Low SNR Capacity of Noncoherent Fading Channels

Wireless wideband channels typically include both time and frequency selective fading. One approach to modeling such channels is to partition the frequency band into narrow subbands, so that the fading is flat, but time-varying, within each subband. If the width of the subbands is approximately the coherent bandwidth of the channel, then they will experience approximately independent fading. The flat fading models used in this paper can be considered to be models for communication over a subband of a wideband wireless fading channel. The peak-power constraints that we impose on the signals can then be viewed as burstiness constraints in both the time and frequency domain for wideband communication, similar to those of [10,14]. However, in this paper, we consider hard peak constraints, rather than fourth moment constraints as in [10,14], and we consider the use of multiple antennas.

The recent work of Srinivasan and Varanasi [13] is closely related to this paper. It gives low SNR asymptotics of the capacity of MIMO channels with no side information for block fading channels, with peak and average-power constraints, with the peak constraints being imposed on individual antennas. One difference between [13] and this paper is that we assume continuous fading rather than block fading. In addition, we provide upper bounds on capacity rather than only asymptotic bounds as in [12,13]. We assume, however, that the fading processes are Rayleigh distributed, whereas the asymptotic bounds do not require such distributional assumption. The work of Rao and Hassibi [12] is also related to this paper. It gives low SNR asymptotics of the capacity of MIMO channels with no side information for block fading channels, but the peak constraints are imposed on coefficients in a particular signal representation, rather than as hard constraints on the transmitted signals.

The model in this paper considers both a peak constraint and an average power constraint. Upper bounds are given on the capacity which are valid for any ratio of these constraints, but the low SNR asymptotics focuses only on the case where the ratio is constant. The ratio is also held constant in the asymptotic analysis of Srinivasan and Varinasi [13]. The paper of Wu and Srikant [18] focuses on the asymptotic capacity and error exponent for a fixed peak constraint, as the average power goes to zero. The paper of Zheng et. al [19] considers a general scaling of the peak constraint to average power constraint, with the scaling depending also on the coherence time. For a fixed ratio of peak constraint to average power constraint, the capacity scales quadratically as SNR converges to zero, whereas for a fixed peak constraint, the capacity scales linearly with capacity as SNR converges to zero. Cases between these two extremes are investigated in [19]. For wideband cellular systems using OFDM modulation, the peak constraint is usually expressed in the time domain, because of the limitations on the linear range of transmit power amplifiers. In such case, the peak power constraint in a particular frequency is not severe, so letting the peak constraint be constant or letting it converge to zero more slowly than the average power may be most appropriate. In cases in which interference with other users within the same band is especially important, for example for use of unlicensed or secondary spectrum, a peak constraint of the same order of magnitude as the average power constraint, as considered in this paper, may be the most relevant. The papers [13,18,19] consider block fading channels, whereas a stationary, correlated fading channel model is adopted here.

The capacity of noncoherent stationary flat fading channels at high SNR was studied in [20][21][22][23], and the capacity of delay spread channels at high SNR was recently studied in [24]. For regular fading processes [20] demonstrated a connection between the high-SNR capacity growth and the error in predicting the fading process from noiseless observations of its past, whereas for nonregular fading [22] demonstrated such a connection to the error in predicting the fading process from noisy observations of its past in the low observation noise regime. In this paper we point to an analogous connection between the low SNR asymptotic capacity and the error in predicting the fading process from very noisy observations of its past. We show that these prediction errors in the high observation noise regime determine the asymptotic low SNR capacity of SISO channels and MIMO channels with sum

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