Ahmet Kirac

Theory and design of optimum FIR compaction filters

The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N<M and N=/spl infin/ have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix, respectively. We deal with the more difficult case of M<N</spl infin/. For the two-channel case and for a restricted but important class of random processes, we give an analytical solution for the compaction filter that is characterized by its zeros on the unit circle. This also corresponds to the optimal two-channel FIR filter bank that maximizes the coding gain under the traditional quantization noise assumptions. With a minor extension, this can also be used to generate optimal wavelets. For the arbitrary M-channel case, we provide a very efficient suboptimal design method called the window method. The method involves two stages that are associated with the above two special cases. As the order increases, the suboptimality becomes negligible, and the filter converges to the ideal optimal solution. We compare the window method with a previously introduced technique based on linear programming.

Results on optimal biorthogonal filter banks

Optimization of filter banks for specific input statistics has been of interest in the theory and practice of subband coding. For the case of orthonormal filter banks with infinite order and uniform decimation, the problem has been completely solved in recent years. For the case of biorthogonal filter banks, significant progress has been made recently, although a number of issues still remain to be addressed. In this paper we briefly review the orthonormal case, and then present several new results for the biorthogonal case. All discussions pertain to the infinite order (ideal filter) case. The current status of research as well as some of the unsolved problems are described.

Theory of cyclic filter banks

We introduce the fundamentals of cyclic multirate systems and filter banks and present a number of important differences between the cyclic and noncyclic (traditional) cases. Some of the additional freedom offered by cyclic systems is pointed out, and a number of open issues are summarized.

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist.

On existence of FIR principal component filter banks

In this paper we have two interesting results. One is of theoretical interest and the other practical. The theoretical result is that the optimum FIR orthonormal filter bank of a fixed finite degree that maximizes the coding gain does not always contain an optimum compaction filter. In other words, in general, there does not exist a principal component filter bank (PCFB) of a given nonzero degree. This is sharply in contrast to the cases of transform coders and ideal subband coders where the existence of PCFBs are assured by their very construction. The practical result of the paper is that constraining the filter corresponding to the largest subband variance to be a compaction filter does not result in a significant loss of performance for practical input signals. Since there exist very efficient methods to design FIR compaction filters and since the best completion of the filter bank given the first filter is trivially done by a KLT, we see that this is an extremely efficient method despite the fact that it is suboptimum.

Results on lattice vector quantization with dithering

The statistical properties of the error in uniform scalar quantization have been analyzed by a number of authors in the past, and is a well-understood topic today. The analysis has also been extended to the case of dithered quantizers, and the advantages and limitations of dithering have been studied and well documented in the literature. Lattice vector quantization is a natural extension into multiple dimensions of the uniform scalar quantization. Accordingly, there is a natural extension of the analysis of the quantization error. It is the purpose of this paper to present this extension and to elaborate on some of the new aspects that come with multiple dimensions. We show that, analogous to the one-dimensional case, the quantization error vector can be rendered independent of the input in subtractive vector-dithering. In this case, the total mean square error is a function of only the underlying lattice and there are lattices that minimize this error. We give a necessary condition on such lattices. In nonsubtractive vector dithering, we show how to render moments of the error vector independent of the input by using appropriate dither random vectors. These results can readily be applied for the case of wide sense stationary (WSS) vector random processes, by use of iid dither sequences. We consider the problem of pre- and post-filtering around a dithered lattice quantifier, and show how these filters should be designed in order to minimize the overall quantization error in the mean square sense. For the special case where the WSS vector process is obtained by blocking a WSS scalar process, the optimum prefilter matrix reduces to the blocked version of the well-known scalar half-whitening filter.

On the minimum phase property of prediction-error polynomials

We provide a simple proof of the minimum phase property of the optimum linear prediction polynomial. The proof follows directly from the fact that the minimized prediction error has to satisfy the orthogonality principle. Additional insights provided by this proof are also discussed.

Cyclic LTI systems and the paraunitary interpolation problem

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. Since the frequency domain is a uniform discrete grid, there is more freedom in theoretical and design aspects. The basics of cyclic (L) multirate systems and filter banks have already appeared in the literature, and important differences between the cyclic and noncyclic cases are known. Since there is a strong connection between paraunitary filter banks and orthonormal wavelets, some deeper questions pertaining to cyclic (L) paraunitary matrices are addressed in this paper. It is shown that cyclic (L) paraunitary matrices do not in general have noncyclic paraunitary FIR interpolants, though IIR interpolants can always be constructed. It is shown, as a consequence, that cyclic paraunitary systems cannot in general be factored into degree one nonrecursive paraunitary building blocks. The connection to unitariness of the cyclic state space realization is also addressed.

Optimal nonuniform orthonormal filter banks for subband coding and signal representation

It has been shown that principal component filter banks (PCFB) are optimum orthonormal filter banks for subband coding for the uniform case in which all decimation ratios are the same. In this paper we present the analogous results for the nonuniform case. In contrast to the uniform case where there is only one PCFB, there are more than one PCFB in the nonuniform case. For a fixed set of decimation ratios, there are as many PCFB as the total number of permutations of the decimation ratios. For a fixed number of channels M, we show that there are finitely many sets of decimation ratios that a nonuniform filter bank can have. We show that one of the PCFB corresponding to a particular set of decimation ratios with a particular permutation is an optimum M-channel nonuniform orthonormal filter bank for subband coding. As in the uniform case, the results are valid at arbitrary bit rates.

Efficient design methods of optimal FIR compaction filters for M-channel FIR subband coders

We propose algorithms for the design of FIR compaction filters, which find applications in FIR subband coders. The techniques produce compaction gains that are very close to that of optimal compaction filters, for any fixed filter order and input autocorrelation. The main theme of the paper is the design of multistage FIR compaction filters based on an iterated linear programming approach. The theory behind this is presented followed by design examples and comparisons. Also, a noniterative algorithm that is much faster than other iterative optimization techniques (e.g. linear programming) is mentioned.