Oversampling PCM techniques and optimum noise shapers for quantizing a class of nonbandlimited signals

Abstract

We consider the efficient quantization of a class of nonbandlimited signals, namely, the class of discrete-time signals that can be recovered from their decimated version. The signals of interest are modeled as the output of a single FIR interpolation filter (single band model) or, more generally, as the sum of the outputs of L FIR interpolation filters (multiband model). By definition, these nonbandlimited signals are oversampled, and it is therefore reasonable to expect that we can reap the same benefits of well-known efficient A/D techniques that apply only to bandlimited signals. Indeed, by using appropriate multirate models and reconstruction schemes, we first show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of the signals. We also show that we can achieve a substantial decrease in bit rate by appropriately decimating the signals and then quantizing them. To further increase the effective quantizer resolution, noise shaping is introduced by optimizing preflters and postfilters around the quantizer. We start with a scalar time-invariant quantizer and study two important cases of linear time invariant (LTI) filters, namely, the case where the postfilter is the inverse of the prefilter and the more general case where the postfilter is independent from the prefilter. Closedform expressions for the optimum filters and average minimum mean square error are derived in each case for both the single band and multiband models. Due to the statistical nature of the signals of interest, the class of noise shaping filters and quantizers is then enlarged to include linear periodically time varying (LPTV)M filters and periodically time-varying quantizers of period M. Because the general (LPTV)M case is difficult to track analytically, we study two special cases in great detail and give complete solutions for both the single band and multiband models. Examples are also provided for performance comparisons between the LTI case and the corresponding (LPTV)M one. ©.