On Outage Probability and Diversity-Multiplexing Tradeoff in MIMO Relay Channels

Fading MIMO relay channels are studied analytically, when the source and destination are equipped with multiple antennas and the relays have a single one. Compact closed-form expressions are obtained for the outage probability under i.i.d. and correlated Rayleigh-fading links. Low-outage approximations are derived, which reveal a number of insights, including the impact of correlation, of the number of antennas, of relay noise and of relaying protocol. The effect of correlation is shown to be negligible, unless the channel becomes almost fully correlated. The SNR loss of relay fading channels compared to the AWGN channel is quantified. The SNR-asymptotic 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, and is 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. There is however a 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. The results are extended to the multi-relay channels with selection relaying and typical outage events are identified.Comment: accepted by IEEE Trans. on Comm., 201

Asymptotic Capacity and Optimal Precoding Strategy of Multi-Level Precode & Forward in Correlated Channels

We analyze a multi-level MIMO relaying system where a multiple-antenna transmitter sends data to a multipleantenna receiver through several relay levels, also equipped with multiple antennas. Assuming correlated fading in each hop, each relay receives a faded version of the signal transmitted by the previous level, performs precoding on the received signal and retransmits it to the next level. Using free probability theory and assuming that the noise power at the relay levels - but not at the receiver - is negligible, a closed-form expression of the end-to-end asymptotic instantaneous mutual information is derived as the number of antennas in all levels grow large with the same rate. This asymptotic expression is shown to be independent from the channel realizations, to only depend on the channel statistics and to also serve as the asymptotic value of the end-to-end average mutual information. We also provide the optimal singular vectors of the precoding matrices that maximize the asymptotic mutual information : the optimal transmit directions represented by the singular vectors of the precoding matrices are aligned on the eigenvectors of the channel correlation matrices, therefore they can be determined only using the known statistics of the channel matrices and do not depend on a particular channel realization.Comment: 5 pages, 3 figures, to be published in proceedings of IEEE Information Theory Workshop 200

Capacity and Power Scaling Laws for Finite Antenna MIMO Amplify-and-Forward Relay Networks

In this paper, we present a novel framework that can be used to study the capacity and power scaling properties of linear multiple-input multiple-output (MIMO) $d\times d$ antenna amplify-and-forward (AF) relay networks. In particular, we model these networks as random dynamical systems (RDS) and calculate their $d$ Lyapunov exponents. Our analysis can be applied to systems with any per-hop channel fading distribution, although in this contribution we focus on Rayleigh fading. Our main results are twofold: 1) the total transmit power at the $n$th node will follow a deterministic trajectory through the network governed by the network's maximum Lyapunov exponent, 2) the capacity of the $i$th eigenchannel at the $n$th node will follow a deterministic trajectory through the network governed by the network's $i$th Lyapunov exponent. Before concluding, we concentrate on some applications of our results. In particular, we show how the Lyapunov exponents are intimately related to the rate at which the eigenchannel capacities diverge from each other, and how this relates to the amplification strategy and number of antennas at each relay. We also use them to determine the extra cost in power associated with each extra multiplexed data stream.Comment: 16 pages, 9 figures. Accepted for publication in IEEE Transactions on Information Theor

Ergodic Capacity Analysis of Amplify-and-Forward MIMO Dual-Hop Systems

This paper presents an analytical characterization of the ergodic capacity of amplify-and-forward (AF) MIMO dual-hop relay channels, assuming that the channel state information is available at the destination terminal only. In contrast to prior results, our expressions apply for arbitrary numbers of antennas and arbitrary relay configurations. We derive an expression for the exact ergodic capacity, simplified closed-form expressions for the high SNR regime, and tight closed-form upper and lower bounds. These results are made possible to employing recent tools from finite-dimensional random matrix theory to derive new closed-form expressions for various statistical properties of the equivalent AF MIMO dual-hop relay channel, such as the distribution of an unordered eigenvalue and certain random determinant properties. Based on the analytical capacity expressions, we investigate the impact of the system and channel characteristics, such as the antenna configuration and the relay power gain. We also demonstrate a number of interesting relationships between the dual-hop AF MIMO relay channel and conventional point-to-point MIMO channels in various asymptotic regimes.Comment: 40 pages, 9 figures, Submitted to to IEEE Transactions on Information Theor

Performance Analysis of Optimal Single Stream Beamforming in MIMO Dual-Hop AF Systems

This paper investigates the performance of optimal single stream beamforming schemes in multiple-input multiple-output (MIMO) dual-hop amplify-and-forward (AF) systems. Assuming channel state information is not available at the source and relay, the optimal transmit and receive beamforming vectors are computed at the destination, and the transmit beamforming vector is sent to the transmitter via a dedicated feedback link. Then, a set of new closed-form expressions for the statistical properties of the maximum eigenvalue of the resultant channel is derived, i.e., the cumulative density function (cdf), probability density function (pdf) and general moments, as well as the first order asymptotic expansion and asymptotic large dimension approximations. These analytical expressions are then applied to study three important performance metrics of the system, i.e., outage probability, average symbol error rate and ergodic capacity. In addition, more detailed treatments are provided for some important special cases, e.g., when the number of antennas at one of the nodes is one or large, simple and insightful expressions for the key parameters such as diversity order and array gain of the system are derived. With the analytical results, the joint impact of source, relay and destination antenna numbers on the system performance is addressed, and the performance of optimal beamforming schemes and orthogonal space-time block-coding (OSTBC) schemes are compared. Results reveal that the number of antennas at the relay has a great impact on how the numbers of antennas at the source and destination contribute to the system performance, and optimal beamforming not only achieves the same maximum diversity order as OSTBC, but also provides significant power gains over OSTBC.Comment: to appear in IEEE Journal on Selected Areas in Communications special issue on Theories and Methods for Advanced Wireless Relay