Cooperative Diversity in Wireless Relay Networks with. Multiple-Antenna Nodes

Transcription

1 Cooperative Diversity in Wireless Relay Networks with Multiple-Antenna Nodes YINDI JING AND BABAK HASSIBI Department of Electrical Engineering California Institute of Technology asadena, CA 95 Abstract In [], the idea of of space-time coding devised for multiple-antenna systems is applied to the problem of communication over a wireless relay network, a strategy called distributed space-time coding, to achieve the cooperative diversity provided by antennas of the relay nodes In this paper, we extend the idea of distributed space-time coding to wireless relay networks with multiple-antenna nodes and fading channels We show that for a wireless relay network with M antennas at the transmit node, N antennas at the receive node, and a total of R antennas at all the relay nodes, provided that the coherence interval is long enough, the high SNR pairwise error probability E) behaves as ) ) min{m,n}r MR if M N and log /M if M = N, where is the total power consumed by the network Therefore, for the case of M N, distributed space-time coding achieves the same diversity as decode-and-forward without any rate constraint on the transmission For the case of M = N, the penalty is a factor of log /M which, compared to, becomes negligible when is very high We also show that for a fixed total transmit power across the entire network, the optimal power allocation is for the transmitter to expend half the power and for the relays to share the other half with the power used at every relay proportional to the number of antennas it has Introduction It is known that multiple antennas can greatly increase the capacity and reliability of a wireless communication link in a fading environment using space-time coding [, 3, 4, 5] Recently, with the increasing interest in ad hoc This work is supported in part by the National Science Foundation under grant nos CCR-3388 and CCR-36554, by the David and Lucille ackard Foundation, and by Caltech s Lee Center for Advanced Networking

2 networks, researchers have been looking for methods to exploit spatial diversity using the antennas of different users in the network [6, 7, 8, 9,,, ] In [8], the authors exploit spatial diversity using the repetitionbased and space-time cooperative algorithms The mutual information and outage probability of the network are analyzed However, in their model, the relay nodes need to decode their received signals In [9], a network with a single relay under different protocols is analyzed and second order spatial diversity is achieved In [], the authors use space-time codes based on the Hurwitz-Radon matrices and conjecture a diversity factor around R/ from their simulations Also, the simulations in [] show that the use of Khatri-Rao codes lowers the average bit error rate This work follows the strategy of [], where the idea of space-time coding devised for multiple-antenna systems is applied to the problem of communication over a wireless relay network In [], the authors consider wireless relay networks in which every node has a single antenna and the channels are fading, and use a cooperative strategy called distributed space-time coding by applying a linear dispersion space-time code [] among ) the relays It is proved that without any channel knowledge at the relays, a diversity of R can be log log log achieved, where R is number of relays and is the total power consumed in the whole network This result is based on the assumption that the receiver has full knowledge of the fading channels Therefore, when the total transmit power is high enough, the wireless relay network achieves the diversity of a multiple-antenna system with R transmit antennas and one receive antenna, asymptotically That is, antennas of the relays work as antennas of the transmitter although they cannot fully cooperate and do not have full knowledge of the transmit signal Compared with the other widely used cooperative strategy, decode-and-forward, since no decoding is needed at the relays, distributed space-time coding saves both time and energy, and more importantly, there is no rate constraint on the transmission In this paper, we extend the idea of distributed space-time coding to wireless relay networks whose nodes have multiple antennas, and analyze the achievable diversity As in [], the focus of this paper is on the pairwise error probability E) analysis We investigate the achievable diversity gain in a wireless relay network by having the relays cooperate distributively By diversity gain, or diversity in brief, we mean the negative of the exponent of the SNR or transmit power in the E formula at the high SNR regime This definition is consistent with the diversity definition in multiple-antenna systems [5, 3] It determines how fast the E decreases with Although having the same name, the distributed space-time coding idea in [] is different from that in [8] Similar ideas for networks with one and two relays have appeared in [9, ] log E However, this definition is different from the formal diversity definition, lim, given in [4] It is more precise log in the sense that it can describe the E behavior in more detail For example, if E [c+f )], the same diversity c will be

3 the SNR or transmit power We use the same two-step transmission method in [, 5, 6], where in one step the transmitter sends signals to the relays and in the other the relays encode their received signals into a linear dispersion space-time code and transmit to the receiver For a wireless relay network with M antennas at the transmitter, N antennas at the receiver, and a total of R antennas at all the relay nodes, our work shows that when the coherence interval is ) long enough, a diversity of min{m, N}R, if M N, and MR log log M log, if M = N can be achieved, where is the total power used in the network With this two-step protocol, it is easy to see that the error probability is determined by the worse of the two steps: the transmission from the transmitter to the relays and the transmission from the relays to the receiver Therefore, when M N, distributed space-time coding is optimal since the diversity of the first stage cannot be better than MR, the diversity of a multiple-antenna system with M transmit antennas and R receive antennas, and the diversity of the second stage cannot be better than NR When M = N, the penalty on the diversity because the relays cannot fully cooperate and do not have full knowledge of the signal is R log log log When is very high, it is negligible Therefore, with distributed space-time coding, wireless relay networks achieve the same diversity of multiple-antenna systems, asymptotically The paper is organized as follows In the following section, the network model and the generalized distributed space-time coding is explained in detail The E is first analyzed in Section 3 based on which an optimum power allocation between the transmitter and the relays with respect to minimizing the E are given in Section 4 In Section 5, the diversity for the network with an infinite number of relays is discussed Then the diversity for the general case is obtained in Section 6 Section 7 is the conclusion and discussion roofs of some of the technical theorems are given in the appendices Wireless Relay Network System Model We first introduce some notation For a complex matrix A, Ā, A t, and A denote the conjugate, the transpose, and the conjugate transpose of A, respectively det A, rank A, and tr A indicate the determinant, rank, and trace of A, respectively I n denotes the n n identity matrix and m,n is the m n matrix with all zero entries obtained for any function f satisfying lim f ) = Also, this more precise definition is used to emphasize that the slope of the loge vs logsnr curve changes with the SNR 3

4 We often omit the subscripts when there is no confusion log indicates the natural logarithm indicates the Frobenius norm and E indicate the probability and the expected value gx) = Ofx)) means that lim x gx) fx) is a constant hx) = ofx)) means that lim x gx) fx) = Consider a wireless network with R + nodes which are placed randomly and independently according to some distribution There is one transmit node and one receive node All the other R nodes work as relays This is a practical appropriate model for many sensor networks By using a straight-forward TDMA technique, it can be generalized to ad hoc wireless networks with multiple pairs of transmitters and receivers 3 The transmitter has M transmit antennas, the receiver has N receive antennas, and the i-th relay has R i antennas, which can be used for both transmission and reception Since the transmit and received signals at different antennas of the same relay can be processed and designed independently, the network can be transformed to a network with R = R i= R i single-antenna relays by designing the transmit signal at every antenna of every relay according to the received signal at that antenna only 4 Therefore, without loss of generality, in the following, we assume that every relay has a single antenna 5 Therefore, the network can be depicted by Figure Denote the channels from the M antennas of the transmitter to the i-th relay as f i, f i,, f Mi, and the channels from the i-th relay to the N antennas at the receiver as g i, g i,, g in Here, we only consider the fading effect of the channels by assuming that f mi and g in are independent complex Gaussian with zero-mean and unit-variance This is a common assumption for networks in urban areas or indoors when there is no line-of-sight, or situations where the distances between the relays and the transmitter/receiver are about the same We make the practical assumption that the channels f mi and g in are not known at the relays 6 What every relay knows is only the statistical distribution of its local connections However, we do assume that the receiver knows all the fading coefficients f mi and g in Its knowledge of the channels can be obtained by sending training signals from the relays and the transmitter We use the block-fading model [3] by assuming a coherence interval T, that is, the time during which f mi and g in keep constant 7 The information bits are thus encoded into T M matrices s = [s,, s M ], 3 However, this straightforward TDMA scheme may not be optimal 4 This is one possible scheme In general, the signal sent by one antenna of a relay can be designed using received signals at all the antennas of the relay However, as will be seen later, this simpler scheme achieves the optimal diversity asymptotically although a general design may improve the coding gain of the network 5 The main reason is to highlight the diversity results by simplifying notation and formulas 6 For the i-th relay to know f mi, training from the transmitter and estimation at the i-th relay are needed It is even less practical for the i-th relay to know g in, which needs feedback from the receiver 7 From the two-step protocol that will be discussed in the following, we can see that we only need f mi to keep constant for the 4

5 f f R r r relays t t g g N transmitter f M f MR r R t R g R g RN receiver Figure : Wireless relay network with M antennas at the transmitter and N antennas at the receiver where s m, a T -dimensional vector, is the signal sent by the m-th transmit antenna For the power analysis, s is normalized as E tr s s = M ) To send s to the receiver, the same two-step strategy in [] is used here As shown in Figure, in step one, which is from time to T, the transmitter sends T/Ms with T/Ms m being the signal sent by the m-th antenna from time to T Based on ), the average total power used at the transmitter for the T transmissions is T The received signal at the i-th relay at time τ is denoted as r i,τ We denote the additive noise at the i-th relay at time τ as v i,τ In step two, which is from time T + to T, the i-th relay sends t i,,, t i,t We denote the received signal and noise at the n-th receive antenna of the receiver at time T + τ by x τn and w τn Assume that the noises are independent complex Gaussian with zero-mean and unit-variance, that is, the distribution of v i,τ, w τ,n is iid CN, ) We use the following notation: v i = v i, v i, r i = r i, r i, t i = t i, t i, w n = w n w n x n = x n x n, v i,t r i,t t i,t w T n x T n where v i, r i, and t i are the noise, the received signal, and the transmit signal at the i-th relay, w n is the noise at the n-th antenna of the receiver, and x n is the received signal at the n-th antenna of the receiver v i, r i, t i, w n, and x n are all T -dimensional column vectors first step of the transmission and g in to keep constant for the second step It is thus good enough to choose T as the minimum of the coherence intervals of f mi and g in 5

6 Since f mi and g in keep constant for T transmissions, clearly r i = M T/M f mi s m + v i = T/Msf i + v i ) m= and R x n = g in t i + w n 3) i= [ ] t where we have defined f i = f i f i f Mi Distributed Space-Time Coding We want the relays in the network cooperate in a way such that their antennas work as transmit/receive antennas of the same user to obtain diversity There are two main differences between the wireless network in Figure and a point-to-point multiple-antenna communication system analyzed in [5, 3] The first is that in a multiple-antenna system, antennas of the transmitter can cooperate fully while in the network, the relays do not communicate with each other and can only cooperate in a distributive fashion The other difference is that in a wireless network, every relay just has a noisy version of the transmit signal s Therefore, a crucial issue is how should every relay help the transmission, or more specifically, how should the i-th relay design its transmit signal t i based on its received signal r i One of the most widely used strategies is called decode-and-forward, see, eg, [6, 8]), in which the i-th relay fully decodes its received signal, and then encodes the information again and transmits the newly encoded signal If the transmission rate is sufficiently low so that all the R relays are able to successfully decode, the system can act as a multiple-antenna system with R transmit antennas and N receive antennas Therefore the communication from the relays to the receiver can achieve a diversity of NR However, if some nodes decode incorrectly, they will forward incorrect information to the receiver, which can significantly harm the decoding at the receiver Therefore, to use decode-and-forward and obtain the maximum diversity NR requires that the transmission rate is low enough so that all the relays can decode correctly Thus, the decode-and-forward requires a substantial reduction of rate, especially for large R, and so we will therefore not consider it here There are other disadvantages of decode-and-forward Because of the decoding complexity, it causes both extra time delay and energy consumption In this paper, we will use the cooperative strategy called distributed space-time coding proposed in [] Thus, design the transmit signal at relay i as t i = + A ir i, 4) 6

7 a linear function of its received signal 8 A i is a T T unitary matrix As in [], while within the framework of linear dispersion codes A i can be arbitrary, to have a protocol that is equitable among different users and among different time instants we set A i to be unitary This simplifies the analysis of the E considerably, as will be seen in the following section Because E tr ss = M, f mr, v r,j are CN, ), and f mr, s, v r,j are independent, the average received power at relay i can be calculated to be E r i r i = E T/Msf i + v i ) T/Msf i + v i ) = + )T Therefore, the average transmit power at relay i is E t i t i = + E A ir i ) A i r i ) = + E r i r i = T, which explains our normalization in 4): is the average transmit power for one transmission at every relay Let us now focus on the received signal Clearly from ) and 3), f g n T [ ] f g n R x n = A + )M s A s A R s + g in A i v i + w + i= f R g Rn By defining X = g i = [ [ ] [ x x x N, S = A s A s A R s f g g i g i g in ], H = ], f g, 5) f R g R and W = [ + R i= g ia i v i + w ] R + i= g in A i v i + w N, 6) 8 In general, the transmit signal at the relays can be designed as any function of their received signals Although only linear functions are used here, we conjecture that the diversity result cannot be improved using other more complicated designs Also, more complicated designs make the noise analysis intractable 7

8 the system equation can be written as T X = SH + W 7) M + ) Let us examine the dimension of each matrix The received signal matrix X is T N S which is actually a linear space-time code is T MR since the A i are T T, s is T M, and there are R of them f i is M and g i is N Therefore, the equivalent channel matrix H is RM N W which is the equivalent noise matrix is T N As in [], S works like the space-time code in a multiple-antenna system It is called the distributed space-time code since it has been generated distributively by the relays without having access to the transmit signal The power constraint on S is: E tr S S = E R tr s A i A is = RE tr s s = RM Therefore, the code has the same normalization as a T M R unitary matrix i= 3 airwise Error robability To analyze the E, we have to determine the maximum-likelihood ML) decoding rule This requires the conditional probability density function DF) X s k ), where s k S and S is the set of all possible transmit signal matrices Theorem Given that s k is transmitted, define [ S k = A s k A s k A R s k ] Then conditioned on s k, the rows of X are independently Gaussian distributed with the same variance I N + + GG, where G = g g R g N g RN The t-th row of X has mean T M +) [S k] t H with [S k ] t the t-th row of S k In other words, X s k ) = q )e π NT det I tr X T M +) S kh I N + q + GG X T M +) S kh 8) T N + + GG 8

9 roof: See Appendix A In view of the above theorem, we should emphasize that for a wireless relay network with multiple antennas at the receiver, the columns of X are not independent although the rows of X are The covariance matrix of each row I N + + GG is not diagonal in general) That is, the received signals at different antennas are not independent, whereas the received signals at different times are This is the main reason that the E analysis in the new model is much more difficult than that of the network in [], where X had only a single column With X s i ) in hand, we can obtain the ML decoding and thereby analyze the E The result follows Theorem ML decoding and the E Chernoff bound) Define R W = I N + + GG The ML decoding of the relay network is arg min tr s k ) T X M + ) S kh R W ) T X M + ) S kh 9) With this decoding, the E of mistaking s k by s l, averaged over the channel realization, has the following upper bound: s k s l ) E e T 4M+ ) tr S k S l ) S k S l )HR f mi,g in W H ) roof: See Appendix B 4 ower Allocation The main purpose of this work is to analyze how the E decays with transmit power or with the receive SNR at the high SNR regime The total power used in the whole network is = + R Therefore, one natural question is how to allocate power between the transmitter and the relays if is fixed In this section, we find the optimum power allocation such that the E is minimized Because of the expectations over f mi and g in, and also the dependency of R W on g in, the exact minimization of formula ) is very difficult Therefore, similar to the argument in [], we recourse to an asymptotic argument for R Notice that the i, j)-th entry of R GG is R R r= g riḡ rj When R, according to the law of large numbers, the off-diagonal entries of R GG goes to zero while the diagonal entries approach with probability since R r= g ri has a gamma distribution with both mean and variance R Therefore, it is reasonable to ) + R + I N assume R GG I N for large R, which is the same as R W 9

10 Therefore, from ), s k s l ) E e f mi,g in T 4M+ +R ) tr S k S l ) S k S l )HH Since S k, S l, and H are independent of and, minimizing the E is now equivalent to maximizing T 4M+ +R ) This is exactly the same power allocation problem that appeared in [] Therefore, with the same argument, we can conclude that the optimum solution is to set = and = R ) That is, the optimum power allocation is such that the transmitter uses half the total power and the relays share the other half When the number of relay nodes are large, which is the case for many sensor networks, every relay spends only a very small amount of power to help the transmitter Note that as discussed in Section, for the general network where the i-th relay has R i antennas, the antennas are treated as R i different relays in Figure Therefore, it is easy to see that for this multiple-antennarelay-node case, the optimum power allocation is such that the transmitter used half the total power as before, but every relay uses power that is proportional to its number of antennas That is = R at the i-th relay is i R j= R j With this power allocation, at high, T 4M+ +R ) T T 6MR Therefore, the E satisfies and the power used s k s l ) E e 6MR tr S k S l ) S k S l )HH ) f mi,g in It is easy to see that the expected receive SNR of the system is T 4+ +R ) Therefore, this optimal power allocation also maximizes the expected receive SNR We should emphasis that this result is only valid for the wireless relay network described in Section, in which all channels are assumed to be iid Rayleigh fading It may not be optimal if these assumptions are not met 5 Diversity Analysis for R 5 Basic Results As mentioned earlier, to obtain the diversity, we have to compute the expectations over f mi and g in in ) We shall do this rigorously in Section 6 However, since the calculation is detailed and gives little insight, in this section, we give a simple asymptotic derivation for the case where the number of relay nodes approaches

11 infinity, that is R As discussed in the previous section, when R is large, we can make the approximation ) R W + R + I N and then obtain formula ) Denote the n-th column of H as h n From 5), h n = G n f, [ ] t where we have defined G n = diag {g n I M,, g Rn I M } and f = f f R Therefore, from ), s k s l ) E e T 6MR tr H S k S l ) S k S l )H f mi,g in e T N 6MR n= h n S k S l ) S k S l )h n f mi,g in = E e T 6MR f [ N n= G n S k S l ) S k S l )G n]f f mi,g in = E Since f is white Gaussian with mean zero and variance I RM, [ ] s k s l ) E det I RM + T N G gin 6MR ns k S l ) S k S l )G n 3) Similar to the multiple-antenna case [5, 3] and the case of wireless relay networks with single-antenna nodes [], the full diversity condition can be obtained from 3) It is easy to see that if S k S l drops rank, the exponent of in the right side of 3) increases That is, the diversity of the system decreases Therefore, at high transmit power, the Chernoff bound is minimized, which is equivalent to saying that the diversity is maximized, when S k S l is full-rank, or equivalently, dets k S l ) S k S l ) for all S k S l S Since the distributed space-time code S k and S l are T MR, there is no point in having MR larger than the coherence interval T Thus, in the following, we will always assume T M R Assume that the code is fully-diverse From 3), roughly speaking, the larger the positive matrix S k S l ) S k S l ), the smaller the upper bound This improvement in the E because of the code design is called coding gain Since the focus of this paper is on the diversity provided by the independent transmission routes from the antennas of the transmitter to the antennas of the receiver via the relays, the coding gain design, or the code optimization, is not an issue Denote the minimum singular value of S k S l ) S k S l ) by σ min From the full diversity of the code, σmin > Therefore, the right side of 3) can be further upper bounded as: ] s k s l ) E det [I RM + T N σ min G gin n 6MR G n n= ) R M = E + T N σ min g in gin 6MR i= Define g i = N n= g in Since g in are iid CN, ), g i are iid gamma distributed with DF pg i ) = N )! gn i e g i Therefore, s k s l ) N )! R [ n= n= + T ) M ] R σ min 6MR x x N e x dx

12 By defining y = + T σ min 6MR 6MR x, which is equivalent to x = y ), and changing the variable in the T σmin integral from x to y, we have ) 6MR NR s k s l ) N )! R T σmin e When the transmit power is high, that is,, e s k s l ) 6MR T σ min 6MR T σ min l= [ y ) N y M Therefore, ) 6MR NR N N N )! R T σmin l The following theorem can be obtained by calculating the integral e 6MR T σ min y dy ] R R y l M e 6MR T σ min y dy Theorem 3 Diversity for R ) Assume that R, T MR, and the distributed space-time code is full diverse For large total transmit power, by looking at only the highest order term of, the E of mistaking s k with s l has the following upper bound: s k s l ) N )! R ) 6MR min{m,n}r T σ min N ) R M N NR if M > N ) MR if M = N log /M N M )! R MR if M < N 4) Therefore, the diversity of the wireless relay network is min{m, N}R d = MR M log log log ) if M N if M = N 5) roof: See Appendix C 5 Discussion With the two-step protocol, it is easy to see that regardless of the cooperative strategy used at the relay nodes, the error probability is determined by the worse of the two transmission stages: the transmission from the transmitter to the relays and the transmission from the relays to the receiver The E of the first stage cannot be better than the E of a multiple-antenna system with M transmit antennas and R receive antennas, whose optimal diversity is MR, while the E of the second stage can have diversity no larger than NR Therefore, when M N, according to the decay rate of the E, distributed space-time coding is optimal For the case of M = N, the penalty on the decay rate is just R log log log, which is negligible when is high Therefore,

13 distributed space-time coding is better than decode-and-forward since it achieves the optimum diversity gain without the rate constraint needed for decode-and-forward If we use the diversity definition in [4], since lim log log log obtained =, diversity min{m, N}R can be The results in Theorem 3 are obtained by considering only the highest order term of in the E formula When analyzing the diversity gain, it is not only the highest order term that is important, but also how dominant it is Therefore, we should analyze the contributions of the second highest order term and also other terms of compared to that of the highest one This is equivalent to analyzing how large the total transmit power should be to have the terms given in 4) to dominate The following remarks are on this issue They can be observed from the proof of Theorem 3 in Appendix C Remarks: If M N >, from 7) and 9), the second highest order term of in the E formula behaves as min{m,n}r+ The difference between the highest and the second highest order terms of in the E is a factor Therefore, the highest order term is dominant when In other words, contributions of the second highest order term and other lower order terms are negligible when If M = N, from 8), the second highest order term of in the E formula is M )! R 6MR T σ min M R M )! R 6MR T σ min ) MR log R MR, which has one less log compared with the the highest order term Therefore, the highest order term, ) MR ) log /M MR, is dominant if and only if log, which is a much stronger condition than When is not very large, contributions of the second highest order term and even other lower order terms are not negligible 3 If M N =, from 6) and 3), the second highest order term of in the E formula behaves as min{m,n}r log The difference between the highest and the second highest order terms is a log factor Therefore, the highest order term given in 4) is dominant if and only if log This condition is weaker than the condition log in the previous case, however, it is still stronger than the normally used condition Thus, although the situation is better than the case of M = N, still when is not large enough, the second highest order term and even other lower order terms in the E formula are not negligible 3

14 6 Diversity Analysis for the General Case 6 A Simple Derivation The diversity analysis in the previous section is based on the assumption that the number of relays is very large In this section, analysis on the E and the diversity results for the general case, networks with any number of relays, are given As discussed in Section 3, the main difficulty of the E analysis lies in the fact that the noise covariance matrix R W is not diagonal From ), we can see that one way of upper bounding the E is to upper bound R W Since R W = I N + + GG, R W tr R W )I N = N + + n= ) R g in I N = Therefore, from ) and using the power allocation given in ), s k s l ) E e f mi,g in i= 8MNR+ NR N + + N n= i= T Nn= Ri= g in ) tr H S k S l ) S k S l )H ) R g in I N when If the space-time code is fully diverse, using similar argument in the previous section, s k s l ) E gin R i= + T σ min 8MNR g i + NR R i= g i ) M, where, as before, σ min is the minimum singular value of S k S l ) S k S l ) and g i = N n= g in Calculating this integral, the following theorem can be obtained Theorem 4 Divesity for wireless relay network) Assume that T M R and the distributed space-time code is full diverse For large total transmit power, by looking at the highest order terms of, the E of mistaking s k by s l satisfies: s k s l ) N )! R ) 8MNR min{m,n}r T σ min [ M R NM N)] NR if M > N ) ) + R log /M MR if M = N N [ N + N M )!] R MR if M < N 6) Therefore, the same diversity as in 5) is obtained roof: See Appendix D Although the same diversity is obtain as in the R case There is a factor of N in formula 6), which does not appear in 4) This is because we upper bound R W by tr R W )I N, whose expectation is N times 4

15 the expectation of R W, while in the previous section we approximate R W by its expectation This factor of N can be avoided by finding tighter upper bounds of R W In the following subsection, we analyze the maximum eigenvalue of R W Then in the Subsection 63, a E upper bound using the maximum eigenvalue of R W is obtained 6 The Maximum Eigenvalue of Wishart Matrix Denote the maximum eigenvalue of R GG as λ max Since G is a random matrix, λ max is a random variable We first analyze the DF and the cumulative distribution function CDF) of λ max If entries of G are independent Gaussian distributed with mean zero and variance one, or equivalently, both the real and imaginary parts of every entry in G are Gaussian with mean zero and variance, R GG is known as the Wishart matrix While there exists explicit formula for the distribution of the minimum eigenvalue of a Wishart matrix, surprisingly we could not find non-asymptotic formula for the maximum eigenvalue Therefore, we calculate the DF and CDF of λ max from the joint distribution of all the eigenvalues of R GG in this section The following theorem has been proved Theorem 5 G is an N R matrix whose entries are iid CN, ) The DF of the maximum eigenvalue of R GG is p λmax λ) = R RN λ R N e Rλ N det F, 7) n= ΓR n + )Γn) where F is an N ) N ) Hankel matrix whose i, j)-th entry equals f ij = λ λ t) t R N+i+j e Rt dt The CDF of the maximum eigenvalue of R GG is λ max λ) = R RN N n= ΓR n + )Γn) det F, 8) where F is an N N Hankel matrix whose i, j)-th entry equals f ij = λ tr N+i+j e Rt dt roof: In Appendix E A theoretical analysis of the DF and CDF from 7) and 8) appears quite difficult To understand λ max, we plot the two functions in figures and 3 for different R and N Figure shows that the DF has a peak at a value a bit larger than As R increases, the peak becomes sharper An increase in N shifts the peak right However, the effect is smaller for larger R From Figure 3, the CDF of λ max grows rapidly around λ = and 5

16 rλ max = λ) DF of λ max of Wishart matrix R= N= R= N=3 R= N=4 R=4 N= R=4 N=3 R=4 N=4 rλ max < λ) CDF of λ max of Wishart matrix R= N= R= N=3 R= N=4 R=4 N= R=4 N=3 R=4 N= λ λ Figure : DF of the maximum eigenvalue of R GG Figure 3: CDF of the maximum eigenvalue of R GG becomes very close to soon after The larger R, the faster the CDF grows Similar to the DF, an increase in N results in a right shift of the CDF However, as R grows, the effect diminishes This verifies the validity of the approximation GG RI N in Section 6 for large R In the following corollary, we give an upper bound on the DF This result is used to derive the diversity result for general R in the next subsection Corollary The DF of the maximum eigenvalue of R GG can be upper bounded as where p λmax λ) C λ RN e Rλ, 9) C N R RN = N n= ΓR n + )Γn) N n= R N + n )R N + n)r N + n + ) ) is a constant that depends only on R and N roof: From the proof of Theorem 5, F is a positive semidefinite matrix Therefore det F N n= f nn From 7), f nn can be upper bounded as f nn we have λ λ t) t R N+n dt = det F Thus, 9) is obtained R N + n )R N + n)r N + n + ) λr N+n+, N N n= R N + n )R N + n)r N + n + ) λrn R+N 6

17 63 Diversity Results for the General Case If the maximum eigenvalue of R GG is λ max, the maximum eigenvalue of R W = I N + + GG is + ) R + λ max, and therefore R W + R + λ max I N From ) and using the power allocation given in ), we have s k s l λ max = c) E e T 4M+ + Rλmax) tr S k S l ) S k S l )HH f mr,g rn E e f mr,g rn T 8+λmax)MR tr S k S l ) S k S l )HH The only difference of the above formula with formula ) is that the coefficient in the constant in the denominator of the exponent is 8 + λ max ) now instead of 6 This makes sense since c as R Therefore, using an argument similar to the proof of Theorem 3, at high total transmit power, by looking at the highest order terms of, s k s l λ max = c) N )! R The following theorem can thus be obtained [ ] 8 + c)mr min{m,n}r T σ min N ) R M N NR if M > N ) MR if M = N log /M N M )! R MR if M < N ) Theorem 6 Divesity for wireless relay network) Assume that T M R and the distributed space-time code is full diverse For large total transmit power, by looking at the highest order terms of, the E of mistaking s k by s l can be upper bounded as: s k s l ) ) Ĉ 8MR min{m,n}r N )! R T σmin N ) R M N NR if M > N ) MR if M = N log /M N M )! R MR if M < N, ) where min{m,n}r Ĉ = C i= min{m, N}R i min{m, N}R + i )! R min{m,n}r+i Therefore, the same diversity as in 5) is obtained roof: s k s l ) = s k s l λ max = c)p λmax c)dc 7 C c RN e Rc s k s l λ max = c)dc

18 using 9) in Corollary From ), s i s i ) C N )! R ) 8MR min{m,n}r T σmin c RN e Rc + c) min{m,n}r dc N M N ) R NR if M > N log M ) R if M = N N M )! R MR if M < N Since c RN e Rc + c) min{m,n}r dc = ) is obtained min{m,n}r i= min{m, N}R i R min{m, N} + i )! R R min{m,n}+i, 7 Conclusion and Discussion In this paper, we generalize the idea of distributed space-time coding to wireless relay networks whose transmitter, receiver, and/or relays can have multiple antennas We assume that the channel information is only available at the receiver The ML decoding at the receiver and E of the network are analyzed We have shown that for a wireless relay network with M antennas at the transmitter, N antennas at the receiver, a total of R antennas at all the relay nodes, and a coherence interval no less than MR, an achievable diversity is min{m, N}R, if ) M N, and MR log log M log, if M = N, where is the total power used in the whole network This result shows the optimality of distributed space-time coding according to the diversity gain It also shows the superiority of distributed space-time coding to decode-and-forward We also show that for a fixed total transmit power across the entire network, the optimal power allocation is for the transmitter to expend half the power and for the relays to share the other half such that the power used by every relay is proportional to the number of antennas it has There are several directions for future work that can be envisioned In this paper, We have obtained an achievable diversity for the wireless relay network using the idea of distributed space-time coding To achieve this diversity, the distributed space-time code { [ S k = ] } A s k A s k A R s k s k S should be full diverse That is, dets k S l ) S k S l ) for all s k s l S In addition to the diversity gain, the coding gain design or the code optimization is also an important issue since it effects the actually perfor- 8

19 mance of the system From 3), the larger S k S l ) S k S l ), the better the coding gain Therefore, possible design criterion are max min sk s l S dets k S l ) S k S l ) and max min sk s l S σs k S l ) S k S l ), where σa) indicates the minimum singular value of A More analysis is needed for this problem However, it is beyond the scope of this paper Another important problem is the non-coherent case In this work, we assume that the receiver knows all the channel information, which needs training from both the transmitter and the relay nodes For networks with high mobility, this is not a practical assumption Therefore, it should be interesting to see whether differential space-time coding technique can be generalized to this network The network model can be generalized in many ways too The network analyzed in this paper has only one transmitter and receiver pair When there are multiple transmitter-and-receiver pairs, interference plays an important role Smart detection and/or interference cancellation techniques will be needed Also, in our model, only the fading effect of the channels is considered The achievable diversity when channel path-loss is in consideration is another important problem A roof of Theorem roof: It obvious that since H is known and W is Gaussian, the rows of X are Gaussian We only need to show that the rows of X are uncorrelated and that the mean and variance of the t-th row are T +)M [S k] t H and I N + + GG, respectively The t, n)-th entry of X can be written as T R M T R x tn = f mi g in a i,tτ s k,τm + M + ) + i= m= τ= i= τ= T g in a i,tτ v iτ + w tn, where a i,tτ is the t, τ)-th entry of A i and s k,τm is the τ, m)-th entry of s k With full channel information at the receiver, T R M E x tn = M + ) i= m= τ= T f mi g in a i,tτ s k,τm 9

20 Therefore, the mean of the t-th row is T M +) [S k] t H Since v i, w n, and s k are independent, Cov x t n, x t n ) = E x t n E x t n )x t n E x t n ) = R T R T E g i n + a i,t τ v r τ ḡ i n ā i,t τ v i τ + E w t n w t n = + = δ t t i = τ = i = τ = R i= τ= + T a i,t τ ā i,t τ g in ḡ in + δ n n δ t t ) R g in ḡ in + δ n n r= = δ t [ ] t g + n g Rn ḡ n ḡ Rn + δ n n The fourth equality is true since A i are unitary Therefore, the rows of X are independent since the covariance of x t n and x t n is zero when t t It is also easy to see that the variance matrix of each row is I N + + GG Therefore, [X] t s k ) = )e π N det I N + + GG h q i tr X T M +) S kh I N + q i hx t + GG T M +) S kh t, from which 8) can be obtained B roof of Theorem roof: It is straightforward to obtain the ML decoding formula 9) from formula 8) For any λ >, the E of mistaking s k by s l has the following Chernoff upper bound [8, 3]: s k s l ) E e λlog X s l ) log X s k )) Since s k is transmitted, X = T M +) S kh + W From 8), log x s l ) log x s k ) [ T = tr M + ) S k S l )HR W H S k S l ) T + M + ) S k S l )HR W W ] T + M + ) W R W H S k S l )

21 From the proof of Theorem, W is also Gaussian distributed with mean zero and its variance is the same as that of X Therefore, s k s l ) h q q i T E f mi,g in,w e λtr M +) S k S l )HR W H S k S l ) + T M +) S k S l )HR W W + T M +) W R W H S k S l ) = E = E f mi,g in e λ λ) T M+ ) tr S k S l )HR tr W H S k S l ) e e λ λ) T M+ ) tr S k S l ) S k S l )HR W H f mi,g in q λ T M +) S k S l )H+W R W π NT det R W q λ T M +) S k S l )H+W Choosing λ =, which maximizes λ λ) and therefore minimizes the right side of the above formula, ) is obtained dw C roof of Theorem 3 roof: Define I = N l= N l We first give three integral equalities that will be used later u u x n e µx dx = e uµ n k= n! k! e µx x n+ dx = )n+ µn Ei µu) n! e µx u x u k y l M e 6MR T σ min y dy, u >, Rµ >, n =,,, 3) µ n k+ + e µu u n To calculate I, we discuss the following cases separately n k= ) k µ k u k, µ >, n =,, 4) n n k) dx = Ei µu), Rµ >, u 5) C Case I: M < N In this case, I = N l=m N l y l M e 6MR T σ min y dy + N M + M l= N l e 6MR T σ min y y dy y M l) e 6MR T σ min y dy

22 Using equalities 3-5) with u =, µ = 6MR, and n = l M or n = M l, T σmin I = N l=m N l ) 6MR l M+) l M)! T σmin + N log M M + N + lower order terms of l M l l= By only looking at the highest order term of, which is in the first term with l = N, we have ) 6MR N M) I = N M )! T σmin + o N M)) Therefore, s k s l ) = ) [ 6MR NR 6MR N )! R T σmin N M )! T σmin [ ] N M )! R ) 6MR MR N )! MR + o MR T σ min ) N M) + o ) MR ) ] R While analyzing the performance of the system at high transmit power, not only is the highest order term of important but also how fast other terms decay with respect to it Therefore, we should also look at the second highest order term of To do this, we have to consider two different cases If N = M +, I = = ) 6MR [ T σmin + M Ei 6MR )] T σmin + O) ) 6MR + M log + O) T σ min Therefore, s k s l ) M! R ) 6MR MR T σ min RM + MR M! R The second highest order term of in the E behaves as If N > M +, 6MR I = N M )! T σmin = 6MR T σ min 6MR T σ min log MR+ ) MR+ ) log log MR+ + o MR+ 6) = MR+ log log log ) N M) ) N M ) 6MR + N )N M )! T σmin + o N M ) ) N M) [ N M )! + N )N M )! 6MR T σmin + o )]

23 Therefore, ) N M )!R 6MR MR s k s l ) N )! R T σmin MR + N )N M )N M )! R 6MR N )! R T σmin C Case II: M = N ) MR+ MR+ + o MR+ ) 7) In this case, I = e 6MR T σ min y y dy + N l= N l y M l) e 6MR T σ min y dy Using 5) with µ = 6MR and u =, and 4) with u = and n = M l, we have T σmin I = log + N l= N < log + N + lower order terms of l + lower order terms of M l Therefore, s k s l ) M )! R 6MR T σ min ) MR log R MR + N R M )! R ) MR 6MR log R T σ min MR + o ) log R MR 8) Also, the second highest order term of in the E behaves as logr RM and so on and the next term has one log less C3 Case III: M > N In this case, I = M l= N l y M l) e 6MR T σ min y dy Using4) with u =, µ = I = 6MR, and n = M l, T σmin N l= N l + lower order terms of M l 3

24 Thus, s k s l ) = R ) 6MR NR N N N ) R T σmin l= l M l + o) R N 6MR N ) NR N )! l M l T σmin NR + o NR) l= We can further upper bound the E to get a simpler formula Notice that M l M N Thus, s k s l ) R N 6MR N ) NR M N)N )! l= l T σmin NR [ N ] R ) 6MR NR M N)N )! T σmin NR As discussed before, we also want to see how dominant the highest order term of given in the above formula is If M > N +, M l > N + N ) = From 4), Therefore, I < N M N N 6MR M N)M N ) T σmin [ N ] R 6MR s k s l ) M N)N )! T σmin ) NR [ NR + R M N ) + o ) 6MR T σ min The second highest order term in the E behaves as If M = N +, NR+ I < N + 6MR ) log T σmin + O Therefore, s k s l ) RN ) N )! R 6MR T σ min ) NR NR + R )N ) R N )! R 6MR T σ min which indicates that the second highest order term in the E behaves as ] ) NR+ + o NR+ 9) ) NR+ ) log log NR+ + o NR+, 3) log NR+ = R NR+ log log log D roof of Theorem 4 roof: Since g i have DF pg i ) = N )! gn i e g i, s k s l ) R T i,,i r, r= i < <i r R 4

25 where T i,,i r = N )! R the i, i r -th integrals are from x to, others are from to x r R r R i= + T σ min 8MNR g i + NR R i= g i ) M g N e gi dg dg R and x is any positive real number Let us calculate T,,r first x ) M x R T,,r = N )! R + T σ min g i x x 8MNR } {{ }} {{ } i= + R NR i= g gi N e gi dg dg R i < = N )! R N )! R x x r i= T σ min 8MNR ) T σ rm min γ R r N, x) 8MNR x x g i + R r NR x + NR r i= g i x + R r NR x + NR x i ) M g N e gi dg dg r R i=r+ ) rm r r g i i= i= i g N i e gi dg r+ dg R e gi g M N+ i dg dg r, where γn, x) is the incomplete gamma function [7] We should choose x so that the diversity is maximized Define x = β α, where β is a positive constant and α is any real constant The value of β doesn t affect T σ α the diversity Here, to have the E result consistent with formula 4) in Section 6, we set β = min 8MNR) Therefore, choosing the optimal in the sense of maximizing the diversity) x is equivalent to choosing the optimal α If α >, the r = term in the E upper bound is N )! R γr N, α ) = + o) Therefore, having α positive is not optimal according to diversity Similarly, if α =, x = The r = term in the E upper bound, N )! R γ R N, ), is a constant Therefore, α should be negative Thus, γn, x) = N xn + ox N ) = N βn αn + o αn ) We are only interested in the highest order term of When is large, R r NR x is negligible compared with Therefore, T,,r where we have defined Λ = ) T σ rm+αnr r) min N )! R N R r rm+αnr r) Λ, 8MNR x x + NR ) rm r r g i i= i= 5 e g i g M N+ i dg dg r

26 Consider the expansion of A + a k i= i) λ into monomial terms: + NR ) a r g i = i= a j= l < <l j k i,,i j Ci,, i j ) NR) i + +i j g i l g i l g i j l j, im a where j denotes how many g i are present, l,, l j are the subscripts of the g i that appears, i m indicates that g lm is taken to the i m -th power, and finally ) ) ) k k i k i i j Ci,, i j ) = counts how many times the term g i l g i l g i j l j appears in the expansion Thus, where Λ = r j= l < <l j r i i i,,i j im r Ci,, i j )Λj; l,, l j ; i,, i j ) i j Λj; l,, l j ; i,, i j ) = NR) i + +i j j m= x e g lm g M N+ im l m dg lm ) i i,i j x e g i g M N+ i dg i From 3) 5), While, α <, and n >, x λ n e λ dλ = n! + o), e λ x λ dλ = α) log + olog ), and x e λ λ n+ dλ = n β n αn + o αn ) Therefore, the highest order term of in Λ is the j = term If we only keep the highest oder term of in Λ, r e g i M N) β rm N) rαm N) if M > N r Λ = i= x gi M N+ dg i α) r log r if M = N N M )! r if M < N From the symmetry of g,, g R, we have T i,,i r = T,,r Therefore, R s k s l ) R T,,r r= r R R T σ ) rm+αnr r) min N )! R r= r N R r rm+αnr r) 8MNR T σ rαm N) min M N) 8MNR) rαm N) if M > N r α) r log r if M = N N M )! r if M < N 6

27 We should choose a negative α such that the exponent of the highest order term of in the above formula is minimized In other words, if we denote the exponent of the r-th term as fr), choose a α < such that max r fr) is minimized If M > N, fr) = rm + αnr r) rαm N) = αnr rm + α) If α, fr) is an increasing function of r Thus, max r fr) = fr) = αm N)R MR, which is minimized when α equals its maximum If α, fr) is an decreasing function of r Thus, max r fr) = f) = αnr, which is minimized when α equals its minimum Therefore, we should set α = Therefore, s k s l ) ) R N + M N N )! R ) 8MNR NR [ NR = T σ min M/N M N)N )! ] R ) 8MNR NR NR T σ min N If M < N, fr) = αnr rnα + M N ) By similar argument, we should set α = M N Thus, N s k s l ) + N M )!) R ) 8MNR MR N )! R T σmin MR ) If M = N, fr) = αnr rn α + log log N log Using similar argument, the optimal choice of α is s k s l ) log log log Therefore, [ 8MNR N )! R T σmin N + ) R 8MNR N )! R T σ min ) log log log N + log log N log ) MR ) MR log /M ) ] R 8MNR T σ min ) ) MR MR log /M E roof of Theorem 5 roof: We first give a theorem that will be needed later Theorem 7 Define Λ = λ,, λ N ) For any function f, g, and h, N dλ fλ i ) det V g Λ) det V h Λ) = N! det F gh, i= 7

28 where V g Λ) = g λ ) g λ N ) h λ ) h λ N ), V hλ) = g N λ ) g N λ N ) h N λ ) h N λ N ) g t) [ F gh = ft) h t) h N t) g N t) ] dt and Define G as a complex Gaussian matrix whose entries real and imaginary parts have mean zero and variance one Denote the ordered eigenvalue of G G as λ λ λ N It is well-known that the eigenvalues have the following joint distribution [9]: where C = λ,, λ N ) = C N i= λ R N i e λ i i<j N λ i λ j ), 3) RN Q N n= ΓR n+)γn) is a constant Denote the ordered eigenvalues of R GG as λ λ λ N Therefore, λ i = Rλ i The joint distribution of λ,, λ N is therefore λ,, λ N ) = λ,, λ N ) dλ dλ N dλ dλ N = det [diag {R,, R}] Rλ,, Rλ N ) N = R) N C Rλ i ) R N e Rλ i [Rλ i λ j )] i= = CR) RN N i= λ R N i e Rλ i i<j N i<j N λ i λ j ) To get the DF of λ, we have to do the integral over λ,, λ N Define fx) = λ x) x R N e Rx and g i x) = h i x) = x i = = λ max = λ) = λ = λ) λ λ λ N λ, λ,, λ N )dλ dλ N N )! λ λ λ, λ,, λ N )dλ dλ N 8

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