Soft Information Transformation for

Coded Bi-directional Relay System

Qingjun Jin

The Graduate School

Yonsei University

Department of Electrical and Electronic Engineering

Soft Information Transformation for

Coded Bi-directional Relay System

Qingjun Jin

A Dissertation Submitted to the

Graduate School of Yonsei University

in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE

Supervised by

Professor Hong-Yeop Song, Ph.D.

Department of Electrical and Electronic Engineering

The Graduate School

YONSEI University

June 2011

This certifies that the master thesis

of Qingjun Jin is approved.

______________________________

Thesis Supervisor: Hong-Yeop Song

______________________________

Kwang Soon Kim

_____________________________

Sooyong Choi

The Graduate School

Yonsei University

June 2011

Acknowledgments

First and above all, thank God for the inspiration, wisdom and perseverance that

he has been bestowed upon me during the course of this work, and indeed,

throughout my life.

I want to express my deep gratitude to my parents, Jin Hongjun and Li

Dianshu. Thank you for your patience, for giving me confidence and especially

for giving me the chance to pursue my studies in my own way.

I am deeply grateful to my advisor professor Hong-Yeop Song for his great

guidance, endless support and sustained encouragement. His constant advice

and interest in my thesis are sincerely appreciated.

I am very thankful to my colleagues in the coding and crypto lab for the

helpful discussions along and for many good memories. Very special thanks to

Ki-Hyeon Park，Mi-Young Nam and JinSoo Park for the assistance of my M.S

journey.

I would like to thank my uncle Jin Mingjun and my aunt Han Mingzi

including other relatives for their unconditional love and support. And I owe

special thanks to my fellow brothers and sisters in Christ from Seoul Shinan

Church, for their continual pray and encouragement.

Finally, I would like to thank all of people who have helped me during my

M.S journey.

Qingjun Jin

June 2011

i

Contents

List of Figures iii

Abstract v

1 Introduction 1

1.1 Motivation …………………………………………………….... 1

1.2 An Overview …………………………………………………… 3

2 Previous works for BTC and CBR 4

2.1Block Turbo Codes ……………………………………………... 4

2.1.1 Property of BTC ………………………………………… 5

2.1.2 Chase-Pyndiah algorithm ……………………………….. 6

2.1.3 Distance Based Algorithm ………………………………. 9

2.1.4 BTC’s application in cooperative system ……………… 12

2.2 Coded Bi-directional Relaying……………………………….. 13

2.2.1 Introduction …………………………………………... 13

2.2.2 Gaussian Channel ……………………………………... 15

3 Soft Transformation 17

3.1 Motivation of soft transformation ……………………………... 17

ii

3.1.1 The problem of direct SISO ……………………………… 17

3.1.2 Proposed Technique – Soft Transformation ……………… 19

3.2 Posteriori Probabilistic Approaches …………………………….. 20

3.3 LLR Interpretation ………………………………………………. 21

3.4 Simulation Results ………………………………………………. 23

3.5 The Cause of SNR Loss …………………………………………. 25

3.6 Soft Transformation in flat fading channel ……………………… 26

4 Conclusions 32

Bibliography 34

Abstract (in Korean) 39

iii

List of Figures

2.1 The construction of Block Turbo Codes……………………………….. 5

2.2 The Block diagram of BTC encoder ………………………………….. 6

2.3 The Block diagram of BTC decoder ………………………………….. 8

2.4 The performance of (64, 51, 6) EBCH Block Turbo Codes ………….. 9

2.5 The Confidence Value of (64, 51, 6) EBCH Block Turbo Codes ….…. 11

2.6 BER performance of (64, 51, 6) EBCH Block Turbo Codes on a Gaussian

channel using BPSK signaling at iteration 4 …………………………… 11

2.7 Uncoded Bi-directional Relaying ……………………………………… 14

2.8 Three-Step Coded Bi-directional Relaying ……………………………. 14

2.9 Two-Step Coded Bi-directional Relaying ..……………………………. 15

3.1 Virtual node CBR ……………………………………………………… 19

3.2 Proposed Decoder ……………………………………………………… 21

3.3 BER curve with SNR, using extended BCH product code (64,51,6)2… 24

3.4 FER curve with SNR, using extended BCH product code (64,51,6)2… 24

3.5 Normalized Absolute Value of Average LLR Comparison .…………… 26

3.6 Performance comparison in flat fading channel with 0dB channel power

ratio, using extended BCH product code(64,51,6)2 …………………… 30

iv

3.7 BER Performance comparison in flat fading channel with 5dB channel

power ratio, using extended BCH product code(64,51,6)2 …………… 30

3.8 FER Performance comparison in flat fading channel with 5dB channel

power ratio, using extended BCH product code(64,51,6)2 …………… 31

v

ABSTRACT

Soft Information Transformation for Coded

Bi-directional Relay System

Qingjun Jin

Department of Electrical

and Electronic Eng.

The Graduate School

Yonsei University

This paper proposes to transform received bit from multiple access channel to

a soft value as it was received from virtual node in the Coded Bi-directional

Relaying (CBR) system. The soft transformation technique is based on the same

posteriori probability. After the soft transformation is implemented, the multiple

access channel (MAC) model which is sophisticated to analyze could change into

point-to-point channel model. The simulation result states that the two-step CBR

combined with Soft-Input-Soft-Output (SISO) decoder is more attractive,

because the SNR loss is only 0.8 dB when compare with the three-step CBR

vi

combined with SISO decoder. The simulation results in flat fading channel

indicate that then SNR loss could negligible when there is a strong link which is

favorable enough.

Key words: Coded Bi-directional Relaying, Soft Information Relaying, Block Turbo

codes

1

Chapter 1

Coded Bi-directional Relaying [2], [3], [8] has drawn researchers’ attention in

recent years. Network coding introduced by [1], is used in such system, which is

beneficial to improve the throughput.

When the SNR is low, conventional network coding does not exhibit good

error correction capabilities. Many research works have been focusing on

approach of designing and detecting the channel and network codes jointly.

Hausl and Dupraz [4] consider network and channel coding jointly for improving

the error performance in the case of multiple access relay channels. Joint network

and channel coding with regular LDPC code has been considered using tanner

graph in [5]. Recently, Guo et.al. [6] have proposed a practical method to

combine non-binary channel coding with network coding together with iterative

decoding scheme to jointly exploit the redundancy of the two schemes.

Introduction

1.1 Motivation

２

There have been a lot of work on coding for the bidirectional relay problem as

well ([10]-[14]). In [10], Katti et.al, showed the usefulness of network coding for

this problem. A scheme called analog network coding was recently proposed in

[11]. Notice that if 𝑋𝐴 and 𝑋𝐶 are codewords of the same linear code, 𝑋𝐴⊕𝑋𝐶

is also a valid codeword from the same code. Hence, the relay can decode

𝑋𝐴⊕𝑋𝐶 and transmit this during the broadcast phase. It is quite interesting to

note that if random codes were used instead of linear codes, 𝑋𝐴⊕𝑋𝐶 cannot be

decoded at the relay. Erez et.al has mentioned that minimum distance decoding of

linear codes, which is not practical [15]. It is well known fact

Hard-Input-Hard-Output (HIHO) decoding cannot utilize channel information

sufficiently. On the other hand, the analog network mapping at the relay node is

two-to-one correspondence. And it is multiple access channel, which is more

sophisticated to analyze than point-to-point channel.

Fortunately, under the condition of the same posteriori probability, the

received bit could be transformed to the soft value that was received from

point-to-point channel. After the transformation is applied, the channel decoding

and network encoding procedure ends simultaneously. This paper proposes the

soft transformation technique in two-step CBR to enhance both error correction

capability and throughput.

３

In Chapter 2, we discuss previous works for Block Turbo Codes and Coded

Bidirectional Relaying system. Block Turbo Codes will be used as error

correcting codes in the Coded Bi-directional Relaying System. In Chapter 3, we

will show the derivation of soft transformation and its performance in Coded

Bi-directional Relaying System. The simulation results will be analyzed as well.

Finally in Chapter 4, we will present a general conclusion of the master thesis.

The future work will also be mentioned in this part.

1.2 An Overview

４

Chapter 2

Block turbo codes can be obtained by serial concatenating two simple linear

block codes, such as BCH codes [7]. Two systematic linear block codes with

parameters (n1, k1, δ1) and (n2, k2, δ2) are considered. The n, k and δ stands

for code length, number of information bits, and minimum distance, respectively.

Then the parameter of the block turbo codes is (n1 ∗ n2, k1 ∗ k2, δ1 ∗ δ2) and

the construction is shown in Fig.2.1.

Previous works for BTC and CBR

2.1Block Turbo Codes

５

Fig2.1: The construction of Block Turbo Codes

The block diagram of are shown in Fig.2.2. It is clear that the encoding

complexity of Block Turbo Codes is linear. If two systematic linear block codes

with parameters (n1, k1, δ1) and (n2, k2, δ2) are considered, the minimum

distance of BTC is δ1 ∗ δ2[16].

Proof: There is a (n1, k1, δ1) column codeword that has minimum weight. The

others are all zero column codeword. After interleaving and encoding, there are

δ1 nonzero row codeword. So the minimum distance of BTC is δ1 ∗ δ2.

2.1.1 Property of BTC

2k

2n

1k

1n

Checks

on rows

Checks on columns

Checks on

checks

６

Fig.2.2: The Block diagram of BTC encoder

Extended linear codes are preferred for binary Block Turbo Codes, because

minimum distance is increased by δ1+δ2 + 1 with only extra even parity bits on

each row (column).

Moreover, Block Turbo Codes are extremely flexible in terms of rate, and

tradeoff between error performance and complexity. The structure of Block

Turbo Codes is also well suited to wireless communication. Due to independence

of each row (column) codeword, parallel decoding is available to Block Turbo

Codes. Besides, each row (column) codeword could be decoded by elementary

decoder. Both Chase-Pyndiah algorithm and Distance based decoding algorithm

are featured by the two advantages just mentioned, which will be illustrated

detailedly in the next two sections respectively.

Let us consider the transmission of binary elements {0, 1} coded by linear block

code C with parameters (n, k, δ) on Gaussian channel using binary symbols {-1,

+1}. We shall consider the following mapping of the symbols 0→1 and 1→-1.

2.1.2 Chase-Pyndiah algorithm

Linear code ),( 11 kn

Interleaver Linear code ),( 22 kn

７

The observation R = (𝑟1, … , 𝑟𝑙, … , 𝑟𝑛) at the output of the Gaussian channel for

a transmitted codeword E = (𝑒1, … , 𝑒𝑙, … , 𝑒𝑛) is given by

R = E + G

where components 𝑔𝑙 of G = (𝑔1, … , 𝑔𝑙, … , 𝑔𝑛) are additive white Gaussian

noise samples of standard deviation σ.

Pyndiah proposes near-optimum decoding of block turbo codes [9]. At first,

the list of candidate codewords should be obtained [17].We should determine

p = ⌊𝛿 2⁄ ⌋ least reliable elements of R based on log-likelihood 𝜌𝑙 =

|ln(Pr *𝑒𝑙=+1/𝑟𝑙+

Pr *𝑒𝑙=−1/𝑟𝑙+)| = (

2

𝜎2)𝑟𝑙 . Then form 2𝑝 test patterns from hard decided

codeword H by adding a single 1 among the p least reliable positions, then two 1s

among the p positions,…, and finally p 1s among the p positions. Using algebraic

decoder, decode the test codewords. Until now, it is the traditional chase type II

algorithm.

After the test codewords are obtained, there are three possible events can

happen.[18]

1. Two or more codewords are found;

2. One codeword is found; or

3. No codeword is found.

In the last event, the extrinsic information is zero. For the first event, save two

closet modulated codeword �̅� and �̅� to the received sequence. Then the

extrinsic information 𝜔𝑙 = 𝑥𝑙 ∑ 𝑟𝑚𝑥𝑚𝑛𝑥𝑚≠𝑦𝑚𝑚=1,𝑚≠𝑙

. For the second situation,

８

𝜔𝑙 = 𝛽𝑥𝑙 and βis the reliability factor. For next iteration, the soft input of SISO

decoder 𝑟𝑙′ = 𝑟𝑙 + 𝛼𝜔𝑙 and α is scaling factor. For BPSK modulation, the

values of scaling factor and reliability factor were computed up to four iterations,

as [9]

α = (0.0 0.2 0.3 0.5 0.7 0.9 1.0 1.0)

β = (0.2 0.4 0.6 0.8 1.0 1.0 1.0 1.0)

The block diagram of this algorithm is shown in Fig.2.3:

Fig 2.3: The Block diagram of BTC decoder

９

Fig 2.4: The performance of (64, 51, 6) EBCH Block Turbo Codes

Block Turbo Codes that uses extended BCH codes (64, 51, 6) as constituent

codes has been simulated. Then number of test patterns are set to 8 and the

maximum iteration number are set to 4. The BER and FER performance are

shown in Fig.2.4.

The Chase-Pyndiah algorithm depends on the performance of Chase type II

algorithm. So Chase-Pyndiah algorithm is appropriate for decoding high code

rate Block Turbo Codes, but becomes very inefficient as the hamming distance

2.1.3 Distance Based Algorithm

１０

increases. Distinguished from Chase-Pyndiah algorithm which relies on

list-based decoding, Distance Based Decoding has been presented in [19].

Distance Based Decoding generates extrinsic information from the distance of

decision codeword x̅ , which is the primary difference from Chase-Pyndiah

algorithm.

Once decision codeword x̅ is found, it will be evaluated by a confidence value

Φ, which is defined as the probability that the decoder makes correct decision

given received sequence R, written as Φ = Pr *x̅ = E|R+ that could be calculated

from Euclidian Distance. However, this method only works at the first iteration

where there is no extrinsic information. Side effects of the extrinsic information

have to be removed from the calculation of distance in later iterations. A solution

is made based on the study of Euclidean distance property. Define destructive

Euclidean distance, denoted Distdes, as the sum of Euclidean distance where the

noise vector has a different polarity than the decision vector, written as

Distdes = ∑ (rl − xl)2

l∈DES

, where DES = *l|(rl − xl)xl < 0+

Taking BTC (64, 51, 6)2 as an example, the figure was generated by simulation

for 10000 samples of decoding results. As can be seen in Fig.2.5, all resulting

curves are similar to each other. Therefore, for practical considerations, the

influence of the variable t, Eb/No, and p could be omitted and treat the

confidence value Φ as a function of destructive Euclidean distance Dist𝑑𝑒𝑠 ,

written as φ ≈ f(Dist𝑑𝑒𝑠) .

１１

Fig 2.5: The Confidence Value of (64, 51, 6) EBCH Block Turbo Codes [19]

Fig2.6: BER performance of (64, 51, 6) EBCH Block Turbo Codes on a

Gaussian channel using BPSK signaling at iteration 4

１２

As shown in [19], through derivation, the extrinsic information could be

computed as

ωl = xl(σ2

2ln(

ϕ + exp .2rlxlσ2/

1 − ϕ) − rlxl)

The simulation results are shown in Fig.2.6. The “traditional” represents the

algorithm introduced in [20].

As can be seen, Distance Based Decoding improves performance over the

traditional algorithm by approximately 0.2 dB at a BER of 10−5 with 16 test

patterns (p = 4), approximately 0.5 dB at a BER of 10−5 with 8 test patterns (p =

3), and approximately 0.7 dB at a BER of 10−5 with 4 test patterns (p = 2). In

conclusion, Distance Based Decoding outperforms Chase-Pyndiah algorithm

with the same test patterns.

Block Turbo Codes is very attractive in cooperative system because not only

does it have good error correcting performance but also it has a simple block

structure. For decode and forward technique, Xia et.al have considered

distributed cooperative Block Turbo Codes in multi-source multi-relay wireless

network [21]. Besides, Pyndiah has analyzed the multiple source diversity [24] of

Block Turbo Codes in the noisy relay [22].

The inter-user channel between the source and the relay in the decode and

2.1.4 BTC’s application in cooperative system

１３

forward scheme is always assumed to be so good that the decoding at the relay is

considered to be reliable. Nevertheless, in practical, this assumption is too

optimistic. If errors are corrupted at the relay, the conventional decode and

forward technique will be degraded by erroneous incremental redundancy.

Obiedat et.al have proposed soft decode and forward technique for distributed

Block Turbo Codes [23].

A lot of attentions have been focused on utilization of Block Turbo Codes in

one-way relay. It is interesting to think over implementation of Block Turbo

Codes in Coded Bi-directional Relaying.

As illustrated in [8], the communication scenario in Bi-directional Relaying

system is that the node A has packets destined to C and vice versa. And there is a

constraint that packets have to transmit through the relay node B. Let XA and XC

is the packet that is sent from node A and C respectively. As shown in the Fig.2.7,

the uncoded bidirectional consumes four time slots. The Fig.2.8 considers the

three-step coded bi-directional relaying. At the first time slot of three-step coded

bi-directional relaying, the node A sends XA to the relay node B. Then, node C

sends XC to node B. At the last time slot, node B broadcasts the network coded

2.2 Coded Bi-directional Relaying

2.2.1 Introduction

１４

information to node A and C. Thus to transfer the same amount of data, the

three-step scheme improve throughput 33% over the uncoded bi-directional

relaying. Alternatively, node A and C can simultaneously transmit over the

multiple-access channel, as shown in Fig.2.9. For the two-step scheme, node A

and C send XA and XC to node B simultaneously at first. Then node B

broadcasts the network coded information to both node A, and C. Obviously,

throughput improvement of two-step scheme is 100% over uncoded

bi-directional relaying.

Fig 2.7: Uncoded Bi-directional Relaying

Fig2.8: Three-Step Coded Bi-directional Relaying

１５

Fig 2.9: Two-Step Coded Bi-directional Relaying

Assumed that the wireless channel is real Gaussian channel, such that for the

two-step scheme.

At first time slot, multiple-access channel at the relay B;

YB = XA + XC + ZB

At next time slot, it is broadcasting channel:

YA = XB + ZA

YC = XB + ZC

where Z is the Gaussian noise, X is transmit symbol，Y is the received value. And

it is assumed that

P(XA = 1) = P(XA = −1) =1

2，

2.2.2 Gaussian Channel

１６

P(XC = 1) = P(XC = −1) =1

2

For simplicity, assume the Gaussian noise is i. i. d and with zero mean and the

same variance σ2.

Several works [25] [15] have been considered the usage of lattice codes for

two-way relaying channel. A lattice Λ is discrete subgroup of the Euclidean

space with ordinary addition operation. If λ1, λ2 ∈ Λ , then λ1 − λ2 ∈ Λ and

λ1 + λ2 ∈ Λ. The lattices introduce algebraic structure for the codewords in the

continuous channel.

Obviously, linear error correcting codes is good candidate of the lattice code.

Narayan has mentioned usage of linear error correcting codes such as hamming

codes as well. However, only ML decoding is considered in [25], which is not

practical. It is clear that HIHO decoding could be considered after denoising

maps [26]. Next chapter will present soft transformation technique for SISO

decoding.

１７

Chapter 3

As mentioned in 2.2.2, a lot of engineers’ attention has been focused on the

Coded Bi-directional Relaying system. Most of them analyze the multiple

access channel directly. Because a two-to-one mapping is accomplished at the

relay node, it will to interesting to think over transform the received value into a

soft value as it was received from point-to-point channel. At first, I want start

this chapter with motivation of soft transformation.

It is familiar result that the log-likelihood-ratio in point-to-point AWGN channel

is 2

𝜎2𝑟𝑗, where 𝜍2 is noise variance and 𝑟𝑗 is the jth received bit. Now let us to

compute the log-likelihood-ratio of the multiple-access-relay-channel of Coded

Soft Transformation

3.1 Motivation of soft transformation

3.1.1 The problem of direct SISO

１８

Bi-directional Relaying system.

BPSK modulation is used in the CBR system and the symbols are mapped as

follows: 0 → 1,1 → −1. Then, without Gaussian noise, the received symbol at

the relay belongs to the set {-2, 0, 2}. {-2, 2} indicate that the information bit sent

from node A and C are the same. On the other hand, {0} implies that the

information bit sent from node A and C are different. Since the relay node

broadcasts network coded symbol, then, *−2, 2+ → 1 and 0 → −1.

Then, the log-likelihood-ratio could be calculated as:

LLR = log(P(XB = 1|YB)

P(XB = −1|YB))

P(XB = 1|YB) =1

4∗,P(YB|XA=XC=1)+P(YB|XA=XC=−1)-

1

4∗,P(YB|XA=XC=1)+P(YB|XA=XC=−1)+P(YB|XA=−1,XC=1)+P(YB|XA=1,XC=−1)-

P(XB = −1|YB) =1

4∗,P(YB|XA=−1,XC=1)+P(YB|XA=1,XC=−1)-

1

4∗,P(YB|XA=XC=1)+P(YB|XA=XC=−1)+P(YB|XA=−1,XC=1)+P(YB|XA=1,XC=−1)-

Where,

P(YB|XA = XC = 1) =1

√2πσe−12σ2

(YB−2)2

P(YB|XA = XC = −1) =1

√2πσe−12σ2

(YB+2)2

P(YB|XA = −1, XC = 1) =1

√2πσe−12σ2

YB2

P(YB|XA = 1, XC = −1) =1

√2πσe−12σ2

YB2

１９

LLR = ln

(

1 +12 ∗ ,exp(−

2𝜍2(1 − 𝑌𝐵)) + exp(−

2𝜍2(1 + 𝑌𝐵))-

1 + (12 ∗ ,exp (−

2𝜍2(1 − 𝑌𝐵)) + exp (−

2𝜍2(1 + 𝑌𝐵))-)−1

)

This result is different from LLR of point-to-point AWGN channel. However,

it is inspiring to transform the received bit to the desired bit based on the

concept of network coding.

The symbols that relay node broadcasts are network coded information.

Conceptually, the relay node can treat the received vector as an output of a BSC,

over which a “virtual” transmitter sends the network coded message [8]. Then the

relay node tries to reliably decode the network coded message, rather than the

individual message from node A or C. In this way, MAC channel could be

changed into point-to-point channel, as shown in the following Figure.3.1.

Fig 3.1: Virtual node CBR

3.1.2 Proposed Technique – Soft Transformation

２０

SISO decoding performance is influenced by the log-likelihood ratio. When

the two posteriori probability is the same, the log-likelihood ratio is also identical.

So it could be concluded that the SISO decoding performance that YB could be

achieved, is the same as SISO decoding performance of YB′ . Therefore, it is

reasonable to calculate YB′ based on the same posteriori probability, which will

be illustrated in detail in next section.

YB′ could be viewed as it was sent from the “Virtual” node V with BPSK

modulation, which encodes XA and XC in advance. The posteriori probability of

YB′ could be computed as

P(XB = 1|YB′ ) =

P(YB′ |XV = 1)P(XV = 1)

P(YB′ |XV = −1)P(XV = −1) + P(YB

′ |XV = 1)P(XV = 1)

Then the result is

P(XB = 1|YB′ ) =

1

1 + exp(−2YB′ / σ2)

And,

P(XB = 1|YB) = 1

1 + (12 ∗ 0exp(−

2σ2(1 − YB)) + exp(−

2σ2(1 + YB))1)−1

There are only one variable in the formulation of both P(XB = 1|YB) and

P(XB = 1|YB′ ) . So they could be represented as P(XB = 1|YB) = f(YB) and

P(XB = 1|YB′ ) = g(YB

′ ). Under the condition of the same posteriori probability

P(XB = 1|YB) = P(XB = 1|YB′ ), the SISO decodable soft bit YB

′ = g−1(f(YB))

3.2 Posteriori Probabilistic Approaches

２１

is:

YB′ = (

σ2

2) ∗ ln(

1

2∗ [exp(−

2

σ2(1 − YB)) + exp(−

2

σ2(1 + YB))])

The same result is obtained, under the condition of P(XB = −1|YB) =

P(XB = −1|YB′ ). Using the log-max approximation ln(ea + eb) ≈ max(a, b),

we could simplify the result to the following form:

YB′ = (

σ2

2) ∗ ln (

1

2) + |YB| − 1

After the transformation procedure is end, the obtained soft value can be fed

into the SISO decoder directly. The following Figure 3.2 exhibits the block

diagram of proposed decoder.

Fig 3.2: Proposed Decoder

As illustrated in Section 3.1.2, when the posteriori probability is the same, the

LLR is identical as well. If we consider the constraint that the LLR is identical,

can the same result be obtained? The answer is positive.

3.3 LLR Interpretation

２２

LLR(YB) = ln(

1 +12 ∗ ,exp(−

2𝜍2(1 − 𝑌𝐵)) + exp(−

2𝜍2(1 + 𝑌𝐵))-

1 + (12 ∗ ,exp(−

2𝜍2(1 − 𝑌𝐵)) + exp(−

2𝜍2(1 + 𝑌𝐵))-)−1

)

And

LLR(YB′ ) =

2

𝜍2YB′

Then under the condition that LLR(YB) = LLR(YB′ ),

YB′ = (

σ2

2) ∗ ln(

1

2∗ [exp(−

2

σ2(1 − YB)) + exp(−

2

σ2(1 + YB))])

We could find out that this result is the equation obtained in previous section.

The same result could also be obtained, if we consider |𝑌𝐵| instead of 𝑌𝐵 as

random variable. Without Gaussian noise, when |𝑌𝐵| is 0, 𝑋𝐵 is 1 and when

|𝑌𝐵| is 2, 𝑋𝐵 is -1. The probability P(𝑋𝐵 = 1||𝑌𝐵|) is half normal distributed.

And the probability P(𝑋𝐵 = −1||𝑌𝐵|) is approximately Gaussian distributed.

So the LLR is:

LLR = ln (

1𝜍√2𝜋 exp (−

𝑌𝐵2

2𝜍2)

1

√2𝜋𝜍2exp (−

(|𝑌𝐵| − 2)2

2𝜍2)) = ln (

1

2) + (

2

σ2) (|YB| − 1)

Obviously, it is exactly the same as the result that is obtained in the previous

section.

２３

Both nodes A and C send Block Turbo Codes to node B. Due to the linear

property of Block Turbo Codes, the code word that relay node broadcasts is also

Block Turbo Codes. The extended BCH product code (64,51,6)2 is considered

in this simulation. Chase-Pyndiah algorithm is implemented with 8 test patterns

and 4 iterations. The noise variance is modified according to the code rate. The

BER performance and FER performance are shown in Fig.3.3 and Fig.3.4,

respectively. The BER and FER performance could be an indicator for

throughput performance.

The simulation results exhibit that the coding gain at BER (10−4) is 2.5dB,

compare to the algebraic decoding. And SNR loss of SISO two-step scheme is

less than 0.8dB, when compare with the three-step scheme. In section 3.1.2, it is

mentioned that MARC is transformed into point-to-point channel. Nevertheless,

there is still 0.8dB SNR loss, which is resulted from different LLR representation.

When take the time slot saving and decoding complexity into account, we could

conclude that the SISO two-step scheme outperforms the other two schemes.

3.4 Simulation Results

２４

Fig 3.3: BER curve with SNR, using extended BCH product code (64,51,6)2

Fig 3.4: FER curve with SNR, using extended BCH product code(64,51,6)2

２５

As the result obtained in previous section, there is 0.8dB gap between three-step

scheme and two-step scheme. On the other hand, in section 3.1.2, it is mentioned

that MAC is transformed into point-to-point channel. Those two statement seems

contradict to each other. However, if the constellation of MAC is considered, it

could be stated that SNR loss is reasonable.

At first, I want to start with a simple example. Zero mean Gaussian noise is

used in this simulation. Due to the symmetric property of Gaussian distribution

and zero mean, the probability of positive noise and negative noise. The

probability of information bit -1 and +1 are identical to each other as well. If we

send +1 and negative noise is corrupted, the reliability of this information bit will

be decreased. When positive noise is corrupted, the reliability of this information

bit will be increased. And converse for information bit -1. So the probability of

that “harmful” noise is corrupted is 1/2. For the MAC, when the information

symbol is 0, there is only “harmful” noise could be corrupted. And the “harmful”

noise for symbol 2 and symbol -2 is negative noise and positive noise,

respectively. The probability that harmful noise could be corrupted in the MARC

channel is 3/4. So it could be concluded that the noise in the MAC is more

“harmful”. Then, the average LLR in different SNR is shown in Fig.3.5. It could

be observed normalized absolute value of average LLR in two-step scheme is

3.5 The Cause of SNR Loss

２６

always smaller than 1. In the SNR region from 0dB to 4dB, the normalized

absolute value of average LLR is only 80% of BPSK channel, which is the

substantial reason that results in 0.8dB SNR loss.

Fig 3.5: Normalized Absolute Value of Average LLR Comparison

Soft transformation provides only 0.8dB SNR loss with throughput

improvement as shown in Fig.3.3 and Fig.3.4. It is interesting to investigate soft

transformation in flat fading channel, which is more practical than AWGN

3.6 Soft Transformation in flat fading channel

２７

channel. The channel model is as follows:

At first time slot, multiple-access channel at the relay B;

YB = αAXA + αCXC + ZB

At next time slot, it is broadcasting channel:

YA = αAXB + ZA

YC = αCXB + ZC

Where α~Rayleigh(1) is channel coefficient, Z is additive white Gaussian

noise.

For simplicity, BPSK modulation is used. So α is real value. Without

Gaussian noise, the received symbol at the relay belongs to the set *αA +

αC, αA − αC, −αA + αC, −αA − αC+ . *αA + αC, −αA − αC+ indicate that the

information bit sent from node A and C are the same. On the other hand,

*αA − αC, −αA + αC+ implies that the information bit sent from node A and C

are different. Since the relay node broadcasts network coded symbol, then,

*αA + αC, −αA − αC+ → 1 and *αA − αC, −αA + αC+ → −1 . Then we could

calculate the posteriori probabilities.

P(YB|XA = XC = 1, αA, αC) =1

√2πσe−12σ2

(YB−(αA+αC))2

P(YB|XA = XC = −1, αA, αC) =1

√2πσe−12σ2

(YB+(αA+αC))2

P(YB|XA = −1, XC = 1, αA, αC) =1

√2πσe−12σ2

(YB+(αA−αC))2

P(YB|XA = 1, XC = −1, αA, αC) =1

√2πσe−12σ2

(YB−(αA−αC))2

２８

The probability P(XB = 1|YB, αA, αC) could be calculated by Bayes rule. It

is assumed that YB′ is received from virtual channel. We could assume virtual

channel is AWGN channel, because it does not exist in reality. So the

probability P(XB = 1|YB′ ) =

1

1+exp(−2YB′ /σ2)

. Under the condition of the same

posteriori probability P(XB = 1|YB, αA, αC) = P(XB = 1|YB′ ) , YB

′ could be

represented by YB, αA, αC as the following equation:

YB′ = −αA ∗ αC −

σ2

2ln (

exp .2αAYBσ2

/ + exp .2αCYBσ2

/

exp (2(αA + αC)YB

σ2) + 1

)

Using the log-max approximation ln(ea + eb) ≈ max(a, b), we could simplify

the result to the following form:

YB′ = −αA ∗ αC −max(αAYB, αCYB) +

σ2

2ln (exp (

2(αA + αC)YBσ2

) + 1)

YB′ could be fed into SISO decoder without any modification. Simulation

parameters are the same as Section 3.4. Channel power ratio is defined as E,αA

2 -

E,αC2 -.

When channel power ratio is 0dB, BER and FER performance is shown in

Fig3.6. We could find out that there is about 3.0dB SNR loss between two-step

scheme and three-step scheme in flat fading channel. When compare with SNR

loss in AWGN channel, the SNR loss is enlarged by nearly 4 times. As

illustrated in Section 3.5, the normalized absolute value of average LLR in

multiple access channel is poorer than BPSK channel. The corrupted fading

coefficient degrades the performance of multiple access channel more severely.

２９

When channel power ratio is 5dB, BER and FER performance is shown in

Fig3.7 and Fig 3.8, respectively. Strong link means that the average channel

power is relative high. Weak link means that the average channel power is

relative low. The simulation results exhibit that the SNR loss is reduced to

1.3dB, when compare with the performance with 0dB channel power ratio.

Moreover, the performance difference between strong link and weak link is

negligible. There is only 0.2dB coding gain for three-step scheme when channel

power ratio increases from 0dB to 5dB. The network coding performance at the

relay node is determined by the weak link [27]. So it is not difficult to

understand why the coding gain is only 0.2dB. On the other hand, there is

nearly 2.0dB coding gain for two-step scheme when channel power ratio

increases from 0dB to 5dB. The substantial reason why there is 2.0dB coding

gain is that the normalized absolute value of average LLR in two-step scheme is

increased by incremental channel power ratio. Asymmetric channel condition is

common in practical. For example, the link between base station and relay is

usually better than the link between relay and mobile phone. So the simulation

results also indicate that the SNR loss between three-step scheme and two-step

scheme is negligible, when channel power ratio is large.

３０

Fig 3.6: Performance comparison in flat fading channel with 0dB channel power

ratio, using extended BCH product code(64,51,6)2

Fig 3.7: BER Performance comparison in flat fading channel with 5dB channel

power ratio, using extended BCH product code(64,51,6)2

３１

Fig 3.8: FER Performance comparison in flat fading channel with 5dB channel

power ratio, using extended BCH product code(64,51,6)2

３２

Chapter 4

This paper proposes to transform the received value in two-step CBR to the soft

value as it was received from virtual node. The simulation results in Section 3.4

show that the SNR loss between three-step scheme and two-step scheme is only

0.8dB. With 0.8dB SNR loss, one time slot could be saved, which results in

throughput improvement. The cause of SNR loss is also analyzed. SNR loss is

caused by changed constellation. The probability that noise reduces reliability in

MAC channel is higher than point-to-point channel. When taking the time slot

saving and decoding complexity into account, we could conclude that the SISO

two-step scheme outperforms the other two schemes.

The performance of soft transformation is also investigated with different

power ratio in flat fading channel. When channel power ratio is 5dB, the SNR

loss is only 1.3dB, which is much smaller than the SNR loss of symmetric

channel condition. Asymmetric channel condition is common in practical. The

Conclusions

３３

simulation results indicate two-step scheme with soft transformation could be

benefited more from the better channel condition. If the channel condition is

favorable enough, the SNR loss could be negligible.

The obtained results show that soft transformation based on the same

posteriori probability is a promising solution to enhance both error correction

capabilities and throughput in CBR. However, it is not proved that there is no or

negligible information loss in the soft transformation procedure, which will be

left as future work. It will be also interesting to extend the proposed technique

to soft incremental redundancy.

３４

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３９

국문요약

부호화된 양방향 릴레이 시스템을 위한 소프트 정보

전환에 관한 연구

본 논문에서는 부호화된 양방향 릴레이 시스템에서 후천확률을

기반으로 한 소프트 정보전환방법을 제안하였다. 소프트 정보전환

방법은 멀티펄 억세스 채널에서 받은 값을 대응된 가상 노드에서

전환하는 방법을 말한다. 소프트 정보를 전환함으로써 다중접근 채널

모델이 점대점 채널로 전환된다. AWGN 채널에서의 실험결과는 3단계

방안과 비교했을 때 2단계 방안의 SNR 손실이 0.8dB임을 보여준다.

Throughput과 복잡도를 고려하였을 때 2단계 방안이 3단계 방안보다

성능이 좋다고 할 수 있다. 플랜패이딩채널에서의 실험결과는 한

링크의 채널상황이 좋았을 때 SNR loss가 줄어든다는 것을 보여준다.

핵심 되는 말: 부호화된 양방향 릴레이, 소프트 정보 전환, 블록 터보

코드