The Second-Order Coding Rate of the MIMO Quasi-Static Rayleigh Fading Channel

Abstract

The second-order coding rate of the multiple-input multiple-output (MIMO) quasi-static Rayleigh fading channel is studied. We tackle this problem via an information-spectrum approach and statistical bounds based on recent random matrix theory techniques. We derive a central limit theorem (CLT) to analyze the information density in the regime where the block-length n and the number of transmit and receive antennas K and N , respectively, grow simultaneously large. This result leads to the characterization of closed-form upper and lower bounds on the optimal average error probability when the coding rate is within O(1/ √ nK) of the asymptotic capacity. Index Terms-Finite block-length, second-order coding rate, error probability, quasi-static fading channel, block-fading channel , MIMO, information spectrum, random matrix theory.

Gallager Bound for MIMO Channels: Large- $N$ Asymptotics

“…Our large deviation result is valid for all normalized rates 0 < r < r erg . When we evaluate the error exponent for small |r erg − r| ≪ 1, our results match with the upper bound obtained by [9]. While the asymptotic limit of large antenna numbers is somewhat idealized, it is known from other works, e.g.…”

“…These results are of a central-limittheoretic nature, in that they are valid for large blocklengths with the rate converging to the ergodic rate at a fixed error probability. Similar results were obtained for MIMO systems in [9], where the number of antennas also goes to infinity at a fixed ratio with T . In contrast to the Gallager bound this approach does not capture the tails of the error probability, i.e.…”

“…Applying techniques from random matrix theory, we obtain analytic expressions of the error exponent when the length of the codeword increases to infinity at a fixed ratio with the antenna array dimensions. Analyzing its behavior at rates close to the ergodic rate, we find that the Gallager error bound becomes asymptotically close to an upper error bound obtained recently by Hoydis et al 2015. We also obtain an expression for the Gallager exponent in the case when the codelength spans several Rayleigh fading blocks, hence taking into account the situation when the channel varies during each transmission.…”

Saddlepoint Approximations for Short-Packet Wireless Communications

“…The codewords satisfy the power constraint 3 2 The assumption that H and W take on independent realizations over successive coherence intervals is critical for the results obtained in this paper, namely, Proposition 1 in Section IV and the saddlepoint expansions presented in Section V that follow from it. In contrast, the assumption that H is zero-mean Gaussian is not critical since Proposition 1 applies to any fading distribution for which (17) and (18) are satisfied. 3 While in the information and communication theory literature, it is more common to impose a power constraint per codeword X L , practical systems typically require a per-coherence-interval constraint.…”

“…Indeed, it can be shown that the family of random variables (29) implies that the first condition (17) required for Proposition 1 and Corollary 2 is satisfied. Regarding the second condition (18), it can be observed that V s (ρ)…”