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On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach, (extended version) arXiv:0710.4051. Julien Dumont was born in Flers
 and France Télécom Labs
, 1979
"... Abstract—In this paper, the capacityachieving input covariance matrices for coherent blockfading correlated multiple input multiple output (MIMO) Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the opti ..."
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Cited by 25 (13 self)
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Abstract—In this paper, the capacityachieving input covariance matrices for coherent blockfading correlated multiple input multiple output (MIMO) Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding. In the asymptotic regime where the number of transmit and receive antennas converge to infinity at the same rate, new results related to the accuracy of the approximation of the average mutual information are provided. Based on the accuracy of this approximation, an attractive optimization algorithm is proposed and analyzed. This algorithm is shown to yield an effective way to compute the capacity achieving matrix for the average mutual information and numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information. Index Terms—Multiple input multiple output (MIMO) Rician channels, ergodic capacity, large random matrices, capacity achieving covariance matrices, iterative waterfilling. I.
Fluctuations of eigenvalues and second order Poincaré inequalities
, 2007
"... Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified t ..."
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Cited by 25 (3 self)
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Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out; some of them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.
A CLT for Informationtheoretic statistics of Gram random matrices with a given variance profile
, 2008
"... Consider a N × n random matrix Yn = (Y n ij) where the entries are given by Y n σij(n) ij = √ X n n ij, the Xn ij being centered, independent and identically distributed random variables with unit variance and (σij(n); 1 ≤ i ≤ N,1 ≤ j ≤ n) being an array of numbers we shall refer to as a variance ..."
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Cited by 10 (5 self)
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Consider a N × n random matrix Yn = (Y n ij) where the entries are given by Y n σij(n) ij = √ X n n ij, the Xn ij being centered, independent and identically distributed random variables with unit variance and (σij(n); 1 ≤ i ≤ N,1 ≤ j ≤ n) being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable log det (YnY ∗ n + ρIN) where Y ∗ is the Hermitian adjoint of Y and ρ> 0 is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4 th moment of the Xij’s differs from the 4 th moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.
A Central Limit Theorem for the SINR at the LMMSE Estimator Output for Large Dimensional Signals
, 2008
"... This paper is devoted to the performance study of the Linear Minimum Mean Squared Error estimator for multidimensional signals in the large dimension regime. Such an estimator is frequently encountered in wireless communications and in array processing, and the Signal to Interference and Noise Ratio ..."
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Cited by 5 (5 self)
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This paper is devoted to the performance study of the Linear Minimum Mean Squared Error estimator for multidimensional signals in the large dimension regime. Such an estimator is frequently encountered in wireless communications and in array processing, and the Signal to Interference and Noise Ratio (SINR) at its output is a popular performance index. The SINR can be modeled as a random quadratic form which can be studied with the help of large random matrix theory, if one assumes that the dimension of the received and transmitted signals go to infinity at the same pace. This paper considers the asymptotic behavior of the SINR for a wide class of multidimensional signal models that includes general multiantenna as well as spread spectrum transmission models. The expression of the deterministic approximation of the SINR in the large dimension regime is recalled and the SINR fluctuations around this deterministic approximation are studied. These fluctuations are shown to converge in distribution to the Gaussian law in the large dimension regime, and their variance is shown to decrease as the inverse of the signal dimension.
On the capacity achieving covariance matrix for frequency selective MIMO channels using the asymptotic approach
 IEEE Trans. Inf. Theory
, 2011
"... Abstract—In this contribution, an algorithm for evaluating the capacityachieving input covariance matrices for frequency selective Rayleigh MIMO channels is proposed. In contrast with the flat fading Rayleigh case, no closedform expressions for the eigenvectors of the optimum input covariance matr ..."
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Cited by 3 (1 self)
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Abstract—In this contribution, an algorithm for evaluating the capacityachieving input covariance matrices for frequency selective Rayleigh MIMO channels is proposed. In contrast with the flat fading Rayleigh case, no closedform expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding. In this paper, it is proposed to optimize (w.r.t. the input covariance matrix) a large system approximation of the average mutual information derived by Moustakas and Simon. The validity of this asymptotic approximation is clarified thanks to Gaussian large random matrices methods. It is shown that the approximation is a strictly concave function of the input covariance matrix and that the average mutual information evaluated at the argmax of the approximation is equal to the capacity of the channel up to a O(1/t) term, where t is the number of transmit antennas. An algorithm based on an iterative waterfilling scheme is proposed to maximize the average mutual information approximation, and its convergence studied. Numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information. Index Terms—Ergodic capacity, frequency selective MIMO channels, large random matrices I.
ON THE ERGODIC CAPACITY OF FREQUENCY SELECTIVE MIMO SYSTEMS EQUIPPED WITH MMSE RECEIVERS: AN ASYMPTOTIC APPROACH.
"... This paper is devoted to the study of the ergodic capacity of frequency selective MIMO systems equipped with a MMSE receiver when the channel state information is available at the receiver side and when the second order statistics of the channel taps are known at the transmitter side. As the express ..."
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Cited by 1 (0 self)
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This paper is devoted to the study of the ergodic capacity of frequency selective MIMO systems equipped with a MMSE receiver when the channel state information is available at the receiver side and when the second order statistics of the channel taps are known at the transmitter side. As the expression of this capacity is rather complicated and difficult to analyse, it is studied in the case where the number of transmit and receive antennas converge to + ∞ at the same rate. In this asymptotic regime, the main results of the paper are related to the design of an optimal precoder in the case where the transmit antennas are correlated. It is shown that the left singular eigenvectors of the optimum precoder coincide with the eigenvectors of the mean of the channel taps transmit covariance matrices, and its singular values are solution of a certain maximization problem. 1.
1 On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach
, 2008
"... In this contribution, the capacityachieving input covariance matrices for coherent blockfading correlated MIMO Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are avai ..."
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In this contribution, the capacityachieving input covariance matrices for coherent blockfading correlated MIMO Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closedform expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed by numerical techniques. As the corresponding optimization algorithms are not very attractive, an approximation of the average mutual information is evaluated in this paper in the asymptotic regime where the number of transmit and receive antennas converge to + ∞ at the same rate. New results related to the accuracy of the corresponding large system approximation are provided. An attractive optimization algorithm of this approximation is proposed and we establish that it yields an effective way to compute the capacity achieving covariance matrix for the average mutual information. Finally, numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information, while being much more computationally attractive.
Asymptotic Performance of Linear Receivers in 1 MIMO Fading Channels
, 810
"... Linear receivers are considered as an attractive lowcomplexity alternative to optimal processing for multiantenna MIMO communications. In this paper we characterize the performance of MIMO linear receivers in two different asymptotic regimes. For fixed number of antennas, we investigate the Divers ..."
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Linear receivers are considered as an attractive lowcomplexity alternative to optimal processing for multiantenna MIMO communications. In this paper we characterize the performance of MIMO linear receivers in two different asymptotic regimes. For fixed number of antennas, we investigate the DiversityMultiplexing Tradeoff (DMT), which captures the outage probability (decoding blockerror probability) in the limit of high SNR. For fixed SNR, we characterize the outage probability for a large (but finite) number of antennas. As far as the DMT is concerned, we report a negative result: we show that both linear ZeroForcing (ZF) and linear Minimum MeanSquare Error (MMSE) receivers achieve the same DMT, which is largely suboptimal even though outer coding and decoding is performed across the antennas. We also provide an approximate quantitative analysis of the different behavior of the MMSE and ZF receivers at finite rate and nonasymptotic SNR, and show that while the ZF receiver achieves poor diversity at any finite rate, the MMSE receiver error curve slope flattens out progressively, as the coding rate increases. When SNR is fixed and the number of antennas grows large, we show that the mutual information at the output of a MMSE or ZF linear receiver has fluctuations that converge in distribution to a Gaussian random variable, whose mean and variance can be characterized in closed form. This analysis extends to the linear receiver case a result that was previously obtained for the optimal receiver. Simulations reveal that the asymptotic analysis captures accurately the outage behavior of systems even with a
1 Asymptotic Mutual Information Statistics of SeparatelyCorrelated Rician Fading MIMO Channels ∗
, 712
"... Precise characterization of the mutual information of MIMO systems is required to assess the throughput of wireless communication channels in the presence of Rician fading and spatial correlation. Here, we present an asymptotic approach allowing to approximate the distribution of the mutual informat ..."
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Precise characterization of the mutual information of MIMO systems is required to assess the throughput of wireless communication channels in the presence of Rician fading and spatial correlation. Here, we present an asymptotic approach allowing to approximate the distribution of the mutual information as a Gaussian distribution in order to provide both the average achievable rate and the outage probability. More precisely, the mean and variance of the mutual information of the separatelycorrelated Rician fading MIMO channel are derived when the number of transmit and receive antennas grows asymptotically large and their ratio approaches a finite constant. The derivation is based on the replica method, an asymptotic technique widely used in theoretical physics and, more recently, in the performance analysis of communication (CDMA and MIMO) systems. The replica method allows to analyze very difficult system cases in a comparatively simple way though some authors pointed out that its assumptions are not always rigorous. Being aware of this, we underline the key assumptions made in this setting, quite similar to the assumptions made in the technical literature using the replica method in their asymptotic analyses. As far as concerns the convergence of the mutual information to the Gaussian distribution, it is shown that it holds under some mild technical conditions, which are tantamount to assuming that the spatial correlation structure has no asymptotically dominant eigenmodes. The accuracy of the asymptotic approach is assessed by providing a sizeable number of numerical results. It is shown that the approximation is very accurate in a wide variety of system settings even when the number of transmit and receive antennas is as small as a few units.
A CLT on the SNR of Diagonally Loaded MVDR Filters ∗
, 2012
"... Abstract: This paper studies the fluctuations of the signaltonoise ratio (SNR) of minimum variance distorsionless response (MVDR) filters implementing diagonal loading in the estimation of the covariance matrix. Previous results in the signal processing literature are generalized and extended by c ..."
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Abstract: This paper studies the fluctuations of the signaltonoise ratio (SNR) of minimum variance distorsionless response (MVDR) filters implementing diagonal loading in the estimation of the covariance matrix. Previous results in the signal processing literature are generalized and extended by considering both spatially as well as temporarily correlated samples. Specifically, a central limit theorem (CLT) is established for the fluctuations of the SNR of the diagonally loaded MVDR filter, under both supervised and unsupervised training settings in adaptive filtering applications. Our secondorder analysis is based on the NashPoincaré inequality and the integration by parts formula for Gaussian functionals, as well as classical tools from statistical asymptotic theory. Numerical evaluations validating the accuracy of the CLT confirm the asymptotic Gaussianity of the fluctuations of the SNR of the MVDR filter.