The future use of cellular radio systems incorporating relay links is viewed [1-3] as key to the satisfaction of ever-increasing demands for greater capacity, while maintaining installation and operating costs at manageable levels. However, because of the scarcity of published propagation measurements on relay channels, channel models that have heretofore been used to support the claimed advantages of systems incorporating relays must be considered as only provisional until more measured results are available. This paper therefore reports radio propagation measurements that were made to observe quasi-simultaneous characteristics on direct and indirect (relay) radio links in base-relay-mobile terminal configurations that result in an emulation of radio links in urban microcells. Statistics for both wide and narrowband channel parameters, models for narrowband fading, and comparisons of results for direct and indirect links pertinent to both amplify and forward (AF) and decode and forward (DF) relay systems are reported.

A procedure for the analysis of narrowband fast fading data with statistically non-stationary characteristics is proposed and applied to data obtained from wideband radio propagation experiments involving two-hop relay channels at 2.25 GHz. The reported experiments were conducted from two channel sounder base station sites in Ottawa to enable a study of the characteristics of relay links in urban microcell environments and of vehicle-to-vehicle communications in relay-based cellular networks. It is shown that empirically-derived cumulative probability distributions for narrowband fast fading on relay links during time intervals when such fading exhibits quasi-stationary characteristics conform closely to a Double-Rician model, the derivation of which is detailed. Analysis of the Double-Rician distribution shows that Double-Rician fading is deeper than Rician fading, unless the K-ratios of the Rician links comprising the relay link are highly dissimilar (having 10 dB difference or more). In such cases, fading on the relay link has a distribution that is very similar to that of the constituent link with the smallest K-ratio. While, in general, relaying is meant to increase average received SNR, it is shown that when the fast fading on the links constituting a relay channel is i.i.d., N-hop relaying reduces the specular-to-random power ratio on the relay link by N times, or significantly increases the depth of fading. This result indicates a need for significant gains at relay stations in some cases.

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.

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 Telatar's 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.

This paper provides a partial answer to the question: "what is the best angular density of multipath in MIMO channels?" using the size-asymptotic theory of Toeplitz matrices for uniform 2-D and 3-D antenna arrays. A Kronecker-type approximation of the array correlation structure is proposed and used to find the angular densities that completely eliminate correlation between any elements of antenna arrays and thus maximize the asymptotic MIMO capacity for a broad class of fading distributions. At half-wavelength spacing, the best angular density is shown to be non-uniform, which implies that the popular Clarke's (Jake's) model does not represent the best case scenario. The asymptotic results are validated via Monte-Carlo simulations, and a number of practical guidelines for antenna design and optimal orientation are provided.

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.

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.

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.

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), wherentis the number of Tx antennas.

Diversitymultiplexing tradeoff (DMT) presents a compact framework to compare various MIMO systems and channels in terms of the two main advantages they provide (i.e. high data rate and/or low error rate). This tradeoff was characterized asymptotically (SNR-> infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse (2003). The SNR-asymptotic DMT overestimates the finite-SNR one (R. Narasimhan, 2006). In this paper, using the recent results on the size-asymptotic (in the number of antennas) outage capacity distribution, we derive and analyze the finite-SNR DMT for a broad class of channels (not necessarily Rayleigh fading). Systems with unequal number of Tx and Rx antennas exhibit qualitatively-different behavior from those with equal number of antennas: while the size-asymptotic DMT of the latter converges to the SNR- asymptotic DMT as the SNR grows, that of the former does not. However, the size-asymptotic DMT does provide an accurate approximation of the true DMT at low to moderately-high SNR, even for modest number of antennas, and hence is complementary of the SNR-asymptotic DMT of Zheng and Tse. Combining these two, a new DMT is obtained that is accurate over the whole SNR range. A number of generic properties of the DMT that hold at any SNR, for any number of antennas (i.e. not only asymptotic, either in size or in SNR) and for any fading channel are given. In particular, we demonstrate that the linear interpolation of the DMT for fractional multiplexing gain in (Zheng and Tse 2003) does not hold at finite SNR. Extensive Monte-Carlo simulations validate the analysis and the conclusions.

Diversity-multiplexing tradeoff (DMT) presents a compact framework to compare various MIMO systems and channels in terms of the two main advantages they provide (i.e. high data rate and/or low error rate). This tradeoff was characterized asymptotically (SNR-> infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse [1]. The asymptotic DMT overestimates the finite-SNR one [2]. In this paper, using the recent results on the asymptotic (in the number of antennas) outage capacity distribution, we derive and analyze the finite- SNR DMT for a broad class of channels (not necessarily Rayleigh fading). Based on this, we give the convergence conditions for the asymptotic DMT to be approached by the finite-SNR one. The multiplexing gain definition is shown to affect critically the convergence point: when the multiplexing gain is defined via the mean (ergodic) capacity, the convergence takes place at realistic SNR values. Furthermore, in this case the diversity gain can also be used to estimate the outage probability with reasonable accuracy. The multiplexing gain definition via the high-SNR asymptote of the mean capacity (as in [1]) results in very slow convergence for moderate to large systems (as l/ln(SNR) circ2) and, hence, the asymptotic DMT cannot be used at realistic SNR values. For this definition, the high-SNR threshold increases exponentially in the number of antennas and in the multiplexing gain. For correlated keyhole channel, the diversity gain is shown to decrease with correlation and power imbalance of the channel. While the SNR-asymptotic DMT of Zheng and Tse does not capture this effect, the size-asymptotic DMT does.

Keyhole MIMO channels were predicted theoretically and also observed experimentally. However, they are not often encountered in practice since the assumption of a single propagation eigenmode is only a rough approximation of real propagation environments. To overcome this problem, the paper extends the single-keyhole channel model by introducing a "multi-keyhole channel", which includes a number of statistically independent keyholes. Correlated full-rank and rank-deficient multi-keyhole channels are considered in detail. It is shown that under some general conditions the full-rank multi-keyhole channel is asymptotically Rayleigh fading, if the number of keyholes is large. When the number of both Tx and Rx antennas is large, the asymptotic capacity of a rank-deficient multi-keyhole channel is a sum of the capacities of the equivalent single-keyhole channels. The outage capacity distribution of both full-rank and rank-deficient multi-keyhole channels is asymptotically Gaussian. Based on the asymptotic capacity analysis, full ordering scalar measure of MIMO channel correlation and power imbalance is introduced

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