Optimal orthonormal subband coding and lattice quantization with vector dithering

Citation

Kirac, Ahmet (1999) Optimal orthonormal subband coding and lattice quantization with vector dithering. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ec82-t391. https://resolver.caltech.edu/CaltechETD:etd-02202008-104935

Abstract

In the digital era that we live in, efficient coding of signals is an unquestionable need. This thesis is about one of the most useful and popular technique of digital coding: subband coding. Subband coding and its cousin wavelet-based coding are now the preferred methods for not only speech, but also audio, image, and video signals. Subband coding involves a linear part which is a filter bank, and a nonlinear part which is usually a uniform scalar quantization of each of the subbands. Subband coders are classified according to the type of filter bank used for its transform. This thesis is mainly about orthonormal subband coding. The ability of an orthonormal filter bank to decompose the signal into components that have a diverse set of signal energies is an indicator of its efficiency for subband coding. Such a diversity in the set of the subband energies is fully utilized by a process called bit allocation. The traditional results on the optimality of a filter bank for given input statistics assume that the quantizers operate at high bit rates.

This thesis presents optimality results under more general quantizer models without assuming high bit rates. This is accomplished by revealing the relationship between the problems of optimal orthonormal subband coding and principal component representation of signals. The latter is done using what is called a principal component filter bank (PCFB). A PCFB is one that compacts most of the energy of a signal into smaller subsets of subbands. To date, there has not been significant theoretical developments in the field of optimal nonuniform subband coding, although the successful techniques of wavelet-based coding are among the state of the art in practice. Such techniques utilize a form of a nonuniform filter bank with a certain structure which makes it efficient for its implementation. In this thesis, we provide optimality results for the nonuniform orthonormal subband coding as well. As in the uniform case, the principal component representation of signals continues to play the key role. We introduce nonuniform PCFB's and link them to the optimal subband coding problem. A PCFB, in particular, contains a filter that compacts most of the signal energy into one single channel: energy compaction filter. The thesis goes into details of designing such filters optimally. In particular, we propose an analytical method in the two-channel case and a very efficient window method in the arbitrary M—channel case. Multistage design of compaction filters has also been worked out.

Finally we extend the analysis of uniform scalar quantization to multiple dimensions. We provide an exact statistical relationship between a lattice quantizer noise and its input vector. We then extend the idea of dithering to the vector case. Dithering is a means of statistically rendering the quantization noise independent of the input. We address the optimal choice of a lattice for a given dimension and also optimal pre- and post-filtering of a dithered lattice quantizer.

Thesis Files