Georgy Levin

On the Outage Capacity Distribution of Correlated Keyhole MIMO Channels

Keyhole multiple-input-multiple-output (MIMO) channels have recently received significant attention since they can model, to a certain extend, some practically important propagation scenarios and also relay channels in the amplify-and-forward mode. This paper investigates instantaneous signal-to-noise ratio (SNR) and outage capacity distributions of spatially correlated keyhole MIMO channels with perfect channel state information (CSI) at the receive end and with or without CSI at the transmit end. For a small number of antennas, the impact of correlation on the capacity distribution can be characterized by the effective average SNR. This SNR, as well as the outage capacity, decreases with correlation. For a large number of transmit (receive) antennas, the keyhole channel is asymptotically equivalent (in terms of capacity) to the Rayleigh diversity channel with a single transmit (receive) antenna and multiple receive (transmit) antennas. The outage capacity of the keyhole channel is upper-bounded by that of the equivalent Rayleigh diversity channel. When the number of both transmit and receive antennas is large, the outage capacity distribution of the keyhole channel is asymptotically Gaussian. In some cases, the asymptotic Gaussian approximation is accurate already for a reasonably small number of antennas. The perfect transmit CSI is shown to bring a fixed SNR gain. A more general channel model with multiple keyholes is proposed. For a large number of antennas, the capacity of a multikeyhole channel is a normally distributed sum of the capacities of single keyhole channels. The fact that, despite the strong degenerate nature of the keyhole channel, its outage capacity distribution is asymptotically normal indicates that Gaussian distribution has a high degree of universality for the capacity analysis of MIMO channels.

On physically-based normalization of MIMO channel matrices

Various normalizations of the MIMO channel matrix are discussed from a physical perspective. It is demonstrated that the physics of antenna arrays and propagation channel should be taken into account when normalization is chosen, so that SNR has proper physical meaning, the conclusions are physical and correspond to realistic systems. The antenna array geometry and the transmission strategy (coherent/non-coherent) limits the choice of normalization and determines how the capacity and other performance metrics scale with the number of antennas, which is more pronounced for densely-populated antenna arrays. This is especially important for an asymptotic analysis, when the number of antennas increases to infinity. Limitations of such analysis from the physical perspective are pointed out.

Finite-SNR Diversity-Multiplexing Tradeoff via Asymptotic Analysis of Large MIMO Systems

Diversity-multiplexing tradeoff (DMT) was characterized asymptotically (SNR- > infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse . The SNR-asymptotic DMT overestimates the finite-SNR one . This paper outlines a number of additional limitations and difficulties of the DMT framework and discusses their implications. Using the recent results on the size-asymptotic (in the number of antennas) outage capacity distribution, the finite-SNR, size-asymptotic DMT is derived for a broad class of fading distributions. The SNR range over which the finite-SNR DMT is accurately approximated by the SNR-asymptotic one is characterized. The multiplexing gain definition is shown to affect critically this range and thus should be carefully selected, so that the SNR-asymptotic DMT is an accurate approximation at realistic SNR values and thus has operational significance to be used as a design criterion. The finite-SNR diversity gain is shown to decrease with correlation and power imbalance in a broad class of fading channels, and such an effect is described in a compact, closed form. Complete characterization of the outage probability (or outage capacity) requires not only the finite-SNR DMT, but also the SNR offset, which is introduced and investigated as well. This offset, which is not accounted for in the DMT framework, is shown to have a significant impact on the outage probability for a broad class of fading channels, especially when the multiplexing gain is small. The analytical results and conclusions are validated via extensive Monte Carlo simulations. Overall, the size-asymptotic DMT represents a valuable alternative to the SNR-asymptotic one.

From Multi-Keyholes to Measure of Correlation and Power Imbalance in MIMO Channels: Outage Capacity Analysis

An information-theoretic analysis of a multi-keyhole channel, which includes a number of statistically independent keyholes with possibly different correlation matrices, is given. When the number of keyholes or/and the number of Tx/Rx antennas is large, there is an equivalent Rayleigh-fading channel such that the outage capacities of both channels are asymptotically equal. In the case of a large number of antennas and for a broad class of fading distributions, the instantaneous capacity is shown to be asymptotically Gaussian in distribution, and compact, closed-form expressions for the mean and variance are given. Motivated by the asymptotic analysis, a simple, full-ordering scalar measure of spatial correlation and power imbalance in MIMO channels is introduced, which quantifies the negative impact of these two factors on the outage capacity in a simple and well-tractable way. It does not require the eigenvalue decomposition, and has the full-ordering property. The size-asymptotic results are used to prove Telatars conjecture for semi-correlated multi-keyhole and Rayleigh channels. Since the keyhole channel model approximates well the relay channel in the amplify-and-forward mode in certain scenarios, these results also apply to the latter.

On the outage capacity distribution of correlated keyhole MIMO channels

Keyhole MIMO channels, which were predicted theoretically and also observed experimentally, have recently received significant attention as they may appear in some practically-important propagation scenarios. This paper concentrates on a capacity study of such channels. Closed-form expressions for the instantaneous SNR and outage capacity distributions of a spatially correlated keyhole MIMO channel are given. The case of non-singular correlation matrices with distinct eigenvalues is considered in detail. When the number of Tx (Rx) antennas is large, the correlated keyhole channel tends asymptotically to the Rayleigh diversity channel with a single Tx (Rx) and multiple Rx (Tx) antennas. The outage capacity at low outage probabilities and the diversity order of the keyhole channel is upper-bounded by that of the equivalent Rayleigh diversity channel. The asymptotic outage capacity distribution, when the numbers of Tx and Rx antennas are both large, is Gaussian under general conditions on the correlation (the average SNR affects the mean and the correlation affects the variance). The Gaussian approximation is accurate already for a reasonably small number of antennas. Using the single-parameter exponential correlation matrices, we show that the outage capacity at low outage probabilities decreases with correlation

Comments on "Asymptotic eigenvalue distributions and capacity for MIMO channels under correlated fading

A stronger and general sufficient condition for the asymptotic normality of MIMO channel eigenvalues and its capacity is given. Physical interpretation of this condition is discussed. Simple alternative conditions, which do not require eigenvalue decomposition, are proposed. It is demonstrated that some popular correlation matrix models satisfy these conditions. In many cases, the convergence to the asymptotic normality is at least as 1/radic(nt), where nt is the number of Tx antennas.

Statistical analysis of a measured MIMO channel

In this paper we discuss an approach to a rigorous statistical analysis of a measured indoor MIMO channel. The channel characteristics such as outage capacity and channel gain distributions, transmit and receive correlations, and the frequency response are statistically analysed under the constraint of limited data available. We also compare the measured channel to the known analytical MIMO channel models in terms of the mean and outage capacity distributions. As the result, we show, in a statistically rigorous way, that the measured channel is a nondegenerate frequency-selective Rayleigh MIMO channel with significant TX and RX correlations.

Diversity-Multiplexing Tradeoff in the Low-SNR Regime

An extension of the popular diversity-multiplexing tradeoff framework to the low-SNR (or wideband) regime is proposed. The concept of diversity gain is shown to be redundant in this regime since the outage probability is SNR-independent and depends on the multiplexing gain and the channel power gain statistics only. The outage probability under the DMT framework is obtained in an explicit, closed form for a broad class of channels. The low and high-SNR regime boundaries are explicitly determined for the scalar Rayleigh-fading channel, indicating a significant limitation of the SNR-asymptotic DMT when the multiplexing gain is small.

Diversity-multiplexing tradeoff and outage probability in MIMO relay channels

MIMO single-relay fading channels are studied, where the source and destination are equipped with multiple antennas and the relay has a single one. Compact closed-form expressions are obtained for the outage probability under i.i.d. and correlated Rayleigh-fading links. Insightful high-SNR approximations are derived, which show the impact of the number of antennas, correlation, relay noise, relaying protocol, etc. Diversity-multiplexing tradeoff (DMT) is obtained for a broad class of fading distributions, including, as special cases, Rayleigh, Rice, Nakagami, Weibull, which may be non-identical, spatially correlated and/or non-zero mean. The DMT is shown to depend not on a particular fading distribution, but rather on its polynomial behavior near zero. It turns out to be the same for the simple “amplify-and-forward” protocol and more complicated “decode-and-forward” one with capacity achieving codes, i.e. the full processing capability at the relay does not help to improve the DMT. However, we also emphasize significant difference between the SNR-asymptotic DMT and the finite-SNR outage performance: while the former is not improved by using an extra antenna on either side, the latter can be significantly improved and, in particular, an extra antenna can be traded-off for a full processing capability at the relay.

Statistical approach to MIMO capacity analysis in a fading channel

Theoretical analysis of the MIMO outage capacity distribution for an uncorrelated Rayleigh fading channel reveals that the Gaussian distribution is a good approximation, especially for large a number of antennas. The question as to whether the measured channel capacity distribution is close to the theoretical one has not been answered yet in a satisfactory way. We address this problem by developing a statistically-rigorous procedure (in terms of hypothesis testing) for the analysis of the mean and outage capacities of MIMO channels (both, theoretical models and measured) with the main goal being to compare the theoretical and measured capacity distribution, especially the validity of Gaussian approximation for the measured outage capacity. As we demonstrate, there is a tight lower bound on the amount of measured data necessary to provide a unique answer with high confidence probability. Additionally, based on the procedure above, we develop guidelines for future measurements and Monte-Carlo simulations in terms of accuracy.