Optimum quantization of a class of non-bandlimited signals

We consider the quantization of a special class of non bandlimited signals, namely the class of discrete time signals that can be recovered from their decimated version. Similar to sigma-delta modulation ideas, we show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of these signals. We then consider noise shaping by optimizing a pre- and post filter around the quantizer and develop a closed form expression for the coding gain of the scheme under study. The use of an orthonormal filter bank as a sophisticated quantizer is investigated and several examples are provided

Optimum low cost two channel IIR orthonormal filter bank

In this paper, we statistically optimize a well known class of IIR two channel orthonormal filter banks parameterized by a single coefficient when subband quantizers are present. The optimization procedure is extremely simple and very fast compared for example to the linear programming method used in the FIR case to achieve similar compaction (coding) gains. The special form of the filters assure the existence of a zero at π which can be important for some wavelet applications and eliminate some of the major concerns that arise in the FIR design case. Finally, the compaction gain obtained is high and numerically very close to two (ideal case) for low pass spectra, high pass spectra and certain cases of multiband spectra. For these cases, the use of higher order IIR filters does not increase the compaction (coding) gain

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

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 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). 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. We first show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of the signals. 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 prefilters 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. Closed form expressions for the optimum filters and average minimum mean square error are derived in each case for both the single band and multiband models. 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. We study two special cases in great detail

The role of the discrete-time Kalman-Yakubovitch-Popov lemma in designing statistically optimum FIR orthonormal filter banks

We introduce a new approach to design FIR energy compaction filters of arbitrary order N. The optimization of such filters is important due to their close connection to the design of an M-channel orthonormal filter bank adapted to the input signal statistics. The novel procedure finds the optimum product filter Fopt(Z)=H opt(Z)Hopt(Z^-1) corresponding to the compaction filter Hopt(z). The idea is to express F(z) as D(z)+D(z^-1) and reformulate the compaction problem in terms of the state space realization of the causal function D(z). For a fixed input power spectrum, the resulting filter Fopt(z) is guaranteed to be a global optimum due to the convexity of the new formulation. The new design method can be solved quite efficiently and with great accuracy using recently developed interior point methods and is extremely general in the sense that it works for any chosen M and any arbitrary filter length N. Finally, obtaining Hopt(z) from F opt(z) does not require an additional spectral factorization step. The minimum phase spectral factor can be obtained automatically by relating the state space realization of Dopt(z) to that of H opt(z)

Statistically optimum pre- and postfiltering in quantization

We consider the optimization of pre- and postfilters surrounding a quantization system. The goal is to optimize the filters such that the mean square error 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 postfilters 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 postfilters in the form 1+αz-1 and 1/(1+γz^-1 ). In specific, we study two cases: 1) FIR prefilter, IIR postfilter and 2) IIR prefilter, FIR postfilter. For each case, we obtain a mean square error expression, optimize the coefficients α and γ and provide some examples where we compare the coding gain performance with the case of α=γ. In the last section, we assume that the quantization system is an orthonormal perfect reconstruction filter bank. To apply the optimum preand postfilters derived earlier, the output of the filter bank must be wide-sense stationary 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

Globally optimal FIR filters with applications in source and channel coding

In this paper, we derive a novel formulation to solve an important FIR filter optimization problem. The problem has received considerable attention in the past because it appears in a wide variety of disciplines. The newly proposed method finds the globally optimal solution to the problem and provides several other advantages over previous optimization techniques

Optimum pre- and postfilters for quantization

We consider the optimization of pre- and post filters surrounding a uniform quantizer such that the mean square error due to quantization is minimized. 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. Using these optimum solutions, we obtain a coding gain expression for the system under study. We then repeat the same analysis with first order pre- and post filters in the form 1+αz^-1 and 1/(1+γz^-1) providing some examples where we compare coding gain performance with the case of α=γ

The design of optimum filters for quantizing a class of non bandlimited signals

We consider the efficient quantization of a class of non bandlimited signals, namely the class of discrete time signals that can be recovered from their decimated version. By definition, these signals are oversampled and it is reasonable to expect that we can reap the same benefits of well known efficient A/D conversion techniques. Indeed, by using appropriate multirate 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. To further increase the effective quantizer resolution, noise shaping is introduced by optimizing linear time invariant (LTI) and linear periodically time varying (LPTV)M pre- and post-filters around the quantizer. Closed form expressions for the optimum filters and the minimum mean squared error are derived for each case

A state space approach to the design of globally optimal FIR energy compaction filters

We introduce a new approach for the least squared optimization of a weighted FIR filter of arbitrary order N under the constraint that its magnitude squared response be Nyquist(M). Although the new formulation is general enough to cover a wide variety of applications, the focus of the paper is on optimal energy compaction filters. The optimization of such filters has received considerable attention in the past due to the fact that they are the main building blocks in the design of principal component filter banks (PCFBs). The newly proposed method finds the optimum product filter Fopt(z)=Hopt(Z)Hopt (z^-1) corresponding to the compaction filter Hopt (z). By expressing F(z) in the form D(z)+D(z^-1), we show that the compaction problem can be completely parameterized in terms of the state-space realization of the causal function D(z). For a given input power spectrum, the resulting filter Fopt(z) is guaranteed to be a global optimum solution due to the convexity of the new formulation. The new algorithm is universal in the sense that it works for any M, arbitrary filter length N, and any given input power spectrum. Furthermore, additional linear constraints such as wavelets regularity constraints can be incorporated into the design problem. Finally, obtaining Hopt(z) from Fopt(z) does not require an additional spectral factorization step. The minimum-phase spectral factor Hmin(z) can be obtained automatically by relating the state space realization of Dopt(z) to that of H opt(z

Statistical optimization of multirate systems and orthonormal filter banks

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