  We address the arbitrarily varying channel (AVC) with colored Gaussian noise.
The work consists of three parts. First, we study the general discrete AVC with
fixed parameters, where the channel depends on two state sequences, one
arbitrary and the other fixed and known. This model can be viewed as a
combination of the AVC and the time-varying channel. We determine both the
deterministic code capacity and the random code capacity. Super-additivity is
demonstrated, showing that the deterministic code capacity can be strictly
larger than the weighted sum of the parametric capacities.
  In the second part, we consider the arbitrarily varying Gaussian product
channel (AVGPC). Hughes and Narayan characterized the random code capacity
through min-max optimization leading to a "double" water filling solution.
Here, we establish the deterministic code capacity and also discuss the
game-theoretic meaning and the connection between double water filling and Nash
equilibrium. 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
scalar codes is suboptimal for the AVGPC.
  Finally, we establish the capacity of the AVC with colored Gaussian noise,
where double water filling is performed in the frequency domain. The analysis
relies on our preceding results, on the AVC with fixed parameters and the
AVGPC.
