Statistically Optimum Pre - and Post Filtering in Quantization,
Abstract
We consider the optimization of pre- and post filters surrounding a quantization system. The goal is to optimize the filters such that the mean square error due to quantization is minimized under the key constraint that the quantization noise variance is directly proportional to the variance of the quantization system input. Unlike some previous work, the postfilter is not restricted to be the inverse of the prefilter. With no order constraint on the filters, we present closed form solutions for the optimum pre- and post filters when the quantization system is a uniform quantizer. Using these optimum solutions, we obtain a coding gain expression for the system under study. The coding gain expression clearly indicates that, at high bit rates, there is no loss in generality in restricting the postfilter to be the inverse of the prefilter. We then repeat the same analysis with first order pre- and post filters in the form 1 + alphaz-1 and 1/(1 + gammaz-1). In specific, we study two cases : (a) FIR prefilter, IIR postfilter and (b) IIR prefilter, FIR postfilter. For each case, we obtain a mean square expression, optimize the coefficients alpha and gamma and provide some examples where we compare coding gain performance with the case of alpha = gamma. In the last section, we assume that the quantization system is an orthonormal perfect reconstruction filter bank. To apply the optimum pre- and post filters derived earlier, the output of the filter bank must be WSS which, in general, is not true. We provide two theorems, each under a different set of assumptions, that guarantee the wide sense stationarity of the filter bank output. We then propose a suboptimum procedure to increase the coding gain of the orthonormal filter bank.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1996
- Accession Number
- ADA323664
Entities
People
- Jamal Tuqan
- Palghat Vaidyanathan
Organizations
- California Institute of Technology