Efficient reconstruction of band-limited sequences from nonuniformly decimated versions by use of polyphase filter banks

An efficient polyphase structure for the reconstruction of a band-limited sequence from a nonuniformly decimated version is developed. Theoretically, the reconstruction involves the implementation of a bank of multilevel filters, and it is shown that how all these reconstruction filters can be obtained at the cost of one Mth band low-pass filter and a constant matrix multiplier. The resulting structure is therefore more general than previous schemes. In addition, the method offers a direct means of controlling the overall reconstruction distortion T(z) by appropriate design of a low-pass prototype filter P(z). Extension of these results to multiband band-limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized, along with generalizations to the multidimensional case. Design examples are included to demonstrate the theory, and the complexity of the new method is seen to be much lower than earlier ones

Coding gain in paraunitary analysis/synthesis systems

A formal proof that bit allocation results hold for the entire class of paraunitary subband coders is presented. The problem of finding an optimal paraunitary subband coder, so as to maximize the coding gain of the system, is discussed. The bit allocation problem is analyzed for the case of the paraunitary tree-structured filter banks, such as those used for generating orthonormal wavelets. The even more general case of nonuniform filter banks is also considered. In all cases it is shown that under optimal bit allocation, the variances of the errors introduced by each of the quantizers have to be equal. Expressions for coding gains for these systems are derived

Optimum nonuniform transmultiplexer design

This paper considers an optimum nonuniform FIR transmultiplexer design subject to specifications in the frequency domain. Our objective is to minimize the sum of the ripple energy for all the individual filters, subject to the specifications on amplitude and aliasing distortions, and to the passband and stopband specifications for the individual filters. This optimum nonuniform transmultiplexer design problem can be formulated as a quadratic semi-infinite programming problem. The dual parametrization algorithm is extended to the design of this nonuniform transmultiplexer problem. If the lengths of the filters are sufficiently long and the set of decimation integers is compatible, then our algorithm guarantees that the solution obtained will give rise to the global minimum, and the required specifications are satisfied

Vector space framework for unification of one- and multidimensional filter bank theory

A number of results in filter bank theory can be viewed using vector space notations. This simplifies the proofs of many important results. In this paper, we first introduce the framework of vector space, and then use this framework to derive some known and some new filter bank results as well. For example, the relation among the Hermitian image property, orthonormality, and the perfect reconstruction (PR) property is well-known for the case of one-dimensional (1-D) analysis/synthesis filter banks. We can prove the same result in a more general vector space setting. This vector space framework has the advantage that even the most general filter banks, namely, multidimensional nonuniform filter banks with rational decimation matrices, become a special case. Many results in 1-D filter bank theory are hence extended to the multidimensional case, with some algebraic manipulations of integer matrices. Some examples are: the equivalence of biorthonormality and the PR property, the interchangeability of analysis and synthesis filters, the connection between analysis/synthesis filter banks and synthesis/analysis transmultiplexers, etc. Furthermore, we obtain the subband convolution scheme by starting from the generalized Parseval's relation in vector space. Several theoretical results of wavelet transform can also be derived using this framework. In particular, we derive the wavelet convolution theorem

Classical sampling theorems in the context of multirate and polyphase digital filter bank structures

The recovery of a signal from so-called generalized samples is a problem of designing appropriate linear filters called reconstruction (or synthesis) filters. This relationship is reviewed and explored. Novel theorems for the subsampling of sequences are derived by direct use of the digital-filter-bank framework. These results are related to the theory of perfect reconstruction in maximally decimated digital-filter-bank systems. One of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error. The reconstruction filters turn out to be multiplierless and of the FIR (finite impulse response) type. These ideas are extended to the case of two-dimensional signals by use of a Kronecker formalism. The subsampling of bandlimited sequences is also considered. A sequence x(n ) with a Fourier transform vanishes for |ω|&ges;Lπ/M, where L and M are integers with L<M, can in principle be represented by reducing the data rate by the amount M/L. The digital polyphase framework is used as a convenient tool for the derivation as well as mechanization of the sampling theorem

Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain

Convolution theorems for filter bank transformers are introduced. Both uniform and nonuniform decimation ratios are considered, and orthonormal as well as biorthonormal cases are addressed. All the theorems are such that the original convolution reduces to a sum of shorter, decoupled convolutions in the subbands. That is, there is no need to have cross convolution between subbands. For the orthonormal case, expressions for optimal bit allocation and the optimized coding gain are derived. The contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum. With one of the convolved sequences taken to be the unit pulse function,,e coding gain expressions reduce to those for traditional subband and transform coding. The filter-bank convolver has about the same computational complexity as a traditional convolver, if the analysis bank has small complexity compared to the convolution itself

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

Generalizations of the sampling theorem: Seven decades after Nyquist

The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well-known form is Shannon's uniform-sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed

One- and two-level filter-bank convolvers

In a recent paper, it was shown in detail that in the case of orthonormal and biorthogonal filter banks we can convolve two signals by directly convolving the subband signals and combining the results. In this paper, we further generalize the result. We also derive the statistical coding gain for the generalized subband convolver. As an application, we derive a novel low sensitivity structure for FIR filters from the convolution theorem. We define and derive a deterministic coding gain of the subband convolver over direct convolution for a fixed wordlength implementation. This gain serves as a figure of merit for the low sensitivity structure. Several numerical examples are included to demonstrate the usefulness of these ideas. By using the generalized polyphase representation, we show that the subband convolvers, linear periodically time varying systems, and digital block filtering can be viewed in a unified manner. Furthermore, the scheme called IFIR filtering is shown to be a special case of the convolver