|Date:||Wed, March 13, 2013|
|Place:||Research I Seminar Room|
Abstract: Analog-to-digital (A/D) conversion is the process by which signals are replaced by bit streams to allow for digital storage, transmission, and processing using modern computers. Typically, A/D conversion is thought of as being composed of sampling and quantization. Sampling consists of collecting inner products of the signal with appropriate (deterministic or random) vectors. Quantization consists of replacing these inner products with elements from a finite set. Often, quantization is followed by compression or encoding, in order to reduce the size of the digital data-set. A good A/D scheme allows for accurate reconstruction of the original object from its quantized (and compressed) samples. In this talk, we discuss methods for quantizing and encoding oversampled signals. We focus on three types of signals: bandlimited functions, finite dimensional signals, and sparse signals. In all cases, we employ Sigma-Delta quantization, and investigate the reconstruction error as a function of the bit-rate. For bandlimited functions and finite dimensional signals, we propose and investigate encoding algorithms for the Sigma-Delta bit-stream and show that they yield near-optimal error rates when coupled with suitable reconstruction algorithms. In particular, in the finite dimensional setting, the near-optimality of Sigma-Delta encoding applies to measurement vectors from a large class that includes both deterministic and sub-Gaussian random vectors. These results, in turn, have implications for compressed sensing, which we also discuss.