Capacity of noncoherent time-selective Rayleigh-fading channels

The capacity of noncoherent time-selective Rayleigh-fading channels is studied under various models for the variations in time. The study includes both single-input and single-output (SISO) and multiple-input and multiple-output (MIMO) systems. A block-fading model is first considered where the channel changes correlatively over each block period of length T, and independently across blocks. The predictability of the channel is characterized through the rank Q of the correlation matrix of the vector of channel gains in each block. This model includes, as special cases, the standard block-fading model where the channel remains constant over block periods (Q = 1), and models where the fading process has finite differential entropy rate (Q = T). The capacity is initially studied for long block lengths and some straightforward but interesting asymptotes are established. For the case where Q is kept fixed as T → ∞, it is shown that the non-coherent capacity converges to the coherent capacity. For the case where both T, Q → ∞, with Q/T being held constant, a bound on the capacity loss due to channel unpredictability is established. The more interesting scenario of large signal-to-noise ratio (SNR) is then explored in detail. For SISO systems, useful upper and lower bounds on the large SNR asymptotic capacity are derived, and it is shown that the capacity grows logarithmically with SNR with a slope of _T/ ^T-Q, for Q < T. Next, in order to facilitate the analysis of MIMO systems, the rank-Q block-fading model is specialized to the case where each T-symbol block consists of Q subblocks of length L, with the channel remaining constant over each subblock and changing correlatively across subblocks. For this model, it is shown that the log SNR growth behavior of the capacity is the same as that of the standard block-fading model with block length L. Finally, the SISO and MIMO channel models are generalized to allow the fading process to be correlated across blocks in a stationary and ergodic manner. It is shown that the log SNR growth behavior of the capacity is not affected by the correlation across blocks.