The Distortion-Rate Function of Sampled Wiener Processes

The minimal distortion attainable in recovering the waveform of a continuous-time Wiener process from an encoded version of its uniform samples is considered. We first introduce a combined sampling and source coding problem for the Wiener process under uniform sampling, and prove an associated source coding theorem that characterizes the minimal distortion from samples. We then derive a closed-form expression for this minimal distortion, given in terms of a reverse waterfilling expression involving the asymptotic eigenvalue density of a sequence of finiterank operators. This expression allows us to determine the minimal distortion attainable for any prescribed sampling rate and number of bits per time unit. In additional, it can be used to determine the ratio between bitrate and sampling rate where the distortion due to sampling is of the same magnitude as the distortion due to lossy compression. For example, we show that using approximately 1 bit per sample, the distortion is about 1.12 times that of the distortionrate function of the Wiener process at the same bitrate. We also derive an upper bound on the minimal distortion, which is obtained by a sub-optimal compress-and-estimate remote source coding scheme. This bound implies that as the sampling rate goes to infinity, compress-and-estimate is sub-optimal up to a linear factor in the sampling interval.