Distributed Turbo Coding

7/30/2019 Distributed Turbo Coding

1/5

Distributed Turbo Coding Using Log-Likelihood Thresholding

for Cooperative Communications

Ghaleb Al-Habian, Ali Ghrayeb, Mazen Hasna and Adnan Abu-Dayya

ECE Dept, Concordia Univ, Montreal, Quebec, CanadaEE Dept, Qatar University, Doha, Qatar

{g alhabi,aghrayeb}@ece.concordia.ca,{hasna,adnan}@qu.edu.qa

Abstract1In cooperative communications, error propagationat relays reduces the diversity order of the system. To combat thateffect, we present a novel technique to control error propagationat the relays, which is presented in the context of a relaycooperating with a source to communicate with a destinationusing a turbo code. The relay calculates log-likelihood ratio(LLR) values for the bits sent from the source. These value aresubjected to a threshold to selectively forward bits that are mostreliable and discard bits that are less so, resulting in less errorspropagating to the destination. We derive upper bounds on thebit-error rates for our proposed system and use them to optimizethe threshold at the relay, showing that our system provides abetter trade off between prevented errors and blocked correctbits. We compare our scheme with controlling error propagationusing only a cyclic redundancy code(CRC) check at the relay,forwarding analog LLR values, and employing no error controlat the relay at all. Based on system simulations, we show that ourproposed scheme provides a significant diversity gain comparedto other techniques.

I. INTRODUCTION

Cooperative communications have recently emerged to

achieve diversity in faded wireless links, namely, cooperative

diversity. Early works on cooperative communications sug-

gested two modes of operation for a cooperating relay [1]:amplify-and-forward (AF)where the relay just amplifies the

signal (subject to a power constraint) without decoding it and

forwards it to the destination, and decode-and-forward (DF)

where the relay detects and demodulates the signal and then

re-modulates it and forwards it to the destination. While DF is

prone to error propagation due to decoding errors, AF requires

the destination to have full channel state information (CSI) of

the source-relay and relay-destination channels.

More Recently, a few other relaying protocols were proposed

such as estimate-and-forward (EF), and compress-and-forward

(CF) [2], and bit-by-bit thresholding [3]. These protocols were

shown to improve the end-to-end performance. However, all

of these protocols where designed and analyzed for the caseof uncoded transmission, often called memoryless relaying,

where no channel coding was used at any point in the

transmission.

Schemes that use channel coding in cooperative systems are

usually called distributed coding schemes; examples of which

include coded cooperation [4], and distributed turbo coding

1This work was supported in part by a 3G project funded by Qatar Telecom(Qtel), and NSERC Grant N00858

(DTC) [5], [6]. Previous work that discussed distributed coding

over relay channels often assumed error-free relaying that

the relay can make correct decisions on the bits received and

hence is forwarding correct code bits [5], [6]. This assumption

is impractical, since even small error rates at the relay will

reduce the diversity gain and might cause an error floor in end-

to-end bit error rate [6]. A number of remedies were proposed

for relay networks utilizing channel coding, ranging from a

simple cyclic redundancy code (CRC) check at the relay [4];preventing it from forwarding if CRC fails. However, a single

error in a coded frame would trigger a CRC failure at the

relay, and hinder a significant number of correct bits to pass

on to the destination. Alternatively, the authors in [6], [7],

propose to calculate a reliability measure of received bits and

forward that to the destination along with the bits. This grants

the destination some flexibility in deciding on the bits. While

both use the log-likelihood ratio (LLR) as that measure, the

authors in [6] assumes the relay can transmit these LLRs as

unconstrained analog values to the destination. The authors in

[7] display a rate of expansion of as much as 3 bits per code

bit to relay a quantized value of that measure; that in addition

to extra processing at both the relay and destination.In this paper, we propose to calculate LLRs for bits received

at the relay, and then forwarding only reliable, hard-decided

bits to the destination. We establish reliability of the bits by

means of a threshold at the relay. We derive upper bounds

on the end-to-end bit error rate of the system under general

conditions and show the impact of a threshold on such bit

error rate. Consequently, we find the optimum threshold to be

set at the relay based on minimizing bit error rate. The result

of which is minimizing error propagation at the relay while

gaining significant diversity at the destination. Using computer

simulations, we show the performance of our technique oper-

ating in a DTC context under various conditions. We also show

the performance of other techniques used (described earlier)

under the same conditions, demonstrating the strength of the

proposed technique vis-a-vis other techniques.

The rest of this paper is organized as follows. Section II

presents the system model and then describes the operation

of the system throughout different stages of the cooperation

strategy. Section III analyzes the performance of the system.

Section IV discusses optimizing the threshold with minimum

end-to-end bit error rate as a target. Section V presents

simulation results, and Section VI concludes the paper.

1005978-1-4244-2941-7/08/$25.00 2008 IEEE Asilomar 2008

Authorized licensed use limited to: QATAR UNIVERSITY. Downloaded on December 27, 2009 at 04:15 from IEEE Xplore. Restrictions apply.


2/5

I I . PROPOSED SYSTEM

The design of the system presented next aims at enabling

the use of our remedy in a coded cooperative scenario. In

our system, the source encodes N information bits using aserially-concatenated convolutional code (SCCC), with two

recursive systematic convolutional (RSC) codes as constituent

encoders (denoted by E1, E2 for the outer and inner encoders,respectively). A complete system diagram is shown in Fig. 1.

Following the definitions on the diagram, let b = [b1, . . . , bN]be the frame of information bits to be encoded, where bits

take values {1}, and p = [p1, . . . , pN] be the parity bitsadded by E1. Hence c = [c1, . . . , c2N] = [b1, p1 . . . , bN, pN]will be the output of E1. As shown in the diagram, u = (c)or the interleaved c, and is the input to E2. Similarly, letl = [l1, . . . , l2N] be the parity bits added by E2. Hence, theoutput of E2 is x = [x1, . . . , x4N] = [u1, l1, . . . , u2N, l2N].For the rest of the paper, we assume the modulation used

throughout is binary phase shift keying (BPSK). Although the

scheme is expandable to multiple relays, we assume a single

cooperating relay to simplify the analysis. The output of theSCCC is transmitted by the source over two successive stages:

The broadcast stage, followed by the cooperation stage.

A. The Broadcast Stage

During this stage, the source transmits a punctured version

ofx to both the destination and relay. We elect to use a code

rate equal to Rc1 =13 , which is accomplished by transmitting

one of the systematic bits and both code bits in x (3 bits

per information bit). Thus, the modulated signal transmitted

from the source is expressed as {y[n]} = {l1, u2, l2, l3, . . .}.During this stage, the signals received at the destination and

the cooperating relay can be expressed as

rSD [n] =

Rc1EbhSD [n]y[n] + nSD [n], (1)

rSR [n] =

Rc1EbhSR [n]y[n] + nSR [n], (2)

where n = 1, 2, . . . , N / Rc1 , rSD and rSR are the signalsreceived at the destination and the relay, respectively, hSDand hSR are the fading coefficients for the source-destinationand source-relay channels, respectively, Eb is the energytransmitted per bit from the source, nSD and nSR are complexwhite Gaussian noise (AWGN) signals with per-dimension

power N0/2.At the relay, a SISO decoder is used (D2, here a maximum a

posteriori decoder, otherwise known as a MAP decoder) which

is matched to E2 to obtain soft estimates of u, denoted byLu. Specifically, these soft estimates are log-likelihood ratios(LLRs), formally defined in (3).

Lui = logeP(ui = 1|hSR , rSR)

P(ui = 0|hSR , rSR), (3)

LLRs are used as reliability measures of the individual bits;

increasing in absolute value as the reliability of the bit in-

creases.

B. The Cooperation Stage

In this stage, the source transmits to the destination u. Thus,

the code rate for this stage is Rc2 =12

. The relay has from

the previous stage soft estimates ofu. Consequently, the relay

can either: forward hard decisions based on the sign of the

soft estimates (which is simple DF), run a CRC check over

the frame (discarding the frame if CRC failed, cf. [4]), forward

the LLRs (after normalizing their power) to the destination, or

employ a threshold T such that the relay transmits only bitsthat have associated LLRs with absolute value exceeding a

threshold value. To prevent correct bits from being blocked, we

set the threshold to operate only after a CRC check fails, such

that no bits are blocked when we know the frame has been

successfully decoded. The relay then stays silent (transmits

zero energy) in place of the unreliable bits, and the hard

decisions of the reliable ones. We note here that we assume the

relay does not allocate the energy of the unreliable bits (which

are nulled) to the reliable ones, effectively lowering the total

transmit power. For reason of tractability of the analysis we

ignore this effect. During this stage, we can express the signals

received at the destination from the source and relay as

rSD [n] =

Rc2Eb/2hSD [n]u[n] + nSD [n], (4)

rRD[n] =

Rc2Eb/2hRD[n]u[n] + nRD[n], (5)

respectively, where n = N/Rc1 + 1,N/Rc1 +2, . . . , (N/Rc1 + N/Rc2), and with variable definitionssimilar to that for (1) and (2). u[n] is the modulated outputof the relay, u[n] is the modulated output of the source. Wedivide Eb by 2 to maintain a constant energy per bit acrossboth stages.

C. Decoding At The Destination

The destination receives both the broadcast stage frame (1),

and the cooperation stage frames (4)-(5). It then combines the

two copies of the cooperation stage frame using maximum

ratio combining (MRC). We remark that since the frame

received from the relay can contain incorrect bits, MRC

combining can be constructive or destructive for the bit in

question. However, as the energy contributed from the wrong

bit begins to increase, the total energy will reach zero after

which further increase in the wrong bits energy will not have

an impact on the probability of error. We use this concept in

system analysis in the next section.

The destination then multiplexes the combined frame with

the broadcast stage frame to get the complete coded frame.

Afterward, an iterative decoder (described in detail in [8],)

decodes the frame and produces the information bits. See Fig.

1 for a complete diagram of the system. Consequently, using

such a setup will increase the overall system code rate to1

(1/Rc1)+(1/Rc2)= 1/5. We remark that the use our proposed

technique is not limited to DTC schemes, since it is potentially

applicable in other distributed coding scenarios as well.

Throughout this paper, we assume all receiving nodes have

perfect knowledge of CSI. We also simulate the system

assuming Rayleigh-faded channels, with the source-destination

1006



3/5

Fig. 1. System block diagram,E1, E2 are the constituent encoders, D2 is a SISO decoder matched to E2, and is an interleaver

and relay-destination channels exhibiting quasi-static fading

and the source-relay channel exhibiting block fading.

I I I . PERFORMANCE ANALYSIS

To analyze the performance of our system, we derive upper

bounds on the end-to-end bit error rate. Referring to (1),(4),

and (5), the instantaneous received SNR at the output of the

multiplexer at the destination can be expressed as (cf. [9])

D[n] =

2Rc1SD [n] + Rc2SD [n] + A[n]Rc2RD[n]

+,

(6)

where ()+ denotes the bigger of the contents of the parenthe-ses and zero, n is the discrete time index for the total period ofboth stages, SD [n] =

EbN0

|hSD [n]|2 , RD[n] =

EbN0

|hRD[n]|2

and A[n] distinguishes a wrong decoded bit from a correctone, formally defined as

A[n] =

+1, u[n] = u[n]

1, u[n] = u[n]. (7)

Since we assumed quasi-static fading for the SD and RD

channels, SD [n] = SD , RD[n] = RD. Hence, assumingan all-zero codeword was transmitted, the probability of the

destination decoding a codeword of weight d bits (also calledthe pairwise error probability, or PEP),conditioned over the

instantaneous SNRs SD , RD, can be found as

P(d|SD , RD) = Q

{n}d

D[n]

. (8)

We split d into d1+d2 = d, where d1, d2 refer to the weight ofthe error event during the first and second stage, respectively.

Since the relay is contributing only during the second stage,

we can split d2 = dr + d, where dr indicates the weight ofbits(in the error event) receiving contribution from the relay

during the second stage, and d equals to the number of bitsreceiving contribution from the source only during the second

stage; due to possible bits being nulled at the relay.

Since the relay can forward wrong bits, we further split dr =dc+de, where dc, de equal to the number of bits correctly and

wrongly relayed from the relay in the error event, respectively.

The expression in (8) can be expanded as

P(d|SD , RD, dc, de) =

Q

(2d1Rc1 + d2Rc2) SD + (dc de) Rc2RD

+.(9)

Observing the piece-wise nature of P(d|SD , RD), weneed to evaluate the conditional PEP for different values of

SD , RD, dc, de, namely

P(d|SD , RD, dc, de) =

Q(0) = 12 , de > dc and

RD > SD

Q

()

, otherwise,

(10)

where is defined as

= 2d1Rc1 + d2Rc2

deRc2 dcRc2. (11)

To obtain the average PEP, we integrate (9) over the

joint PDF of (SD , RD). Assuming that the fades ex-perienced by source-destination and relay-destination chan-

nels are independent, p() =1 exp

, SD =

EbN0

EhSD

|hSD |

2

, RD =EbN0

EhRD

|hRD|

2

Without going into the details of the derivation (we refer the

interested reader to [10] for the complete proof), (9) becomes,

by substituting Craigs formula for the Q function (cf. [9]),integrating over the PDFs and some algebraic manipulation

P(d|dc, de) =

1

/20

(1+s1)(1+)d +

1

2

1

1+ , dc < de1

/20

1

1+s1

1

1+s2

d, dc de

,

(12)

where s1, s2 are defined as

s1 =(2d1Rc1 + (d2) Rc2)

2sin2 , s2 =

(dc de) Rc22sin2

,

respectively. We note from (12) that the PEP can tend to a

constant, given that de > dc, as tends to . We also observethat in all cases where de > 0 the resultant PEP will increase.

1007



4/5

Note that the PEP expression given in (12) is conditional on

de, dc (or de, dr : dr = dc + de) which are specific to a groupof error events of weight d. Thus, we need to sum the PEP overthe probability of an error word of weight d having dr, dc, deas components, as

P(d) =

d2dr=0

drde=0 P(d|d

e, dr = dc + de)pde(de)pdr(dr).(13)

Assuming a uniform distribution of relayed/error bits over the

forwarded frame, the PDF functions for dr, de are equal to

pdr(dr) =

d2dr

2Nd2dRdr

2NdR

, pde(de) =drde

dRdrdEde

dRdE

, (14)where dR, dE represent the total number of forwarded bitsfrom the relay and the number of which are wrong, respec-

tively. Finally, assuming maximum-likelihood (ML) decoding

of the received codeword at the destination, the resultant bit

error rate can be upper-bounded by (cf. [9])

Pb(e) 20 dB. Comparingboth thresholds, we can see the genie-aided threshold still

outperforms the proposed CSI-based threshold.

VI. CONCLUSION

In this paper, we have presented a novel technique to

mitigate error propagation in cooperative communications. Our

proposed system relied on soft estimates of bits, and used

them to block unreliable bits from being forwarded to the

destination. We developed upper bounds on the performance

of the system and showed that they converge to the actual

performance. We compared our technique with just using CRC

at the relay, with simple DF, and analog LLR forwarding and

displayed significant improvement in diversity and bit error

rate. While analog LLR forwarding and simple DF caused

an error floor in the end-to-end bit error rate of the system,CRC lost too much diversity by discarding the whole frame,

and our proposed technique was able to circumvent both

disadvantages.

REFERENCES

[1] D. Chen and J. N. Laneman, Modulation and demodulation for coop-erative diversity in wireless systems, IEEE Trans. Wireless Commun.,vol. 5, pp. 17851794, Jul. 2006.

[2] G. Kramer, M. Gastpar, and P. Gupta, Cooperative strategies andcapacity theorems for relay networks, IEEE Trans. Inf. Theory, vol. 51,pp. 30373063, Sep. 2005.

0 5 10 15 20 25 3010

6

105

104

103

102

101

100

in dB

End-to-end

BER

simple DF

analog LLR relaying

Simple CRC

Bound on CSIbased threshold

CSIbased thresholding

genieaided Thresholding

Bound on errorfree relaying

error free relaying

Fig. 2. Bit error rate vs. , for SR fixed at 6 dB. Analog LLR relayingwas implemented according to [6]

0 5 10 15 20 25 3010

6

105

104

103

102

101

10

0

in dB

End-to-endBER

Simple DF

Simple CRC

analog LLR relaying

Bound on CSIbased threshold

CSIbased threshold

genieaided threshold

Bound on errorfree relaying

error free relaying

Fig. 3. Bit error rate vs. , for SR fixed at 9 dB. Analog LLR relayingwas implemented according to [6]

[3] H. V. Khuong and H. Y. Kong, LL R-based decode-and-forward proto-col for relay networks and closed-form ber expressions, IEICE Trans.Fundamentals, vol. E89A, pp. 18321841, Jun. 2006.

[4] T. Hunter and A. Nosratinia, Diversity through coded cooperation,IEEE Trans. Wireless Commun., vol. 5, pp. 283289, Feb. 2006.

[5] B. Zhao and M. Valenti, Distributed turbo coded diversity for relaychannel, Electronics Letters, vol. 39, pp. 786787, May 2003.

[6] Y. Li, B. Vucetic, T. F. Wong, and M. Dohler, Distributed turbo codingwith soft information relaying in multihop relay networks, IEEE J. Sel.

Areas Commun., vol. 24, pp. 20402050, Nov. 2006.[7] Y. Hairej, A. Darmawan, and H. Morikawa, Cooperative diversity usingsoft decision and distributed decoding, in Proc. Mobile and WirelessCommunications Summit, Nagoya, Japan, Jul. 2007, pp. 15.

[8] T. M. Duman and A. Ghrayeb, Coding for MIMO CommunicationSystems. New York: Wiley, 2008.

[9] M. Elfituri, W. Hamouda, and A. Ghrayeb, A convolutional-based codedcooperation scheme for relay channels, IEEE Trans. Veh. Technol.,accepted for publication, May 2008.

[10] G. Al-Habian, A. Ghrayeb, M. Hasna, and A. Abu-Dayya, A noveltechnique for mitigating error propagation in cooperative communicationnetworks, submitted to IEEE Trans. on Communications. [Online].Available: http://users.encs.concordia.ca/g alhabi/jrnl.pdf

1009

Distributed Turbo Coding

Documents

Transcript of Distributed Turbo Coding