UC San Diego

Gaussian distribution is often assumed for the signals in the analysis of and in the design of communication systems and signal processing systems, although Gaussian signals can never be realized in practice. Indeed, Gaussian distribution has proven optimal in many problems of communication and signal processing, e.g., the channel input with a Gaussian distribution achieves the channel capacity of a communication channel. Moreover, many problems of communication and signal processing are mathematically tractable when the Gaussian signal distribution is assumed. This dissertation is concerned with the performance of the systems (or algorithms) in which the actual signal distribution is not Gaussian. In particular, we study the performance loss of an optimal system with non-Gaussian signals in comparison with the system performance with the optimal Gaussian signals. In addition, when the actual signal distribution is non- Gaussian, we investigate the performance of the practical algorithm that has been derived under the assumption that the signal distribution is Gaussian. Two well-known problems in communication and signal processing are investigated. First, we study a communication problem, in particular, the power allocation problem that minimizes the outage probability over a slow Rayleigh fading channel or maximizes the mutual information over a fast Rayleigh fading channel, where the channel input is equiprobable QAM signal constellations. The mercury/water-filling (MWF) power allocation is optimal for this problem, while the water-filling (WF) power allocation is optimal if the channel input is Gaussian rather than QAM signals. We show that WF performs close to MWF as long as the constellation size is appropriately chosen, more specifically, the MWF performance itself is not limited by having too small a constellation size. In addition, we study a simple practical power allocation policy, uniform power allocation with thresholding (UPAT) that assigns nonzero constant power only to a subset of the fading blocks. The UPAT can significantly alleviate the feedback overhead and the complexity compared to MWF and WF. We show that the optimal UPAT, namely, the UPAT with the optimal threshold, performs close to MWF as long as the constellation size is large enough. Next, we study a signal processing problem, in particular, the asymptotic performance limits of reliably recovering the support of block-sparse signals (including scalar-sparse signals as a special class) through an arbitrarily distributed random measurement matrix (including Gaussian) in a noisy setting. Sharp sufficient and necessary conditions for asymptotically reliable support recovery are derived in terms of the signal dimension, the number of nonzero blocks, the block size, the number of measurements, the distribution of the random measurement matrix, and signal-to-noise ratio (SNR) of each nonzero block. The results reveal the effect of the distribution of the random measurement matrix on the number of measurements required for asymptotically reliable support recovery. They also unveil how much we can potentially reduce the number of measurements required for asymptotically reliable support recovery, when a signal is block-sparse and its structure is known, by making use of the block-sparsity structure compared to treating the signal as being scalar-sparse