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The heat channel is defined by a linear time-varying (LTV) filter withadditive white Gaussian noise (AWGN) at the filter output. The continuous-timeLTV filter is related to the heat kernel of the quantum mechanical harmonicoscillator, so the name of the channel. The channel's capacity is given inclosed form by means of the Lambert W function. Also a waterfilling theorem inthe time-frequency plane for the capacity is derived. It relies on a specificSzego theorem for which an essentially self-contained proof is provided.Similarly, the rate distortion function for a related nonstationary source isgiven in closed form and a (reverse) waterfilling theorem in the time-frequencyplane is derived. Finally, a second closed-form expression for the capacity ofthe heat channel based on the detected perturbed filter output signals ispresented. In this context, a precise differential connection between channelcapacity and the normalized optimal detection error (NODE) is revealed. ThisC-NODE relationship is compared with the well-known I-MMSE relationshipconnecting mutual information with the minimum mean-square error (MMSE) ofestimation theory.

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