Performance Analysis of Coded Cooperation Based on Distributed … · 2016-08-19 · 2 Network Optimization Center, Network Department,China Mobile Group Hebei Co. Ltd.,Shijiazhuang

Manuscript received February 15, 2016; revised July 18, 2016. Corresponding author email: [email protected].

doi:10.12720/jcm.11.7.667-676

667

Journal of Communications Vol. 11, No. 7, July 2016

©2016 Journal of Communications

Performance Analysis of Coded Cooperation Based on

Distributed Turbo Codes with Multiple Relays

Jing An1 and Chang Li

2

1 Department of Information Engineering, Shijiazhuang Vocational Technology Institute, Shijiazhuang050081, China

2 Network Optimization Center, Network Department,

China Mobile Group Hebei Co. Ltd.,

Shijiazhuang 050035, China

Email: {anj.sjz, lichang.sjz}@hotmail.com

Abstract—Diversity is an effective technique to enhance link

quality and enable multiple users to pool their resources to form

a virtual antenna array that obtains spatial diversity. In this

paper, we propose a generalized Distributed Turbo Codes

(DTC)-based coded cooperation protocol for two-hop relay

networks with an arbitrary number of relays. This scheme aims

at achieving improved diversity over the classical coded

cooperation method in Rayleigh fading channels. We derive a

finite range single integral solution for the outage probability,

which characterizes the coded performance with multiple relays

at various rates. We also develop a closed-form expression for

the Pairwise Error Probability (PEP) and a tight upper bound for

the Bit Error Rate (BER) using DTC. The results demonstrate

the merits of DTC-based coded cooperation with multiple relays,

under various relay and uplink channel conditions, and show

that the proposed scheme is able to achieve the full diversity

order. Moreover, the analytical upper bounds are validated with

simulation results. Index Terms—Bit error rate, channel coding, coded

cooperation, distributed turbo code, fading channels, multiple

relays, outage probability, pairwise error probability

I. INTRODUCTION

In order to provide reliable transmission at high data

rates and offer a variety of multimedia services in future

wireless communication systems, diversity has been

considered as an effective technique in combating

detrimental effects of channel fading caused by multi-

path propagation and Doppler spread. However, in

cellular network, due to the limited size, cost and

hardware limitations, it may not be possible equip a

mobile terminal with multiple transmit antennas.

Recently, the cooperative communications technique is

emerging to enable a new form of spatial diversity in

wireless systems. Single antenna mobiles in a multi-user

environment share their antennas and form a virtual

multiple antenna transmitter that is able to achieve

transmit diversity. The recent surge of research interests

in cooperative communications is subsequent to the

pioneering work of Sendonaris et al. [1], [2]. Shortly

afterwards, Laneman et al. [3]–[5] proposed several

cooperative protocols and analyzed their performance in

both ergodic as well as quasistatic channels. Among the

cooperative protocols they proposed and analyzed are

Decode and Forward (DF), Amplify and forward (AF),

and adaptive methods that switch between the two. In

parallel to the work of Laneman, an alternative

framework, dubbed coded cooperation was proposed, in

which cooperative signaling is integrated with channel

coding [6]–[9]. The basic idea is that each user, instead of

repeating the received bits (either via amplification or

decoding), tries to transmit incremental redundancy for

its partner. In the original coded cooperation strategy [7],

users transmit only their partners’ data in the cooperation

frame whenever possible, which can achieve full

diversity and lead to significant performance

improvements in slow fading channels. However, this

strategy is unable to achieve diversity gains, and suffers

from cooperation imbalances between users operating in

fading environments. Then, space time coded cooperation

is proposed to achieve improved diversity over the

original coded cooperation protocol and reduce the

impact of cooperation imbalances [10]. However,

synchronization for space-time cooperation especially in

Time Division Multiple Access (TDMA) systems

becomes a serious problem because the symbol-level

synchronization between users in the uplink cannot be

guaranteed.

In [11], Elfituri et al. proposed a convolutional coding

based distributed coded cooperation scheme for multiple

relay channels. The authors derived simplified closed-

form expressions for the outage probability in the case of

error free and erroneous relaying. However, their

analytical results (i.e., (28) and (34) in [11]) are flawed

and do not match with the presented simulation curves

(e.g., Fig. 7 and Fig. 9 in [11]). This is because that the L-

fold multiple integral in (27) of [11] is overly simplified

into L single integrals.

To further explore the cooperative spatial diversity and

cooperative coding gains, various distributed coding

schemes have been proposed, such as convolutional

codes [8], [12], distributed space time coding [13], [14],

distributed low density parity check codes (D-LDPC)

[15], [16] as well as the distributed turbo coding (DTC)

scheme [17]-[19].

As the idea of relay networks attracted attention,

researchers began to investigate their information

theoretic aspects. In particular, determining achievable

rate regions was investigated under a number of different

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assumptions [19]-[24]. [25] proposed an extended

distributed turbo coding scheme for half-duplex relay by

transferring information via the timing of the relay-

receive and relay transmit phases to the destination.

However, most existing DTC schemes are only designed

for a relay networks with a single relay node. In [26] a

generalized hybrid relaying scheme was proposed for

multiple relays network with any number of relays.

However, the hybrid relaying protocol including the AF

and DF strategy is too complex [27]-[38]. Therefore, it is

very important to develop a DTC scheme for the general

relay network with any number of relays.

In literature [39], they develop a closed-form

expression for the Pairwise Error Probability (PEP) and a

tight upper bound for the Bit Error Rate (BER) using

DTC.

In this work, we consider the multiple-relay Rayleigh

fading channels, and study the outage probability and

BER of the coded cooperation protocol based on DTC

with multiple relays. We aim to derive a finite range

single integral solution for the outage probability, and a

closed-form expression for the Pairwise Error Probability

(PEP) of the proposed DTC-based coded cooperative

protocol. In comparison with the original coded

cooperation protocol, the proposed strategy based on

DTC with multiple relays is capable of providing the

added benefits of improved diversity gains, and reducing

the impact of cooperation imbalances.

This paper is organized as follows. Section 2 describes

the system model for the proposed coded cooperation

scheme. Section 3 presents a mathematical probability

model and derives a finite range single integral solution

for the outage probability. Section 4 evaluates the bit

error probability of DTC-based coded cooperation with

an arbitrary number of relays when Binary Phase-Shift

Keying (BPSK) modulation is used. Section 5 presents

simulation results and discussions. Conclusion is drawn

in Section 6.

S D

R

.

.

.

R

R

M

S D

R

.

.

.

R

R

M

Frame 1 Frame 2 Fig. 1. System model of coded cooperation with multiple relays.

II. SYSTEM MODEL

In our model, we consider the scenario of one source

with the help of M relays. We design a distributed

coding scheme at both the source and relays, where the

encoding process is divided into two frame transmissions.

In what follows, we denote by s , ir and d the source,

thi relay, and destination nodes, respectively. Consider

the multiple-relay channel shown in Fig. 1, where the

data is sent from s to d with the assistance of M relay

nodes.

As illustrated in Fig. 1, both the source and relay nodes

employ convolutional coding to protect information bits.

There are K bits in each source block, and the length of

the codeword is N . In the distributed coding scheme

under consideration here, the -bitN codeword is

partitioned into two successive frames (i.e., frame 1 and

frame 2) of 1N and 2N bits ( 1 2N N N ) with rate 1R

and 2R [6], [7], using a Rate-Compatible Punctured

Convolutional (RCPC) code puncturing matrix [8]. We

can quantify the level of cooperation with the parameter

1 1N N R R (1)

which indicates the portion of the total channel codeword

allocated for the first frame.

In frame 1, the source broadcasts the first 1N bits to

both the relays and destination. If a relay correctly

decodes the message received from the source, it re-

encodes the message into the original -bitN codeword.

Otherwise, the relay becomes silent. In the second frame,

the source and the relays whose Cyclic Redundancy

Checks (CRCs) are validated transmit the second 2N bits

to the destination, note the transmit energy is split equally

among all the active relays. The received copies of the

second part are combined using Maximal Ratio

Combining (MRC), and the information bits in the first

and second frames are decoded by a Viterbi decoder at

the receiver.

A. Transmission Protocols

In the following analysis, we analyze the performance

of the proposed cooperative coding scheme based upon

the DTC over slow fading channels, where fading

coefficients remain constant during each frame

transmission interval.

For the first frame, the signals that are received at the

relay and destination nodes are given by

, , , ,s d s d s d s dy h E x n (2)

, , , ,i i i is r s r s r s ry h E x n (3)

where x denotes the modulated source signal, and

1,2,...,i M . ,s dh and , is rh represent the fading channel

coefficients from s to d and s to ir , respectively. ,s dE

and , is rE are the transmitted signal power for the

corresponding links, while ,s dn and ,s dn represent complex

Additive White Gaussian Noise (AWGN) over the

corresponding channels.

Let L be the number of relays that are used for

cooperation in the second phase (i.e., these relays

correctly decode the received message). Accordingly, the

received signals at the destination node are given by

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,

, , ,ˆi

i i i

r d

r d r d r d

Ey h x n

L (4)

, , , ,s d s d s d s dy h E x n (5)

where x̂ denotes the relay transmitted signal, and 1 L is

the ratio for maintaining the same average power

compared to the single relay scenario. That is, the

transmit energy is split equally among all the active

relays in the second phase, each of the relays transmits

with the same power. Note every two frames have a

different L .

SourceCRC

RSC2πViterbi

Decoder

Systematic bits

Parity 1

R1

Sys+Parity 1

RSC1

RSC2π Parity 2

RSC2πViterbi

Decoder

Parity 2RM

Sys+Parity 1

M

R

C

Parity 2

Turbo

Iterative

DecoderParity 2

Sys+Parity 1

...

Sys bits

Sys bits

Frame 2

Frame 1

C

T

N

Fig. 2. Distributed turbo encoding in a coded cooperation scheme.

B. Distributed Turbo Codes with Multiple Relays

The implementation of coded cooperation using DTC

is illustrated in Fig. 2. The turbo codes employ two

constituent Recursive Systematic Convolutional (RSC)

codes with interleaving. The source and relay share the

same random interleaver, shown as . To reduce

implementation complexity, the relay employs

conventional Viterbi decoding. Note that the systematic

and the first parity bits are 1N , the second parity bits are

2N . It is possible to have a more flexible level of

cooperation, as well as better performance, by using

punctured turbo codes or Rate Compatible Punctured

Turbo codes (RCPT) [40].

Fig. 2 shows the sketches a block diagram for coded

cooperation scheme. In the first frame, the source

broadcasts the 1N bits (systematic and the first parity bits)

to both the relays and destination. If a relay correctly

decodes the message received from the source using the

Viterbi decoder, it interleaves the source bits and re-

encodes them into 2N bits (the second parity bits).

Otherwise, that relay keeps silent. In the second frame,

the source and the relays whose Cyclic Redundancy

Checks (CRCs) are validated transmit the 2N bits to the

destination. At the destination, the received copies of the

second parity bits are combined using Maximal Ratio

Combining (MRC), and the information bits in the first

and second frame are concatenated (CTN) and then

decoded by a turbo decoder. The low-complexity iterative

decoder offers near-optimum decoding performance for

turbo codes.

III. OUTAGE PROBABILITY ANALYSIS

First, we consider non-cooperative direct transmission

between the source and destination. With quasi-static

fading, the capacity conditioned on channel realization,

characterized by the instantaneous Signal-to-Noise Ratio

(SNR), can be expressed by the classic Shannon formula

2log 1C in b/s/Hz. The channel is in outage if

the conditional capacity falls below a selected threshold

rate R, and the corresponding outage event is

C R , or equivalently, 2 1R . The outage

probability is defined as

2 1

0Pr 2 1

R

R

out rP p d

(6)

where rp denotes the probability density function

(pdf) of random variable . For Rayleigh fading, has an

exponentially distributed with parameter 1/ , where

denotes the mean of SNR of the fading channel and

accounts for the combined effects of large-scale path loss

and shadowing. The outage probability for Rayleigh

fading can thus be evaluated as

2 1

0

1 2 1exp 1 exp

R R

outP d

(7)

A. Outage Probability

The PDF document should be sent as an open file, i.e.

without any data protection.

In this section, we consider the situation where some

of the relays may fail to correctly decode the message

that they received from the source. That is, their CRCs do

not pass. Let L be the number of cooperative relay

stations. Apparently, L ranges from 0 to M. Denote by

the set of indices of the cooperating relays

1 2, , , 1,2, ,Li i i M (8)

Note that the cardinality of is L ( 0 L M ).

As discussed in the previous section, in coded

cooperation user data are transmitted over two successive

frames. In the first frame, each user transmits a rate

1R R codeword. There are two possible cases for the

second frame transmission depending on the number of

relay stations that are able to successfully decode the first

frame. Case I (denoted by = 1): None of the relay

670



stations decodes the source frame successfully. In the

information theoretical sense, this scenario corresponds to

the following event (Note that every two frames may

have a different L ).

, , 2 ,log 1i i is r s r s rC R (9)

where subscripts s and ir denote the transmission from

the source to the thi relay, and 1,2, , .i M In the

second frame, the source transmits additional parity bits.

For a given source, the destination will receive the first

frame and second transmission frames both from the

source. The first frame occupies fraction, whereas the

second frame uses the remaining 1 fraction of the

total bits. These two transmission can thus be viewed as

time sharing between two independent channels, where

the first channel uses a fraction of the time. Thus, we can

write the outage events as

, , 2 ,

2 ,

1 log 1

1 log 1

s d s d s d

s d

C

R

(10)

In case 2 (denoted by 2 ), some of the relays

decode the source frame successfully

, , 2 ,log 1i i is r s r s rC R (11)

where i . On the other hand, for the relays that failed

to decode the source message, we have

, , 2 ,log 1i i is r s r s rC R (12)

where i . In the second frame, both the L (1 L M )

error-free relays and the source transmit additional parity

bits to the destination. The corresponding outage events

are given as

, , 2 ,

2 , ,

2 log 1

1 log 1i

s d s d s d

s d r d

i

C

R

(13)

Since the above two cases are disjoint, we can write

the overall outage probability as in (14). Note that the

first, and second terms correspond to cases 1 and 2,

respectively. It should be emphasized that, in all the two

cases, the destination is in outage. For the case of

Rayleigh fading, (14) can be shown to evolve into (15),

where LS is an integration region given by (16).

In contrast, the L-dimensional region of integration in

(27) of [11] is decomposed into L independent one-

dimensional regions of integration. Therefore, the L-

dimensional integral is overly simplified into L single

integrals. The derived outage probability expressions

given in (28) and (34) in [11] are thus flawed. This is

further evidenced by the fact that the analytical results do

not match the simulation results presented in Fig. 7 and

Fig. 9 in [11].

As can be seen from (15), the outage probability for

DTC-based coded cooperation with multiple relays is a

function of the mean channel signal-to-noise ratio (SNR)

values ( ,s d , , is r and ,ir d ) the allocated rate R , and the

cooperation level . While the mean channel SNR

values and R may often be constrained by underlying

communications channels, is a free parameter that can

be varied to optimize performance.

According to the derivations in Appendix, we can

simplify (15) into (17). Note that we can simplify the

multiple integrals function to the single integral function,

thanks to the independence of different ,ir d .

/

, ,

11

/ /

, , , , ,

1

Pr 2 1 Pr 2 1

Pr 2 1 Pr 2 1 Pr 1 1 2

i

i i i

MR R

out s r s d

i

MR R R

s r s r s d s d r d

K ii i

P

M

L

(14)

/

1 , ,

/ /,,

,

, , , , ,

1 2 1 21 exp 1 exp

1 2 1 2 1 11 exp exp exp exp

i

i

i i iL

R RM

out

i s r s d

R Rr ds d

s d

i is r s r s d s d r d rS

P

Md

L

,

1 ,i

i

M

r d

L i d

d

(15)

1

, , , , , , ,= , 0, 0, , 1 1 2i i i

R

L s d r d s d r d s d s d r d

i

S i

(16)

/

, ,

/ /,

0, , , , ,

1 2 1 21 exp 1 exp

! 1 2 1 2 11 exp exp exp 1 exp

! !

i

i i i

MR R

out

s r s d

M L LR R

s d

s r s r s d s d r d

P

M a

L M L

2 1

,

1

R

LM

s d

L

d

(17)

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B. Asymptotic Analysis and Diversity Order

The PDF document should be sent as an open file, i.e.

without any data protection.

In order to study the asymptotic behavior of the outage

probability at high SNRs, we assume the following re-

parameterization for the mean SNR values among the

source, relay and destination [7]

, ,s d T s d (18)

where T is the ratio of the user transmit power to the

received noise, and ,s d is a finite constant accounting for

large-scale path loss and shadowing effects. Relative

differences in quality between the various channels are

still captured by the ,s d values. Thus, by expressing

outage probability as a function of1 T , and then letting

T (i.e., the high-SNR regime), the diversity order

is given by the smallest exponent of 1 T . Under the

Rayleigh fading assumption, the first term of (15) in the

asymptote of large behaves as (19). The same as the

above, we can obtain (20). For the integral of (15), using

the fact that exp 1x for all 0x x _ 0, we have (21).

This means that each term contributing to the sum in (15)

behaves as 11 M

T

, full diversity order is achieved.

/ /

11 1, , , ,

1 2 1 2 1 2 1 2 11 exp 1 exp 1 exp 1 exp

i i

R R R RM M

Mi is r s d T s r T s d T

(19)

/ /

, ,

1 2 1 2 11 exp exp

i i

R R

M Li is r s r T

(20)

, ,, ,

, , , ,1

, , , , , , , ,

, ,1

, ,

1 1 1 1 1exp exp exp

1 1 1

i i

i i

i i i iL L

i

iL

r d r ds d s d

s d r d s d r dLii i is d s d r d r d s d r d T s d T r dT

S S

s d r dLi is d r dT

S

d d d d

d d

1 1

, ,

1 1 1 1

i

L Lis d r dT T

(21)

IV. BIT ERROR RATE ANALYSIS

The pairwise error probability (PEP) for a coded

system is defined as deciding in favor of code word

1 2( , , , )e Ne e e when code word 1 2( , , , )c Nc c c was

transmitted. Therefore, for a binary code with BPSK

modulation, the PEP of noncooperative transmission can

be written as [10]

( | ) 2 ( )c en

P Q n

(22)

where ( )Q x denotes the Gaussian Q-function [12]. The

instantaneous received SNR values are denoted by vector

γ . The transmitted codeword is c , the erroneously

decoded codeword is e , and the set is the set of all n

for which ( ) ( )c en n , thus d is the Hamming

distance between c and e . For linear codes, the PEP

depends only on d and not the particular code words c

and e , so that the conditional PEP is typically denoted by

( | )P d .

To obtain the unconditional PEP we must average (22)

over the fading distributions

0( ) ( | ) ( )P d P d p d

(23)

where ( )p is the probability density function (PDF) of

.We can obtain an exact solution to (23) using the

techniques of Simon and Alouini. The first step is to use

the following alternative representation for the Gaussian

Qfunction, originally derived by Craig [41]-[42], and then

applied to performance analysis in fading channels in

[43]-[45]

22

20

1( ) exp

2sin

xQ x d

(24)

Substituting (24) in (22) and (23), Using the well

known MGF function method [45], for the case of

Rayleigh fading, we can obtain

12

20

1( ) 1

sin

dP d d

(25)

where is the average SNR. Equation (25) is an exact

expression for the unconditional PEP and can be easily

evaluated with numerical integration techniques.

We can obtain the following upper bound for (25) by

noting that the integrand is maximized when 2sin =1 , so

that

11

( ) 12

P d d

(26)

As can be seen from (26), for large SNRs, the PEP is

inversely proportional to the product of the average SNR

and d .

A. Pairwise Error Probability

In this section, we analyze the PEP of the proposed

coded cooperation protocol. Analogous to the outage

probability analysis, there are two possible cases

depending on how many relay stations are able to

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successfully decode the first frame. Assuming,s d , , is r

and ,ir d are the mean channel SNR values. In the

following, we derive a closed-form expression for the

pairwise error probability. Although the expression is

developed for the specific case of BPSK modulation, the

analysis can be readily extended to accommodate other

types of modulation.

For Case I, none of the relay stations decodes the

source frame successfully, all the relays keep silent. So

that the conditional PEP for the destination is

, , 1 , 2 ,

,

( | , ) 2( )

2

is d r d s d s d

s d

P d Q d d

Q d

(27)

where 1d and 2d are the portions of the error event bits

transmitted respectively, such that 1 2d d d .

The average sequence error probability can be derived

by averaging (2) over the fading coefficients. Let ( )P d

be the average probability of decoding an erroneous code

sequence with weight d, we can obtain

1

2 1,

,20

1 1( ) 1 1

2sin

s d

s d

dP d d d

(28)

For Case II, some of the relays decode the source node

successfully

, , 1 , 2 , 2 ,

, 2 ,

( | , ) 2

2

i i

i

s d r d s d s d r d

i

s d r d

i

P d Q d d d

Q d d

(29)

which can be evaluated as

112 2 ,,

2 20

11

, 2 ,

1( ) 1 1

sin sin

11 1

2

i

i

r ds d

i

s d r d

i

ddP d d

d d

(30)

The BER for a turbo code is bounded by (32).

Substituting ( )P d into (32), we can obtain the BER for

every case. into (32), we can obtain the BER for every

case.

B. Bit Error Rate of Turbo Codes

The bounds for the BER and BLER of turbo codes can

be obtained using the weight enumerating function (WEF)

of the equivalent block code, as shown in [46] - [48]. The

WEF of the overall concatenated code is given based on

the WEF of the constituent codes. We follow the same

method of [47] with a minor modification for a turbo

code with 1C and 2C as the constituent recursive

convolutional codes and an interleaver with size K .

The conditional WEF of a block code, ( )C

WA Z ，gives

all possible code words generated by the set of input

sequences with weight w (note that Z is a dummy

variable). Denote by 1 ( )C

WA Z and 2 ( )C

WA Z the conditional

WEFs of 1C and

2C , respectively.

Then using the probabilistic uniform interleaver the

conditional WEF of the turbo code is [47]

1 1( ) ( )( , )

C C

C W W

W

A Z A ZA Z Y

K

w

(31)

The BER and BLER of the turbo code are obtained

using the union bound argument [47]

, ,

0 0 1

( )K K K

b w z y

z y w

wP a P d

K

(32)

, ,

0 0 1

( )K K K

block w z y

z y w

P a P d

(33)

where , ,w z ya denotes the multiplicity of code words

corresponding to input weight w and parity weights z

and y , and ( )P d is the corresponding PEP expression.

Note that 1d is equal to the summation of the exponents

of w and z , and 2d is equal to the exponent of y .

C. End-to-end Bit Error Rate

We now use the above PEP results to determine the

end to end bit error probabilities for the coded

cooperation protocol. The first step is to calculate the

probabilities of the cooperative cases. The cooperative

case probabilities are determined by the BLER of the first

frame transmission. The BLER for a turbo code is

bounded by (33).

As mentioned before, we parameterize the two cases

by 1,2 and we can express the probability for Case

I ( 1 ) as follows

,

1

( 1)i

M

block r

i

P P

(34)

where , iblock rP denote the BLER for the thi relay.

For case II, the probability is given as

1

, ,

1

( 2) 1i i

M

block r block r

L i i

MP P P

L

(35)

The overall end-to-end BER equals the average of the

BER over the two transmission scenarios as [49]

2

1

b b

i

P P P i

(36)

where bP denote the end-to-end BER.

V. NUMERICAL RESULT

In this section, we present simulation results to

compare various relaying schemes with multiple relays.

All simulations are performed assuming BPSK

modulation and a frame size of 130 symbols over quasi-

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static fading channels. We use a 4-state recursive

systematic convolutional code (RSC) with the code rate

of 1/2 and the generator matrix of (1, 5/7). It is possible

to have flexible code rates through the use of RCPT [40]

codes. The baseline scheme for comparisons is a

noncooperative turbo coded system. The different

channels between the source, relays and destination are

assumed to be independent flat quasi-static Rayleigh

fading channels, where the channel coefficients are fixed

for the duration of the frame and change independently

from one frame to another.

Fig. 3. Outage probability vs. mean SNR for various cooperative

protocols for M=1,2 and 3 relay channels.

Fig. 3 plots the outage probability given by (17) versus

the SNR for four different system configurations at rate

1R b/s/Hz. All channels have equal mean SNRs. As

can be observed from Fig. 3, coded cooperation with

1M is able to achieve around 1 dB gain over repetition

based DF. Moreover, the coding gains increase with the

increase of the number of relays. For example, the gain is

nearly 10 dB for coded cooperation with 1M at the

outage probability of 510 as can be observed from Fig. 3.

By contrast, we see that the repetition based DF scheme

is not very effective in the low SNR regime. This is due

to the fact that the DF codeword is equivalent to

repetition coding, which is relatively inefficient as

opposed to distributed coding. Coded cooperation,

however, performs much better than all the other systems

in Fig. 3 at all SNRs. This is attributed to the coding

gains induced by the coded cooperation protocol.

Fig. 4. Outage probability vs. rate. All the channels have a mean SNR of 10 dB.

Fig. 4 compares the outage probabilities at different

rates for the various cooperative schemes. All the

channels have a mean SNR of 10 dB. In the low rate

regime, both DF and coded cooperation provide

significant improvements, with the latter offering

generally better overall performance. As the rate

increases, the outage probability of the DF protocol will

exceed that of its coded cooperation counterpart. This is

again a manifestation of the inefficiency of the repetition

coding nature of DF. We note that coded cooperation in

the worst scenario always performs at least as well as the

no cooperation and DF schemes.

Fig. 5 plots the outage probabilities versus the number

of relays at the fixed rate of R =1b/s/Hz. Two typical

channel SNR values are considered in Fig. 5, i.e., 10 dB

and 15 dB. As can be observed from the figure, as the

number of relays increases, the outage probability

decreases due to the increased diversity gains. However,

the additional gains obtained from increasing the number

of the relays come at the expense of approximately

linearly increased complexity, resulting in diminishing

returns.

Fig. 5. Outage probability vs. the number of relays. All the channels have the equal average SNRs, i.e., 10 dB or 15 dB.

Fig. 6. Comparing bit error rates of coded cooperation for 1, 2, and 3

relays, and all the channels have equal SNR (varies 0-20 dB).

Fig. 6 shows the comparison of the simulated BER

results and the analytical BER using (36) for the proposed

transmission scheme with L = 1, 2 and 3 relay channels,

assuming all the nodes have the equal mean SNR. As

-5 0 5 10 15 20 25 3010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Mean SNR (dB)

Outa

ge P

robabili

ty

No cooperation (M=0)

DF (M=1)

Coded cooperation (M=1)



0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Rate (b/s/Hz)

Outa

ge P

robabilit

y


DF (M=1)




1 2 3 4 5 6 7 8 9 1010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

The Number of Relays

Outa

ge P

robabili

ty

SNR=10dB

SNR=15dB

0 2 4 6 8 10 12 14 16 18 2010

-6

10-5

10-4

10-3

10-2

10-1

100

Mean SNR (dB)

Bit E

rror

Rate

Bound

Simulation





674



shown in Fig. 3, the analytical results are in close

agreement with our simulated BER results. Additionally,

the diversity gain that is achieved using a different

number of relays is evident in these results. Compared

with the noncooperative scheme, the gain at BER of 510

is from 12 dB (for the case of L =1) to 18 dB (for the

case of L = 3).

Fig. 7 shows the simulation results for the BER of

turbo coded cooperation using two relays under various

channel conditions. For most application scenarios, the

backhaul and cooperative links are more reliable than the

direction transmission link. In this simulation, the mean

SNR of the source to relay and relay to destination

channels are 5 dB, 10 dB or 15 dB higher. That is,

, , ,i is r r d s d +5 dB (or +10 dB, +15 dB). The union

bounds well match the simulation results curve.

Fig. 7. Comparison of the bit error rates of turbo coded cooperation for two relays with di_erent mean SNRs. The x-axis is the SNR from

source to destination. Note that four cases are considered. All the

channel have the same SNR. The mean SNR of the source to relay and relay to destination channels are 5 dB, 10 dB or 15 dB higher than that

direct transmission link.

VI. CONCLUSION

In this paper, we investigated the outage probability

and the BER of coded cooperation based on DTC with

multiple relays in Rayleigh fading channels. A finite

range single integral solution for the outage probability

expressions of the proposed turbo coded cooperation

protocol was derived. Furthermore, we also developed a

closed-form expression for the PEP and a tight upper

bound for the BER using DTC. The analytical upper

bounds were verified with simulation results. Our results

demonstrated the efficacy of the proposed protocol in

comparison with other cooperative strategies, and showed

that our scheme is able to achieve a full diversity order.

APPENDIX A APPENDIX TITLE

The integration region LS in (15) can be rewritten

1

, ,1

,

21

1i

R

r d s d

is d

(A1)

1

, ,1

,

1 21

1i

R

r d s d

s d

aL

(A2)

note that (A2) is true because each of the L cooperative

relays has the same transmission power during the second

frame transmission. Since , 0s d we have

1

,1

,

21

1

R

s d

s d

(A3)

, 2 1R

s d (A4)

We are now ready to simplify the multiple integrals in

(15) as shown in (A5).

1

1

,,

, ,

, , , ,

2 ,,

, ,0 0

, , , ,

2 ,

0, ,

1 1exp exp

1 1exp exp

1exp 1 exp

i

i

i iL

R

i

i

i i

R

r ds d

s d r d

is d s d r d r dS

a r ds d

r d s d

is d s d r d r d

s d

s d s d

d d

d d

,

,i

L

s d

r d

ad

(A5)

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Jing An was born in Hebei Province,

China, in 1980. She received the B.S.

and M.S. degree from the Shijiazhuang

Railway Institute, Shijiazhuang, in 2004

and 2008, respectively, both in computer

science. She is currently working in

Department of Information Engineering,

Shijiazhuang Vocational Technology

Institute. Her research interests are in the broad area of

communications and information theory, particularly coding

and signal processing for multimedia communications systems.

Chang Li was born in Hebei Province,

China, in 1980. He received the B.S.

degree from the Hebei University of

Science and Technology of China,

Shijiazhuang, in 2003, the M.S. degree

from the Shijiazhuang Railway Institute,

Shijiazhuang, in 2006, and the Ph.D.

degree from Beijing University of Posts

and Telecommunications (BUPT), Beijing, in 2011, all in

electrical engineering and communications. He is currently

working in China Mobile Group Hebei Co., Ltd. as senior

engineer. His research interests focus on wireless

communications theories and Technologies, especially the

cooperative technology and channel coding.

Documents

Performance Analysis of Coded Cooperation Based on Distributed … · 2016-08-19 · 2 Network Optimization Center, Network Department,China Mobile Group Hebei Co. Ltd.,Shijiazhuang