On the Capacity of the Heat Channel, Waterfilling in the Time-Frequency Plane, and a C-NODE Relationship

The heat channel is defined by an analog filter and a subsequent measurement of the filter output signal perturbed by additive white Gaussian noise. The filter is related to the heat kernel of the quantum mechanical harmonic oscillator, so the name of the channel. The channel is modeled as an infinite-dimensional vector Gaussian channel and the capacity in terms of average energy of the input signal is derived along with a method of capacity achieving signaling for the continuous-time channel. Then, a characterization of the capacity by water-filling in the time-frequency plane is stated and proved. We compare our findings with a classical capacity result of Gallager. A related problem in rate distortion theory is investigated to some extent. Finally, a second formula for the capacity of the heat channel based on average energy of the measured perturbed filter output signal is derived. The result is interpreted in context of estimation theory and a parallel to the I-MMSE relationship due to Guo et al. is presented connecting the capacity of the heat channel with an estimation error for the output.