DIVERSITY techniques have been widely accepted as

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 7, JULY Modulation and Demodulation for Cooperative Diversity in Wireless Systems Deqiang Chen, Student Member, IEEE, and J. Nicholas Laneman, Member, IEEE Abstract This paper develops a general framework for maximum likelihood (ML) demodulation in cooperative wireless communication systems. Demodulators with piecewise-linear combining are proposed as an accurate approximation of the nonlinear ML detectors for coherent and noncoherent decode-and-forward (DF). The detectors with piecewise-linear combiner not only have certain implementation advantages over the nonlinear ML detectors, but also can lead to tight closed-form approximations for their error probabilities. High SNR approximations are derived based on the closed-form BER expressions. For noncoherent DF, the approximation suggests a different optimal location for the relay in DF than for the relay in amplify-and-forward (AF). A set of tight bounds of diversity order for coherent and noncoherent DF with multiple relays is also provided, and comparison between DF and AF suggests that DF with more than one relay loses about half of the diversity order of AF. Index Terms Modulation, cooperative diversity, fading, biterror rate (BER), relay channel. I. INTRODUCTION DIVERSITY techniques have been widely accepted as one of the effective ways to combat multipath fading in wireless communications []. For some scenarios, in which time or frequency diversity might be difficult to exploit due to delay or bandwidth constraints, multiple transmit or receive antennas at the same terminal are often desirable for providing spatial diversity. However, many wireless applications, such as cellular, ad hoc or sensor networks, preclude the use of multiple antennas due to the size and cost limitations of the terminals. Cooperative diversity [2], [3], [4] avoids the size limitation and provides spatial diversity by allowing the terminals to relay in parallel, thus sharing multiple antennas belonging to different terminals. A. Related Research Cooperative diversity is first proposed in [2], [3], which show that cooperative diversity enlarges the capacity region over non-cooperative transmission for ergodic fading, given that channel state information (CSI) is available to the transmitters. Later on, [5] shows that cooperative diversity does not increase the maximum sum-rate comparing with noncooperative transmission when CSI is unavailable to the transmitters but available to the receivers. A variety of algorithms Manuscript received July 9, 2004; revised January 2, 2005 and April 27, 2005; accepted May 2, The associate editor coordinating the review of this paper and approving it for publication was G. Vitetta. This work has been supported in part by the State of Indiana through the Twenty-First Century Research and Technology Fund. The authors are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN ( {dchen2, jlaneman}@nd.edu) Digital Object Identifier 0.09/TWC /06$20.00 c 2006 IEEE that can achieve full diversity order for cooperative diversity are proposed in [5]. In [6], the diversity order effects of various routing methods and processing schemes at the destination are investigated for cooperative diversity with up to two amplifying relays. The results show that the diversity order is equal to the number of independent links combined at the destination. Thus, a network with M relays can provide diversity order M+. On the other hand, the combination of correlated links does not provide diversity but extra coding gains due to repetition of information. From the modulation and demodulation point of view, [4] develops maximum likelihood (ML) detectors and the corresponding analysis, in terms of bounds on the uncoded bit error rate (BER), for coherent cooperative diversity. By simulation, it is also observed that systems with an amplifying relay appear to perform comparably, if not better, than systems with a decoding relay. The relative simplicity of the amplify-and-forward (AF) protocol has inspired several extensions [7], [8], [9]. A blind relay that does not require instantaneous CSI between the source and relay is proposed in [7], [8]; however, it cannot satisfy an instantaneous power constraint. Building upon methods from [0], [9] provides closed-form approximations of the average symbol error probability (SEP) for general multibranch, multi-hop cooperative diversity at asymptotically high signal-noise-ratio (SNR). It also demonstrates that, in the sense of minimizing SEP with sufficiently high SNR, it is best for a single amplifying relay to be located at the midpoint between the source and destination. B. Motivation Most of the previous work on cooperative diversity assumes that each receiver accurately estimates the fading coefficients affecting its received signals. In slow-fading scenarios [], such CSI might be obtained via estimation from training sequences in the protocol headers. However, in fast fading scenarios, CSI may not be accurately obtainable if the fading coefficients vary quickly within the period of one transmission block. Also, as the coherence time decreases, estimation of CSI substantially reduces effective transmission rate since pilot tones must be inserted. In these scenarios, noncoherent modulation and demodulation can be more practical. In addition, noncoherent modulation and demodulation are robust methods for realizing control signaling in wireless networks. To the best of our knowledge, there has not been a comprehensive treatment of noncoherent demodulation for cooperative communication schemes. Moreover, most of the previous research on an uncoded cooperative diversity adopts AF protocols [4], [7], [9]. How-

2 786 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 7, JULY 2006 ever, for AF, when instantaneous CSI is not available to the receivers, satisfying the relay power constraints greatly complicates demodulation as well as analysis [7], [2]. For decode-and-forward (DF), the relay power constraints can be easily satisfied. In addition, DF can be extended to combine with coding techniques and might be easier to incorporate into network protocols. For example, the existence of the relay provides an opportunity for the codeword to be interleaved and thus provide a distributed turbo coded system. This idea is demonstrated in more recent works [3], [4], [5]. In [5], an adaptive DF protocol is proposed in which the relay decides to retransmit or not depending on whether or not it can successfully decode the entire coded sequence. All these works show that DF combined with coding can yield impressive gains. We note that the DF protocols in these extensions are more complex than the uncoded DF protocol studied in this paper; in particular, our uncoded DF performs symbol-bysymbol demodulation and retransmission. Overall, this paper provides more detailed discussion about uncoded DF, and compares with the existing results for AF [4], [7], [9]. C. Summary of Results In the following sections, we develop: ) A general framework for ML detection of both coherent and noncoherent uncoded cooperative diversity. Comparison within one framework shows how different assumptions about CSI and different processing by relays affect the diversity combining at the destination. In general, different assumptions about CSI affect the generation of sufficient statistics, and different processing by relays affects how the sufficient statistics are combined. 2) Detectors with piecewise-linear (PL) combiner, which closely approximate the nonlinear ML detectors for both coherent and noncoherent DF. These detectors have certain implementation advantages, and can provide tight closed-form approximations for the uncoded BER. In principle, these detectors with PL combiner can be extended to noncoherent DF with any number of relays, although obtaining closed-form BER expressions would become very cumbersome. 3) High SNR approximations, which show that noncoherent DF with one relay achieves full diversity order for asymptotically high SNR. These approximations also suggest that the optimal location for a decoding relay can be different from that of an amplifying relay [9]. 4) Tight bounds on the diversity order with multiple decoding relays. The lower bounds are obtained via the Bhattacharyya bound. Assuming the channels between the relays and destination are perfect, i.e., non-fading and noiseless, we obtain tight upper bounds on the diversity order. These bounds show that, for both coherent and noncoherent uncoded cooperative diversity, DF with more than one relay loses about half of the diversity order of AF, which achieves full diversity order. D. Outline An outline of this paper is as follows. Section II describes the system model. Section III presents a unified framework for ML detection. To analyze the performance of noncoherent DF, Section IV presents a PL approximation for the nonlinear ML combiners. The analysis based on the PL approximation leads us to closed-form BER approximations and a more compact, high SNR approximation. Section V presents tight bounds on the diversity order of DF with multiple decoding relays. Numerical simulation results are provided in Section VI. Section VII summarizes the results and concludes the paper. II. SYSTEM MODEL A. Notation In the following sections, random variables are denoted using the sans serif font (e.g., x) while random vectors and sequences are denoted with bold sans serif (e.g., x). A vector is assumed in column form unless otherwise stated. Instead of the more precise p x (x), the probability density function of the random variable x is denoted by p(x). B. Performance Metric For uncoded modulation and demodulation, the bit error probability, or average BER, is a commonly used performance metric. Furthermore, it has been observed that, when average BER is plotted on a logarithm scale versus SNR in db, the curve is approximately linear for high SNR. This has motivated the following high SNR approximation [] P b (c γ) d, () where P b is the average BER of the system, γ is the average SNR, c is called the coding gain and d is called the diversity order. In this paper, we focus on the study of diversity order and define it as d := sup{d R :0< lim P b γ d < }, (2) γ If the diversity order of P b is d and lim P b γ d is finite, () γ is usually tight for moderate and high SNR. In general, the diversity order is at most the number of independent fading coefficients [], and the higher the diversity order is, the better performance the system has in high SNR regimes. The diversity order can also be defined in terms of outage capacity [5]. C. Model To facilitate comparison between coherent and noncoherent demodulation, we focus on binary frequency-shift keying (BFSK). Generally speaking, coherent demodulation assumes instantaneous CSI is available to the receivers, and noncoherent demodulation refers to scenarios in which only the channel statistics are available to the receivers. More specific details on these assumptions will be stated below for the individual cooperative diversity schemes. We adopt a system model similar to that in [4]. As shown in Fig., a source terminal broadcasts to several relays and a destination in the first subchannel. After some processing,

3 CHEN and LANEMAN: MODULATION AND DEMODULATION FOR COOPERATIVE DIVERSITY IN WIRELESS SYSTEMS 787 y 0, R x y,m y 0,M,0 t 0 g (, ) y 0 0,M, x 0 Source y 0,M y M,M Destination y,m,0 y,m, g (, ) t f ( ) + 0 ˆ x 0 y 0,M- R M- x M y M,M,0 Fig.. Block diagram for cooperative diversity with multiple relays. y M,M, g (, ) M t M f ( ) M the relays retransmit signals to the destination in the remaining M orthogonal subchannels. The condition of orthogonality reflects the practical constraint that radio transceivers cannot transmit and receive at the same time in the same frequency band. It also avoids extra interference at the destination by requiring relays to transmit in orthogonal channels. After passing the signals through the appropriate matched filters, we obtain a baseband-equivalent discrete-time model. In the baseband model, the signals received by the destination are y 0,i,0 =( x 0 ) E 0 a 0,i + n 0,i,0, (3) y 0,i, = x 0 E0 a 0,i + n 0,i,, where i =,..., M correspond to the relays and i =M corresponds to the destination. The third subscript 0, denotes the two frequency subbands for BFSK signaling. The source symbols are x 0 {0, }, ande 0 is the average sample energy of the source. Finally, a 0,i models the effect of the frequency nonselective fading between the source and terminal i, and n 0,i,0, n 0,i, capture the effects of additive white Gaussian noise (AWGN) and other interference. For AF, the relays simply amplify and transmit the signal received from the source without forming an estimate of the transmitted signal. Thus, the destination receives y i,m,0 = y 0,i,0 β i a i,m + n i,m,0, (4) y i,m, = y 0,i, β i a i,m + n i,m,, where: i =,..., M denotes the signal from relay R i ; a i,m is the fading coefficient between the destination and relay i; n i,m,0 and n i,m, represent AWGN. β i is the amplifying factor at relay i. For our purposes, a detailed expression of β i is not required; however, it depends on the power constraint and assumptions of CSI at the relays [4], [8]. For DF, the relays spend extra effort in demodulating the signals received from source, forming an estimate of the transmitted signal, and retransmitting their estimate. Therefore, DF can more easily satisfy the power constraint than AF. Suppose the signals received by relays from (3) are demodulated into x i {0, }, then the destination receives y i,m,0 =( x i ) E i a i,m + n i,m,0, (5) y i,m, = x i Ei a i,m + n i,m,. from the relays for i =,..., M. In this paper, the fading coefficients a i,j for the link between the terminals i and j are modeled as mutually independent, zero mean, circularly symmetric complex Gaussian random variables with variances σi,j 2, and the additive noises n i,j,0 Fig. 2. relays. General detector structure for cooperative diversity with multiple and n i,j, are modeled as white, mutually independent, zero mean, complex Gaussian random variables with variance N 0. The instantaneous SNR of the link between the terminals i and j is defined as γ i,j := a i,j 2 E i /N 0 ; and the average SNR is defined as γ i,j := σ 2 i,j E i/n 0. III. MAXIMUM LIKELIHOOD DETECTION All ML detectors in this paper can be implemented as shown in Fig 2. This detector structure extends immediately to all coherent and noncoherent binary modulation formats by properly defining functions g i (y i,m,0, y i,m, ),f i (t i ). Thus, it provides a unified framework for analysis. It might facilitate changes between different transmission formats according to system requirements. Another purpose of using this general structure is to show the similarities and differences of various cooperative diversity schemes. A. DF Assuming that the destination receiver knows the average BER at various relays, and using the conditional independence of the channel outputs given channel inputs, an ML detector for noncoherent DF cooperative diversity can be shown to employ the functions γ i,m g i (y i,m,0, y i,m, )= ( y i,m,0 2 y i,m, 2 ), ( + γ i,m )N 0 (6) for i =0,,..., M, and f i (t i )=ln ( ɛ i)e ti + ɛ i ɛ i e ti +( ɛ i ), (7) for i =,..., M,whereɛ i is the average probability of error at relay R i, which uses a conventional envelope detector. For DF with coherent BFSK, an ML detector can employ functions g i (y i,m,0, y i,m, )= 2Re{(y i,m,0 y i,m, )ai,m } E i (8) N 0 for i =0,,..., M, andf i (t i ) as in (7). In this case, our general detector structure can be simplified to the detector structure in [4]. We also assume that the destination knows the average BER at each individual relay, and each relay implements coherent demodulation.

4 788 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 7, JULY 2006 B. AF For coherent AF with BFSK, following methods similar with [4], an ML detector can be realized by defining Ei g i (y i,m,0, y i,m, )= 2Re{(y i,m,0 y i,m, )ai,m a 0,i }β i ( + βi 2 a i,m 2 )N 0 (9) T i and f i (t i )=. (0) It is assumed that the destination knows CSI between the source and relays as well as CSI between the relays and destination. For noncoherent AF with power constraints, we have been unable to obtain a simple, closed-form expression for an ML detector that fits into the block diagram in Fig. 2. See [2] for more details. Thus, it is not clear that the discussion above can extend immediately to noncoherent AF. C. Discussion As we can see, for DF cooperative diversity, the difference between coherent and noncoherent demodulation is in the generation of sufficient statistics by the functions g i (y i,m,0, y i,m, ) in (6) and (9); they share a common sigmoid function f i (t i ) to limit the error propagation due to the potential demodulation mistakes at the relays. It can be easily observed that (8) and (9) are essentially in the same form if a i,m a 0,i β i are viewed as the equivalent fading coefficients and ( + βi 2 a i,m 2 )N 0 is regarded as the equivalent noise for coherent AF cooperative diversity. The comparison of ML detectors for these three different schemes within one framework offers an interpretation of this detector structure from detection and estimation theory. The matched filter outputs are processed by a function g i (y i,m,0, y i,m, ) to produce sufficient statistics. The form of functions g i (y i,m,0, y i,m, ) depends upon the coherent and noncoherent assumptions. The form of f i (t i ) depends on the signal processing schemes at the relays. For DF, the function f i (t i ) essentially limits the influence of its input, i.e., sufficient statistic, using the statistical knowledge of the transmission link. An analysis of coherent AF has been presented in [4], [7], [9]. Analysis of DF diversity transmission is challenging since it must treat the nonlinear behavior of (7), which significantly complicates the effort of obtaining a closed form solution for BER. In the sequel we develop a piecewise-linear approximation to f i (t i ), from which we also derive good approximations to the ML detector performance. IV. DETECTOR WITH PIECEWISE-LINEAR COMBINER AND ITS PERFORMANCE As noted in [4] and shown in Fig. 3, f i (t i ) essentially clips the input to the values of ± ln[ɛ i /( ɛ i )] for large inputs, and is approximately linear between these extreme values for small inputs. Here we approximate (7) with a piecewise-linear function as shown in Fig. 3. This approximation suggests an alternative detector, which we call the detector with piecewise-linear (PL) combiner, that might be f i (t i ) T i T i t i Fig. 3. Plots of the sigmoid function and its approximation. more amenable to practical implementation. Furthermore, as we will see, this detector s performance provides a tight upper bound on the error probability of the ML detector. Although this detector applies to both coherent and noncoherent DF, the following sections focus on noncoherent DF. Mathematically, the approximation can be expressed as, T i for t i T i f i (t i ) = f PL (t i )= t i for T i t i T i, () T i for t i T i where T i =ln[( ɛ i )/(ɛ i )], andɛ i < /2. Detectors using the approximation () rather than (7) are called detectors with PL combiner. We note that the diversity combiner resulting from this piecewise-linear approximation relates to the clipped-linear combiners in [6], where the basic idea is to limit the impact of partial-band interference by clipping the output of envelope detectors with respect to the signal output voltage. In our scenario, we limit the impact of uncertainty in the decisions at the relays. Since the detector in [6] employs knowledge of the probability with which the interference appears, our approximations might provide a mechanism for optimizing the clipping level in that context as well. A. Closed-Form BER Although the PL approximation extends to the case of multiple relays [2], we focus in this section on the case of one relay to obtain a closed-form BER expression. For noncoherent DF, the detector with PL combiner is t 0 + f PL (t ) 0 0, (2) where t 0 and t are the outputs of g i (y i,m,0, y i,m, ) from (6). By the total probability law, the average BER can be written T i

5 CHEN and LANEMAN: MODULATION AND DEMODULATION FOR COOPERATIVE DIVERSITY IN WIRELESS SYSTEMS 789 as P b =Pr{t 0 T < 0 x 0 =0} Pr{t < T x 0 =0} +Pr{t 0 + T < 0 x 0 =0} Pr{t >T x 0 =0} +Pr{t 0 + t < 0 T t T, x 0 =0} Pr{ T t T x 0 =0} (3) The distribution of t 0 and t can be obtained by noticing that both t 0 and t are the difference between two exponential random variables with different rates. However, given x 0 =0, the rates for t 0 are fixed, but the rates for t also depend on whether the relay demodulates correctly. Moreover, t 0 and t are conditionally independent given x 0. Thus, after determining the probability density functions of these two random variables, applying the total probability law conditional on the decision of the relay and performing the integral, we obtain the complicated, but closed-form, expression for BER P b = A + A 2 + A 3, (4) where A,A 2,A 3 correspond to the three terms in (3) and can be expressed as A =h(λ 0,λ 0,T ) [( ɛ )h(λ,λ, T ) + ɛ h(λ,λ, T )], A 2 =h(λ 0,λ 0, T ) {( ɛ )[ h(λ,λ,t )] + ɛ [ h(λ,λ,t )]}, A 3 =( ɛ ) q(λ 0,λ 0,λ,λ, T ) [h(λ,λ, 0) h(λ,λ, T )] + ɛ q(λ 0,λ 0,λ,λ, T (5) ) [h(λ,λ, 0) h(λ,λ, T )] +( ɛ ) n(λ 0,λ 0,λ,λ,T ) [h(λ,λ,t ) h(λ,λ, 0)] + ɛ n(λ 0,λ 0,λ,λ,T ) [h(λ,λ,t ) h(λ,λ, 0)], where λ 0 =,λ =,λ 0 γ 0,2 γ =+,λ,2 γ =+, (6) 0,2 γ,2 and the average BER at the relay for noncoherent BFSK is ɛ =/(2+ γ 0, ). The functions in (5) are defined as follows: r 0 e rt for t 0 h(r 0,r,t):= r 0 + r r e r0t for t 0, r 0 + r r r 3 q(r,r 2,r 3,r 4,t):= (r + r 2 )(r 2 + r 3 ) e(r2+r3)t e r3t, r 2 r 4 n(r,r 2,r 3,r 4,t):= (r + r 2 )(r + r 4 ) e (r+r4)t e r4t. (7) Note that h(r 0,r,t) is the probability distribution function of the difference between two exponential random variables with parameters r and r 0, respectively. As we will see, the error probability (4) of the detector with PL combiner provides a tight approximation of the error probability of the nonlinear ML detector in all cases we consider. This observation suggests that we can use the much simpler () rather than the more complex (7) in the detector while maintaining the benefits of diversity. In [2], this technique has been extended to the cases in which the system has two parallel relays or has two antennas at the destination. We shall refer to the BER approximation simply as the BER for DF cooperative diversity due to its tightness, however, it should be kept in mind that the BER approximation from the PL technique is actually an upper bound. The detector with PL combiner is also applicable to coherent DF, and results in an integral expression for the BER [2]. B. High SNR Approximation Although (4) provides a close approximation of the performance of noncoherent cooperative diversity with a decoding relay, it is too complicated to provide more insight about the system. Therefore, we develop an approximation of (4) suitable for high SNR. Examining (6) and (4), we note that three SNRs, namely γ 0,2, γ 0,, γ,2, parameterize performance. Our high SNR approximation allows all three parameters to become large, but keeps them in fixed proportions to the other. This constraint of proportionality is for the purpose of accounting for the effects of network geometry and power allocation on performance. Specifically, we assume γ 0,2 = k γ, γ 0, = k 2 γ, γ,2 = k 3 γ, (8) where γ can be the average single-hop SNR and k,k 2,k 3 are constants related to the network geometry, power allocation, and the path loss model. The high SNR approximation can be obtained by substituting (8) into (6), expressing (4) and (5) as a function of / γ,k,k 2,k 3, expanding in a power series around the point / γ =0, and dropping all but the first term of the series as an asymptotic approximation. The result is ( P b γ ln ) k 2 +ln γ. (9) k k 3 k k 2 As we will see, (9) provides a very tight approximation to the simulation results as SNR becomes large. Based on this high SNR approximation, optimization with respect to the location of the relay and/or the power allocation between the source and relay can be performed. This paper shows that cooperative diversity with different relay locations provides diversity gain even though the power allocation is not at all optimal. Assuming both the source and relay have the same individual power constraint, the simplest power allocation strategy is equal power sharing and this will be the focus of the following discussion. With equal power sharing, geometric interpretation of (9) shows that there exists an asymmetry with respect to the location of the relay in the performance of noncoherent DF. This asymmetry is further confirmed in [7] for coherent DF from the perspective of outage capacity, and contrasts to results for coherent cooperative diversity with an amplifying relay. For The key step is to make use of the approximation (c 2 r + c 3 ) /(cr) ln(c 2r) for asymptotically high r. c r

6 790 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 7, JULY 2006 coherent AF, the BER is shown to be approximately [9] P b γ ( 2 + ), (20) k k 2 k 3 which indicates that the optimum location for the amplifying relay is halfway between the source and destination. The asymmetry in (9) suggests the location for the noncoherent decoding relay to maximize the system performance might be different from that of coherent amplify-and-forward. The optimum location for DF depends on the SNR value and tends to be closer to the source than to the relay for equal power sharing. For the case of one relay, numerical minimization of (9) shows that, for all practical SNR values, the midpoint is close to the optimum location with equal power sharing. In [2], this high SNR analysis is extended to the case of two parallel decoding relays. It turns out that the diversity order obtained for cooperative diversity with two parallel decoding relays is exactly 2 rather than 3, and there is again an asymmetry in the BER expression with respect to the position of the relays. These observations motivate Section V, where we examine whether they can generalize, regardless the number of relays. V. DIVERSITY ORDER OF DF WITH MULTIPLE RELAYS Although the PL approximation can greatly simplify the ML detectors for DF cooperative diversity, the complexity of analyzing its performance grows exponentially in the number of relays [2]. Therefore, this section proposes a set of tight bounds on the diversity order of DF, using techniques whose complexity does not increase with the number of relays. The main results can be summarized as the following theorem. Theorem : For uncoded cooperative diversity with either coherent or noncoherent DF and ML detection, the diversity order satisfies if M is even, and (M + )/2 d M/2+ (2) d =(M+)/2 (22) if M is odd. The lower bound of Theorem is obtained by invoking the Bhattacharyya upper bound on error probability. We develop the upper bound of Theorem by assuming the links between the relays and destination do not suffer any fading or AWGN. In other words, the transmission between any relay and the destination is free of errors. For DF, the assumed system is equivalent to a M branch diversity system, in which the i =,..., M branches provide hard decisions that are combined with the soft input of the i =0branch. We refer to such an equivalent system as a mixed diversity system, in contrast to hard or soft diversity combining. We further assume that the hard decision branches in the corresponding mixed system have the same probability of error ɛ. This can be achieved by choosing ɛ =min{ɛ i } for i =,...M, whereɛ i is the probability of error at the relay R i in the original DF cooperative diversity. Thus, the BER for such a mixed system can provide a lower bound of BER for the original cooperative diversity and is refereed to as the mixed lower bound in the sequel. The details for obtaining these bounds are presented in the Appendix The tightness of Theorem can be verified by comparing with results from the PL techniques for the case of noncoherent DF. If M=2, (2) predicts that the diversity order is between.5 and 2, which coincides with the observation from Section IV-B that the diversity order is 2. If M=3,which corresponds to cooperative diversity with two decoding relays, (22) predicts that diversity order is 2, which is consistent with results in [2]. Therefore, in terms of diversity order, Theorem yields similar conclusions as results based on the closed-form BER from the PL approximation. Also, we observe that cooperative diversity with one DF relay achieves full diversity order, and cooperative diversity with more than one DF relays does not achieve full diversity order. In short, when M is large, Theorem states that for both coherent and noncoherent DF with M parallel relays, the diversity order is roughly M/2, rather than M. Note that for coherent AF with M relays, it is shown in [9] that the full diversity order M can be achieved. The loss of about half diversity order in DF with multiple relays is due to the hard decisions made at the relays. As with receive antenna diversity, hard decisions before diversity combination at the destination causes loss of information and reduces diversity order []. We would like to point out that our discussion is limited to coherent and noncoherent DF, as well as coherent AF. Due to the difficulty illustrated in [2], a comprehensive comparison between noncoherent AF and other protocols is still not available. VI. NUMERICAL RESULTS This section provides numerical simulation results for cooperative diversity with one relay. The simulation conditions follow the same lines as in [4]. Specifically, the coordinates of the whole communication network are normalized by the distance d 0,2 between the source and destination transceivers, and the positive direction is defined as from the source to the destination. Without loss of generality, the source is assumed to be located at (0, 0), and the destination located at (, 0). For simplicity of exposition, the relay is assumed to be located at (l, 0). In general, the coordinates of the relay can be arbitrary. We refer to direct transmission from the source to destination as single-hop, and transmission in which the source transmits to the relay and the relay demodulates and retransmits to the destination as dual-hop. Comparing with cooperative diversity, the destination for dual-hop does not directly receive signals from the source and therefore does not achieve diversity gain. The fading variances σi,j 2 are assigned using a path-loss model in the form of σi,j 2 d v i,j,whered i,j is the distance from node i to node j, andv is a constant value, chosen to be 4 in our setup. The total network energy per transmitted bit is also normalized to. Specifically, we set E 0 = for single-hop transmission; and for diversity transmission, we assign equal sharing of power among the transmitters, i.e., E 0 =E =/2. We note that this power allocation does not consider the channel conditions and need not be optimal in general. Figs. 4 6 show the simulated average BER for noncoherent DF for relay locations (0., 0), (0.5, 0) and (0.9, 0),

7 CHEN and LANEMAN: MODULATION AND DEMODULATION FOR COOPERATIVE DIVERSITY IN WIRELESS SYSTEMS 79 Average Error Performance NonCoherent BFSK DF Coherent BFSK DF Coherent BFSK AF NonCoherent BFSK DF (Closed-form BER) Single Hop DualHop Average Error Performance NonCoherent BFSK DF Coherent BFSK DF Coherent BFSK AF NonCoherent BFSK DF (Closed-form BER) Single Hop DualHop Fig. 4. Error probability performance of cooperative diversity for v =4 and normalized geometries with the relay located at (0.,0), i.e., close to the source Fig. 6. Error probability performance of cooperative diversity for v =4 and normalized geometries with the relay located at (0.9,0), i.e., close to the destination. Average Error Performance NonCoherent BFSK DF Coherent BFSK DF Coherent BFSK AF NonCoherent BFSK DF (Closed-form BER) Single Hop DualHop Average Error Probability Simulation PL approximation High SNR approximation Bhattacharrya Upper Bound Mixed Lower Bound Fig. 5. Error probability performance of cooperative diversity for v =4and normalized geometries with the relay located at (0.5,0), i.e., halfway between source and destination Fig. 7. Curves for the high SNR approximation (9), Bhattacharyya upper bound (23), mixed lower bound (27), and closed-form BER (4) from the PL approximation for noncoherent DF cooperative diversity. The relay is located at (0.,0), i.e., closed to the source. respectively. These results indicate that there is an apparent decrease in slope for cooperative diversity transmission in all three cases. Furthermore, the curves for cooperative diversity transmission also demonstrate certain shifts to the left, corresponding to various coding gains, which combat path-loss. These coding gains vary depending on the location of the relay. Among the three scenarios we simulated, the case with the relay located at (0.5, 0) has better performance than the other two cases, given the same SNR. The closed-form error probability of the noncoherent detector with PL combiner (4) is also presented in Figs Its curves essentially overlap the simulation results of the nonlinear ML detector. Since the error probability of the detector with PL combiner is in closed form and consists of only elementary functions, it is also convenient for estimating the performance of the ML detector, rather than resorting to expensive Monte-Carlo simulations. Figs. 4 6 also display simulation results for the average BER for DF with coherent BFSK. The shifts among the curves vary slightly with the position of the relay, though, in general, coherent BFSK is about 3 db better than noncoherent BFSK. Note that, at present, we have no simple, closedform expression for the BER of coherent DF; our results suggest that the approximation (4), plus a suitable shift of 3 db, provides a good approximation for the performance of coherent DF with BFSK. For coherent BPSK, a shift of 3 db from coherent BFSK is expected and confirmed by numerical results. The BER curves for coherent BFSK with AF are also plotted in Figs In general, cooperative diversity with AF appears to perform at least as well as DF. The curves for the high SNR approximation (9) are presented in Figs Note that k =/2,k 2 = d 4 0, /2,k 3 = /2 according to the normalized path loss model, and γ is the average SNR for single hop transmission. In all scenarios, d 4,2

8 792 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 7, JULY Simulation PL approximation High SNR approximation Bhattacharrya Upper Bound Mixed Lower Bound Relay at (0.,0) Relay at (0.5,0) Relay at (0.9,0) TX,3RX Average Error Probability Average Error Probability Fig. 8. Curves for the high SNR approximation (9), Bhattacharyya upper bound (23), mixed lower bound (27), and closed-form BER (4) from the PL approximation for noncoherent DF cooperative diversity. The relay is located at (0.5,0), i.e., halfway between source and destination Fig. 0. Curves for the BER of noncoherent DF cooperative diversity with two relays. Both relays have the same location. The BER curve for optimum diversity combining of noncoherent BFSK through three independent Rayleigh fading channels is also shown for comparison. Average Error Probability Simulation PL approximation High SNR approximation Bhattacharrya Upper Bound Mixed Lower Bound Fig. 9. Curves for the high SNR approximation (9), Bhattacharyya upper bound (23), mixed lower bound (27), and closed-form BER (4) from the PL approximation for noncoherent DF cooperative diversity. The relay is located at (0.9,0), i.e., close to the destination. (9) corresponds well with simulation results when SNR is higher than 5 db. The curves for the Bhattacharyya upper bound and mixed lower bound (27) are also in the plots. It can be observed that the Bhattacharyya upper bound is loose. However, the curve for the mixed lower bound (27) provides a very tight approximation for noncoherent DF cooperative diversity when the relay is close to the destination, since the relay is more likely to make mistakes in this scenario. Moreover, the mixed lower bound is still good even when the relay is located halfway between the source and destination. It is not very surprising to see that the mixed lower bound is quite loose when the relay is close to the source, as there are fewer errors caused by transmission between the source and relay than between the relay and destination. Fig. 0 presents simulation curves for cooperative diversity with two parallel relays. The BER curve for a one transmit, three receive antenna system, having diversity order 3, is also shown for comparison. The loss of diversity order for cooperative diversity with two DF relays can be clearly observed when the relays are relatively far away from the source. The loss of diversity order is less apparent if the relays are close to the source due to the smaller number of decision errors made on the source-relay link. VII. CONCLUSION AND SUMMARY This paper compares some aspects of the performance of uncoded cooperative diversity with different relay processing, namely DF and AF, and different demodulations, namely coherent and noncoherent. Because several previous works have addressed coherent AF, this paper s main contribution is its focus on DF processing and noncoherent demodulation. For purposes of comparison, this paper develops a general framework for ML demodulation in cooperative diversity. A simple piecewise-linear combiner is proposed as an accurate approximation of the nonlinear ML combiner for DF, and leads to tight, closed-form BER approximations for the noncoherent ML combiner. This paper also develops tight bounds on diversity order for DF, revealing that DF loses about half of the diversity order compared to AF. APPENDIX Theorem shows a set of tight bounds on the diversity order of DF with multiple relays. In this appendix, we prove these results using the Bhattacharyya upper bound and the mixed lower bound mentioned in Section V. We focus on the case of noncoherent DF; extension to coherent DF is straightforward. A. Lower Bound On Diversity Order A lower bound of the diversity order can be developed based upon the Bhattacharyya upper bound on the BER. The explicit expression of the Bhattacharyya upper bound for the uncoded

9 CHEN and LANEMAN: MODULATION AND DEMODULATION FOR COOPERATIVE DIVERSITY IN WIRELESS SYSTEMS 793 demodulation in our setting is quite involved. Therefore, we provide a more concise development involving the following steps: making use of the conditional independence of signals coming from different branches given x 0 ; substituting the distribution of signals y i,m,0, y i,m, ; utilizing the Gaussian inequality, i.e., a + b + c a + b + c; assuming γ 0,M = c 0 γ, γ 0,i = c i γ; and approximating the average BER at each relay as ɛ i /(k γ 0,i ) []. The end result is lim γ P b γ (M+)/2 2 M k c 0 M i= kc i, (23) where k = 2 corresponds to noncoherent demodulation at the relays. Similar procedures also apply to coherent DF, yielding (23) with k =. Overall, the diversity order for DF cooperative diversity is therefore lower bounded according to d (M + )/2. (24) Note that a more precise development without approximating the average BER of the relays yields the same result in (24). B. Upper Bound On Diversity Order As discussed in Section V, the mixed lower bound on the BER can provide an upper bound on diversity order. For noncoherent DF, the corresponding mixed system assumes that ɛ is available to the diversity combiner. The ML detector can be reduced to M t 0 + ln[δ(x i )( ɛ)+δ(x i )ɛ] i= M i= ln[δ(x i )ɛ + δ(x i )( ɛ)] 0 0, (25) where t 0 = g 0 (y 0,M,0, y 0,M, ) from (6) and δ(x) is defined as δ(x) = { for x =0, 0 for x =. (26) Because we already developed the distribution of t 0 in Section IV, the BER for this mixed system can be obtained in the following way: assume the number of hard-decision errors is known, obtain the conditional BER, and then average. Note that when M is odd, the case that there are the same number of 0 s and s in the hard decoding branches needs to be handled differently. The tedious procedure yields the noncoherent mixed lower bound, ( ) (M 2)/2 M ( ɛ) j ɛ M j j j=0 + γ ( ) M 2k 0,M ɛ γ 0,M 2+ γ 0,M ɛ P b + M j=m/2 (M 3)/2 + j=0 ( M j ( ɛ ɛ ) ( ɛ) j ɛ M j 2+ γ 0,M ) (M 2k)(+/ γ0,m) for M even, ( ) M ( ɛ) j ɛ M j j + γ ( 0,M ɛ 2+ γ 0,M ɛ M j=(m+)/2 ( ( M (M )/2 M j ( ɛ ɛ ) ) M 2k γ 0,M ) ( ɛ) j ɛ M j 2+ γ 0,M ) (M 2k)(+/ γ0,m) + for M odd. (27) When M is even, the dominant terms in (27) correspond to j =(M 2)/2 and j =M/2. WhenM is odd, the dominant term in (27) corresponds to j =(M )/2. To provide a more concise expression, we assume ɛ /(kc γ), γ 0,M = c 2 γ, and obtain the high SNR approximation ( ) ( ) M/2 M ln(kc γ) (M 2)/2 kc c 2 γ ( ) +M/2 ( ) M/2 [ M + + ln(kc ] γ) M/2 kc c 2 γ +M/2 c 2 γ P b ( for M ) even, ( ) (M )/2 M (M )/2 c 2 kc γ (M+)/2 for M odd (28) where k =2again corresponds to noncoherent BFSK. As before, this procedure can be extended to the coherent case, yielding (28) with k =. Thus, an upper bound on diversity order is [( ɛ)ɛ] (M )/2 2+ γ 0,M { M/2+ for M even d (M + )/2 for M odd. (29) Combining (24) and (29) yields Theorem. Note that, as with (24), (29) can be developed more precisely without using the high SNR approximation for the relay BER.

10 794 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 7, JULY 2006 REFERENCES [] J. G. Proakis, Digital Communications. McGraw-Hill, Inc., 995. [2] A. Sendonaris, E. Erkip, and B. Aazhang, User cooperation diversity, part I: system description, IEEE Trans.Commun., vol. 5, no., pp , Nov [3], User cooperation diversity, part II: implementation aspects and performance analysis, IEEE Trans.Commun., vol. 5, no., pp , Nov [4] J. N. Laneman and G. W. Wornell, Energy-efficient antenna-sharing and relaying for wireless networks, in Proc. IEEE Wireless Communications and Networking Conference (WCNC), [5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: efficient protocols and outage behavior, IEEE Trans. Inform. Theory, vol. 50, no. 2, pp , Dec [6] M. Yuksel and E. Erkip, Diversity in relaying protocols with amplify and forward, in Proc. IEEE GLOBECOM, [7] M. O. Hasna and M.-S. Alouini, A performance study of dual-hop transmissions with fixed gain relays, in Proc. ICASSP, vol. 4, 2003, pp. IV [8], A performance study of dual-hop transmissions with fixed gain relays, IEEE Trans. Wireless Commun., vol. 3, no. 6, pp , Nov [9] A. Ribeiro, X. Cai, and G. B. Giannakis, Symbol error probabilities for general cooperative links, IEEE Trans. Wireless Commun., vol. 4, no. 3, pp , May [0] Z. Wang and G. B. Giannakis, A simple and general parameterization quantifying performance in fading channels, IEEE Trans. Commun., vol. 5, no. 8, pp , Aug [] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice-Hall, Inc., 996. [2] D. Chen, Noncoherent communication theory for cooperative diversity in wireless networks, Master s thesis, University of Notre Dame, [3] B. Zhao and M. C. Valenti, Distributed turbo coded diversity for the relay channel, IEE Electronics Letters, vol. 39, no. 0, pp , May [4] R. Liu, P. Spasojevic, and E. Soljanin, User cooperation with punctured turbo codes, in Proc. Allerton Conference on Comm. Contr. Computing [5] M. Janani, A. Hedayat, T. E. Hunter, and A. Norsatinia, Coded cooperation in wireless communications: space-time transmission and iterative decoding, IEEE Trans. Signal Processing, vol. 52, no. 2, pp , Feb [6] C. M. Keller and M. B. Pursley, Clipped diversity combining for channels with parital-band interference part I: clipped-linear combining, IEEE Trans. Commun., vol. 35, no. 2, pp , Dec [7] J. N. Laneman, Network coding gain of cooperative diversity, in Proc. IEEE Military Comm. Conf. (MILCOM), Nov Deqiang Chen Deqiang Chen (M 03) received B.S. and M.E. degrees in Mechatronics from the University of Science and Technology of China, HeFei, AnHui, P.R.China, in 999 and 2002, respectively. Since 2003, he has been with the Department of Electrical Engineering, University of Notre Dame, where his current research interests lie in wireless communications and networks, signal processing and information theory. J. Nicholas Laneman J. Nicholas Laneman (S 93-M 02) received B.S. degrees (summa cum laude) in Electrical Engineering and in Computer Science from Washington University, St. Louis, MO, in 995. At the Massachusetts Institute of Technology (MIT), Cambridge, MA, he earned the S.M. and Ph.D. degrees in Electrical Engineering in 997 and 2002, respectively. Since 2002, Dr. Laneman has been on the faculty of the Department of Electrical Engineering, University of Notre Dame, where his current research interest lie in wireless communications and networking, information theory, and detection & estimation theory. From 995 to 2002, he was affiliated with the Department of Electrical Engineering and Computer Science and the Research Laboratory of Electronics, MIT, where he held a National Science Foundation Graduate Research Fellowship and served as both a Research and Teaching Assistant. During 998 and 999 he was also with Lucent Technologies, Bell Laboratories, Murray Hill, NJ, both as a Member of the Technical Staff and as a Consultant, where he developed robust source and channel coding methods for digital audio broadcasting. His industrial interactions have led to five U.S. patents. Dr. Laneman received the MIT EECS Harold L. Hazen Teaching Award in 200, the ORAU Ralph E. Powe Junior Faculty Enhancement Award in 2003, and the NSF CAREER Award in He is a member of IEEE, ASEE, and Sigma Xi.