In many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set ofNindependent, identically distributed measurements of anMdimensional random vector the maximum likelihood estimate is the sample covariance matrix. Here we consider the case whereNMsuch that this estimate is singular (noninvertible) and therefore fundamentally bad. We present a radically new approach to deal with this situation based on the idea of dimensionality reduction through an ensemble of isotropically random unitary matrices. We obtain two estimates cov and invcov which are estimates for the covariance matrix and the inverse covariance matrix respectively. Both estimates retain the original eigenvectors while altering the eigenvalues. We have a closed form analytical expression for cov and invcov in terms of the eigenvector/eigenvalue decomposition of the sample covariance. We motivate the use of invcov through applications to linear estimation, supervised learning, and high-resolution spectral estimation. We also compare the performance of these estimators with other more conventional methods.

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.

Fourth-generation (4G) systems are expected to support data rates of the order of 100 Mb/s in the outdoor environment and 1 Gb/s in the indoor/stationary environment. In order to support such large payloads, the radio physical layer must employ receiver algorithms that provide a significant increase in spectrum efficiency (and, hence, capacity) over current wireless systems. Recently, an explosion of multiple-input-multiple-output (MIMO) studies have appeared with many journals presenting special issues on this subject. This has occurred due to the potential of MIMO to provide a linear increase in capacity with antenna numbers. Environmental considerations and tower loads will often restrict the placing of large antenna spans on base stations (BSs). Similarly, customer device form factors also place a limit on the antenna numbers that can be placed with a mutual spacing of 0.5 wavelength. The use of cross-polarized antennas is widely used in modern cellular installations as it reduces spacing needs and tower loads on BSs. Hence, this approach is also receiving considerable attention in MIMO systems. In order to study and compare various receiver architectures that are based on MIMO techniques, one needs to have an accurate knowledge of the MIMO channel. However, very few studies have appeared that characterize the cross-polarized MIMO channel. Recently, the third-generation partnership standards bodies (3GPP/3GPP2) have defined a cross-polarized channel model for MIMO systems but this model neglects the elevation spectrum. In this paper, we provide a deeper understanding of the channel model for cross-polarized systems for different environments and propose a composite channel impulse model for the cross-polarized channel that takes into account both azimuth and elevation spectrum. We use the resulting channel impulse response to derive closed-form expressions for the spatial correlation. We also present models to describe the dependence of cross-polarization discrimination (XPD) on distance, azimuth and elevation and delay spread. In addition, we study the impact of array width, signal-to-noise ratio, and antenna slant angle on the mutual information (MI) of the system. In particular, we present an analytical model for large system mean mutual information values and consider the impact of elevation spectrum on MI. Finally, the impact of multipath delays on XPD and MI is also explored.

Differential unitary space–time modulation (DUSTM) and its earlier nondifferential counterpart, USTM, permit high-throughput multiple-input multiple-output (MIMO) communication entirely without the possession of channel state information by either the transmitter or the receiver. For an isotropically random unitary input we obtain the exact closed-form expression for the probability density of the DUSTM received signal, permitting the straightforward Monte Carlo evaluation of its mutual information. We compare the performance of DUSTM and USTM through both numerical computations of mutual information and through the analysis of low- and high-signal-to-noise ratio (SNR) asymptotic expressions. In our comparisons the symbol durations of the equivalent unitary space–time signals are equal to$T$. For DUSTM the number of transmit antennas is constrained by the scheme to be$M = T/2$, while USTM has no such constraint. If DUSTM and USTM utilize the same number of transmit antennas at high SNRs the normalized mutual information of the two schemes expressed in bits/s/Hz are asymptotically equal, with the differential scheme performing somewhat better. At low SNRs the normalized mutual information of DUSTM is asymptotically twice the normalized mutual information of USTM. If, instead, USTM utilizes the optimum number of transmit antennas then USTM can outperform DUSTM at sufficiently low SNRs.

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

Channel information at the transmitter is known to significantly increase information throughput. However, practical constraints in the opposite link limit the possible amount of feedback. In this paper we describe a way to calculate the capacity of a simple, yet relevant feedback scheme. In particular, we consider the case of a two-antenna transmitter and a single antenna receiver, and limit the amount of feedback to 2 bits. This case is of practical interest because it is included in the UMTS standards. We determine the transmission weights maximizing the outage mutual information, and find that they interpolate between beamforming and orthogonal coding. Further, the case of the feedback bits having finite and generally unequal probabilities of error is also considered. In the latter case, the optimal relative phase between the two transmitting antennas deviates from their symmetric /spl plusmn//spl pi//spl plusmn//spl pi//4 value. This effect is important, if the bits are fed back at different times and thus their reliability is not the same (one feedback bit being more stale than the other).

The use of multi-antenna arrays has been predicted to provide substantial throughput gains for wireless communication systems. However, these predictions have to be assessed in realistic situations, such as correlated channels and in the presence of interference. We present a model which provides expressions for the average and the variance of the distribution of the mutual information of multi-antenna systems with arbitrary correlations and interferers. This model, even though it is based on analytic calculations in the limit of large antenna numbers, produces extremely accurate results, even for small arrays. We use this model to optimize over the input signal covariance with channel covariance feedback and thus to calculate closed-loop capacities We analyze in detail two specific examples and also compare simulations to establish its validity. This method provides a simple tool to analyze the statistics of throughput for arrays of any size.

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.

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).

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.