Capacity of noncoherent time-selective Rayleigh-fading channels

We study the capacity of noncoherent time-selective Rayleigh fading channels under various models for the variations in time. Our study includes both single-input and single-output (SISO) and multiple-input and multiple-output (MIMO) systems. We first consider a block fading model 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). We first study the capacity for long block-lengths and establish some straightforward but interesting asymptotes. For the case where Q is kept fixed as T → ∞, we show that the noncoherent capacity converges to the coherent capacity. For the case where both T, Q → ∞, with Q/T being held constant, we establish a bound on the capacity loss due to channel unpredictability. We then study the more interesting scenario of large signal-to-noise ratio (SNR). For SISO systems, we derive useful upper and lower bounds on the large SNR asymptotic capacity, and prove that the capacity grows logarithmically with SNR with a slope of T −Q T, for Q < T. Next, in order to facilitate the analysis of MIMO systems, we specialize the rank Q block fading model 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, we show that the log SNR growth behavior is the same as that of the standard block fading model with block length L. We then generalize the SISO and MIMO channel models to allow the fading process to be correlated across blocks in a stationary and ergodic manner. We show that the log SNR growth behavior of the capacity is not affected by the correlation across blocks.