Capacity Analysis of MIMO Systems with Unknown Channel State Information

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1 Capacity Analysis of MIMO Systems with Unknown Channel State Information Jun Zheng an Bhaskar D. Rao Dept. of Electrical an Computer Engineering University of California at San Diego Abstract We consier a mobile wireless communication system compose of M transmit an N receive antennas operating in a faing environment. Assuming channel state information is unavailable to the transmitter an the receiver, a capacity upper boun of the unknown MIMO channel uner the assumption of restricte input istributions is provie. By analyzing the propose capacity upper bouns, we reenforce the avantages of using an orthogonal pilot structure which minimizes the mean square estimation error, in that it also maximizes the capacity upper bouns. Interestingly, the capacity upper boun is shown to be a monotonically ecreasing function with respect to the number of pilot symbols T τ. Numerical evaluations of the capacity upper boun further emonstrate that the capacity gain is insignificant when the number of pilot symbols T τ ecreases below M, suggesting an optimum training uration of M time slots. I. INTRODUCTION Communication systems using multiple antennas at both the transmitter an the receiver has recently receive increase attention ue to its capability of proviing great capacity increases in a wireless faing environment, as reporte by Telatar an Foschini. However, the capacity analysis provie is base on the unerlying assumption that the faing channel coefficients between each transmit an receive antenna pairs are perfectly known at the receiver without any cost, which is not a reasonable assumption for most practical communication systems especially when the faing channel is changing fast. Marzetta an Hochwal provie in the capacity analysis of an unknown MIMO channel with a finite coherent time interval T. They showe that the capacity is achieve when the transmitte signal matrix is equal to the prouct of an isotropically istribute unitary matrix times a ranom iagonal matrix with real, nonnegative iagonal elements. Furthermore, Zheng an Tse compute the asymptotic capacity of this channel at high signal to noise ratios. However, in practice not only fining the optimal input istribution is an involve task an requires numerical optimizations, but also there are no known space-time coes that can approach this capacity. Hence, this paper takes a more pragmatic approach an focuses on systems that are able to take avantages of the existing channel estimation algorithms an This research was supporte by CoRe grant No. -9 sponsore by Ericsson. the powerful forwar error correction coing techniques, like turbo or LDPC coes. Hassibi an Hochwal propose in a channel moel that separates one coherent block into two phases: training an ata. Base on the two phase channel moel, as well as by applying the MMSE channel estimation algorithm, they provie a capacity lower boun for the unknown MIMO channel, an prove that the optimal number of training symbols is equal to T τ = M when the training an ata powers are allowe to vary. The capacity lower boun provie in assumes that the channel estimation LMMSE is obtaine by only using the training symbols, thereby not making use of the channel information containe in the receive ata symbols. Therefore, the lower boun is pessimistic an unable to represent the true capacity or the maximum achievable information rate accurately. Without assuming any specific channel estimation algorithm, we propose in this article a capacity upper boun for the unknown MIMO channel with a two-phase transmitte signal structure given in. By analyzing the propose capacity upper bouns with respect to ifferent system parameters, we show that the orthogonal pilots structure not only minimizes the mean square estimation error, but also maximizes the capacity upper bouns. Furthermore, we also prove that the channel capacity upper boun is a monotonically ecreasing function with respect to the number of pilot symbols T τ, but with insignificant capacity increment when T τ ecreases below the number of transmit antennas M, which is verifie by numerical evaluation. II. SYSTEM MODEL We consier in this article a MIMO system with M transmitter antennas an N receive antennas, signaling through a frequency flat faing channel with i.i. channel coefficient between each transmit an receive antenna pairs. It is assume that the faing coefficient H remains static within a coherent time interval of T symbol perios, an varies inepenently from one coherent time block to another; an each element of H is complex Gaussian istribute. Hence, the signaling moel can be written as Y = X H + w, where Y is a T N receive complex signal matrix, X is a T M transmitte complex signal matrix, H is a M N complex channel matrix, an w is a T N matrix of aitive Gaussian noise. Both matrix H an w have zero mean unit variance inepenent complex Gaussian entries. We also assume that

2 the entries of the transmitte signal matrix X have, on average, the following power constraint, T E tr X H X = ρ, where ρ is the average signal to noise ratio at each receive antenna. The transmitte signal matrix X is further separate into two submatrixes: training followe by ata, which is represente as ρτ /M S X = τ ρ /M, X where S τ is the fixe pilot symbols an X is the information bearing ata symbols, whose structures are given by S τ = s H,, s H Tτ H, Sτ C Tτ M, X = x H,, x H T H, X C T M. Conservation of time an energy leas to the following constraints, tr S H τ S τ = MTτ, E X tr X H X = MT, T = T τ + T, ρt = ρ τ T τ + ρ T. III. CAPACITY ANALYSIS A. Restricte Capacity Upper Boun From the capacity analysis result provie in, we know that not only fining the capacity achieving input istribution is an involve task an requires numerical optimization, but also there are no known viable space-time coes that can approach this capacity. In this section we restrict our attention to a conventional MIMO system having an input signal structure, which is escribe in Section II. It is further assume that the input ata vectors x i follow i.i.. Gaussian istributions, i.e. E x H i x j = δ i,j I M. Although input istribution is not optimize to achieve the maximum mutual information rate, it is a reasonable assumption of a communication system with no channel state information available at the transmitter. Therefore, uner this restricte input istribution assumption, we have the following MIMO capacity upper boun. All the proofs provie are brief ue to length restrictions. Please refer to for more etails. Proposition Mutual information between X an Y are upper boune by, I X; Y IM C = N log + ρ τ + log +ρ IM E X log + ρ τ + ρ M XH X, 7 where the expectation E X is taken with respect to ata X. Proof: First, conitione on any input ata sequences X or X, vec Y is a Gaussian istribute vector of zero mean an variance Σ Y X = Cov vec Y X = I N XX H + I T. 8 Taking expectation of 8 with respect to X, the covariance matrix of vec Y is obtaine as, Σ Y = Cov vec Y = I N Σ + I T, 9 where Σ = ρτ M S τ S H τ ρ I. Due to the fact that Gaussian istribution has the maximum entropy among any vector istributions with the same covariance matrix, entropy hy can be upper boune by h Y log πe NT Σ Y. Therefore, we have the following capacity upper boun I X; Y = h Y h Y X a log πe NT Σ Y E X log πe NT Σ Y X, where the secon term of inequality a is from a irect expansion of the conitional entropy accoring to the efinition. Substituting 8 an 9 into, capacity upper boun 7 can be obtaine in a straightforwar manner. Since the receive signal Y can be viewe as a weighte sum of Gaussian ranom vectors, its istribution is close the Gaussian as long as it contains a large number of inepenent ranom variables accoring to the central limit theorem. Hence, the upper boun is tight an quite likely to be even less than the true unknown MIMO channel capacity provie in. Therefore, maximizing the capacity upper boun is a reasonable approach an will not make the boun become loose ue to the fact that both the capacity upper boun as well as the mutual information I X; Y in 7 increases through the optimization with respect to ifferent system parameters. B. Pilot Structure Optimizations The most commonly use pilots have an orthogonal structure. They are optimal in a sense that they minimize the mean square channel estimation error as well as achieve the Cramer- Rao lower boun. As a straightforwar extension of the single input single output system, the covariance matrix of the MMSE estimation error H = H Ĥ for the unknown MIMO channel is given by C H, H = Cov vec H, vec H = I N I M + ρ τ. It is obvious that the average mean square error of the channel estimation tr C H, H = N tr IM + ρ τ,

3 is minimize when the non-zero eigenvalues of S H τ S τ are all equal. Therefore, the following orthogonal pilot structure, represente as where U is any unitary matrix, an a follows from the fact that X U has the same istribution as X. Further ue to the fact that Σ Q = Σ U H QU, S H τ S τ = T τ I M, T τ M, S τ S H τ = M I Tτ, T τ < M, minimizes the MIMO MMSE channel mean square estimation error. In orer to obtain the optimal pilot structure with respect to the capacity upper boun C, we utilize the following concavity property. Proposition The capacity upper boun obtaine in Proposition is concave with respect to matrix Q = S H τ S τ, i.e., λ C Q + λ C Q C λ Q + λ Q, where λ,. Proof: The Hessian of the capacity upper boun is given by C vec Q = Nρ τ M ln E X where Σ an Σ are given by Σ = I M + ρ τ M Q, Σ Σ Σ Σ, 7 Σ = I M + ρ τ M Q+ ρ M XH X. 8 It can be shown that the Hessian matrix is negative semiefinite, an hence the capacity upper boun is concave with respect to Q. As a irect result of Proposition, we have the following optimal pilot structure. Proposition The optimal pilot structure, which maximizes the capacity upper boun 7, satisfies the following orthogonal conitions Q = S H MT τ IminTτ,M τ S τ =, 9 mint τ, M which is equivalent to. Proof: First, substituting 8 into 7, the capacity upper boun can be represente as C = N T log + ρ + log Σ E X log Σ. The following equality is true E X log Σ Q = E X log U H Σ U IM = E X log + ρ τ M UH Q U + ρ M X U H X U a IM = E X log + ρ τ M UH Q U + ρ M XH X = E X log Σ U H QU, the capacity upper boun is hence invariant uner the following transformation CQ = CU H QU. If the unitary matrix U is set to be compose of the eigenvectors of Q, then accoring to we only nee to focus our attention on the case where Q is a iagonal matrix. Furthermore, it is also true that any permutations on the non-zero iagonal elements of Q will not change the upper boun, C Q = C P H Q P a C P H Q P, K! K! P P where K = mint τ, M, matrix P is any permutation matrix that permutes the first K rows or columns of the non-zero elements of Q, an a follows from the concavity property of the upper boun. At this point, it is evient that the optimal pilot, which achieves the maximum capacity upper boun, has an orthogonal structure given by 9. Therefore, although starting from ifferent perspectives, orthogonal pilots structure not only minimizes the estimation mean square error, but also maximizes the capacity upper bouns. Substituting the optimal structure 9 into the equation 7, we obtain the following capacity upper boun IM C = N T log +ρ E X log + ρ M Λ X H X, where X is of size C M, an Λ is given by + ρ τ T τ Λ = mint τ,m IminTτ,M. I M mintτ,m C. Equal Training an Data Power Allocation For some communication systems, it might not be possible to vary the power uring the training slots an ata slots. Hence the capacity upper boun, assuming training symbol an ata symbol share the same power, is obtaine by substituting the power allocations ρ τ = ρ = ρ into. The capacity upper boun is further optimize with respect to the ata allocation scheme T τ, T, an we have the following important result concerning the epenence on T, the number of ata symbols. Proposition Capacity upper bouns uner equal power allocation schemes are monotonically increasing with respect to the number of ata slots T, i.e. CT = k CT = k, k T M, CT = T CT = T M. 7

4 Proof: We begin with the first part of 7 when T τ M, where equation is reuce to C = N T IM log +ρ E X log + ρ M XH X, 8 where ρ = ρ/ + ρt τ /M. We further separate the ata matrix X into X an x T, X X =, X x C M, x C M. 9 MIMO capacity per Transmission, C/T 7 Orthogonal Pilots, T=, = Orthogonal Pilots, T=, = Orthogonal Pilots, T=, = Orthogonal Pilots, T=, = Ranom Pilots, T=, = Ranom Pilots, T=, = Ranom Pilots, T=, = Ranom Pilots, T=, = by MIMO system Capacity Upper Boun Comparison Then we have the following inequality, IM E X log + ρ M XH X IM = E X log + ρ M a IM E X log + ρ M X H X + x H T x X H X + E x x H x, 8 snrb Figure : Capacity comparison between orthogonal pilot structures an ranom pilot structures uner equal power allocation schemes of a MIMO system with coherent time intervals T =, an ata interval T =,,, where inequality a follows from the fact that log is a concave function. Therefore, using assumption, we can obtain the following result CT = k CT = k fρ, where + ρtτ + /M fρ = log +ρ M log, + ρt τ /M ue to the fact that f an fρ. A similar approach can be use to prove the secon part of 7. D. Optimization over ρ τ, ρ an T τ, T For communication systems where the power allocation can be varie between training an ata symbols, the optimal capacity upper boun is obtaine by solving the following constraine optimization problem C T τ, T = max C ρ τ, ρ, T τ, T, ρτ T τ +ρ T = ρt. ρ τ,ρ However, numerical results provie in Section IV inicate that there is insignificant capacity loss by using equal power allocations, which is much easier for implementation. IV. NUMERICAL AND SIMULATION RESULTS A. Optimal Pilot Structure We know from Section III. B that orthogonal pilot structure not only minimizes the mean square estimation error, but also maximizes the capacity upper boun 7. Fig. emonstrate the sensitivity of the capacity upper boun with respect to the pilots structures. As can be observe from the plot, the capacities using ranom pilot structure, which are enote as ote curves, are inferior to those of applying orthogonal pilot structure. An the capacity loss is significant in moerate to high SNR ranges. mimo capacity per Transmission, C/T T= T= T= by MIMO System Capacity Upper Boun with Equal Power Allocation 8 8 Figure : Capacity upper boun of a MIMO system uner equal power allocation scheme of SNR ρ = B, an with ifferent coherent time intervals T =,,, 7, 8,,, B. Equal Power Allocation The capacity upper bouns of a MIMO system uner equal power allocation scheme with ifferent coherent time T an ata slot allocation T are emonstrate in Fig.. From the plot, we can observe that the capacity upper boun is monotonically increasing with respect to T even for the case T > T M. However, the capacity gain is insignificant for T beyon T M especially when T is larger than M. Therefore, T = T M, T > M is a goo trae-off point between the achievable capacity an implementation complexity. C. Optimal Power Allocation We emonstrate in Fig. the capacity upper boun with

5 mimo capacity per Transmission, C/T..... by MIMO System Capacity Upper Boun with Optimal Power Allocation T= T= T= MIMO capacity per Transmission, C/T 8 Upper bouns, T=, = Upper bouns, T=, = Upper bouns, T=, = Upper bouns, T=, = Lower bouns, T=, = Lower bouns, T=, = Lower bouns, T=, = Lower bouns, T=, = Ieal CSIR by MIMO system Capacity Upper Boun Comparison snrb Figure : Capacity upper boun of a MIMO system uner optimal power allocation scheme of SNR ρ = B, an with ifferent coherent time intervals T =,,, 7, 8,,, Figure : Capacity comparison between lower bouns an upper bouns uner equal power allocation schemes of a MIMO system with coherent time intervals T =, an ata interval T =,,, mimo capacity gain, C/C opt Capacity Gain of by MIMO System uner Optimal Power Allocation T= T= T=. 8 8 Figure : Capacity gain of optimal power allocations over equal power allocations of a MIMO system with SNR ρ = B, uner ifferent coherent time intervals T =,,, 7, 8,,, optimal power allocation schemes for the same MIMO system use in the Fig. with varying coherent time intervals. As a comparison, the capacity gain from using optimal power allocation over equal power allocation scheme is shown in Fig.. We can observe from the plot that, when the number of training symbols T τ is small, there is insignificant capacity loss ue to using equal power allocations, which is much easier for implementation. D. Comparison In Fig., for comparison purpose we plot the capacity lower boun provie in, the capacity upper boun provie in this paper, as well as the MIMO channel capacity with ieal channel state information at the receiver. V. CONCLUSION In this paper, we propose a capacity upper boun of the unknown MIMO channel. Through the analysis of the propose upper boun, we show that orthogonal pilot structure is optimal. It not only minimizes the mean square estimation error, but also maximizes the propose capacity upper boun. We also prove that uner equal power allocation scheme, capacity upper boun is a monotonically increasing function with respect to the number of ata slots T. Through numerical evaluations, we further emonstrate that the capacity increment is insignificant when T is larger than T M, an limite capacity gain can be achieve by using optimal power allocation between training an ata symbols when compare to the simple equal power allocation scheme. REFERENCES E. Telatar, Capacity of multi-antenna Gaussian channels, Eur. Trans. Telecommun., vol., no., pp. 8-9, 999. G. J. Foschini, Layere space-time architecture for wireless communication in a faing environment when using multiple antennas, Bell Labs Technical Journal, vol., no., pp. 9, Autumn, 99. T. L. Marzetta an B. M. Hochwal, Capacity of a mobile multipleantenna communication link in Rayleigh flat faing, IEEE Trans. on Information Theory, vol., Issue., pp. 9-7, Jan, 999. L. Zheng an D. N. C. Tse, Communication on the Grassmann manifol: a geometric approach to the noncoherent multiple-antenna channel, IEEE Trans. on Information Theory, vol. 8, Issue., pp. 9-8, Feb,. B. Hassibi an B. M. Hochwal, How much training is neee in multiple-antenna wireless links?, IEEE Trans. on Information Theory, vol. 9, Issue., pp. 9-9, Apr,. J. Zheng an B. D.Rao, Is Training Necessary? Part I: Capacity Analysis of MIMO Systems with Unknown Channel State Information, in preparation, Apr,.