ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filter-ing steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically re-duces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. 1.

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In this paper, we present the design, implementation and application of several families of fast multiplierless approximations of the discrete cosine transform (DCT) with the lifting scheme, named the binDCT. These binDCT families are derived from Chen's and Loeffler's plane rotation-based factorizations of the DCT matrix, respectively, and the design approach can also be applied to DCT of arbitrary size. Two design approaches are presented. In the first method, an optimization program is de ned, and the multiplierless transform is obtained by approximating its solution with dyadic values. In the second method, a general lifting-based scaled DCT structure is obtained, and the analytical values of all lifting parameters are derived, enabling dyadic approximations with different accuracies. Therefore the binDCT can be tuned to cover the gap between the Walsh-Hadamard transform and the DCT. The corresponding 2-D binDCT allows a 16-bit implementation, enables lossless compression, and maintai...

This paper solves an open problem, namely how to construct perfect reconstruction filter banks with rational sampling factors. Such filter banks have N branches, each one having a sampling factor of p i q i and their sum equals to one. In this way, the well-known theory of filter banks with uniform band splitting is extended to allow for non-uniform divisions of the spectrum. This can be very useful in the analysis of speech and music. The theory relies on two transforms, 1 and 2. While Transform 1, when applied, leads to uniform filter banks having polyphase components as individual filters, Transform 2 results in a uniform filter bank containing shifted versions of same filters. This, in turn, introduces dependencies in design, and is left for future work. As an illustration, several design examples for the ( 2 3 ; 1 3 ) are given. Filter banks are then classified according to the possible ways in which they can be built. It is also shown that some cases cannot be solved even wi...

A novel approach to the design of M-channel pseudo-quadrature mirror filter (QMF) banks is presented. In this approach, the prototype filter is constrained to be a linear-phase spectral-factor of a 2_1Ith band filter. As a result, the overall transfer function of the analysis/synthesis system is a delay. Moreover, the aliasing cancellation {AC) constraint is derived such that all the significant aliasing terms are canceled. Consequently, the aliasing level at the output is comparable to the stopband attenuation of the prototype filter. In other words, the only error at the output of the analysis/synthesis system is the aliasing error which is at the level of stopband attenuation. Using this approazh, it is possible to design a pseudo-QMF bank where the stopband attenuation of the analysis land thus synthesis) filters is on the order of-100 dB. Moreover, the resulting reconstruction error is also on the order of-100 riB. Several examples are included.

It is well known that FIR filter banks that satisfy the perfect-reconstruction (PR) property can be obtained by cosine modulation of a linear-phase prototype filter of length N = 2rnM, where M is the number of channels. In this paper, we present a PR cosine-modulated filter bank where the length of the prototype filter is arbitrary. The design is formulated as a quadratic-constrained least-squares optimization problem, where the optimized parameters are the prototype filler coefficients. Additional regularity conditions are imposed on the filter bank to obtain the cosine-modulated orthonormal bases of compactly supported wavelets. Design examples are given.

The design of an orthogonal FIR quadrature--mirror filter (QMF) bank (H; G) adapted to input signal statistics is considered. The adaptation criterion is maximization of the coding gain and has so far been viewed as a difficult nonlinear constrained optimization problem. In this paper, it is shown that in fact the coding gain depends only upon the product filter P (z) = H(z)H(z \Gamma1 ), and this transformation leads to a stable class of linear optimization problems having finitely many variables and infinitely many constraints, termed linear semi-- infinite programming (SIP) problems. The sought--for, original filter, H(z), is obtained by deflation and spectral factorization of P (z). With the SIP formulation, every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms. The natural regularity properties inherent in the SIP formulation enhance the performance of these algorithms. We present a comprehensive theoretical analysis of ...

In this paper we show an algebraic approach for the design of ladder structures for causal bi-orthogonal filter banks. The key ingredient of the approach is known in literature as Euclid's algorithm. Using this algorithm we derive some strong result on the design freedom for ladder structures. In particular we show that the dimensionality of the problem plays an important role. We end by with some conjectures concerning the extensions to multi-channel and non-causal filter banks. Keywords--- Digital signal processing, bi-orthogonal filter bank, multidimensional, ladder structure, Euclid's algorithm, elementary matrix. I. Notations In this article, we use the following notations. With F we denote any of the sets Q, R or C , i.e. any of the sets of rational, real or complex numbers 1 . With K we denote any of the sets of integer (Z), rational, real or complex numbers. The set of polynomials in m variables with coefficients from K will be denoted by Km . A matrix over Km is a 2 \Thet...

Absrruct-Traditionally, lossless network functions and matrices have played an important role in electrical network theory. Many of the basic mathematical concepts and results pertaining to lossless 5ystems, however, continue to have major applications in modern digital signal processing today. This paper is an attempt at a self-contained exposure to discrete-time losdess \ystems, their properties, and relevance in digital signal processing. I.

In this paper, we develop structures for FIR perfect-rec. nstruction QMF banks which cover a subclass of systems that yield linear-phase analysis filters for arbitrary M. The parameters of these structures can be optimized in order to design analysis filters with minimmu stopband energy which at the same time have linear-phase and satisfy the perfect-reconstruction property. If there are M subbands, then depending upon whether the coefficients h(n) of each analysis filter is symmetric or antlsymmetric, several combinations of filter banks are possible. Some of these permit perfect-reconstruction and some do not. For a given M, we develop a formula for the number of combinatiuns for a subclass of linear-phase perfect-reconstruction structures. As an example, we elaborate on a perfect-reconstruction linear-phase lattice structure for three channels and develop a lattice structure for this case. The lattice structure is such that, regardless of the parameter values, the QMF bank e10oys perfect-reconstruction property while at the same time the analysis filters have linear phase. These parameters can therefore be optimized to obtain analysis filters with good magnitude response, without losing lhe above two features.