On the Optimality of Single-Carrier Transmission in Large-Scale Antenna Systems

Transcription

1 On the Optimality o Single-Carrier Transmission in Large-Scale Antenna Systems Antonios Pitarokoilis, Sai han Mohammed and Erik G. Larsson Linköping University Post Print N.B.: When citing this work, cite the original article. Original Publication: Antonios Pitarokoilis, Sai han Mohammed and Erik G. Larsson, On the Optimality o Single-Carrier Transmission in Large-Scale Antenna Systems, 2012, IEEE Wireless Communications Letters, 1), 4, Copyright: 2012 IEEE. Personal use o this material is permitted. However, permission to reprint/republish this material or advertising or promotional purposes or or creating new collective works or resale or redistribution to servers or lists, or to reuse any copyrighted component o this work in other works must be obtained rom the IEEE. Postprint available at: Linköping University Electronic Press

2 1 On the Optimality o Single-Carrier Transmission in Large-Scale Antenna Systems Antonios Pitarokoilis, Sai han Mohammed, Erik G. Larsson Abstract A single carrier transmission scheme is presented or the requency selective multi-user MU) multiple-input singleoutput MISO) Gaussian Broadcast Channel GBC) with a base station BS) having M antennas and single antenna users. The proposed transmission scheme has low complexity and or M it is shown to achieve near optimal sum-rate perormance at low transmit power to receiver noise power ratio. Additionally, the proposed transmission scheme results in an equalization-ree receiver and does not require any MU resource allocation and associated control signaling overhead. Also, the sum-rate achieved by the proposed transmission scheme is shown to be independent o the channel power delay proile PDP). In terms o power eiciency, the proposed transmission scheme also exhibits an OM) array power gain. Simulations are used to conirm analytical observations. Index Terms Single-Carrier Transmission, Large MIMO. I. INTRODUCTION Multiple-input multiple-output MIMO) systems have attracted signiicant research interest during the last decade due to various advantages they promise, both in single user [1] and multiuser channels [2]. It has been recently shown that the employment o an excess o antennas at the BS very large MIMO) oers unprecedented array and multiplexing gains both in the uplink and in the downlink [3], [4]. The array gain oered by very large MIMO systems allows or power savings that scale as 1/M and 1/ M with perect and imperect channel state inormation CSI) respectively, where M is the number o BS antennas [5]. The multiplexing gains oered by very large MIMO allows tens o users to be allocated the entire system bandwidth simultaneously. This eliminates to a large extent the need or resource allocation and the associated control signaling overhead. Since each user communicates over the whole system bandwidth, even low per user spectral eiciencies can result in very high per user throughput. In a MU-MISO GBC with users and M very large MIMO), a low per user spectral eiciency implies an operating regime where the ratio o the total transmit power to the receiver additive noise power is small. Since MU intererence at each receiver is proportional to the total transmit power, the additive noise dominates over MU intererence and thereore even suboptimal precoding algorithms like beamorming with the conjugate transpose o the channel gain matrix) have near optimal perormance. The authors are with the Department o Electrical Engineering ISY), Linköping University, Linköping, Sweden {antonispit,sai,erik.larsson}@isy.liu.se). This work was supported by the Swedish Foundation or Strategic Research SSF) and ELLIIT. E. G. Larsson is a Royal Swedish Academy o Sciences VA) Research Fellow supported by a grant rom the nut and Alice Wallenberg Foundation. Previous results or very large MIMO systems have only considered requency lat channels [3], [4], [5]. In this paper we consider a MU-MISO requency selective GBC with M. For this channel OFDM OFDMA) is an attractive transmission scheme as it acilitates scheduling in the requency domain and simpliies receiver equalization. However, there is a substantial price to pay or this. OFDM comes at a loss in spectral and power eiciency owing to the insertion o cyclic preix. Moreover, the signals resulting rom OFDM modulation have a very large peak-to-average ratio, requiring the RF power ampliiers to work with a large power backo and in an operating regime where they have low eiciency. For this reason, single-carrier or single-carrier-like modulation schemes like DFT-precoded OFDM are oten used when there are stringent requirements on power eiciency o the RF ampliiers. Single-carrier signals have a much lower peak-toaverage ratio and can be shaped to have constant envelope even in multiuser MIMO systems [6]. The contributions made in this paper are summarized as ollows. 1) We irstly propose a low complexity single carrier transmission scheme or the requency selective MU-MISO GBC. 2) At low total transmit power to receiver noise ratio, the proposed transmission scheme is shown to eectively suppress intersymbol intererence ISI) and MU intererence at each receiver, thereby achieving near optimal sum-rate perormance. 3) Additionally, the proposed scheme does not require any receiver equalization. Also, its simplicity allows or separate, decentralized computation at each BS antenna. 4) An achievable inormation sum-rate is derived or the proposed scheme. This sum-rate is urther shown to be invariant o the channel PDP. 5) In terms o power eiciency, the proposed scheme is shown to exhibit an array power gain proportional to the number o BS antennas. II. SYSTEM MODEL A requency selective MU-MISO downlink channel is considered, with M BS antennas and single antenna users. The channel between the m-th transmit antenna and the k- th user is modeled as a inite impulse response FIR) ilter with L taps. The l-th channel tap is given by d l [k]h l [m,k], where h l [m,k] and d l[k] model the ast and slow varying components, respectively. In this paper we assume a model where h l [m,k] is ixed during the transmission o a block o N symbols and varies independently rom one block to another. However, the slowly varying component i.e. d l [k]) is assumed to be ixed throughout the entire communication. We urther assume h l [m,k] to be i.i.d. CN0,1) distributed. d l [k] 0, l = 0,..., models the PDP o the requency

3 2 selective channel or the k-th user. 1 Let x m [i] be the symbol transmitted rom transmit antenna m at time i. The received signal at user k at time i is then given by y k [i] = m=1 M dl [k]h l[m,k]x m[i l]+n k [i], 1) where n k [i] is the CN0,1) distributed AWGN at the k-th receiver at time i. Deine y[i] = [y 1 [i],...,y [i]] T C to be the vector o received user symbols at time i. Similarly, let x[i] = [x 1 [i],...,x M [i]] T C M be the transmitted vector at time i. Let n[i] = [n 1 [i],...,n [i]] T, with independent components. The received signal vector at time i is given by y[i] = D1/2 l H H l x[i l] + n[i], where D l = diag{d l [1],...,d l []}, and H l C M is a matrix whose m,k)-th element is h l [m,k]. Also the channel PDP or each user is normalized such that d l [k] = 1, k = 1,...,. 2) The BS is assumed to have ull CSI, whereas the users have knowledge o the channel statistics only. 2 Let s k [i] denote the inormation symbol to be communicated to the k-th user at time i. The inormation symbol vector s[i] = [s 1 [i],...,s [i]] T is considered to have i.i.d. CN0,1) components, i.e. E [ s[i]s H [i+j] ] = I δ j, E [ s[i]s T [i+j] ] = 0. In this paper we propose a precoding scheme, where the transmitted vector at time i is given by x[i] = M H l D 1/2 l s[i+l], 3) where ρ = E [ x[i] 2] is the long-term average total power radiated by the BS antennas. In the ollowing, we derive an achievable sum-rate or the proposed precoder in 3). III. ACHIEVABLE SUM-RATE The bounding technique o [7], [8] is used here to obtain an achievable rate. In the ollowing, a set o achievable rates is presented. For notational brevity we deine v l [k] = H l D 1/2 l e k, where e k is the all-zero vector except or the k- th component which is equal to 1. Using 1) and 3) the signal received by user k at time i is given by y k [i] = M where 3 n k[i] = [ E v H l [k]v l [k]] ) s k [i] } {{ } Desired Signal Term + n k[i], Eective Noise Term [ v H l [k]v l [k] E v H l [k]v l [k]] ) s k [i] M Additional Intererence Term IF) 1 PDP determines the distribution o the received power across dierent channel taps. 2 In a time division duplex TDD) system, CSI at the BS can be acquired through uplink training and exploiting the uplink-downlink channel reciprocity. 3 Following [7], [8], we have split the coeicient o the term M v l[k] H v l [k]s k [i] into a sum o its mean value which is known to the receiver) and the deviation around its mean. 4) + M a=1 L a 0 min+a,) l=maxa,0) v H l [k]v l a [k]s k [i a] } {{ } Intersymbol Intererence ISI) + M q=1 a=1 L q k min+a,) l=maxa,0) v H l [k]v l a [q]s q[i a] Multiuser Intererence MUI) +n k [i] 5) AWGN is the eective noise term. This term includes i) the IF term which represents the variation o the desired signal around its mean, ii) the ISI term between the current symbol o user k, i.e. s k [i], and the symbols intended to the same user at other time instances i.e. s k [i+j], j 0), iii) the MUI term due to the inormation symbols intended or other users and, iv) the AWGN term. In the proposed precoder, each user s codeword is long enough such that it spans across multiple coherence intervals. With long codewords, the eective variance o n k [i] is no longer dependent on a particular channel realization but only depends on the channel statistics. From this it ollows that the desired signal s k [i] is uncorrelated with the eective noise n k [i], i.e. E[s k[i]n k [i]] = 0, where the expectation is taken over the channel realizations, the inormation symbols and additive noise. Thereore, with long codewords the channel is eectively an additive noise channel with the noise n k [i] being non-gaussian and uncorrelated to the inormation symbol s k [i]. Further, the user has perect knowledge o its channel statistic and thereore it knows the scaling actor E[ v H l [k]v l [k] ]. Hence, an achievable inormation rate or the channel in 4) is given by considering the worst case uncorrelated additive noise having the same variance as n k [i]. Given that the data signal s[i] is Gaussian, the worst uncorrelated additive noise is circularly symmetric Gaussian distributed with the same variance as n k [i]. Thereore, the ollowing inormation rate is achievable or the k-th user R k = log 2 1+Sk /Var n k[i] )) 6) where S k = E sk [i][ M E[ v H l [k]v l [k] ] ] s k [i] 2 is the average power [ o the desired signal term in 4) and Varn k [i]) = E n k [i] E[n k [i]] 2]. Proposition 1: The variance o n k [i] is invariant o any PDP that satisies 2), and is given by Var n k[i] ) = ρ +1. 7) Proo: Using 5), the eective noise variance is given by Var n k[i] ) = ρ d l a [k]d l [q]+d l [k]d l a [q]) + ρ q=1 a=1 l=a d l [k]d l [q]+1, 8) q=1 where the expectation is taken over the statistics o H l, l = {0,...,}, s[i+a], a = {1 L,...,} and n k [i]. Deine R L such that [] i,j = d j 1 [i] and let 1

4 3 {1} L L denote the matrix with all entries equal to one. Then, 8) can be expressed as Var n k[i] ) = ρ e T k e l a+1 e T l+1 q=1 a=1 l=a + e l+1 e T l a+1 = ρ ) T e q + ρ e T k T e q +1 q=1 e T k1 T e q +1. 9) q=1 From 2) it ollows that e T k1 = [1... 1]. Using this act in 9) completes the proo. It is apparent rom 7) that the variance o the eective noise consists o the variance o the white noise term which is 1) and the variance o the sum o intererence terms which is ρ ). In the ollowing we provide an explanation as to why the variance o the eective noise term is invariant o the PDP. Note that the precoder in 3) is like a matching pre-ilter whose impulse response is a time reversed and complex-conjugated image o the channel impulse response CIR). Due to this special structure o the proposed precoder, n k [i] is composed o terms which consist o all non-zero auto-correlation lags o the CIR or the k-th user ISI term in 5)), as well as all crosscorrelation lags between the CIR o user k and the CIR o the remaining 1) users MUI term in 5)). The eective MUI in y k [i] rom the symbols intended or the q-th user, depends only upon the total power in all channel correlation lags between the CIR s o the k-th and the q-th user. Due to the same channel and inormation symbol statistics or all users, the eective MUI in y k [i] rom each o the remaining 1) users is identical, and is independent o the individual PDPs the total power in the cross-correlation lags depends only upon the total power in the CIR or each user, which is independent o k due to 2)). Further, the useul signal term in y k [i] is proportional to the zero-lag auto-correlation o the CIR or the k- th user. This zero-lag auto-correlation i.e. maximum gain combining o the lags) is proportional to the total channel power gain combining all taps) rom the M BS antennas to the k-th user and is thereore OM). The average power [ o the desired signal term in 4) is given by E sk [i] M E[ v H l [k]v l [k] ] ] s k [i] 2 = ρ M/. Using this act and 7) in 6), the achievable rate R k or user k is given by R k = log 2 1+ρ M/ρ +)). The achievable sum-rate is thereore given by R sumρ,m,) = R k = log 2 1+ ρ ) M. 10) ρ + k=1 For the sake o comparison, we also consider a co-operative upper bound on the sum-capacity o the requency selective GBC. 4 Essentially, we get an upper bound by considering the users to be co-operative, which reduces the MU channel to a single user MIMO channel, with perect CSI at both the transmitter and the receiver. We urther consider transmission 4 The sum-capacity o the MIMO GBC is known. However, or the results reported in this paper, it suices to consider only the co-operative upper bound on the sum-capacity. in time with large blocks block size L), where in each block the last ew transmit vectors are zeros so as to avoid any inter-block intererence. The sum-capacity or this single user MIMO block channel is given by beamorming along the right singular vectors o the eective channel matrix, thus transorming the channel into a set o parallel channels. Gaussian symbols are communicated over the parallel channels and power allocation is given by the waterilling scheme. With i.i.d complex normal entries in H l, it is clear that or ixed, HH l H l M I as M. Thereore or M, the singular values o H l are all roughly equal to M i.e., the power gain or each parallel channel is M). With a uniorm power allocation o ρ / across parallel channels, the co-operative upper bound on the ergodic sum-capacity o the GBC is given by C coopρ,m,) log 2 1+ρ M/). 11) We conclude our analysis with two propositions on the nearoptimality and the array gain o the proposed precoder. Proposition 2: When ρ 1 and M, R sum C coop and the proposed precoder is near-optimal. Proo: Observe that when ρ 1, the eective noise variance, Varn k ) = ρ +1 1 essentially the additive white noise dominates over the intererence terms in 5)). It ollows that, ρ + and thereore the expressions in 10) and 11) are approximately equal. Proposition 3: The proposed precoder exhibits an OM) array power gain. Proo: For the proposed precoder, using 10) the minimum transmit power ρ required to achieve a ixed desired sum-rate R sum with users and M BS antennas is given 1) by ρ M) = 2Rsum/ M+2 Rsum/ 1). Since lim M 1 ρ 1) M ρ M) = 1 > 0 rom [9] it ollows that the proposed 1+2 Rsum/ 1) precoder achieves an OM) array power gain. This implies that or a suiciently large M, ρ M)/ρ 1) 1/M i.e. the total transmitted power can be reduced linearly by increasing the number o BS antennas). A similar analysis o the co-operative sum-capacity see 11)) reveals that the array power gain achieved by a sum-capacity achieving scheme is also OM). IV. SIMULATION RESULTS In the ollowing, representative simulation results are presented, where the perormance o proposed precoder is compared to the co-operative upper bound. Throughout the conducted simulations, the PDP is exponential with L = 4 e and d l [k] = θ k l 3, l = {0,...,3}, where θ i=0 e θ k i k = k 1)/5, k = {1,...,}. As proved in Proposition 1, the achievable sum-rate is invariant o the PDP. Hence, any other PDP which satisies 2), would also yield the same results. Proposition 2 is supported by Fig. 1, where the sum-rate is plotted as a unction o ρ, or M = 50 and = 10. The sum-rate perormance o the proposed precoder is given both by the theoretical expression 10) and via simulations. Similarly, the co-operative sum-capacity upper bound is calculated via simulations and by the approximation in 11). For

5 4 Fig. 1. Sum rate o the proposed precoder and the co-operative sum-capacity upper bound vs ρ, calculated or = 10 users, M = 50 BS antennas. Fig. 2. Minimum required transmit power to achieve a ixed per user inormation rate r = 1 bpcu as a unction o the number o BS antennas. ρ 1 0 db), as can be seen in Fig. 1, the perormance o the proposed precoder is similar to the upper bound, implying optimality. Note that as ρ increases, the intererence terms dominate over the white noise term in 5) and the eective noise variance is thereore ρ +1 ρ. Hence, as ρ the sum-rate o the proposed precoder saturates to the value log 2 1+M/) = bpcu. It can also be seen that the approximation to the sum-capacity upper bound is tight. The analytical result in Proposition 3 is supported by Fig. 2, where or a ixed number o users and a ixed per user rate o 1 bpcu, the minimum total transmit power required is plotted as a unction o the number o BS antennas. In Fig. 2 it is observed that the minimum transmit power required by the proposed precoder can be reduced by roughly 3dB with every doubling in the number o the BS antennas or suiciently large M). This implies the achievability o an OM) array power gain, as stated in Proposition 3. In Fig. 2 it is also observed that or suiciently large values o M the total transmit power required by the proposed precoder is roughly equal to the total transmit power required by a sum-capacity achieving scheme. Further, or the sake o comparison, consider a typical scenario, where OFDM is used. Let ρ OFDM denote the total transmit power or OFDM transmission. Under OFDM transmission with M, it can be shown that the per user ergodic inormation rate in i.i.d. Rayleigh ading channel) is given by r T u T u +T cp log 2 1+ρ OFDM M ), 12) where T cp is the duration o the cyclic preix and T u is the duration o the useul signal. 5 From 12) it ollows that, to achieve an ergodic per user inormation rate o r bpcu the minimum required total transmit power is given by ρ OFDM r) M 2 rt u+t cp)/t u 1 ). For a given desired per user ergodic inormation rate, the additional total transmit power required under OFDM transmission when compared to an optimal GBC sum-capacity achieving scheme is upper bounded by ρ OFDM r)/ρ coop r), where ρ coop r) = 2 r 1)/M is roughly equal to the required transmit power or the co-operative sumcapacity bound to be r bpcu see 11)). The additional transmit power required under OFDM transmission is thereore given by 2r1+Tcp/Tu) 1 2 r 1. Since 2r1+Tcp/Tu) 1 2 r 1 > 1 and the total transmit power required by the proposed precoder is roughly equal to that required by a sum-capacity achieving scheme M and ρ 1 see Proposition 2)), it can be concluded that the proposed precoder is more power eicient than OFDM transmission or large M/. As an example, or a typical IEEE a scenario with T cp = T u /4, a desired per user inormation rate r = 1 bpcu and M, this additional transmit power required under OFDM transmission when compared to the proposed precoder is 1.39 db. The minimum transmit power required under OFDM transmission is also plotted in Fig. 2, where it can be seen that or M > 4 the proposed precoding scheme is more power eicient than OFDM transmission and requires no equalization at the receiver Note that equalization in OFDM receivers requires FFT processing). REFERENCES [1] G. Foschini and M. Gans, On limits o wireless communications in a ading environment when using multiple antennas, Wireless Pers. Comm., vol. 6, pp , Mar [2] D. Gesbert and et al., Shiting the MIMO paradigm, IEEE Sig. Proc. Mag., vol. 24, pp , Sep [3] F. Rusek, D. Persson, B.. Lau, E. G. Larsson, T. L. Marzetta, O. Edors, and F. Tuvesson, Scaling up MIMO: Opportunities and challenges with very large arrays, IEEE Sig. Proc. Mag., to appear, vol. arxiv: v1, [4] T. L. Marzetta, Noncooperative cellular wireless with unlimited numbers o base station antennas, IEEE Trans. Wireless Comm., vol. 9, pp , Nov [5] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, Energy and spectral eiciency o very large multiuser MIMO systems, submitted to the IEEE Trans. Comm., vol. arxiv: , [6] S.. Mohammed and E. G. Larsson, Per-antenna constant envelope precoding or large multi-user MIMO systems, submitted to the IEEE Trans. Comm., vol. arxiv: v1. [7] B. Hassibi and B. M. Hochwald, How much training is needed in multiple-antenna wireless links?, IEEE Trans. In. Theory, vol. 49, pp , Apr [8] T. L. Marzetta, How much training is required or multiuser MIMO?, in Proc. 40th Asilomar Con. ACSSC 06, pp , Nov [9] D. Tse and P. Viswanath, Fundamentals o Wireless Communications. Cambridge, U: Cambridge Univ. Press, Note that in practice, modern wireless standards employ X > 1 OFDM symbols per coherence time interval. Each OFDM symbol consists o N u channel uses or data transmission and N cp channel uses or the cyclic preix. This means that XN cp channel uses in each coherence time interval are used or non-data transmission. In contrast, in the proposed precoding scheme only N cp channel uses per coherence time interval are used or nondata transmission. These N cp channel uses are used or zero-padding at the beginning o each coherence interval.) Note that in practical wireless standards X 10, which implies that the proposed precoder makes better use o available channel bandwidth.