Coherent Multiple Antenna Block Fading Channels at Finite Blocklength
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 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. 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 <inline formula> <tex math notation="LaTeX"$n_{t}\times n_{r} /tex math>/inline formula> and <inline formula> <tex math notation="LaTeX"$n_{r} \times n_{t} /tex math>/inline formula> 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 <inline formula> <tex math notation="LaTeX"$16\times 100$ /tex math>/inline formula> system achieves capacity with codes of length (delay) which is only 60% of the length required for the <inline formula> <tex math notation="LaTeX"$100\times 16$ /tex math>/inline formula> 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.
IEEE Transactions on Information Theory