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 x=(x_1, ..., 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 s=(s_1, ..., s_n) in a causal manner: each s_t (1 ≤ t ≤ n) can only depend on (x_1, ..., x_t). Bob receives y, the sum (over 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 Σ_(t=1)^n x_t^2 ≤ nP and Σ_(t=1)^n s_t^2 ≤ nN. In this work, we characterize the channel capacity for such a channel as the limit superior of the optimal values C_n([P/N]) of a series of optimizations. Upper and lower bounds on C_n([P/N]) 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