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71
Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... 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 filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 480 (7 self)
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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 filtering 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 wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained 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, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
Fast Multiplierless Approximations of the DCT with the Lifting Scheme
 IEEE Trans. on Signal Processing
, 2001
"... 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 rotationbased ..."
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Cited by 60 (10 self)
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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 rotationbased 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 liftingbased 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 WalshHadamard transform and the DCT. The corresponding 2D binDCT allows a 16bit implementation, enables lossless compression, and maintai...
NearPerfectReconstruction PseudoQMF
 IEEE Trans. Signal Processing
, 1994
"... A novel approach to the design of Mchannel pseudoquadrature mirror filter (QMF) banks is presented. In this approach, the prototype filter is constrained to be a linearphase spectralfactor of a 2_1Ith band filter. As a result, the overall transfer function of the analysis/synthesis system is a d ..."
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Cited by 40 (2 self)
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A novel approach to the design of Mchannel pseudoquadrature mirror filter (QMF) banks is presented. In this approach, the prototype filter is constrained to be a linearphase spectralfactor 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 pseudoQMF bank where the stopband attenuation of the analysis land thus synthesis) filters is on the order of100 dB. Moreover, the resulting reconstruction error is also on the order of100 riB. Several examples are included.
Perfect reconstruction filter banks with rational sampling factors
 IEEE Trans. Signal Process
, 1993
"... ..."
The Theory and Design of ArbitraryLength CosineModulated Filter Banks and Wavelets, Satisfying Perfect Reconstruction
 IEEE Trans. Signal Processing
, 1996
"... It is well known that FIR filter banks that satisfy the perfectreconstruction (PR) property can be obtained by cosine modulation of a linearphase prototype filter of length N = 2rnM, where M is the number of channels. In this paper, we present a PR cosinemodulated filter bank where the length of ..."
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Cited by 25 (4 self)
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It is well known that FIR filter banks that satisfy the perfectreconstruction (PR) property can be obtained by cosine modulation of a linearphase prototype filter of length N = 2rnM, where M is the number of channels. In this paper, we present a PR cosinemodulated filter bank where the length of the prototype filter is arbitrary. The design is formulated as a quadraticconstrained leastsquares optimization problem, where the optimized parameters are the prototype filler coefficients. Additional regularity conditions are imposed on the filter bank to obtain the cosinemodulated orthonormal bases of compactly supported wavelets. Design examples are given.
The Role of Linear SemiInfinite Programming in SignalAdapted QMF Bank Design
, 1995
"... The design of an orthogonal FIR quadraturemirror 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 t ..."
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Cited by 19 (6 self)
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The design of an orthogonal FIR quadraturemirror 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 soughtfor, 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 ...
The Role of Lossless Systems in Modern Digital Signal Processing: A Tutorial
 IEEE Transactions on Education
, 1989
"... AbsrructTraditionally, 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 ..."
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Cited by 16 (0 self)
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AbsrructTraditionally, 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 selfcontained exposure to discretetime losdess \ystems, their properties, and relevance in digital signal processing. I.
A New Design Algorithm for TwoBand Orthonormal Rational Filter Banks and Orthonormal Rational Wavelets
 IEEE Trans. Signal Process
, 1998
"... In this paper, we present a new algorithm for the design of orthonormal twoband rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedur ..."
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Cited by 10 (0 self)
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In this paper, we present a new algorithm for the design of orthonormal twoband rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedure, which explains its exponential convergence and adaptability under various linear constraints (e.g., regularity). Although the filters obtained from this algorithm are suboptimally designed, they show excellent frequency selectivity.
On the Realizability of BiOrthogonal, MDimensional 2Band Filter Banks
, 1995
"... In this paper we show an algebraic approach for the design of ladder structures for causal biorthogonal 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. ..."
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In this paper we show an algebraic approach for the design of ladder structures for causal biorthogonal 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 multichannel and noncausal filter banks. Keywords Digital signal processing, biorthogonal 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...