Channel Coding for Gaussian Channels With Mean and Variance Constraints
Adeel Mahmood, Aaron B. Wagner · IEEE Transactions on Information Theory · 2025
We consider channel coding for Gaussian channels with the recently introduced mean and variance cost constraints. Through matching converse and achievability bounds, we characterize the optimal first- and second-order performance. The main technical contribution of this paper is an achievability scheme which uses random codewords drawn from a mixture of three uniform distributions on (n−1)-spheres of radiiR1,R2andR3, whereRi=O( √n) and |Ri−Rj| = O(1). To analyze such a mixture distribution, we prove a lemma giving a uniformO(logn) bound, which holds with high probability, on the log ratio of the output distributionsQcciandQccj, whereQcciis induced by a random channel input uniformly distributed on an (n− 1)-sphere of radiusRi. To facilitate the application of the usual central limit theorem, we also give a uniformO(logn) bound, which holds with high probability, on the log ratio of the output distributionsQcciandQ∗i, whereQ∗iis induced by a random channel input with i.i.d. components.