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تسجيل مستخدم جديدQuadratically Constrained Channels with Causal Adversaries
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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 Alices 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 Alices transmission $mathbf{x}$ and James jamming vector $mathbf{s}$, and is required to reliably estimate Alices message from this sum. In addition, Alice and Jamess 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 Alices 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.
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