The Water Filling Game
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We consider the Gaussian arbitrarily varying product channel (GAVPC), and the arbitrarily varying channel (AVC) with colored Gaussian noise. The random code capacity is obtained by solving an optimization min-max problem, which is interpreted as a two-player zero-sum game, played by the user and the jammer, where their strategies are the power allocations. The optimal power allocations are given by a "double" water filling solution, where the jammer performs water filling first, attempting to whiten the overall noise as much as possible, and then the user performs water filling taking into account the total interference power, which is contributed by both the channel noise and the jamming signal. As in the case of the standard Gaussian AVC, the deterministic code capacity is discontinuous in the input constraint, and depends on which of the input or state constraint is higher. As opposed to Shannon's classic water filling solution, it is observed that deterministic coding using independent codes is suboptimal for the GAVPC. We further extend our results to the AVC with colored Gaussian noise, where double water filling is performed in the frequency domain.
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