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This paper investigates the asymptotic expansion for the maximum rate offixed-length codes over a parallel Gaussian channel with feedback under thefollowing setting: A peak power constraint is imposed on every transmittedcodeword, and the average error probability of decoding the transmitted messageis non-vanishing as the blocklength increases. It is well known that thepresence of feedback does not increase the first-order asymptotics of thechannel, i.e., capacity, in the asymptotic expansion, and the closed-formexpression of the capacity can be obtained by the well-known water-fillingalgorithm. The main contribution of this paper is a self-contained proof of anupper bound on the second-order asymptotics of the parallel Gaussian channelwith feedback. The proof techniques involve developing an information spectrumbound followed by using Curtiss' theorem to show that a sum of dependent randomvariables associated with the information spectrum bound converges indistribution to a sum of independent random variables, thus facilitating theuse of the usual central limit theorem. Combined with existing achievabilityresults, our result implies that the presence of feedback does not improve thesecond-order asymptotics.

A Tight Upper Bound on the Second-Order Coding Rate of the Parallel Gaussian Channel with Feedback