We derive bounds on the noncoherent capacity of wide-sense stationary uncorrelated scattering (WSSUS) channels that are selective both in time and frequency, and are underspread, i.e., the product of the channel's delay spread and Doppler spread is small. For input signals that are peak constrained in time and frequency, we obtain upper and lower bounds on capacity that are explicit in the channel's scattering function, are accurate for a large range of bandwidth and allow to coarsely identify the capacity-optimal bandwidth as a function of the peak power and the channel's scattering function. We also obtain a closed-form expression for the first-order Taylor series expansion of capacity in the limit of large bandwidth, and show that our bounds are tight in the wideband regime. For input signals that are peak constrained in time only (and, hence, allowed to be peaky in frequency), we provide upper and lower bounds on the infinite-bandwidth capacity and find cases when the bounds coincide and the infinite-bandwidth capacity is characterized exactly. Our lower bound is closely related to a result by Viterbi (1967).The analysis in this paper is based on a discrete-time discrete-frequency approximation of WSSUS time-and frequency-selective channels. This discretization explicitly takes into account the underspread property, which is satisfied by virtually all wireless communication channels.

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We derive bounds on the noncoherent capacity of a very general class of multiple-input multiple-output channels that allow for selectivity in time and frequency as well as for spatial correlation. The bounds apply to peak-constrained inputs; they are explicit in the channel's scattering function, are useful for a large range of bandwidth, and allow to coarsely identify the capacityoptimal combination of bandwidth and number of transmit antennas. Furthermore, we obtain a closed-form expression for the first-order Taylor series expansion of capacity in the limit of infinite bandwidth. From this expression, we conclude that in the wideband regime: (i) it is optimal to use only one transmit antenna when the channel is spatially uncorrelated; (ii) rank-one statistical beamforming is optimal if the channel is spatially correlated; and (iii) spatial correlation, be it at the transmitter, the receiver, or both, is beneficial.Index Terms-Noncoherent capacity, MIMO systems, underspread channels, wideband channels. Contributions:We consider a point-to-point MIMO channel model where each component channel between a given transmit antenna and a given receive antenna is underspread [13] and satisfies the standard wide-sense stationary uncorrelatedscattering (WSSUS) assumption [14]; hence, our channel model allows for selectivity in time and frequency. We assume that the component channels are spatially correlated according to the separable correlation model [8], [9] and that they are characterized by the same scattering function; furthermore, the transmit signal is peak constrained. On the basis of a discrete-time, discretefrequency approximation of said channel model that is enabled by the underspread property [15], we obtain the following results:

In this chapter, we are interested in the ultimate limit on the rate of reliable communication over Rayleigh-fading channels that satisfy the wide-sense stationary (WSS) and uncorrelated scattering (US) assumptions and are underspread [Bel63,Ken69]. Therefore, the natural setting is an information-theoretic one, and the performance metric is channel capacity [CT91, Gal68].The family of Rayleigh-fading underspread WSSUS channels (reviewed in Chapter 1) constitutes a good model for real-world wireless channels: their stochastic properties, like amplitude and phase distributions match channel measurement results [SB07,Sch09]. The Rayleigh-fading and the WSSUS assumptions imply that the stochastic properties of the channel are fully described by a two-dimensional power spectral density (PSD) function, often referred to as scattering function [Bel63]. The underspread assumption implies that the scattering function is highly concentrated in the delay-Doppler plane.To analyze wireless channels with information-theoretic tools, a system model, not just a channel model, needs to be specified. A system model is more comprehensive than a channel model because it defines, among other parameters, the transmit-power constraints and the channel knowledge available at the transmitter and the receiver. The choice of a 1