Quadratically Constrained Channels with Causal Adversaries

TL;DR

This paper characterizes the capacity of a communication channel with a causal adversarial jammer under quadratic power constraints, revealing that non-uniform power allocation can outperform uniform strategies in certain scenarios.

Contribution

It provides the first characterization of the capacity for channels with causal jamming adversaries under quadratic constraints, including analytical and numerical bounds.

Findings

Optimal capacity characterized as a limit of optimization problems

Non-uniform power allocation can achieve higher rates than uniform allocation

Bounds on capacity are provided both analytically and numerically

Abstract

We consider the problem of communication over a channel with a causal jamming adversary subject to quadratic constraints. A sender Alice wishes to communicate a message to a receiver Bob by transmitting a real-valued length-$n$ codeword $\mathbf{x}=x_1,...,x_n$ through a communication channel. Alice and Bob do not share common randomness. Knowing Alice's encoding strategy, an adversarial jammer James chooses a real-valued length-n noise sequence $\mathbf{s}=s_1,..,s_n$ in a causal manner, i.e., each $s_t (1<=t<=n)$ can only depend on $x_1,...,x_t$. Bob receives $\mathbf{y}$, the sum of Alice's transmission $\mathbf{x}$ and James' jamming vector $\mathbf{s}$, and is required to reliably estimate Alice's message from this sum. In addition, Alice and James's transmission powers are restricted by quadratic constraints $P>0$ and $N>0$. In this work, we characterize the channel capacity for such a channel as the limit superior of the optimal values of a series of optimizations. Upper and lower bounds on the optimal values are provided both analytically and numerically. Interestingly, unlike many communication problems, in this causal setting Alice's optimal codebook may not have a uniform power allocation - for certain SNR, a codebook with a two-level uniform power allocation results in a strictly higher rate than a codebook with a uniform power allocation would.