Abstract-We study the high-SNR capacity of generic MIMO Rayleigh block-fading channels in the noncoherent setting where neither transmitter nor receiver has a priori channel state information but both are aware of the channel statistics. In contrast to the well-established constant block-fading model, we allow the fading to vary within each block with a temporal correlation that is "generic" (in the sense used in the interferencealignment 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 blockfading 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.

Entropy and differential entropy are important quantities in information theory. A tractable extension to singular random variables-which are neither discrete nor continuoushas not been available so far. Here, we present such an extension for the practically relevant class of integer-dimensional singular random variables. The proposed entropy definition contains the entropy of discrete random variables and the differential entropy of continuous random variables as special cases. We show that it transforms in a natural manner under Lipschitz functions, and that it is invariant under unitary transformations. We define joint entropy and conditional entropy for integer-dimensional singular random variables, and we show that the proposed entropy conveys useful expressions of the mutual information. As first applications of our entropy definition, we present a result on the minimal expected codeword length of quantized integerdimensional singular sources and a Shannon lower bound for integer-dimensional singular sources.Index Terms-Information entropy, rate distortion theory, Shannon lower bound, singular random variables, source coding.

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Abstract-We derive a lower bound on the capacity pre-log of a temporally correlated Rayleigh block-fading multiple-input multiple-output (MIMO) channel with T transmit antennas and R receive antennas in the noncoherent setting (no a priori channel knowledge at the transmitter and the receiver). In this model, the fading process changes independently across blocks of length L and is temporally correlated within each block for each transmit-receive antenna pair, with a given rank Q of the corresponding correlation matrix. Our result implies that for almost all choices of the coloring matrix that models the temporal correlation, the pre-log can be lower-bounded bythat R is sufficiently large. The widely used constant block-fading model is equivalent to the temporally correlated block-fading model with Q = 1 for the special case when the temporal correlation for each transmit-receive antenna pair is the same, which is unlikely to be observed in practice. For the constant block-fading model, the capacity pre-log is given by T (1 − T /L), which is smaller than our lower bound for the case Q = 1. Thus, our result suggests that the assumptions underlying the constant blockfading model lead to a pessimistic result for the capacity pre-log.