A theory of minimizing distortion in reconstructing a stationary signal fromits compressed samples at a given bitrate is developed. We first analyze theoptimal sampling frequency required in order to achieve the optimaldistortion-rate tradeoff for a stationary bandlimited signal. To this end, weconsider a combined sampling and source coding problem in which an analogGaussian source is reconstructed from its encoded samples. We study thisproblem under uniform filter-bank sampling and nonuniform sampling withtime-varying pre-processing. We show that for processes whose energy is notuniformly distributed over their spectral support, each point on thedistortion-rate curve of the process corresponds to a sampling frequencysmaller than the Nyquist rate. This characterization can be seen as anextension of the classical sampling theorem for bandlimited random processes inthe sense that it describes the minimal amount of excess distortion in thereconstruction due to lossy compression of the samples, and provides theminimal sampling frequency $f_{DR}$ required in order to achieve thatdistortion. We compare the fundamental limits of combined source coding andsampling, which we coin analog-to-digital compression, to the performance inpulse code modulation (PCM), where each sample is quantized by a scalarquantizer using a fixed number of bits.
展开▼
