Coherent multiple-antenna block-fading channels at finite blocklength

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. 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. 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 capacities of $n_t\times n_r$ and $n_r \times n_t$ antenna configurations coincide (under fixed received power), the coding delay can be quite sensitive to this switch. For example, at the received SNR of $20$ dB the $16\times 100$ system achieves capacity with codes of length (delay) which is only $60\%$ of the length required for the $100\times 16$ system. Another interesting implication is that for the MISO channel, the dispersion-optimal 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.