The Distortion-Rate Function of Sampled Wiener Processes

TL;DR

This paper derives a closed-form expression for the optimal tradeoff among sampling rate, bitrate, and distortion when reconstructing a Wiener process from quantized samples, revealing performance loss due to sampling constraints.

Contribution

It provides a novel closed-form formula for the distortion-rate tradeoff of sampled Wiener processes, linking spectral properties to encoding performance.

Findings

Approximate 20% increase in distortion when using one bit per sample.

The ratio between the optimal sampled and un-sampled distortion functions depends only on bits per sample.

Nearly optimal encoding performance is achievable without knowledge of the sampling rate.

Abstract

We consider the recovery of a continuous-time Wiener process from a quantized or lossy compressed version of its uniform samples under limited bitrate and sampling rate. We derive a closed form expression for the optimal tradeoff among sampling rate, bitrate, and quadratic distortion in this setting. This expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite-rank operators associated with the optimal estimator of the Wiener process from its samples. We show that the ratio between this expression and the standard distortion rate function of the Wiener process, describing the optimal tradeoff between bitrate and distortion without a sampling constraint, is only a function of the number of bits per sample. For example using one bit per sample on average, the expected distortion is approximately 1.2 times the standard distortion rate function, indicating a performance loss of about 20% due to sampling. We next consider the distortion when the continuous-time process is estimated from the output of an encoder that is optimal with respect to the discrete-time samples. We show that while the latter is strictly greater than the distortion under optimal encoding, the ratio between the two does not exceed 1.027. We therefore conclude that nearly optimal performance is attained even if the encoder is unaware of the sampling rate and encodes the samples without taking into account the continuous-time underlying process.