# The Arbitrarily Varying Channel with Colored Gaussian Noise

## TL;DR

This paper investigates the capacity of arbitrarily varying channels with colored Gaussian noise, extending classical models by incorporating frequency domain analysis and game-theoretic insights, and demonstrating the suboptimality of scalar coding.

## Contribution

It introduces capacity results for AVCs with colored Gaussian noise using double water filling in the frequency domain, connecting game theory and capacity analysis.

## Key findings

- Deterministic and random code capacities are characterized for various AVC models.
- Double water filling in frequency domain is optimal for AVC with colored Gaussian noise.
- Scalar coding is suboptimal for the arbitrarily varying Gaussian product channel.

## Abstract

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.