Quadratically Constrained Channels with Causal Adversaries

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, ldots, x- n) through a communication channel. Alice and Bob do not share common randomness. Knowing Alice's encoding strategy, a jammer James chooses a real-valued length-n adversarial noise sequence mathbf s=(s- 1, ldots, s- n) in a causal manner: each s- t (1leq tleq n) can only depend on (x- 1, ldots, x- t). Bob receives y, the sum (over mathbb R) of Alice's transmission x and James' jamming vector 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 such that sum- t=1 nx- t 2leq nP and sum- t=1 n s- t 2leq nN. In this work, we characterize the channel capacity for such a channel as the limit superior of the optimal values C- nleft(frac P Nright) of a series of optimizations. Upper and lower bounds on C- nleft(frac P Nright) 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.