Rate-distortion behavior at low distortion for densely sampled Gaussian data

It is well known that for discrete-time, stationary sources, most lossy source coding techniques have operational rate-distortion functions that approach the Shannon ratedistortion function with respect to squared error to within an additive constant as distortion approaches zero. With the goal of investigating similar phenomena for continuous-time sources, this paper investigates the low-distortion performance of distributed coding of continuous-time, stationary, Gaussian sources based on high-rate sampling. It is found that for bandlimited sources and nonbandlimited sources whose spectra have sufficiently light, e.g., exponentially decreasing, tails, distributed source coding is asymptotically as good as centralized coding in the small distortion regime. On the other hand, for spectra with tails that decay as a power (greater than one) of frequency, it is found that for small distortions the distributed rate-distortion function is a constant times larger than the Shannon rate-distortion, where the constant decreases as the power increases. For example, it is approximately 1.2 when the power is 2. The conclusion is that for a stationary Gaussian source and asymptotically small distortion, the ratio of the distributed to centralized rate-distortion function is a function of the weight of the tail of the source spectrum. In the process of finding the ratio, the low distortion form of the centralized rate-distortion function is found for sources whose spectra have exponential and power law tails.