Waterfilling theorems in the time-frequency plane for the heat channel and a related source

The capacity of the heat channel, a linear time-varying (LTV) filter with additive white Gaussian noise (AWGN), is characterized by waterfilling in the time-frequency plane. Similarly, the rate distortion function for a related nonstationary source is characterized by reverse waterfilling in the time-frequency plane. The source is formed by the white Gaussian noise response of the same LTV filter as before. The proofs of both waterfilling theorems rely on a specific Szegö theorem for a positive definite operator associated with the filter. An essentially self-contained proof of the Szegö theorem is given. The waterfilling theorems compare well with classical results of Gallager and Berger. In case of the nonstationary source it is observed that the part of the classical power spectral density (PSD) is taken by the Wigner-Ville spectrum (WVS).