1}

logPr{ 0}

k

k

dd

==

is called a priori probability and [1]

1

1 1' 0

1

1 1' 0

( , ', ) ( ') ( )( ) log

( , ', ) ( ') ( )

jk k k

m m je k

jk k k

m m j

y m m m md

y m m m m

γ α β

γ α β

−=

−=

Λ =∑∑∑

∑∑∑. (12)

( )e kdΛ is the extrinsic information. It is a function of the redundant information supplied by the previous component decoder. It does not contain the information decoder input dk.

Consider the information is transmitted over the asymmetric Z-channel with the transfer probability p, as shown in figure 4, we obtain

Pr{ | 1} ,Pr{ | 0}

k k k

k k

x d xx d p

==

= (13)

We can use the above equations and factorize ( )kdΛ to

Pr{ 1}( ) log log ( ).Pr{ 0}

k kk e k

k

d xd dd p

=Λ = + + Λ

= (14)

B. Iterative Decoding for the Z-Channel The decoder produces extrinsic information [11], which is

interleaved (or de-interleaved) and used to produce an improved estimate of the a priori probabilities (Pr {dk = 1} or Pr {dk = 0}) of the information sequence for the other component decoder.

At the beginning of decoding (the first half-iteration), decoder DEC1 estimates the LLR, 1( )kdΛ . Similarly to equation (14), the log likelihood ratio for decoder DEC1 can be decomposed into

1 1Pr{ 1}( ) log log ( )Pr{ 0}

k kk e k

k

d xd dd p

=Λ = + + Λ

=. (15)

The extrinsic information 1 ( )e kdΛ (see figure 3) is generated by component decoder DEC1 and 1 ( )e kdΛ% is the interleaved extrinsic information that is taken as the a priori probability estimate for decoder DEC2. Because,

1Pr{ 1}( ) logPr{ 0}

ke k

k

ddd

=Λ =

=% , (16)

and

Pr{ 1} 1 Pr{ 0}k kd d= = − = , (17)

we obtain ( )

( )

1

1Pr{ 1}

1

e k

e k

d

k d

ede

Λ

Λ= =

+

%

% , (18)

and

( )1

1Pr{ 0} .1 e k

k dd

eΛ= =

+% (19)

After the first half-iterative decoding, decoder DEC2 estimates the LLR, 2 ( )kdΛ ,

22 2

Pr{ 1}( ) log log ( )Pr{ 0}

k kk e k

k

d xd dd p

=Λ = + + Λ

=. (20)

By substituting the a priori probabilities from equation (16) in equation (20), we get

22 1 2( ) ( ) log ( )k

k e k e kxd d dp

Λ = Λ + + Λ% . (21)

The extrinsic information 2 ( )e kdΛ is generated by component decoder DEC2 and 2 ( )e kdΛ% is the interleaved extrinsic information that can be used as the estimates of the a priori probabilities for decoder DEC1 as in equation (16). The log likelihood ratio for decoder DEC1 can be written as

1 2 1( ) ( ) log ( )kk e k e k

xd d dp

Λ = Λ + + Λ% (22)

Let the extrinsic information produced by decoder j, at the rth iteration be denoted by ( )( )r

je kdΛ , {1,2}j∈ . Let the log likelihood ratio produced by decoder j at the rth iteration be denoted by ( )( )r

j kdΛ . The turbo iterative decoding algorithm can be performed as follows:

1. The extrinsic information input to decoder DEC1 is initialized, ( )(0)

2 0e kdΛ = .

2. For iterations r = 1,2…I, where I is the total number of iterations:

(a) The LLR ( )1 ( )r

kdΛ and ( )2 ( )r

kdΛ are generated

for decoder DEC1 and DEC2, respectively, by equation (6).

(b) The extrinsic information ( )1 ( )re kdΛ is

generated for decoder DEC1 by computing the equation (22) as

( ) ( ) ( 1)1 1 2( ) ( ) log ( )r r rke k k e k

xd d dp

−Λ = Λ − −Λ% , (23)

(c) The extrinsic information ( )2 ( )r

e kdΛ is generated for decoder DEC2 by computing the equation (21) as

( ) ( ) ( )22 2 1( ) ( ) log ( )r r rk

e k k e kxd d dp

Λ = Λ − −Λ% . (24)

(d) r is increased by 1. If r I≤ , turns to step (a), else turns to step (e).

A hard decision on dk is made based on ( )2 ( )I

kdΛ .

V. SIMULATION RESULTS AND PERFORMANCE ANALYSIS Many simulations are carried out to evaluate the

performance of turbo codes with the proposed decoding algorithm over the Z-channel. The simulations are performed on PC of Celeron 2.53GHz/ 512MB RAM/ Windows XP Professional/ VC++6.0. All simulation results are given in the form of BER versus the channel transfer probability p.

To investigate the performance of turbo codes with different code rates over the Z-channel, turbo codes with component codes having octal generator polynomials (37, 21) are employed, where the matching interleaver [12] of length N = 65526 and the asymmetric Z-channel are adopted. The simulation results are shown in Figures 5 and 6 with overall code rates 1/2 and 1/3, respectively.

0.020.040.060.080.10.120.140.1610-7

10-6

10-5

10-4

10-3

10-2

10-1

100

p

BE

R

Uncoded1 iteration2 iterations3 iterations4 iterations5 iterations6 iterations10 iterations

Figure 5. Performance of turbo codes over Z-channel with code rate R = 1/2

As shown in the figures, the BER curves decrease with the

increase of iteration numbers. When BER is higher (BER ≥ 10-

2), iteration numbers has no significant influence on the BER performance. However, in the low BER region (BER < 10-2),

0.150.20.250.30.350.40.450.510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

p

BE

R

Uncoded1 iteration2 iterations3 iterations4 iterations5 iterations6 iterations10 iterations

Figure 6. Performance of turbo codes over Z-channel with code rate R = 1/3

the BER curves are descending rapidly with the decrease of the channel transfer probability p. It is obvious that the iterative decoding can improve the BER performance by exchanging the extrinsic information. But figures 5 and 6 also show that the performance improvements become smaller and smaller with the increase of iteration numbers. This is because with the increase of iteration numbers, the correlation between the extrinsic information and the input information increases accordingly. Therefore the error correction capability with the aid of the extrinsic information becomes weak, and any further iterative decoding results in very little improvement in performance. Therefore, it is important to devise an efficient criterion to stop the iteration process and prevent unnecessary computations in order to shorten decoding delay [13].

It is also shown in figures 5 and 6 that decreasing the code rates can improve the performance of the turbo codes. For instance, when the number of iterations is 6 and the BER = 2×10-6, p is 0.136 for the code rate R = 1/2, while p is as high as 0.4 for R = 1/3. Therefore, the more parity bits are punctured, the smaller channel transfer probability p is needed, which means a better channel required.

VI. CONCLUSION Based on the characteristics of asymmetric Z-channel and

turbo codes, a modified MAP decoding algorithm for the turbo codes over the asymmetric Z-channel is proposed for the first time. Our simulations show that at the BER of 10-5 and

iteration 6, the required transfer probability p is 0.138 over the Z-channel. It is obvious that the proposed decoding algorithm of turbo codes on the asymmetric Z-channel can obtain excellent BER performance.

REFERENCES [1] C. Berrou, A. Glavieux and P. Thitimajshima, “Near Shannon limit

error-correcting coding and decoding: Turbo-codes”, in Proc. of ICC’93, Geneva, Switzerland, May 1993, pp. 1064-1070.

[2] S. Benedetto and G. Montorsi, “Unveiling turbo-codes: Some results on parallel concatenated coding schemes”, IEEE Transactions on Information Theory, Mar. 1996, 42(2): 409-428.

[3] C. Berrou, “The ten-year-old turbo codes are entering into service”, IEEE Communications Magazine, Aug. 2003, 41(8): 110-116.

[4] Z. Qin and Kah Chan Teh, “Iterative reduced-complexity multiuser detection based on Chase decoding for synchronous turbo-coded CDMA system”, IEEE Journal on Selected Areas in Communications, Jan. 2006, 24(1): 200-208.

[5] G. Zhu and F. Alajaji, “Joint source-channel turbo coding for binary Markov sources”, IEEE Transactions on Wireless Communications, May 2006, 5(5): 1065-1075.

[6] J. Sun and M.C. Valenti, “Joint synchronization and SNR estimation for turbo codes in AWGN channels”, IEEE Transactions on Communications, Jul. 2005, 53(7): 1136 – 1144.

[7] H. Bouzekri and S.L. Miller, “An upper bound on turbo codes performance over quasi-static fading channels”, IEEE Communications Letters, Jul. 2003,7(7): 302-304.

[8] T. Klove, P. Oprisan and B. Bose, “Diversity combining for the Z-channel”, IEEE Transactions on Information Theory, Mar. 2005, 51(3): 1174-1178.

[9] S.W. Golomb, “The limiting behavior of the Z-Channel”, IEEE Transactions on Information Theory, May 1980, 26(3): 372.

[10] I.S. Moskowitz, S. J. Greenwald and M. H. Kang, “An analysis of the timed Z-channel”, IEEE Transactions on Information Theory, Nov. 1998, 44(7): 3162-3168.

[11] Vucetic Branka and Yuan Jinhong, “Turbo codes: principles and applications”, Boston, Kluwer Academic Publishers, 2004, pp. 78-165.

[12] F. Said, A.H. Aghvami and W.G. Chambers, “Improving random interleaver for turbo codes”, Electronics Letters, Dec. 1999, 35(25): 2194-2195.

[13] Xingcheng Liu, Guangzhao Zhang and Jian Yin, “New iteration stop criterion of turbo codes”, Journal on Communications of China, Sep. 2001, 22(9): 13-19.

[14] Xingcheng Liu, Yan Hu and Lin Zhang, “Design of a Deterministic Interleaver for Turbo Codes and Its Applications in Self-Similar Traffic Streams”, Chinese Journal of Electronics, Apr. 2004, 13(2): 342-345.

[15] Donghua Liu, “The principal and application technology of turbo codes”, Beijing: Publishing House of Electronics Industry, Jan. 2004, pp. 23-29, 120.