Quadratically Constrained Channels with Causal Adversaries - Li, Tongxin; Dey, Bikash Kumar; Jaggi, Sidharth; Langberg, Michael; Sarwate, Anand D. (2018 IEEE International Symposium on Information Theory (ISIT), 2018)
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 \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}^{n}x_{t}^{2}≤ nP and \sum_{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}\left(\frac{P}{N}\right) of a series of optimizations. Upper and lower bounds on C_{n}\left(\frac{P}{N}\right) 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.