Fading Channels: Capacity and Channel Coding Rate in the Finite-Blocklength Regime

Information-theoretic studies on the fundamental limits of communication over wireless fading channels typically rely on simplifying assumptions, such as perfect channel state information (CSI), infinite blocklength, and vanishing probability of error. Although these assumptions are reasonable for most of the current wireless communication systems, they may be inaccurate for next-generation wireless systems. Indeed, next-generation wireless systems will need to support a much wider range of features, such as ultra-high data rate, extremely low latency, and low energy consumption, for which the assumptions listed above may not be valid. This thesis investigates the fundamental limits of fading channels under a set of assumptions that are more relevant for future wireless systems. First, we characterize the capacity of Rayleigh block-fading multiple-input multiple-output (MIMO) channels with no a priori CSI at the transmitter and the receiver in the high signal-to-noise ratio regime. We show that unitary space time modulation, which is capacity-achieving for MIMO systems with a small number of antennas, is not capacity-achieving when the total number of antennas exceeds the coherence time of the fading channel, a situation that is relevant for large-MIMO systems. We also provide the input distribution that achieves the capacity of large-MIMO fading channels. Second, we study the maximal achievable rate for a given blocklength and error probability over MIMO quasi-static fading channels, subject to different power constraints on the transmitted codewords: the short-term (i.e., per-codeword) power constraint and the long-term (i.e., average-over-all-codeword) power constraint. For channels subject to a short-term power constraint, we prove that outage capacity---despite being an asymptotic quantity---is a sharp proxy for the finite-blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion---a quantity that measures the backoff from capacity in the finite-blocklength regime---is shown to be zero regardless of whether the fading realizations are available at the transmitter and/or the receiver. The situation is drastically different when a long-term power constraint is present. In this case, if the transmitter has perfect CSI, then the outage capacity is higher than in the short-term power constraint case. Approaching the outage capacity, however, requires codes with much longer blocklengths. In both cases, we develop easy-to-evaluate approximations for the maximal achievable rate and demonstrate their accuracy by comparison to nonasymptotic achievability and converse bounds. Finally, we investigate the minimum energy required to transmit $k$ information bits with a given reliability over a MIMO Rayleigh block-fading channel, with and without CSI at the receiver. It is well known that the ratio between the minimum energy per bit and the noise level converges to $-1.59$ dB as $k$ goes to infinity, regardless of whether CSI is available at the receiver or not. We show that lack of CSI at the receiver causes a slowdown in the speed of convergence to $-1.59$ dB as $k\to\infty$ compared to the case of perfect receiver CSI. Specifically, in the no-CSI case, the gap to $-1.59$ dB is proportional to $((\log k) /k)^{1/3}$, whereas when perfect CSI is available at the receiver, this gap is proportional to $1/\sqrt{k}$.