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In this paper, we propose a linear precoder for the downlink of a multi-user MIMO system with multiple users that potentially act as eavesdroppers. The proposed precoder is based on regularized channel inversion (RCI) with a regularization parameter α and power allocation vector chosen in such a way that the achievable secrecy sum-rate is maximized. We consider the worst-case scenario for the multi-user MIMO system, where the transmitter assumes users cooperate to eavesdrop on other users. We derive the achievable secrecy sumrate and obtain the closed-form expression for the optimal regularization parameter α LS of the precoder using large-system analysis. We show that the RCI precoder with α LS outperforms several other linear precoding schemes, and it achieves a secrecy sum-rate that has same scaling factor as the sum-rate achieved by the optimum RCI precoder without secrecy requirements. We propose a power allocation algorithm to maximize the secrecy sum-rate for fixed α. We then extend our algorithm to maximize the secrecy sum-rate by jointly optimizing α and the power allocation vector. The jointly optimized precoder outperforms RCI with α LS and equal power allocation by up to 20 percent at practical values of the signal-to-noise ratio and for 4 users and 4 transmit antennas.

We consider multiple-input multiple-output (MIMO) transmit beamforming systems with maximum ratio combining (MRC) receivers. The operating environment is Rayleigh-fading with both transmit and receive spatial correlation. We present exact expressions for the probability density function (p.d.f.) of the output signal-to-noise ratio (SNR), as well as the system outage probability. The results are based on explicit closed-form expressions which we derive for the p.d.f. and c.d.f. of the maximum eigenvalue of double-correlated complex Wishart matrices. For systems with two antennas at either the transmitter or the receiver, we also derive exact closed-form expressions for the symbol error rate (SER). The new expressions are used to prove that MIMO-MRC achieves the maximum available spatial diversity order, and to demonstrate the effect of spatial correlation. The analysis is validated through comparison with Monte-Carlo simulations. 1 I. INTRODUCTION Multiple-input multiple-output (MIMO) antenna technology can provide significant improvements in capacity [1-4] and error performance [5] over conventional single-antenna technology, without requiring extra power or bandwidth. When channel knowledge is available at both the transmitter and receiver, MIMO transmit beamforming with maximum-ratio combining (MRC) receivers [6] is particularly robust against the severe effects of fading. This robustness is achieved by steering the transmitted signal along the maximum eigenmode of the MIMO channel, resulting in the maximization of the signal-to-noise ratio (SNR) at the MRC output. Recently, MIMO-MRC has been investigated in uncorrelated and semi-correlated channel scenarios (i.e. where correlation occurs at only one end of the transmission link, or not at all). A key to deriving analytical performance results is to statistically characterize the SNR at the output of the MRC combiner. In [7-11], uncorrelated Rayleigh fading was considered, and the output SNR statistical properties were derived based on maximum eigenvalue statistics of complex central Wishart matrices. In [12], uncorrelated Rician channels were characterized using maximum eigenvalue properties of complex noncentral Wishart matrices. Semi-correlated Rayleigh channels were considered in [13], utilizing properties of semi-correlated Wishart matrices. In this paper we consider double-correlated Rayleigh channels, by first deriving results for the eigenvalue statistics of double-correlated complex Wishart matrices. In practice, doublecorrelated channels (i.e. with correlation at both the transmitter and receiver) commonly occur due to, for example, insufficient scattering around both the transmit and receive terminals, or to closely spaced antennas with respect to the wavelength of the signal. While there are numerous statistical results on general Wishart matrices, there are almost no results for the eigenvalue statistics in the case of double-correlated Wishart matrices. In [14], the joint probability density function (p.d.f.) of the eigenvalues of such matrices wa...

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This paper investigates the achievable sum rate of multiple-input multiple-output (MIMO) wireless systems employing linear minimum mean-squared error (MMSE) receivers. We present a new analytic framework which unveils an interesting connection between the achievable sum rate with MMSE receivers and the ergodic mutual information achieved with optimal receivers. This simple but powerful result enables the vast prior literature on ergodic MIMO mutual information to be directly applied to the analysis of MMSE receivers. The framework is particularized to various Rayleigh and Rician channel scenarios to yield new exact closed-form expressions for the achievable sum rate, as well as simplified expressions in the asymptotic regimes of high and low signal to noise ratios. These expressions lead to the discovery of key insights into the performance of MIMO MMSE receivers under practical channel conditions.

Abstract-In millimeter wave cellular communication, fast and reliable beam alignment via beam training is crucial to harvest sufficient beamforming gain for the subsequent data transmission. In this paper, we establish fundamental limits in beam-alignment performance under both the exhaustive search and the hierarchical search that adopts multi-resolution beamforming codebooks, accounting for time-domain training overhead. Specifically, we derive lower and upper bounds on the probability of misalignment for an arbitrary level in the hierarchical search, based on a single-path channel model. Using the method of large deviations, we characterize the decay rate functions of both bounds and show that the bounds coincide as the training sequence length goes large. We go on to characterize the asymptotic misalignment probability of both the hierarchical and exhaustive search, and show that the latter asymptotically outperforms the former, subject to the same training overhead and codebook resolution. We show via numerical results that this relative performance behavior holds in the non-asymptotic regime. Moreover, the exhaustive search is shown to achieve significantly higher worst-case spectrum efficiency than the hierarchical search, when the pre-beamforming signal-to-noise ratio (SNR) is relatively low. This study hence implies that the exhaustive search is more effective for users situated further from base stations, as they tend to have low SNR.

In this paper, we study physical layer security for the downlink of cellular networks, where the confidential messages transmitted to each mobile user can be eavesdropped by both (i) the other users in the same cell and (ii) the users in the other cells. The locations of base stations and mobile users are modeled as two independent two-dimensional Poisson point processes. Using the proposed model, we analyze the secrecy rates achievable by regularized channel inversion (RCI) precoding by performing a large-system analysis that combines tools from stochastic geometry and random matrix theory. We obtain approximations for the probability of secrecy outage and the mean secrecy rate, and characterize regimes where RCI precoding achieves a nonzero secrecy rate. We find that unlike isolated cells, the secrecy rate in a cellular network does not grow monotonically with the transmit power, and the network tends to be in secrecy outage if the transmit power grows unbounded. Furthermore, we show that there is an optimal value for the base station deployment density that maximizes the secrecy rate, and this value is a decreasing function of the signal-to-noise ratio.Comment: submitted to IEEE Transactions on Communications, July 201