Information-theoretic analysis of noncoherent block-fading channels and singular random variables

Abstract

Characterizing the capacity, i.e., the maximally possible throughput, of a given communication channel is an important problem in information theory. In this dissertation, we study the capacity of block-fading channels in the noncoherent setting where neither the transmitter nor the receiver has a priori channel state information but both are aware of the channel statistics. We show that the number of degrees of freedom-which characterizes the channel capacity at high signal-to-noise ratio-conforms to an intuitive "dimension counting" argument based on the noiseless receive vectors, and we analyze two special settings in more detail. First, extending the well-established constant block-fading model, we consider the class of generic multiple-input multiple-output (MIMO) Rayleigh block-fading channels. In these channels,we allow the fading to vary within each block with a temporal correlation that is "generic" in the sense used in the interference-alignment literature. We show that the number of degrees of freedom of a generic MIMO Rayleigh block-fading channel with T transmit antennas and block length N is given by T(1 -1/N) provided that T < N and the number of receive antennas is at least T(N - 1)/(N - T). A comparison with the constant block-fading channel (where the fading is constant within each block) shows that, for large block lengths, generic correlation increases the number of degrees of freedom by a factor of up to four. Furthermore, we consider an oversampled continuous-time, time-selective, Rayleigh blockfading channel. Here, we show that sampling the filtered channel output at twice the symbol rate results in a significant increase in the number of degrees of freedom. The noiseless receive vectors of noncoherent block-fading channels and certain random variables arising in other applications are singular random variables, i.e., neither discrete nor continuous. This fact motivates the consideration of information-theoretic properties of integer-dimensional singular random variables in the second part of the thesis. For these random variables, no satisfactory definition of entropy is available. We provide a definition of entropy and show that it is a promising and useful extension of the established concepts of entropy and differential entropy. As possible applications of the proposed entropy definition, we present two new results in source coding. We show that the minimal expected binary codeword length of a quantized integer-dimensional singular random variable can be characterized by the proposed entropy to within an accuracy of one bit. Furthermore, we present a lower bound on the rate-distortion function of an integer-dimensional singular source; this bound depends on the source only via the entropy of the source.