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 noncoherent 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-Q→T, for Q\n\n\t\t