Diversity-Fidelity Tradeoff in Transmission of Analog Sources over MIMO Fading Channels

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1 Diversity-Fidelity Tradeo in Transmission o Analog Sources over MIMO Fading Channels Mahmoud Taherzadeh, Kamyar Moshksar and Amir K. Khandani Coding & Signal Transmission Laboratory Dept. o Elec. and Comp. Eng., University o Waterloo Waterloo, ON, Canada, NL 3G1 Tel: , Fax: {taherzad, kmoshksa, khandani}@cst.uwaterloo.ca Abstract In this paper, the theoretical limits on the asymptotic perormance o joint source-channel coding over MIMO ading channels are investigated. Similar to the concept o Diversity-Multiplexing Tradeo DMT in Digital MIMO systems, a new measure or the asymptotic high- perormance o MIMO source-channel codes is introduced, which is called Diversity-Fidelity Tradeo, and the optimal tradeo is characterized. Also, the problem o constructing robust MIMO sourcechannel codes, which are not sensitive to the knowledge o the, is investigated and a semi-robust scheme is proposed. I. INTRODUCTION In many applications, such as voice and multimedia transmission in cellular and wireless LAN environments, transmission o analog sources over wireless channels is needed. Results o the research in the past decade shows that using multiple-antenna systems can substantially improve the rate and the reliability o communication in wireless ading environments. However, until now, most o the research has been ocused on the transmission o digital data over multiple-input multiple-output MIMO channels, not the transmission o analog sources. For the simpler case o Gaussian channel, without considering the delay limitations, Shannon s source-channel coding separation theorem ensures the optimality o separately designing source and channel codes. However, or the case o a limited delay, even or the Gaussian channel, several articles have shown that joint source-channel codes have a better perormance, compared to the separately designed source and channel codes which are called tandem codes. Thus, designing eicient joint source-channel codes is a key solution or the transmission o analog sources. Although in recent years there has been a colossal amount o research in MIMO communications, the research on MIMO joint source-channel coding is still in its early stages. In [1] and [] some digital and hybrid digital-analog techniques are examined or joint source-channel coding over MIMO channels, and some bounds on the asymptotic exponents o the average distortion are presented. However, the asymptotic exponents o the average distortion is not a good measure or the perormance evaluation o joint source-channel coding schemes in ading environments. The reason is that the average distortion is dominated by rare cases o very bad channels. In these environments, it is more inormative to analyze the probability that distortion is greater than a certain level, instead o averaging very large unusual distortions. Unlike [1] and [] which are ocused on analyzing the asymptotic exponents o the average distortion, here, we analyze the asymptotic exponents o the probability o having a large distortion. This measure, which we call diversityidelity tradeo, can be seen as an analog version o the well-known diversity-multiplexing tradeo which is proved to be very useul in evaluating various digital space-time coding schemes. Another concern o this paper is to design robust or semirobust source-channel codes or MIMO ading channels. It is known that digital joint source-channel coding is very sensitive to the mismatch in the estimation o the channel signal-to-noise ratio. In many cases the exact signalto-noise ratio is not known at the transmitter, and may vary over a large range o values. Two examples o this scenario are transmitting an analog source over a quasistatic ading channel and/or multicasting it to dierent users with dierent channel gains. Thereore, or these kinds o applications, it is very important to have a joint sourcechannel scheme which is robust or a wide range o. To avoid the saturation eect o digital coding, hybrid digital-analog codes are proposed in the past. Although these codes can provide asymptotic gains or high over the simple repetition code, they suer rom a threshold eect. Indeed, when the becomes less than a certain threshold, the perormance o these systems degrades severely. Thereore, the parameters o these methods should be chosen according to the operating, hence, these methods are still very sensitive to the errors in the estimation o. Also, although the perormance o the system is not saturated or the high unlike digital codes, the scaling o the end-to-end distortion is ar rom the theoretical bounds. In [3], or the case o Gaussian channel, a joint sourcechannel coding scheme is introduced which has a robust perormance over the entire range o. Unlike the previous methods, the method asymptotically achieves the optimal scaling o the signal-to-distortion-ratio SDR without being aected by the threshold eect. In this paper we generalize the idea o this scheme or the case o MIMO channels to construct semi-robust joint source-channel codes or point-

2 to-point MIMO systems. II. SYSTEM MODEL We consider a communication system where an analog source o Gaussian independent samples with variance σs is to be transmitted over a, N r block ading MIMO channel where and N r are the number o transmit and receive antennas respectively. Every m samples o the source stacked in a vector x s are transmitted over n channel uses. The channel matrix H is assumed ixed during this period and changes independently or the next n channel uses. We call the ratio η = n m the expansion/contraction actor o the system. In a general setting, the communication strategy consists o source/channel coding and source/channel decoding. As a result o source channel coding, x s is mapped into a n space-time matrix X which in turn is received at the receiver side as an N r n matrix Y where Y = HX + W in which is the average signal to noise ratio at each receive antenna, and W is the additive noise matrix at the receiver whose entries are taken to be CN 0, 1. At the receiver side, the source/channel decoder yields an estimation o x s out o Y as x s. For a speciic channel realization H, the distortion measure is DH = E xs { x s x s H}. 1 For any speciic strategy, we deine the idelity event as A = {H : DH > } and we call the idelity exponent. For speciic values o η, and N r, we deine: d = Pr{A} lim log log We call d the diversity, and denote its maximum over all possible source-channel coding schemes as d. III. DIVERSITY-FIDELITY TRADEOFF In this section, we characterize the tradeo between the idelity exponent and the optimum diversity d. A. Upper bound on d We recall rom [4] that i we denote the eigenvalues o HH H log λi by λ i, setting α i = log, we have1 : p α. = i 1+ N r α i. 3 On the other hand, in a system o tandem coder, i.e., separate source coder and channel coder, we have: Pr{error H} n[ 1 α i + r] where R = r log is the transmission rate over the channel, i.e., R = log M n, and M is the number o quantized points in the output o the source coder. 4 1 In this paper, we use a. = b to denote that a and b are asymptotically equivalent. To get an upper bound on d, we consider the case o delay unlimited where m, n and n m = η is a constant. Also, we assume that the transmitter has perect knowledge o the channel matrix H, and thereore, one may talk about the capacity o this MIMO channel which is given by [4]: R = sup log deti + Σ:trΣ m m HΣHH 5 log deti + HH H. 6 We know that the source rate is R s = ηr. On the other hand, the Distortion-Rate unction o the source is DR s ; H = e Rs. Thereore: DR s ; H 1 deti + HH H. 7 η Let us denote the -idelity event as A here. Thus, we obtain: Pr{A } = Pr{DR s ; H > } 8 1 Pr{ deti + HH H η > } 9 which can be written as: i=n Pr{A } Pr{ 1 + λ i < η }. = Pr{ i i=1 As a result, based on 3 and 10, we get: Pr{A } α 1 α i + < }. 10 η i 1+ Nt N r α i d α 11 where = { α : i 1 α i + < η }. Based on [4], we have: α i 1+ Nt N r α i d α = d ub 1 where d ub = min α i 1+ Nt N r α i. According to the results in [4], or integer values o η, this can be calculated as: d ub = η N r. 13 η Clearly, i we let Pr{A }. = d, we have d d ub. Consequently, d ub is an upper bound on d.

3 B. Achievability o the optimal tradeo We show that or any given bandwidth expansion η = n m, the upper bound or d is tight and it can be achieved by a amily o delay-limited digital space-time codes {C k }. We map mnt samples o the source to an nnt -dimensional vector s = [s 1 s nn t ] T, and construct the MIMO codeword by using n consecutive space-time matrices {C k }, chosen rom a DMT-achieving amily o digital codes or example, space-time code with the non-vanishing determinant property, e.g. Perect Codes [5] [6] [7]. For the modulating signal x s = x 1,, x mn t, consider x i + 1 = 0.b i,1 b i, b i,3. Let b = b i+j 1mNt i,j. For the code C k, we construct s 1, s,, s N where N = nnt as s 1 = 0.b 1 b N+1 b N+1 b k 1N+1, 14 s = 0.b b N+ b N+ b k 1N+, 15 s N = 0.b N b N b 3N b kn, 16 Theorem 1 For the proposed amily o codes, i we use C k or k 1 > η k, or integer values o d = η N r η η. Proo: Because the amily o codes C k are obtained rom a lattice with non-vanishing determinant property, using ML decoding, they achieve the optimal diversity-multiplexing tradeo. Thus, or this amily o codes, P error r. = Nt rnr r 17 R log. or integer values o r, the normalized rate r = In this scheme, code C k has rate R k = k and is used or k 1 > η k. Thereore, r =. η, hence, P error r. = η N r η. 18 When there is no error in decoding C k, the kn = knnt bits b 1,, b kn are decoded correctly, hence the irst k n m = knη bits o x i can be reconstructed without error or 1 i m, hence D =. knη. =. Thereore, { } Pr D > P error r. = η N r η 19 = d η N r. 0 η d =N r =, η= =4, η=1 =4, η=1.5 =4, η= = N r =, η=1 = N r =4, η= idelity exponent Fig. 1. Diversity-Fidelity Tradeo or dierent numbers o antennas and dierent bandwidth expansion actors IV. ROBUST SPACE-TIME SOURCE-CHANNEL CODING In the scheme presented in the previous section, or the achievability o the optimal tradeo, dierent codes should be used or dierent values o distortion D and the channel. In this section, we consider the problem o designing robust or semi-robust codes which can work or the entire range o. The simplest non-trivial case is the case two transmit antennas and one receive antenna. In this special case, using Alamouti scheme at the transmitter and simple Alamouti decoding which is equivalent to zero orcing can easily separate the encoded symbols. Indeed, or the case o η = 1 and a Gaussian source, the Alamouti scheme is optimum, not only in SDR scaling, but in achieving the exact optimal perormance. Theorem Consider a 1 MIMO system and the bandwidth expansion η = 1 a Gaussian i.i.d source x i. I we send the Alamouti scheme to send consecutive source samples x 1, x over two consecutive channel uses, and use the Alamouti detection zero orcing at the transmitter, or any noise variance σ z, the resulting distortion is the minimum possible distortion. Proo: Let H = [h 1 h ] be the channel matrix. When we do not have the channel state inormation CSI at the transmitter, the best power allocation scheme at the transmitter is to assign equal power to the transmit antennas. In this case, or any realization o the channel, the capacity o the 1 channel is equal to [8] CH = 1 log det I + HI H H = log 1 + h1 + h. 1

4 I the source has i.i.d. Gaussian distribution, according to the rate-distortion unction o the Gaussian source, D min = h 1 + h. On the other hand, i we send the analog source samples directly by using the Alamouti scheme, Y = = [y 1 y ] = [ x1 x HX + W = [h 1 h ] x x 1 [ h1 h [x 1 x ] h h 1 ] + [w 1 w ] 3 ] + [w 1 w ] 4 By using the Alamouti decoding which is equivalent to zero orcing, the received signal is equal to: [ x 1 x ] = [x 1 x ] + = D Alamouti = [ [w 1 w] h1 h h h 1 ] 1 5 h 1 + h = D min. 6 For other cases o η 1 using the bandwidth expansion mappings in [3] can achieve the optimal tradeo. However, or other conigurations o MIMO system, no optimum symbol-separating scheme such as the Alamouti scheme which can work with analog symbols is known. Here we propose a semi-robust coding scheme that achieves a diversity order, larger than 1 better than the trivial linear SISO code or MMSE-decoded V-BLAST. We map mk samples o the source to an nk -dimensional vector s = [s 1 s nknt ] T or an inter k, 1 k, using a robust point-to-point joint source-channel code, and construct the MIMO codeword by setting C = L knt I n s where L N t k is the generator o a lattice space-time code with non-vanishing determinant property and mapping it to the entries o n consecutive space-time matrices. To construct the point-to-point joint source-channel code or the bandwidth expansion o η = nn t mk, we generalize the codes presented in the [9] and [3]: For the modulating signal x s = x 1,, x mknt, consider x i + 1 = 0.b i,1 b i, b i,3. Let b i+j 1mk = b i,j or 1 i < mk. We construct s 1, s,, s N where N = nk as s 1 = 0.b 1 0b NN+1 +1 b NN+1 + s = 0.b b 3 0b N+1N+ +1 b N+1N+ + s N = 0.b NN 1 +1 b b NN 1 NN The resulting code C can be seen as successively reining space-time code with ininite layers o reinement. At the receiver, we use simple linear decoding to separate s 1, s,, s N and decode them separately. To analyze the Diversity-idelity tradeo o this single mapping, we irst lower bound the probability that the received eective noise ater zero orcing becomes large. The received signal over n channel uses and N r receive antennas can be represented as a nn r 1 vector y = H I nnt L knt I n s+ w nntn r where w nntn r is the vector representation or the Gaussian noise in n channel uses. At the receiver, we can multiply the pseudoinverse o the matrix H I nnt L knt I n to ind an estimate o s 1, s,, s N. Lemma 1 Consider L N t k as generator o an lattice code with dimension k with non-vanishing determinant property. The probability that the amplitude o eective received noise is larger ε is upper bounded by { Pr w 1 } < cε k+1n r k+1 8 ε or some constant c. Sketch o the proo: Ater multiplying by the pseudoinverse o the equivalent channel matrix, + w = H I nnt L N t k I n w. The probability that w 1 ε can be bounded by the summation o the probability the Frobenius norm o H I Nt L N t k is greater than c 1 ε or a constant c 1 and the probability that w is larger than 1 c 1. we can choose c 1 such that the irst term o the summation becomes dominant. Now, by using the non-vanishing-determinant property o L and the act that the entries o H have independent Gaussian distributions, we can bound the second term by c ε k+1n r k+1, or some constant c. Theorem 3 I we use the proposed scheme, consisting o the SISO source-channel code 7 and a space-time code o rate k constructed by a lattice generator L N t k or an integer k, 1 k min {, N r } with non-vanishing determinant property, and use zero orcing at the receiver, the diversity-idelity tradeo o the scheme is lower bounded by d k + 1N r k + 1 k + 1N r k + 1 k η. 9 Sketch o the proo: Because the bits o the source symbols are almost uniormly distributed among s 1, s,, s N, to have D >, at the decoder, at least one o the modulated symbols s i must see a noise larger than η

5 and or that, we should have w η = 1 η. Now, using the result o lemma 1 and the deinition o d, d k + 1N r k + 1 k + 1N r k + 1 k η. V. CONCLUSIONS Similar to the diversity-multiplexing tradeo which is well known or the high- asymptotic perormance o space-time codes, diversity-idelity tradeo can be used as a benchmark or evaluating various MIMO source-channel codes. This tradeo can be achieved by a simple modiication o the well-known DMT-achieving codes. However, though the optimum tradeo can be achieved or dierent numbers o antennas and arbitrary bandwidth expansion actors, this well-known approach is not robust design o the codes depend on the o the channel. In this paper, a semi-robust MIMO source-channel scheme is proposed that achieves nontrivial diversity and idelity exponents or dierent numbers o antennas. However, the semi-robust scheme that is presented in this paper only achieves the optimum tradeo or N r = 1 or = 1 and in the general case o N r > 1 and > 1, its tradeo curve is suboptimum. Thereore, the problem o achieving the optimum diversity-idelity tradeo by practical and robust codes remains open. REFERENCES [1] D. Gunduz and E. Erkip, Distortion exponents o mimo ading channels, in IEEE Inormation Theory Workshop, p , July 006. [] G. Caire and K. Narayanan, On the distortion exponent o hybrid digital analog space time coding, IEEE Trans. Ino. Theory, vol. 53, pp , August 007. [3] M. Taherzadeh and A. K. Khandani, Robust joint source-channel coding or delay-limited applications, in ISIT 007, NiceFrance, June 007. [4] L. Zheng and D. Tse, Diversity and multiplexing: a undamental tradeo in multiple-antenna channels, IEEE Trans. Ino. Theory, pp , May 003. [5] P. Elia, K. R. Kumar, S. A. Pawar, and P. V. K. H.-F. Lu, Explicit spacetime codes achieving the diversitymultiplexing gain tradeo, IEEE Trans. Ino. Theory, vol. 5, pp , Sep [6] F. Oggier, G. Rekaya, J.-C. Beliore, and E. Viterbo, Perect spacetime block codes, IEEE Trans. Ino. Theory, vol. 5, pp , Sep [7] L. Hsiao-Feng and P. V. Kumar, A uniied construction o space-time codes with optimal rate-diversity tradeo, IEEE Trans. Ino. Theory, vol. 51, pp , May 005. [8] I. E. Telatar, Capacity o multi-antenna gaussian channels, Europ. Trans. Telecommun., pp , Nov [9] M. Taherzadeh and A. K. Khandani, Analog coding or delay-limited applications, in Proceedings o 41st Conerence on Inormation Sciences and Systems, March 007.