Statistical optimization of multirate systems and orthonormal filter banks

Citation

Tuqan, Jamal (1998) Statistical optimization of multirate systems and orthonormal filter banks. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/vx5t-w383. https://resolver.caltech.edu/CaltechETD:etd-02042008-081232

Abstract

The design of multirate systems and/or filter banks adapted to the input signal statistics is a generic problem that arises naturally in variety of communications and signal processing applications. The two main applications we have in mind are the statistical optimization of subband coders for signal compression and the multirate modeling of WSS random processes. These two applications lead naturally to the important concepts of energy compaction filters and principal component filter banks. In this thesis, we study three problems that are directly related to the above mentioned applications. The first problem is motivated by the observation that in the presence of subband quantizers, it is a loss of generality to assume that the synthesis section in a filter bank is the inverse of the analysis section. We therefore consider the statistical optimization of linear time invariant (LTI) pre- and postfilters surrounding a quantization system. Unlike in previous work, the postfilter is not restricted to be the inverse of the prefilter. Closed form expressions for the optimum filters as well as the resulting minimum mean square error (m.m.s.e.) are derived. The importance of the m.m.s.e. expression is that it clearly quantifies the additional gain obtained by relaxing the perfect reconstruction assumption. In the second problem, we study the quantization of a certain class of non bandlimited signals, modeled as the output of L < M interpolation filters where M is the interpolation factor. Using the fact that these signals are oversampled, we show how to decrease substantially the quantization noise variance using appropriate multirate reconstruction schemes. We also optimize a variety of noise shapers, indicating the corresponding additional reduction in the average mean square error for each case. The results of this chapter extend, using multirate signal processing theory, some well known techniques of efficient A/D converters (e.g. sigma-delta modulators) that usually apply only to bandlimited signals. In the last problem, a novel procedure to design globally optimal FIR energy compaction filters is presented. Energy compaction filters are important due to their close connection to orthonormal filter banks adapted to the input signal statistics. In fact, for the two channel case, the problems are equivalent. A special case of compaction filters arise also in applications such as echo cancelation, time varying systems identification, standard subband filter design and optimal transmitter and receiver design in digital communications. The new proposed approach guarantees theoretical optimality which previous methods could not achieve. Furthermore, the new algorithm is: i) extremely general in the sense that it can be tailored to cover any of the above applications. ii) numerically robust. iii) can be solved efficiently using interior point methods. The design of a special class of two channel IIR compaction filters is also considered. We show that, in general, this class of optimum IIR compaction filters, parameterized by a single coefficient, are competitive with very high order optimum FIR filters.

Thesis Files