Aris L. Moustakas

MIMO capacity through correlated channels in the presence of correlated interferers and noise: a (not so) large N analysis

The use of multiple-antenna arrays in both transmission and reception promises huge increases in the throughput of wireless communication systems. It is therefore important to analyze the capacities of such systems in realistic situations, which may include spatially correlated channels and correlated noise, as well as correlated interferers with known channel at the receiver. Here, we present an approach that provides analytic expressions for the statistics, i.e., the moments of the distribution, of the mutual information of multiple-antenna systems with arbitrary correlations, interferers, and noise. We assume that the channels of the signal and the interference are Gaussian with arbitrary covariance. Although this method is valid formally for large antenna numbers, it produces extremely accurate results even for arrays with as few as two or three antennas. We also develop a method to analytically optimize over the input signal covariance, which enables us to calculate analytic capacities when the transmitter has knowledge of the statistics of the channel (i.e., the channel covariance). In many cases of interest, this capacity is very close to the full closed-loop capacity, in which the transmitter has instantaneous channel knowledge. We apply this analytic approach to a number of examples and we compare our results with simulations to establish the validity of this approach. This method provides a simple tool to analyze the statistics of throughput for arrays of any size. The emphasis of this paper is on elucidating the novel mathematical methods used.

Optimizing MIMO antenna systems with channel covariance feedback

We consider a narrowband point-to-point communication system with n/sub T/ transmitters and n/sub R/ receivers. We assume the receiver has perfect knowledge of the channel, while the transmitter has no channel knowledge. We consider the case where the receiving antenna array has uncorrelated elements, while the elements of the transmitting array are arbitrarily correlated. Focusing on the case where n/sub T/=2, we derive simple analytic expressions for the ergodic average and the cumulative distribution function of the mutual information for arbitrary input (transmission) signal covariance. We then determine the ergodic and outage capacities and the associated optimal input signal covariances. We thus show how a transmitter with covariance knowledge should correlate its transmissions to maximize throughput. These results allow us to derive an exact condition (both necessary and sufficient) that determines when beamforming is optimal for systems with arbitrary number of transmitters and receivers.

Communication through a diffusive medium : Coherence and capacity

Coherent wave propagation in disordered media gives rise to many fascinating phenomena as diverse as universal conductance fluctuations in mesoscopic metals and speckle patterns in light scattering. Here, the theory of electromagnetic wave propagation in diffusive media is combined with information theory to show how interference affects the information transmission rate between antenna arrays. Nontrivial dependencies of the information capacity on the nature of the antenna arrays are found, such as the dimensionality of the arrays and their direction with respect to the local scattering medium. This approach provides a physical picture for understanding the importance of scattering in the transfer of information through wireless communications.

A Wideband Spatial Channel Model for System-Wide Simulations

A wideband space-time channel model is defined, which captures the multiple dependencies and variability in multicell system-wide operating environments. The model provides a unified treatment of spatial and temporal parameters, giving their statistical description and dependencies across a large geographical area for three outdoor environments pertinent to third-generation cellular system simulations. Parameter values are drawn from a broad base of recently published wideband and multiple-antenna measurements. A methodology is given to generate fast-fading coefficients between a base station and a mobile user based on the summation of directional plane waves derived from the statistics of the space-time parameters. Extensions to the baseline channel model, such as polarized antennas, are given to provide a greater variety of spatial environments. Despite its comprehensive nature, the models implementation complexity is reasonable so it can be used in simulating large-scale systems. Output statistics and capacities are used to illustrate the main characteristics of the model

Asymptotic Performance of Linear Receivers in MIMO Fading Channels

Linear receivers are an attractive low-complexity alternative to optimal processing for multiple-antenna multiple-input multiple-output (MIMO) communications. In this paper, we characterize the information-theoretic performance of MIMO linear receivers in two different asymptotic regimes. For fixed number of antennas, we investigate the limit of error probability in the high-signal-to noise-ratio (SNR) regime in terms of the diversity-multiplexing tradeoff (DMT). Following this, we characterize the error probability for fixed SNR in the regime of large (but finite) number of antennas.As far as the DMT is concerned, we report a negative result: we show that both linear zero-forcing (ZF) and linear minimum mean- square error (MMSE) receivers achieve the same DMT, which is largely suboptimal even in the case where outer coding and deAcircnot coding is performed across the antennas. We also provide an apAcircnot proximate quantitative analysis of the markedly 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 becomes large, we show that the mutual information at the output of an 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 reAcircnot ceiver case a well-known result previously obtained for the optimal receiver. Simulations reveal that the asymptotic analysis captures accurately the outage behavior of systems even with a moderate number of antennas.

Optimizing multiple-input single-output (MISO) communication systems with general Gaussian channels: nontrivial covariance and nonzero mean

We consider a narrow-band point-to-point communication system with many (input) transmitters and a single (output) receiver (i.e., a multiple-input single output (MISO) system). We assume the receiver has perfect knowledge of the channel but the transmitter only knows the channel distribution. We focus on two canonical classes of Gaussian channel models: (a) the channel has zero mean with a fixed covariance matrix and (b) the channel has nonzero mean with covariance matrix proportional to the identity. In both cases, we are able to derive simple analytic expressions for the ergodic average and the cumulative distribution function (c.d.f.) of the mutual information for arbitrary input (transmission) signal covariance. With minimal numerical effort, we then determine the ergodic and outage capacities and the corresponding capacity-achieving input signal covariances. Interestingly, we find that the optimal signal covariances for the ergodic and outage cases have very different behavior. In particular, under certain conditions, the outage capacity optimal covariance is a discontinuous function of the parameters describing the channel (such as strength of the correlations or the nonzero mean of the channel).

Vector Precoding for Wireless MIMO Systems and its Replica Analysis

This paper studies a nonlinear vector precoding scheme which inverts the wireless multiple-input multiple-output (MIMO) channel at the transmitter so that simple symbol-by-symbol detection can be used in lieu of sophisticated multiuser detection at the receiver. In particular, the transmit energy is minimized by relaxing the transmitted symbols to a larger alphabet for precoding, which preserves the minimum signaling distance. The so-called replica method is used to analyze the average energy savings with random MIMO channels in the large-system limit. It is found that significant gains can be achieved with complex-valued alphabets. The analysis applies to a very general class of MIMO channels, where the statistics of the channel matrix enter the result via the R-transform of the asymptotic empirical distribution of its eigenvalues. Moreover, we introduce polynomial-complexity precoding schemes for binary and quadrature phase-shift keying in complex channels by using convex rather than discrete relaxed alphabets. In case the number of transmit antennas is more than twice the number of receive antennas, we show that a convex precoding scheme, despite its polynomial complexity, outperforms NP-hard precoding using the popular Tomlinson-Harashima signaling.

Capacity and Character Expansions: Moment-Generating Function and Other Exact Results for MIMO Correlated Channels

A promising new method from the field of representations of Lie groups is applied to calculate integrals over unitary groups, which are important for multiantenna communications. To demonstrate the power and simplicity of this technique, a number of recent results are rederived, using only a few simple steps. In particular, we derive the joint probability distribution of eigenvalues of the matrix GGdagger , with G a nonzero mean or a semicorrelated Gaussian random matrix. These joint probability distribution functions can then be used to calculate the moment generating function of the mutual information for Gaussian multiple-input multiple-output (MIMO) channels with these probability distribution of their channel matrices G. We then turn to the previously unsolved problem of calculating the moment generating function of the mutual information of MIMO channels, which are correlated at both the receiver and the transmitter. From this moment generating function we obtain the ergodic average of the mutual information and study the outage probability. These methods can be applied to a number of other problems. As a particular example, we examine unitary encoded space-time transmission of MIMO systems and we derive the received signal distribution when the channel matrix is correlated at the transmitter end

On the Outage Capacity of Correlated Multiple-Path MIMO Channels

The use of multiple-antenna arrays can dramatically increase the throughput of wireless communication systems. Thus, it is important to characterize the statistics of the mutual information for realistic correlated channels. Here, a mathematical approach is presented, using the method of replicas, that provides analytic expressions not only for the average, but also for the higher moments of the distribution of the mutual information for the most general zero-mean Gaussian multiple-input multiple-output (MIMO) channels when the channel is known at the receiver. These channels include multitap delay paths, and channels with covariance matrices that cannot be written as a Kronecker product, such as general dual-polarized correlated antenna arrays. This approach is formally valid for large antenna numbers, in which case all cumulant moments of the distribution, other than the first two, scale to zero. In addition, it is shown that the replica-symmetric result is valid if the variance of the mutual information is positive and finite. In this case, it is shown that the distribution of the mutual information tends to a Gaussian, which enables the calculation of the outage capacity. These results are quite accurate even for few antennas, which makes this approach applicable to realistic situations.

Distributed Learning Policies for Power Allocation in Multiple Access Channels

We analyze the power allocation problem for orthogonal multiple access channels by means of a non-cooperative potential game in which each user distributes his power over the channels available to him. When the channels are static, we show that this game possesses a unique equilibrium; moreover, if the networks users follow a distributed learning scheme based on the replicator dynamics of evolutionary game theory, then they converge to equilibrium exponentially fast. On the other hand, if the channels fluctuate stochastically over time, the associated game still admits a unique equilibrium, but the learning process is not deterministic; just the same, by employing the theory of stochastic approximation, we find that users still converge to equilibrium. Our theoretical analysis hinges on a novel result which is of independent interest: in finite-player games which admit a (possibly nonlinear) convex potential, the replicator dynamics converge to an e-neighborhood of an equilibrium in time O(\log(1/e)).