Information rates of densely sampled Gaussian data
With mean-squared error D as a goal, it is well known that one may approach the rate-distortion function R(D) of a nonbandlimited, continuous-time Gaussian source by sampling at a sufficiently high rate, applying the Karhunen-Loeve transform to sufficiently long blocks, and then independently coding the transform coefficients of each type. Motivated by the question of the efficiency of dense sensor networks for sampling, encoding and reconstructing spatial random fields, this paper studies the following three cases. In the first, we consider a centralized encoding setup with a sample-transform-quantize scheme where the quantization is assumed to be optimal. In the second, we consider a distributed setup, where a spatio-temporal source is sampled and distributively encoded to be reconstructed at a receiver. We show that with ideal distributed lossy coding, dense sensor networks can efficiently sense and convey a field, in contrast to the negative result obtained by Marco et al. for encoders based on time- and space-invariant scalar quantization and ideal Slepian-Wolf distributed lossless coding. In the third, we consider a centralized setup, with a sample-and-transform coding scheme in which ideal coding of coefficients is replaced by coding with some specified family of quantizers. It is shown that when the sampling rate is large, the operational rate-distortion function of such a scheme comes within a finite constant of that of the first case.