Wireless Communications

Information Theory and Reliable Communication

Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.

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Random Matrix Theory and Wireless Communications

Accurate and tractable channel modeling is critical to realizing the full potential of antenna arrays in wireless communications. Current approaches represent two extremes: idealized statistical models representing a rich scattering environment and parameterized physical models that describe realistic scattering environments via the angles and gains associated with different propagation paths. However, simple rules that capture the effects of scattering characteristics on channel capacity and diversity are difficult to infer from existing models. We propose an intermediate virtual channel representation that captures the essence of physical modeling and provides a simple geometric interpretation of the scattering environment. The virtual representation corresponds to a fixed coordinate transformation via spatial basis functions defined by fixed virtual angles. We show that in an uncorrelated scattering environment, the elements of the channel matrix form a segment of a stationary process and that the virtual channel coefficients are approximately uncorrelated samples of the underlying spectral representation. For any scattering environment, the virtual channel matrix clearly reveals the two key factors affecting capacity: the number of parallel channels and the level of diversity. The concepts of spatial zooming and aliasing are introduced to provide a transparent interpretation of the effect of antenna spacing on channel statistics and capacity. Numerical results are presented to illustrate various aspects of the virtual framework.

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Deconstructing multiantenna fading channels

High-rate data communication over a multipath wireless channel often requires that the channel response be known at the receiver. Training-based methods, which probe the channel in time, frequency, and space with known signals and reconstruct the channel response from the output signals, are most commonly used to accomplish this task. Traditional training-based channel estimation methods, typically comprising linear reconstruction techniques, are known to be optimal for rich multipath channels. However, physical arguments and growing experimental evidence suggest that many wireless channels encountered in practice tend to exhibit a sparse multipath structure that gets pronounced as the signal space dimension gets large (e.g., due to large bandwidth or large number of antennas). In this paper, we formalize the notion of multipath sparsity and present a new approach to estimating sparse (or effectively sparse) multipath channels that is based on some of the recent advances in the theory of compressed sensing. In particular, it is shown in the paper that the proposed approach, which is termed as compressed channel sensing (CCS), can potentially achieve a target reconstruction error using far less energy and, in many instances, latency and bandwidth than that dictated by the traditional least-squares-based training methods.

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Compressed Channel Sensing: A New Approach to Estimating Sparse Multipath Channels

Distributed detection strategies for wireless sensor networks are studied under the assumption of spatially and temporally independent and identically distributed (i.i.d.) observations at the sensor nodes. Both intelligent (with knowledge of observation statistics) and dumb (oblivious of observation statistics) sensors are considered. Two types of communication channels are studied: a parallel access channel (PAC) in which each sensor has a dedicated additive white Gaussian noise (AWGN) channel to a decision center, and a multiple-access channel (MAC) in which the decision center receives a coherent superposition of the sensor transmissions. Our results show that the MAC yields significantly superior detection performance for any network power constraint. For intelligent sensors, uncoded (finite duration) communication of local log-likelihood ratios over the MAC achieves the optimal error exponent of the centralized (noise-free channel) benchmark as the number of nodes increases, even with sublinear network power scaling. Motivated by this result, we propose a distributed detection strategy for dumb sensors-histogram fusion-in which each node appropriately quantizes its temporal data and communicates its type or histogram to the decision center. It is shown that uncoded histogram fusion over the MAC is also asymptotically optimal under sublinear network power scaling with an additional advantage: knowledge of observation statistics is needed only at the decision center. Histogram fusion achieves exponential decay in error probability with the number of nodes even under a finite total network power. In principle, a vanishing error probability at a slower subexponential rate can be attained even with vanishing total network power in the limit. These remarkable power/energy savings with the number of nodes are due to the inherent beamforming gain in the MAC

Type-Based Decentralized Detection in Wireless Sensor Networks

This paper develops a general framework for communication over doubly dispersive fading channels via an orthogonal short-time Fourier (STF) basis. The STF basis is generated from a prototype pulse via time-frequency shifts. In general, the orthogonality between basis functions is destroyed at the receiver due to channel dispersion. The starting point of this work is a pulse scale adaptation rule first proposed by Kozek to minimize the interference between the basis functions. We show that the average signal-to-interference-and-noise (SINR) ratio associated with different basis functions is identical and is maximized by the scale adaptation rule. The results in this paper highlight the critical impact of the channel spread factor, the product of multipath and Doppler spreads, on system performance. Smaller spread factors result in lesser interference such that a scale-adapted STF basis serves as an approximate eigenbasis for the channel. A highly effective iterative interference cancellation technique is proposed for mitigating the residual interference for larger spread factors. The approximate eigendecomposition leads to an intuitively appealing block-fading interpretation of the channel in terms of time-frequency coherence subspaces: the channel is highly correlated within each coherence subspace whereas it is approximately independent across different subspaces. The block-fading model also yields an approximate expression for the coherent channel capacity in terms of parallel flat-fading channels. The deviation of the capacity of doubly dispersive channels from that of flat-fading channels is quantified by studying the moments of the channel eigenvalue distribution. In particular, the difference between the moments of doubly dispersive and flat-fading channels is proportional to channel spread factor. The results in this paper indicate that the proposed STF signaling framework is applicable for spread factors as large as 0.01.

Orthogonal time-frequency signaling over doubly dispersive channels

Spatial multiplexing techniques send independent data streams on different transmit antennas to maximally exploit the capacity of multiple-input multiple-output (MIMO) fading channels. Most existing multiplexing techniques are based on an idealized MIMO channel model representing a rich scattering environment. Realistic channels corresponding to scattering clusters exhibit correlated fading and can significantly compromise the performance of such techniques. In this paper, we study the design and performance of spatial multiplexing techniques based on a virtual representation of realistic MIMO fading channels. Since the nonvanishing elements of the virtual channel matrix are uncorrelated, they capture the essential degrees of freedom in the channel and provide a simple characterization of channel statistics. In particular, the pairwise-error probability (PEP) analysis for correlated channels is greatly simplified in the virtual representation. Using the PEP analysis, various precoding schemes are introduced to improve performance in virtual channels. Unitary precoding is proposed to provide robustness to unknown channel statistics. Nonunitary precoding techniques are proposed to exploit channel structure when channel statistics are known at the transmitter. Numerical results are presented to illustrate the attractive performance of the precoding techniques.

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Spatial multiplexing in correlated fading via the virtual channel representation

The dramatic linear increase in ergodic capacity with the number of antennas promised by multiple-input multiple-output (MIMO) wireless communication systems is based on idealized channel models representing a rich scattering environment. Is such scaling sustainable in realistic scattering scenarios? Existing physical models, although realistic, are intractable for addressing this problem analytically due to their complicated nonlinear dependence on propagation path parameters, such as the angles of arrival and delays. In this paper, we leverage a recently introduced virtual representation of physical models that is essentially a Fourier series representation of wide-band MIMO channels in terms of fixed virtual angles and delays. Motivated by physical considerations, we propose a D-connected model for correlated channels defined by a virtual spatial channel matrix consisting of D nonvanishing diagonals with independent and identically distributed (i.i.d.) Gaussian entries. The parameter D provides a meaningful and tractable measure of the richness of scattering. We derive general bounds for the coherent ergodic capacity and investigate capacity scaling with the number of antennas and bandwidth. In the large antenna regime, we show that linear capacity scaling is possible if D scales linearly with the number of antennas. This, in turn, is possible if the number of resolvable paths grows quadratically with the number of antennas. The capacity saturates for linear growth in the number of paths (fixed D). The ergodic capacity does not depend on frequency selectivity of the channel in the wide-band case. Increasing bandwidth tightens the bounds and hastens the convergence of scaling behavior. For large bandwidth, the capacity scales linearly with the signal-to-noise ratio (SNR) as well. We also provide an explicit characterization of the wide-band slope recently proposed by Verdu. Numerical results are presented to illustrate the key theoretical results.

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Capacity scaling and spectral efficiency in wide-band correlated MIMO channels

We consider a wireless network composed of three nodes and limited by the half-duplex and total power constraints. This formulation encompasses many of the special cases studied in the literature and allows for capturing the common features shared by them. Here, we focus on three special cases, namely 1) Relay Channel, 2) Multicast Channel, and 3) Conference Channel. These special cases are judicially chosen to reflect varying degrees of complexity while highlighting the common ground shared by the different variants of the three node wireless network. For the relay channel, we propose a new cooperation scheme that exploits the wireless feedback gain. This scheme combines the benefits of decode-and-forward and compress-and-forward strategies and avoids the idealistic feedback assumption adopted in earlier works. Our analysis of the achievable rate of this scheme reveals the diminishing feedback gain at both the low and high signal-to-noise ratio regimes. Inspired by the proposed feedback strategy, we identify a greedy cooperation framework applicable to both the multicast and conference channels. Our performance analysis reveals several nice properties of the proposed greedy approach and the central role of cooperative source-channel coding in exploiting the receiver side information in the wireless network setting. Our proofs for the cooperative multicast with side-information rely on novel nested and independent binning encoders along with a list decoder.

Ke Liu

Papers

Type-Based Decentralized Detection in Wireless Sensor Networks

Orthogonal time-frequency signaling over doubly dispersive channels

Spatial multiplexing in correlated fading via the virtual channel representation

Capacity scaling and spectral efficiency in wide-band correlated MIMO channels

The Three Node Wireless Network: Achievable Rates and Cooperation Strategies