In this paper, we consider a channel model that is often used to describe mobile wireless scenarios: multiple-antenna additive white Gaussian noise channels subject to random (fading) gains with full channel state information at the receiver. The dynamics of the fading process are approximated by a piecewise-constant process (frequency non-selective isotropic block fading). This paper addresses the finite block-length fundamental limits of this channel model. Specifically, we give a formula for the channel dispersion-a quantity governing the delay required to achieve capacity. The multiplicative nature of the fading disturbance leads to a number of interesting technical difficulties that required us to enhance traditional methods for finding the channel dispersion. Alas, one difficulty remains: the converse (impossibility) part of our result holds under an extra constraint on the growth of the peak-power with blocklength. Our results demonstrate, for example, that while the capacities of nt × nr and nr × nt antenna configurations coincide (under fixed received power), the coding delay can be sensitive to this switch. For example, at the received SNR of 20 dB, the 16 × 100 system achieves capacity with codes of length (delay) which is only 60% of the length required for the 100×16 system. Another interesting implication is that for the MISO channel, the dispersionoptimal coding schemes require employing orthogonal designs such as Alamouti's scheme-a surprising observation considering the fact that Alamouti's scheme was designed for reducing demodulation errors, not improving coding rate. Finding these dispersion-optimal coding schemes naturally gives a criteria for producing orthogonal design-like inputs in dimensions where orthogonal designs do not exist.

It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finiteblocklength channel-coding converse bound by Polyanskiy and Verdú, which involves the ratio of two Neyman-Pearson β functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson β functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verdú's wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.

In this paper we consider a channel model that is often used to describe the mobile wireless scenario: multiple-antenna additive white Gaussian noise channels subject to random (fading) gain with full channel state information at the receiver. Dynamics of the fading process are approximated by a piecewise-constant process (frequency non-selective isotropic block fading). This work addresses the finite blocklength fundamental limits of this channel model. Specifically, we give a formula for the channel dispersion - a quantity governing the delay required to achieve capacity - and present achievability and (partial) converse bounds. Multiplicative nature of the fading disturbance leads to a number of interesting technical difficulties that required us to enhance traditional methods for finding channel dispersion. Knowledge of channel dispersion opens the possibility for studying the impact of channel dynamics, antenna selection rules, etc on the communication rate.

A channel coding achievability bound expressed in terms of the ratio between two Neyman-Pearson β functions is proposed. This bound is the dual of a converse bound established earlier by Polyanskiy and Verdú (2014). The new bound turns out to simplify considerably the analysis in situations where the channel output distribution is not a product distribution, for example due to a cost constraint or a structural constraint (such as orthogonality or constant composition) on the channel inputs. Connections to existing bounds in the literature are discussed. The bound is then used to derive 1) the channel dispersion of additive non-Gaussian noise channels with random Gaussian codebooks, 2) the channel dispersion of an exponential-noise channel, 3) a second-order expansion for the minimum energy per bit of an additive white Gaussian noise channel, and 4) a lower bound on the maximum coding rate of a multiple-input multiple-output Rayleigh-fading channel with perfect channel state information at the receiver, which is the tightest known achievability result.

This work addresses the question of finite block-length fundamental limits of coherently demodulated multi-antenna channels, subject to frequency non-selective isotropic fading. Specifically we present achievability bound for the channel dispersion - a quantity known to determine the delay required to achieve capacity. It is shown that a commonly used isotropic Gaussian input, which is only one of many possible capacity achieving distributions, is suboptimal. Optimal inputs minimizing channel dispersion turn out to include a family of modulation techniques known as orthogonal designs (in particular, Alamouti's scheme). For 8 transmit antennas numerical evaluation shows that up to 40% of additional penalty in delay is incurred by using isotropic codewords (compared to dispersion-optimal architecture exploiting transmit diversity).