Bandwidth-efficient Bit-interleaved Coded Cooperative Communications

Wireless Pers Commun (2008) 46:247–268DOI 10.1007/s11277-007-9420-y

Bandwidth-efficient Bit-interleaved Coded CooperativeCommunications

Hieu T. Nguyen · Ha H. Nguyen · Tho Le-Ngoc

Published online: 16 November 2007© Springer Science+Business Media, LLC. 2007

Abstract A cooperative coding scheme using M-ary modulation for high spectral efficiencyand repetitive information transmission for diversity improvement is proposed. The encoderis a parallel concatenation of two bit-interleaved coded M-ary modulators (BICM) with scal-able repetition of information symbols. As a result, the fraction of repeated information andthe spectral efficiency of M-ary modulation can be used for bandwidth-performance tradeoff.Performance analysis is carried out to verify the performance advantage of the proposed sys-tem. In particular, analytical and simulation results in various block fading scenarios showthat for a given bandwidth efficiency, a proper selection of the fraction of repeated infor-mation and the spectral efficiency of M-ary modulation can provide a large performanceimprovement.

Keywords Cooperative communications · Coded modulation · Performance analysis ·Block fading channel · Adaptive modulation and coding

1 Introduction

Wireless transmission is often error prone due to multipath fading. Diversity offers an effec-tive solution to combat channel fading by providing the receiver with multiple copies of thesame information that are transmitted with independent channel gains. Spatial diversity hasbeen widely accepted as one of the most effective ways to deal with channel fading. Theeffectiveness of spatial diversity has fueled strong interest in employing multiple antennasat both the transmitter and the receiver [1]. Achieving full diversity gain, however, requires

H. T. Nguyen · H. H. Nguyen (B)Department of Electrical & Computer Engineering, University of Saskatchewan, 57 Campus Dr.,Saskatoon SK, Canada S7N 5A9e-mail: [email protected]

T. Le-NgocDepartment of Electrical & Computer Engineering, McGill University, 3480 University St., MontrealQC, Canada H3A 2A7

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248 H. T. Nguyen et al.

Fig. 1 Coded cooperativecommunications

User 1

User 2

Destination

Frame 1 Frame 2

Frame 1 Frame2

1 bits of User 1N 2 bits of User 2N

2 bits of User 1N1 bits of User 2N

multiple antennas be placed sufficiently far apart, which is problematic for size-limited mobileterminals.

An alternative approach to provide spatial diversity without using multiple antennas iscooperative communications. This approach is motivated by the fact that the wireless chan-nels from different users (or nodes) to the destination are likely to fade independently andusers can assist each other with the transmission of their messages. Because information istransmitted multiple times, once from the user itself and then from the partnering (or relay-ing) users, spatial diversity is achieved. Cooperative communications has been recognizedas a very promising solution for future cellular and ad-hoc wireless systems [2–5].

Uncoded cooperative protocols are considered in [2–5]. In these protocols, the informationbits are received and retransmitted to the destination by the partner. Whereas, in the codedcooperative protocols proposed in [6–9], each user decodes the information bits of its partner,re-encodes them and sends only the additional parity bits for the partner to the destination.

As illustrated in Fig. 1, a coded cooperative process takes place over two frame periods. Inframe 1, each user transmits its own N1 bits to the destination. It also listens and decodes hispartner’s frame 1. If the user successfully decodes the information for the partner’s frame 1,indicated by the cyclic redundancy check (CRC), the user switches to the cooperative mode.In this mode the user produces frame 2 consisting of N2 bits for its partner, and transmitsthis frame to the destination. Otherwise, it stays in the non-cooperative mode and transmitsits own frame 2. One can see that the adaptive coded cooperative protocols can thus reducethe effect of error propagation that often happens in the uncoded cooperative protocols. Inthe coded cooperative protocols, the parameter ρ = N2/(N1 + N2) defines the cooperationlevel.

In [6], turbo code (TC) is considered for coded cooperative systems due to its suitableencoding structure [10]. More specifically, since the encoder of a turbo code is a parallelconcatenation of two component codes separated by a random interleaver, the information-bit stream and the first parity-bit stream produced by encoder 1 are combined to form frame1, while encoder 2 is used to produce frame 2 that contains the second parity-bit stream.However, this proposed coded cooperations only consider binary modulation (i.e., BPSK)and transmission of only additional parity bits by the partner [6–9]. In an additive whiteGaussian noise (AWGN) or a fast fading channel, there might be no need to transmit theinformation-bit stream twice. However, in a block fading channel, where the fading coeffi-cients stay constant over some time intervals, repeating the information symbols can be veryhelpful to improve the system performance even when the reduction in the transmission rate

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Bandwidth-efficient Bit-interleaved Coded Cooperative Communications 249

(i.e., bandwidth efficiency) is taken into account. This fact will be confirmed by the resultspresented later in this paper.

From the above observations and motivated by the great demands on high-speed transmis-sion in the current and future wireless networks, this paper introduces a bandwidth-efficientcoded cooperative system in which the information symbols are also repeated by the partner[11]. Though the rate-compatible punctured convolutional (RCPC) codes [10] can be used,this paper focuses on coded cooperative systems where the encoder is a parallel concatenationof two bit-interleaved coded M-ary modulators (BICM) with scalable repetition of informa-tion symbols. As a result, the fraction of repeated information and the spectral efficiency ofM-ary modulation can be used for bandwidth-performance tradeoff. Performance analysis ofthe proposed system is carried out and verified with simulation results. Furthermore, since thecurrent wireless networks (3G) are mainly designed for “bursty” data applications [12,13],link adaptive techniques are preferable to power control techniques (as in the legacy networksthat mainly serve voice services). Here a new link adaptive technique for the proposed codedcooperative system and its potential throughput gain are also discussed.

The remaining of the paper is organized as follows. The proposed coded cooperativescheme for high bandwidth efficiency is described in Sect. 2. Performance analysis is pro-vided in Sect. 3. Section 4 presents analytical and simulation results for different M-PSKand M-QAM systems in various block fading scenarios. Also included in this section is adiscussion of a new link adaptation approach and its advantages. Section 5 gives conclusions.

2 System Description

2.1 Encoder

Consider a two-node cooperative system as illustrated in Fig. 1 in which two users, calledUser 1 and User 2, cooperate with each other to send their information to the third node,called the destination. In this case, User 1 is the partner of User 2 and vice versa. Figure 2shows the block diagram of the encoding process in the proposed bandwidth-efficient codedcooperative system for User 1 (similar for User 2). The coded cooperation happens over twoframes as follows:

CRC SC1M-ary

Mapper

Frame 1

Π

Systembits

Parity bits

BICMdecoder

CRCcheck

Frame 2

BICM

1Π

Frame 1 from User 2

User 1'sinformation bits

SC2M-ary

Mapper

System bits

Parity bits

BICM

2ΠPuncturer

Yes

No

Fig. 2 Encoder in the proposed coded cooperative communications

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– Frame 1: An information block of K bits (including the CRC bits) is fed to the firstsystematic convolutional encoder (SC-1) to produce the first parity-bit sequence. Boththe systematic-bit and the first parity-bit sequences are interleaved by the interleaver �1.Then, the interleaved bits are mapped to the M-ary signal constellation to produce thesymbol sequence for frame 1. Over frame 1, User 1 also receives the transmitted signalof User 2’s frame 1. This implies that User 1 can operate in a full-duplex mode [2–8].Similar process applies to User 2. The encoding carried out in frame 1 is generally knownas bit-interleaved coded modulation (BICM) [14].

– Frame 2: Based on the partner’s received signal in frame 1. Each user implements BICMdecoding to restore the original information for its partner. Whether or not a user suc-cessfully decodes the information for its partner is indicated by the CRC result. If CRCis correct, the decoded information is first interleaved by the interleaver �, then appliedto the second systematic convolutional encoder (SC-2) to produce the second parity-bitsequence. Different from the previously proposed coded cooperative system with BPSKwhere no information bits are repeated by the partner, here the systematic bits togetherwith the parity bits produced by SC-2 are combined and mapped to the M-ary signal con-stellation and sent to the destination in frame 2. Moreover, to provide a flexible spectralefficiency, the portion of the systematic bits repeated in frame 2 is controlled by a punctureras seen in Fig. 2.

For convenience, define the parameter γ , called the repetition level, that represents thepercentage of the systematic bits repeated by the partner as follows:

γ = Number of systematic bits repeated

Total number of systematic bits(1)

By definition, the range of γ is from 0% to 100%. It can be seen that a higher repetition levelindicates a larger number of systematic bits enjoying a diversity order of 2 at the expense oflower transmission efficiency. In order to maintain a given bandwidth efficiency, higher-ordermodulation schemes must be used.

Consider a system using an M-ary modulation scheme with identical SC-1 and SC-2 ofrate rc, and a repetition level γ . Let K be the number of the information bits entering SC-1 andSC-2 of Fig. 2. The frame lengths are simply N1 = K/rc

log2 M symbols and N2 = (K/rc)−(1−γ )Klog2 M

symbols. Recall the definition of the cooperative level ρ = N2/(N1 + N2), it is readilyverified that

ρ = 1 − (1 − γ )rc

2 − (1 − γ )rc. (2)

The bandwidth efficiency of the system is defined as the number of information bits thateach user can transmit over one second and per 1 Hz of bandwidth. It can be determined asfollows:

ηBW = Number of information bits of one user in 2 frames

Total length of the two frames (in symbols)

= K

N1 + N2= K

K [(2/rc) − (1 − γ )]/ log2 M

= rc log2 M

2 − (1 − γ )rc(bits/sec/Hz). (3)

For example, consider the specific case of coded cooperation with identical SC-1 andSC-2 of rate 1/2. A system with 8-PSK constellation and γ = 0%, i.e., a direct extension

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of the system considered in [6], provides a bandwidth efficiency of 1 bit/s/Hz. To maintainthis bandwidth efficiency, a system with γ = 100% (i.e., where all the systematic bits arerepeated in frame 2) needs to use 16-QAM constellation.

It is clear that with higher-order modulation, the minimum distance must be reduced inorder to maintain the same average transmitted power. The reduction of minimum distancein the constellation causes performance loss. However, this loss can be compensated by thediversity gain because more transmitted bits can have a diversity order of 2. As will be shownin this paper, diversity gain provided by increasing the repetition level overcomes the loss inminimum distance due to the increase of the modulation size M , and results in a power andbandwidth-efficient cooperative scheme over block fading channels.

2.2 Decoder

With the turbo-like structure of the encoder in Fig. 2, two iterative decoding methods can beapplied at the receiver. In the first iterative decoding method, described in Fig. 3, iterationsare only carried out between the two SISO channel decoders. In this paper, the SISO decoderemploys the maximum a posterior probability (MAP) algorithm in [16].

In the second method, iterations are implemented not only between the two channel decod-ers but also between the channel decoders and the demappers as illustrated in Fig. 4. Morespecifically, the two demappers simultaneously produce the extrinsic information for thecoded bits based on the received signals corresponding to frame 1 and frame 2, respectively.Then they are interleaved before being applied to SISO-1 and SISO-2. Next, the SISO-1 isfirst activated to produce the extrinsic information for the information bits and the coded bits.The extrinsic information of the coded bits is then interleaved to feedback to demapper 1 forthe next iteration [15]. On the other hand, the extrinsic information of the information bits isalso interleaved and transferred to SISO-2. Finally, similar to SISO-1, SISO-2 produces boththe extrinsic information for the coded bits and information bits to feedback to demapper 2and SISO-1, respectively. In addition, the soft information of the information bits is passedto the decision device to make the hard decisions of the information bits. Clearly, the com-plexity of the second decoding method is much higher than that of the first one, especiallyfor large constellation sizes. This is because the iterations in the second decoder also involvethe M-ary demappers.

Soft M-ary De-mapper & De-interleaver

Soft M-ary De-mapper & De-interleaver

SISO1

SISO2

Frame 1 received signal

Frame 2 received signal

systematic 1L

parity 1L

parity 2L

eL

aLsystematic

systematic

systematic 2

L

L

Π

Decision device

Π

+

L

1−Π

1−Π

eL

aL

Fig. 3 A decoder in the proposed coded cooperative communications: no iterations between the SISO decodersand the M-ary demappers

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Soft M-aryDe-mapper 1

Soft M-aryDe-mapper 2

SISO1

SISO2

Frame 2 receivedsignal

Frame 1 receivedsignal

systematic 1L

parity 1L

parity 2L

)(ieL

)(iaL

systematic

systematic

systematic 2L

L

Π

Decisiondevice

Π+

L

1−Π

1−Π

)(ieL

BICM-ID

BICM-ID

1Π

11

−Π

2Π

11

−Π1

2−Π

)(iaL

)(ceL)(c

aL

)(ceL)(c

aL

Fig. 4 Another decoder in the proposed coded cooperative communications: Iterations are run between theSISO decoders and the M-ary demappers

The simulation results for the above two decoding methods over the block fading channelare shown in Table 1 for the case that both SC-1 and SC-2 are the same systematic convo-lutional code of rate 1/2 and generator polynomials (1, 5/7) in octal form. The interleaversare random interleavers. In addition, the target bandwidth efficiency is 1 bit/s/Hz and it isachieved by using 8-PSK signal constellation. The frame length is chosen such that a blockof K = 132 information bits is carried over every two frames (i.e., the cooperation period).Observe that the number of erroneous bits resulted from the second decoding method issmaller than that of the first decoding method when the same number of frames and signal-to-noise ratio (SNR) are set. However, this difference is relatively small in terms of the biterror rate (BER) while the complexity of the second decoder is significantly higher than thecomplexity of the first one.

Another observation based on the simulation results in Table 1 is that the BER performanceslightly improves from the first iteration to the second iteration in both the decoding meth-ods. For example, Fig. 5 shows that a gain of approximately 0.5 dB is achieved for the casethat iterations are only carried out between the two channel decoders, i.e., the first decodingmethod of Fig. 3. However, almost no performance improvement is obtained after the seconditeration. Two main reasons are as follows. First, the goal of cooperative communicationsis to improve the diversity gain in block fading environment. In block fading environment,the fading coefficients are constant over every two frames of the cooperative process. Whenboth the channels from the two users to the destination are in very deep fades, it makes theinstantaneous SNRs of both frames reduced to a very low level, at which iterative decodingis no longer useful. Second, since the interleaving length of a turbo code used in cooperativecommunications needs to be the same as the frame length, which is typically very short, itsignificantly reduces the error correcting capability of the turbo code.

With the above investigations and observations, the first decoding method, shown in Fig. 3,with only a small number of iterations is chosen for the proposed coded cooperative system.In particular, all the simulation results shown later in this paper are obtained after threeiterations of decoding.

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Table 1 Numbers of erroneous bits with two different iterative decoding methods: RSC codes of rate 1/2 andgenerator polynomials (1, 5/7), bandwidth efficiency is 1 bit/s/Hz and 8-PSK modulation

Iterations Eb/N0 = 5 dB Eb/N0 = 10 dB Eb/N0 = 15 dB

(1,500 frames) (3,000 frames) ( 4,000 frames)

Iteration between SISO decoders only (see Fig. 3)

1 17,443 7,194 1,563

2 16,006 5,893 1,250

3 15,636 5,658 1,133

4 15,373 5,615 1,098

5 15,298 5,545 1,104

6 15,262 5,538 1,114

7 15,161 5,490 1,052

Iteration among SISO decoders and demappers (see Fig. 4)

1 16,876 6,834 1,327

2 15,227 5,644 1,071

3 14,649 5,402 1,014

4 14,516 5,330 1,039

5 14,403 5,217 979

6 14,350 5,210 994

7 14,320 5,183 1,065

Every two frames carry K = 132 information bits. Here Eb/N0 is the information-bit-energy-to-noise-densityratio for both the User 1-destination channel and User 2-destination channel

Let s1[n], n = 1, . . . , L1, and s2[n], n = 1, . . . , L2, be the nth symbols in frame 1 andframe 2, where L1 and L2 are the lengths of frame 1 and frame 2, respectively. With theassumption of orthogonal transmission of two users’ signals, the corresponding baseband-equivalent discrete-time received signals are expressed as:

r1[n] = h10[n]s1[n] + w10[n] (4)

r2[n] = h20[n]s2[n] + w20[n] (5)

where r1[n] and r2[n] are the received signals for User 1 at the destination from User 1and User 2, respectively. The quantities h10[n] and h20[n] are the fading coefficients of thechannels from User 1 and User 2 to the destination (also referred to as User 0), which affectthe nth transmitted symbols of the two frames, respectively. For the assumed block fadingenvironment, the fading coefficients are constant over two frames, i.e., h10[n] = h10 forn = 1, . . . , L1, and h20[n] = h20, n = 1, . . . , L2. Furthermore, both h10 and h20 are modeledas circularly symmetric complex Gaussian random variables (i.e., Rayleigh fading channels)with variances E (1)

s /2 and E (2)s /2 per dimension, respectively. User 1’s channel symbols,

namely {s1[n]}, are drawn from an M-ary signal constellation χ(1) whose average symbolenergy is normalized to unity. Similarly, User 2’s channel symbols, {s2[n]}, belong to an unit-energy M-ary constellation χ(2). The received signals in (4) and (5) are also disturbed by addi-tive circularly symmetric complex Gaussian noises w10[n] and w20[n] of variance N0/2 perdimension. Note that with the above notations, the average symbol-energy-to-noise-densityratios for User 1-destination channel and User 2-destination channel are E (1)

s /N0 and E (2)s /N0,

respectively. Similarly, the average symbol-energy-to-noise-density ratios of the interuser

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12 13 14 15 16 17 18 19 20 21 2210

−5

10−4

10−3

10−2

Eb/N

0 (dB)

BE

R1 iteration2 iterations3 iterations4 iterations

Fig. 5 Error performance with iterations: SC-1 and SC-2 are RSC codes of rate 1/2 and generator polynomials(1, 5/7), bandwidth efficiency is 1 bit/s/Hz, 8-PSK modulation. Here Eb/N0 is the information-bit-energy-to-noise-density ratio for both the User 1-destination channel and User 2-destination channel

channels, namely User 1-User 2 and User 2-User 1 channels, are denoted as E (1,2)s /N0 and

E (2,1)s /N0, respectively. Furthermore, define the related average information-bit-energy-to-

noise-density ratios for the four mentioned channels as E (1)b /N0, E (2)

b /N0, E (1,2)b /N0 and

E (2,1)b /N0, respectively. The relationship between the average information-bit energy and the

corresponding average symbol energy depends on the code rate and the constellation size.As shown in Fig. 3, at the destination, the received signals {r1[n]}L1

n=1 and {r2[n]}L2n=1 are

first applied to the M-ary demappers. In general, since the decoder at the destination workswith both the constellations χ(1) and χ(2) of the two users, the superscripts of the constel-lations are therefore dropped in the following discussion. With the assumption that perfectchannel state information is available at the destination, each demapper first computes thelog-likelihood ratios (LLRs) for all the coded bits, i.e., the bit metrics, as follows:

L(bik[n]|ri [n]) = log

⎛⎝

∑si [n]∈χ1,k

exp[−|ri [n]−hi0si [n]|2

2N0

]

∑si [n]∈χ0,k

exp[−|ri [n]−hi0si [n]|2

2N0

]⎞⎠ (6)

where k = 1, . . . , m = log2 M ; i = 1, 2; n = 1, . . . , Li and bik[n] is the kth coded bit in agroup of m = log2 M bits carried by si [n], the nth symbol of frame i . The subsets χ1,k andχ0,k contain the signal points in the M-ary constellation whose kth labelling bits are “1” and“0”, respectively.

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After bit-metric computation is fulfilled, the bit-metric sequence is deinterleaved anddemultiplexed to separate the systematic-bit metrics from the parity-bit metrics. Next, thesystematic-bit metric sequences produced by the two demappers are combined before appliedto the soft-input soft-output (SISO) decoders. The SISO decoder 1 (SISO-1) produces theextrinsic information for the systematic bits (indicated by the symbol Le in Fig. 3), which isthen passed to the SISO decoder 2 (SISO-2) as shown in Fig. 3. The extrinsic informationprovided by SISO-1 is used in SISO-2 as the a priori information for the systematic-bit stream(indicated by La in Fig. 3). The soft output of SISO-2 is used to make the final hard decisionon the original systematic bits. In addition, the SISO-2 also produces the extrinsic informa-tion on information bit to feedback to the SISO-1 after it is de-interleaved. This informationis used as a a priori information for the systematic-bit stream at the input of the SISO-1 inthe next iteration. The functionality of the SISO module is thoroughly discussed in [16]. Asdiscussed before, almost no performance improvement is obtained after the third iteration.Therefore, only three iterations are used at the receiver side in our investigation.

3 Performance Analysis

Since the proposed coded cooperative communications system is based on bit-interleavedcoded modulation (BICM), to determine its error performance, one needs to first obtain thepairwise error probability (PEP) of M-ary BICM over a point-to-point block fading channel.Although the PEP of M-ary BICM is determined here for the maximum likelihood decoding[17], the analytical results shall be compared with the simulation results of the iterativedecoding system in Fig. 3. This is reasonable as it is well-known that the performance of thesuboptimal iterative decoding approaches that of the maximum likelihood decoding [10].

3.1 Pairwise Error Probability for Point-to-Point M-ary BICM

Let r = (r [1], r [2], . . . , r [L]) be the received vector corresponding to an arbitrary transmit-ted frame and h be the fading coefficient of that frame. If s = (s[1], s[2], . . . , s[L]) is thetransmitted frame of the coded symbols, then

r = hs + w (7)

where w = (w[1], w[2], . . . , w[L]) is the vector of i.i.d. circularly symmetric Gaussianrandom variables.

The pairwise error probability (PEP) for a coded system is defined as the probabil-ity of deciding in favor of a codeword b̂ = (b̂[1], b̂[2], . . . , b̂[N ]) when the codedwordb = (b[1], b[2], . . . , b[N ]) was actually transmitted. For M-ary BICM with an ideal inter-leaver it can be shown that [14] the PEP only depends on the Hamming distance d betweenb and b̂, the constellation χ and the mapping scheme µ. It can be expressed as [17]

PEP(d, χ, µ) = Pr[Pr(b̂|r) > Pr(b|r)|b

]

= Pr

⎡⎣ ∑

j :b[ j]�=b̂[ j]log

(Pr(b̂[ j]|r [n j ])Pr(b[ j]|r [n j ])

)> 0

∣∣∣∣b⎤⎦ (8)

where r [n j ] is the n j th received signal that carries the j th bit in the transmitted codeword.It follows from (7) that the received signal vector r depends on the noise and fading

realizations w and h, respectively. Furthermore, it also depends on the particular modulation

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symbols s, the bit position k of the binary label. For convenience, these random variables aregrouped in a random vector ν � (w, h, s, k). Obviously ν depends on the signal constellationχ and the mapping scheme µ. Without loss of generality, assume that b and b̂ differ in thefirst d positions. Then the PEP can be rewritten as

PEP(d, χ, µ) = Pr

⎛⎝

d∑j=1

� j > 0

⎞⎠ (9)

where the new random variable � j , called the a posteriori log-likelihood ratio, is defined as

� j = logPr(b̂[ j] = b|ν)

Pr(b̂[ j] = b|ν)(10)

where b is either 0 or 1 and b = 1 − b. The a posteriori probability in (10) is computed asfollows:

Pr(b̂[ j] = b|ν) ∝∑

s∈χb,k

exp(−|r [n j ] − hs[n j ]|2) (11)

where χb,k is the subset of all signal points with the kth binary position equal to b.The computation of (9) is obtained in [17] using the saddle point method. In particular, it

is shown in [17] that determining the tail probability in (9) can be conveniently carried outby using the cumulant transform of the random variable � j instead of its probability densityfunction. The cumulant transform of � j is defined as [17]

κ j (p) � log E[ep� j ] = log Eν

[(Pr(b̂[ j] = b|ν)

Pr(b̂[ j] = b|ν)

)p](12)

where Eν denotes the expectation over all the random variables in ν. This expectation can beefficiently computed by using the Gauss-Hermite approximation method in [18].

The PEP calculation using the cumulant transforms is [17]

PEP(d, χ, µ) = 1√2π

∑dj=1 κ ′′

j ( p̂) p̂exp

⎛⎝

d∑j=1

κ j ( p̂)

⎞⎠ (13)

where p̂ is the solution of κ ′j ( p̂) = 0 and it is called the saddle value. It can be proved that

this value exists and is unique. For the binary-input output-symmetric (BIOS) channel, thesaddle point is placed at p̂ = 1

2 . Therefore κ j ( p̂) and κ ′′j ( p̂) in (13) are given as [17]:

κ j ( p̂) = log Eν

⎡⎣

√√√√(

Pr(b̂[ j] = b|ν)

Pr(b̂[ j] = b|ν)

)⎤⎦ (14)

κ ′′j ( p̂) = 1

E[e p̂ν]E

⎡⎣

(log

Pr(b̂[ j] = b|ν)

Pr(b̂[ j] = b|ν)

)2 √Pr(b̂[ j] = b|ν)

Pr(b̂[ j] = b|ν)

⎤⎦ (15)

Recall that the PEP in (13) is for the point-to-point BICM communications system. It isapplied in the next subsection to derive the end-to-end BEP for the proposed BICM codedcooperative systems.

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3.2 End-to-End Bit Error Probability

As thoroughly discussed in [19], the end-to-end bit error probability for the coded coopera-tive systems is also a function of the interuser channel quality. In other words, it depends onwhether or not decoding of the first frame is successful at each user. Similar to [19], define anew random variable that represents the events at both users in decoding the first framesof their partners. In particular, the outcomes of this random variable are defined as follows.

– = 1 when both User 1 and User 2 successfully decode the first frames of their partners.– = 2 when User 1 fails to decode the first frame of User 2 and User 2 succeeds in

decoding the first frame of User 1.– = 3 when User 1 succeeds in decoding the first frame of User 2 but User 2 fails to

decode the first frame of User 1.– = 4 when both User 1 and User 2 fail to decode the first frames of their partners,

respectively.

Conditioned on , the bit error probability (BEP) can be expressed as

P(bit error|) =∫

α1

∫α2

min

⎡⎣1,

1

kc

∞∑d=d f ree

CdPEPcoop(d, χ, µ, ρ, γ, α1, α2|)

⎤⎦

×p(α1)p(α2)dα1dα2 (16)

In (16) d f ree is the free distance of the convolutional code, α1 = |h1|2/N0 and α2 = |h2|2/N0

are the instantaneous signal-to-noise ratios of User 1-destination channel and User 2-desti-nation channel, respectively. The parameter kc is the number of information bits per branchin the code trellis and Cd is the number of bits in error corresponding to the error event on thecode trellis that has Hamming weight d . The functions p(α1) and p(α2) are the probabilitydensity functions of the random variables α1 and α2, respectively. Since Rayleigh fadingchannels are considered, p(α1) and p(α2) are exponential distributions [20]. The functionmin(·) in (16) is used to obtain the tight bound [21,22] on the BEP. Note that the pairwiseerror probability in a cooperative system, PEPcoop in (16), is explicitly shown to depend notonly on d , χ , µ; but also on ρ, γ , α1 and α2.

The PEPcoop expressions for User 1 corresponding to the above four specific outcomes of are provided in the following. Similar expressions hold for User 2.

Case 1 ( = 1) Recall that in this case both users are successful in decoding the first framesof their partners. This means that one codeword is delivered to the destination by two sub-codewords on two independent paths, thus a diversity order of 2 is obtained. Assume thatthe Hamming weight of each sub-codeword is proportional to the frame length, then theconditional PEP is:

PEPcoop(d, χ, µ, ρ, γ, α1, α2| = 1) = 1√2π(d1κ ′′( p̂|α1) + d2κ ′′( p̂|α2)) p̂

× exp(d1κ( p̂|α1) + d2κ( p̂|α2)

)(17)

where d1 and d2 are the Hamming weights of the frame 1 and frame 2, respectively. In thecooperative system without using repetition code [2–5] the summation of d1 and d2 equalsto the Hamming weight d of the full codeword. Whereas in our proposed system, d1 and d2

depend not only on the cooperative level ρ but also on the repetition level γ and the coderate rc. In particular,

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d1 ≈ (1 − ρ)[2 − (1 − γ )rc]d (18)

d2 ≈ ρ[2 − (1 − γ )rc]d (19)

Case 2 ( = 2) In this case, the second frame of User 1 is produced by User 2 on the basisof the decoded results of the first frame of User 1. Therefore,

PEPcoop(d, χ, µ, ρ, γ, α1, α2| = 2)

= exp(d1κ( p̂|α1) + d2κ( p̂|α1) + d2κ( p̂|α2)

)√

2π(d1κ ′′( p̂|α1) + d2κ ′′( p̂|α1) + d2κ ′′( p̂|α2)) p̂(20)

Case 3 ( = 3) In this case, the destination decodes just the first frame received directlyfrom User 1. Hence,

PEPcoop(d, χ, µ, ρ, γ, α1, α2| = 3) = exp(d1κ( p̂|α1))

)√

2π(d1κ ′′( p̂|α1) p̂(21)

Case 4 ( = 4) This case implies no cooperation between the two users and

PEPcoop(d, χ, µ, ρ, γ, α1, α2| = 4) = exp(dκ( p̂|α1)

)√

2π(dκ ′′( p̂|α1) p̂(22)

Finally, the end-to-end bit error probability is obtained by averaging over all four cases:

Pb =4∑

=1

P(bit error|)Pr() (23)

where the probability distribution Pr() of the random variable can be determined as in[19]. For completeness, Appendix A summarizes the derivation of Pr() from the interuserchannel SNR.

4 Results and Discussions

We first consider a cooperative system using two identical SC-1 and SC-2 with rate-1/2,4-state RSC codes of the same generator polynomials (1; 5/7) in octal form, and informationblock length of 132 bits. All the simulation results are obtained with random interleavers.For the target bandwidth efficiency of 1 bit/s/Hz, we select two extreme repetition levels: (i)γ = 0% with 8-PSK, which is a direct extension of the system in [6] and (ii) γ = 100% with16-QAM. Perfect interuser signaling is assumed. For all modulation formats, Gray mappingsare used.

Figure 6 presents the bit-error-rate (BER) performance versus the same average signal-to-noise ratio, Eb/N0, obtained by simulations and numerical analysis, over users’ indepen-dently and identically Rayleigh fading channels, where Eb is the energy per information bit.Similar to [19], we truncate the summations in the computations of the union bounds in (25)and (16) to the first 15 terms of Ad and Cd , respectively. As can be observed, the simulationresults are quite close to the analytical results derived in Sect. 3. Our investigation shows thatif all the terms of Ad and Cd are included in (25) and (16) then the upper bounds are within2–4 dB compared to the simulation results. This is also similar to what observed in [22] and[19] for the case of binary modulation.

Also shown in this figure is the performance of the non-cooperative system that uses thesame RSC code and 8-PSK, and hence has the same spectral efficiency. It is not surprising

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16 17 18 19 20 21 22

10−4

10−3

10−2

Eb(1)/N

0 (dB)

BE

R

Non−cooperative: ρ=0%, 8−PSKProposed: γ=0%,ρ=33%, 8−PSKProposed: γ=100%,ρ=50%, 16−QAMAnalytical bound

Fig. 6 Performance of the proposed coded cooperative systems with bandwidth efficiency of 1 bit/s/Hz,

E(2)b /N0 = E(1)

b /N0

to see that significant coding gains are achieved by the proposed coded cooperative systemsover the non-cooperative one. But what is interesting to observe is that a large coding gain(of about 2 dB at BER of 10−4) is realized by increasing γ from 0% to 100% in the proposedsystem. This observation clearly illustrates the capability of achieving a higher coding gainin the proposed system by repeating more systematic bits.

The results shown in Fig. 7 are obtained for the same systems as in Fig. 6, but withunequal average signal-to-noise ratios: the SNR of User 2 at the destination is 5 dB betterthan that of User 1. This can happen if User 2 is at the location closer to the destination thanUser 1. Other possibility is when the transmission link from User 2 to the destination hasless obstructions than the link from User 1 to the destination. Observe that the coding gainbetween the proposed systems with different repetition levels is still maintained in this case.However, compared to the non-cooperative system, the error performance of the proposedsystems improves quite significantly. For example, at the BER of 10−3, the coding gain ofthe proposed system with γ = 100% over the non-cooperative system increases from 10 dBin Fig. 6 to 13 dB in Fig. 7. This observation means that User 1 obtains more benefit fromthe cooperation with User 2, who has a better channel to the destination.

The end-to-end BER performance depends not only on the quality of the channel fromeach user to the destination but also on the interuser channel quality. As mentioned in Sect. 3,the effect of interuser channel quality on the end-to-end BER performance is quantified bythe probability distribution of the random variable . As an illustrative example, considerthe case to obtain a bandwidth efficiency of 1 bit/s/Hz as in Fig. 6, but with various interuserchannel quality levels indicated by its average information-bit-energy-to-noise-density ratio,E (1,2)

b /N0.

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16 17 18 19 20 21 22

10−4

10−3

10−2

Eb(1)/N

0 (dB)

BE

R

Non−cooperative: ρ=0%, 8−PSKProposed: γ=0%,ρ=33%, 8−PSKProposed: γ=100%,ρ=50%, 16−QAMAnalytical bound

Fig. 7 Performance of the proposed coded cooperative systems with bandwidth efficiency of 1 bit/s/Hz, and

E(2)b /N0 = E(1)

b /N0 + 5 dB

Table 2 Probability distributions of , Pr(), for different values of E(1,2)b /N0

E(1,2)b /N0 (dB) = 1 = 2 = 3 = 4

γ = 0%, ρ = 33%

10 0.517 0.202 0.202 0.079

20 0.935 0.03 0.03 0.005

25 0.980 0.01 0.01 ≈ 0

30 0.996 0.002 0.002 ≈ 0

40 0.999 0.0005 0.0005 ≈ 0

γ = 100%, ρ = 50%

10 0.376 0.237 0.237 0.149

20 0.896 0.05 0.05 0.004

25 0.966 0.017 0.017 ≈ 0

30 0.994 0.003 0.003 ≈ 0

40 0.999 0.0005 0.0005 ≈ 0

Table 2 tabulates Pr() for different values of E (1,2)b /N0 and {γ, ρ} and Fig. 8 shows the

corresponding bit error rates of the proposed system. Note that for a given pair of curves thathave the same marker, the solid curve is obtained from the analysis while the dashed curve

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16 17 18 19 20 21 22

10−4

10−3

10−2

Eb(1)/N

0 (dB)

BE

Rnon−cooperative

γ=0%, ρ=33%, 10dB interuserγ=100%, ρ=50%, 10dB interuserγ=0%, ρ=33%, 20dB interuserγ=100%, ρ=50%, 20dB interuserγ=0%, ρ=33%, 25dB interuserγ=100%, ρ=50%, 25dB interuserγ=0%, ρ=33%, 40dB interuserγ=100%, ρ=50%, 40dB interuserAnalytical bound


E(2)b /N0 = E(1)

b /N0 and various E(1,2)b /N0

represents the simulation results. At first, the simulation and analytical results are in a goodagreement. In general, the proposed scheme provides a better performance than the non-cooperative system for a wide range of signal-to-noise ratios over both the user-destinationand interuser channels. This performance gain increases with the increased interuser channelquality (shown in Fig. 8), corresponding to the increased Pr(= 1) as shown in Table 2. Whenthe interuser channel’s E (1,2)

b /N0 is about 40 dB, the end-to-end performance approaches theperformance corresponding to the ideal interuser channel. Figure 8 also indicates that for avery poor interuser channel quality, e.g., E (1,2)

b /N0 = 10 dB, the BER performance of theproposed system is better with the lower repetition level. This can be explained by the higherfraction in the cooperative mode, e.g., 70% for γ = 0% as compared to 60% for γ = 100%.As the interuser channel quality improves, the BER performance of the proposed systemwith a higher repetition level becomes better. For example, the system BER performancewith γ = 100% approaches that with γ = 0% for E (1,2)

b /N0 = 20 dB, and is better for

E (1,2)b /N0 = 25 dB. As E (1,2)

b /N0 is further increased, the system BER performance withγ = 100% is much better than that with γ = 0%. The results in Fig. 8 are useful for selectingthe optimum value of γ based on the available average interuser channel quality.

Next, consider the bandwidth efficiency of 2 bit/s/Hz. Here SC-1 and SC-2 are selected tobe 4-state, rate-2/3 RSC codes of the same parity check sequences h(0) = 7, h(1) = 3, h(2) = 5in octal form. The information block length of K = 160 bits is used. To achieve this spectralefficiency, the non-cooperative system and the proposed system with γ = 0% can use 16-QAM constellation. On the other hand, the proposed systems with γ = 50% and γ = 100%need to employ 32-QAM and 64-QAM constellations, respectively.

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16 18 20 22 24 26 2810

−6

10−5

10−4

10−3

10−2

Eb(1)/N

0 (dB)

BE

R

Non−cooperative: ρ=0%, 16−QAMProposed: γ =0%, ρ=25%, 16−QAMProposed: γ =50%, ρ=40%, 32−QAMProposed: γ =100%, ρ=50%, 64−QAM


E(2)b /N0 = E(1)

b /N0

The BER performance results presented in Fig. 9 shows that the proposed system withγ = 0% only slightly outperforms the non-cooperative system. This observation is quitedifferent from what observed from Fig. 6 where the gap between these two systems is muchlarger (approximately 8 dB). The explanation is as follows. In order to obtain the higherbandwidth efficiency of 2 bit/s/Hz, a higher code rate of 2/3 needs to be used. When only theadditional parity bits are sent in frame 2 in the proposed system (γ = 0%), the cooperationlevel is only 25%, whereas it is 33% in the case of Fig. 6. This low cooperation level is thusnot good enough to boost the overall system performance in the block fading environment. Infact it can be seen that the diversity orders of the two systems are very similar (indicated bythe similar slopes of the BER curves). This fact also confirms that repeating more informationsymbols (i.e., the systematic bits) in block fading environment is critical.

The situation is, however, completely different for the proposed systems that implementhigher levels of repetition, namely γ = 50% and γ = 100%. As shown in Fig. 9, the BERperformance of both systems significantly improves over that of the previous two systems.In particular, at the BER level of 10−3, a coding gain of about 8 dB can be obtained by theproposed system with γ = 50% over the non-cooperative system. An additional gain ofabout 1 dB is possible by increasing γ from 50% to 100%. It is also interesting to observefrom the slopes of the BER curves in Fig. 9 that the diversity orders of the proposed systemswith γ = 50% and γ = 100% are also very similar. This can be expected from the fact thatthe cooperation levels of ρ = 40% and ρ = 50% in two systems are quite close.

Similar observations are obtained from Fig. 10, where the same performance comparisonsare examined, but for case of asymmetric channels. The performance improvement of User

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16 18 20 22 24 26 2810

−6

10−5

10−4

10−3

10−2

Eb(1)/N

0 (dB)

BE

R

Non−cooperative: ρ=0%, 16−QAMProposed: γ =0%, ρ=25%, 16−QAMProposed: γ =50%, ρ=40%, 32−QAMProposed: γ =100%, ρ=50%, 64−QAM


E(2)b /N0 = E(1)

b /N0 + 5 dB

1 again illustrates that there is a better benefit to cooperate with the user that has a betterchannel to the destination.

Performance comparisons of the proposed coded cooperative systems with various band-width efficiencies and different repetition levels are provided in Fig. 11. The first importantobservation is that it is always advantageous to repeat more information bits in the secondframe at any considered bandwidth efficiency. The second observation is that the systemwith a lower spectral efficiency generally performs better than the one with a higher spectralefficiency. Though this is intuitively expected, it clearly illustrates the design flexibility ofthe proposed system in order to trade between the error performance and the transmissionrate.

It is also of interest to point out that the system corresponding to bandwidth efficiencyof 0.33 bit/s/Hz and γ = 0% uses BPSK and it is exactly the coded cooperative systemconsidered in [6]. As shown in Fig. 11, this bandwidth efficiency can be achieved with theproposed system that uses BPSK in frame 1 and QPSK in frame 2 to implement the repetitionlevel of γ = 100%. The coding gain obtained by deploying such a system over the one in[6] is about 1.5 dB at the BER level of 10−4.

Adaptive modulation and coding (AMC) schemes [23] have been proposed for moreefficient uses of the communications systems’ resources (e.g., power and bandwidth) in atime-varying environment. In addition to the conventional AMC parameters of coding rateand constellation size, the proposed coded cooperative scheme also introduces the new andeffective parameter, repetition level, that gives finer granularity, robust and smooth control.Changing the repetition level is relatively simple without switching encoders and signal map-

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12 14 16 18 20 22 24 2610

−6

10−5

10−4

10−3

10−2

10−1

Eb(1)/N

0 (dB)

BE

R

0.33 b/s/Hz, γ =0%0.33 b/s/Hz, γ =100%1 b/s/Hz, γ =0%1 b/s/Hz, γ =100%2 b/s/Hz, γ =0%2 b/s/Hz, γ =100%

Fig. 11 Performance comparison with different bandwidth efficiencies, E(2)b /N0 = E(1)

b /N0

14 14.5 15 15.5 16 16.5 1710

−4

10−3

10−2

Eb(1)/N

0 (dB)

BE

R

γ =0%, ρ=33%, 16−QAMγ =25%, ρ=38%, 16−QAMγ =50%, ρ=42%, 16−QAMγ =75%, ρ=46%, 16−QAMγ =100%, ρ=50%, 16−QAM

Fig. 12 BER performance for various values of γ , E(2)b /N0 = E(1)

b /N0

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15 15.2 15.4 15.6 15.8 16 16.2 16.4 16.6 16.8 171

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Eb(1)/N

0 (dB)

Thr

ough

put

Fig. 13 Adaptive bandwidth efficiency versus E(2)b /N0 = E(1)

b /N0 with fine granularity

pers at the transmitter side nor decoders and de-mappers at the receiver side. Furthermore,changing coding rate and constellation size introduces coarse steps in both SNR and band-width efficiency. Hence, repetition level is more effective in providing smooth and robustcontrol for fine-granularity adaptation to small environmental variations. As an illustrativeexample, Fig. 10 shows the bit error performance versus SNR curves for various repetitionlevels but with the fixed coding rate and constellation size. It is observed that the higherthe repetition level is, the better the error performance is obtained. To achieve an exampletarget BER of 10−3, changing the repetition level from 0 to 1 can adapt to an Eb/N0 rangefrom 15 dB to 17 dB (in Fig. 12) for a provision of bandwidth efficiency range from 1 to1.32 bit/s/Hz as shown in Fig. 13.

5 Conclusion

A novel bandwidth-efficient coded cooperative system was proposed. Using a very accu-rate and low complexity bounding technique originally developed for bit-interleaved codedmodulation, the performance analysis of the proposed system is also provided to verify itsperformance advantages. In particular, it was demonstrated that using higher-order modu-lation not only can provide a higher bandwidth efficiency but also a means of increasingthe repetition level of the systematic bits. The latter was found to be critical to improve theoverall system performance in block fading environment. In turn, the repetition level γ canbe used as the control parameter when adaptive system is considered to increase the systemthroughput.

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Appendix A

Computing Pr(�)

In order to obtain the probability distribution of the random variable , the block error rate(BLER) is required. As discussed in [19,22,24], the conditional BLER can be bounded as

Pblock(α) ≤ 1 − [1 − PE (α)]K ≤ K PE (α) (24)

where K is the number of trellis branches in the codeword, PE (α) is the error event probabil-ity and α is the instantaneous interuser channel SNR. The quantity PE (α) is upper boundedby

PE (α) ≤∞∑

d=d f ree

Ad P(d|α) (25)

where d f ree denotes the free distance of the trellis code and Ad is the number of the errorevents with Hamming weight d .

Based on (24), the conditional probability distribution of can be determined as follows.

Case 1 ( = 1)

P( = 1|α) = (1 − P(1)block(α))(1 − P(2)

block(α))

≤ (1 − P(1)E (α))K (1 − P(2)

E (α))K

≤ (1 − K P(1)E (α))(1 − K P(2)

E (α)) (26)

where P(1)E and P(2)

E denote the error event probabilities at User 1 and User 2, respectively.

Case 2 ( = 2)

P( = 2|α) = P(1)block(α)(1 − P(2)

block(α))

≤ [1 − (1 − P(1)E (α))K ](1 − P(2)

E (α))K

≤ K P(1)E (α)[1 − K P(2)

E (α)] (27)

Case 3 ( = 3)

P( = 2|α) = (1 − P(1)block(α))P(2)

block(α)

≤ (1 − P(1)E (α))K [1 − (1 − P(2)

E (α))K ]≤ K [1 − K P(1)

E (α)]P(2)E (α) (28)

Case 4 ( = 4)

P( = 2|α) = P(1)block(α)P(2)

block(α)

≤ [1 − (1 − P(1)E (α))K ][1 − (1 − P(2)

E (α))K ]≤ K 2 P(1)

E (α)P(2)E (α) (29)

The unconditional probability distribution of is then computed as

P() =∫

α

P(|α)p(α)dα (30)

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Finally, it should be noted that the limit-before-average technique in [22] can also beused to obtain the tight bound. For example, computing (bounding) P( = 1) with thelimit-before-average technique yields

P( = 1) ≤∫

α

(1 − min[1, P(1)E (α)])K × (1 − min[1, P(2)

E (α)])K p(α)dα (31)

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Author Biographies

Hieu T. Nguyen received the B.Eng. degree from Hanoi University ofTechnology, Hanoi, Vietnam in 2003, the M.Sc. degree from the Univer-sity of Saskatchewan, Saskatoon, Saskatchewan, Canada in 2007, all inelectrical engineering. He is currently a lecturer at the Posts and Telecom-munications Institute of Technology, Hanoi, Vietnam. Hieu Nguyen’sresearch interests are cooperative diversity techniques.

Ha H. Nguyen received the B.Eng degree from Hanoi University ofTechnology, Hanoi, Vietnam, in 1995, the M. Eng degree from AsianInstitute of Technology, Bangkok, Thailand, in 1997, and the Ph.D.degree from the University of Manitoba, Winnipeg, Canada, in 2001,all in electrical engineering. Dr. Nguyen joined the Department of Elec-trical Engineering, University of Saskatchewan, Canada in 2001 andbecome a Full Professor in 2007. He holds adjunct appointments atthe Department of Electrical and Computer Engineering, University ofManitoba, Winnipeg, MB, Canada, and TRLabs, Saskatoon, SK, Canadaand was a Senior Visiting Fellow in the School of Electrical Engineer-ing and Telecommunications, University of New South Wales, Sydney,Australia. His research interests include digital communications, spreadspectrum systems and error-control coding. Dr. Nguyen currently servesas an Associate Editor for the IEEE Transactions on Wireless Commu-nications and the IEEE Transactions on Vehicular Technology. He is aRegistered Member of the Association of Professional Engineers andGeoscientists of Saskatchewan (APEGS).

Tho Le-Ngoc obtained his B.Eng. (with Distinction) in Electrical Engi-neering in 1976, his M.Eng. in Microprocessor Applications in 1978from McGill University, Montreal, and his Ph.D. in Digital Communica-tions in 1983 from the University of Ottawa, Canada. During 1977–1982,he was with Spar Aerospace Limited and involved in the developmentand design of satellite communications systems. During 1982–1985, hewas an Engineering Manager of the Radio Group in the Departmentof Development Engineering of SRTelecom Inc., where he developedthe new point-to-multipoint DA-TDMA/TDM Subscriber Radio SystemSR500. During 1985–2000, he was a Professor at the Department of Elec-trical and Computer Engineering of Concordia University. Since 2000,he has been with the Department of Electrical and Computer Engineeringof McGill University. His research interest is in the area of broadbanddigital communications with a special emphasis on Modulation, Coding,and Multiple-Access Techniques. He is a Senior Member of the Ordredes Ingénieur du Quebec, a Fellow of the Institute of Electrical and Elec-

tronics Engineers (IEEE), a Fellow of the Engineering Institute of Canada (EIC), and a Fellow of the CanadianAcademy of Engineering (CAE). He is the recipient of the 2004 Canadian Award in TelecommunicationsResearch, and recipient of the IEEE Canada Fessenden Award 2005.

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Bandwidth-efficient Bit-interleaved Coded Cooperative Communications

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