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54
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 434 (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 factoriza ..."
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Cited by 50 (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 35 (1 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 Processing
, 1995
"... 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 wellknown theory of filter banks with unifo ..."
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Cited by 34 (0 self)
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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 wellknown theory of filter banks with uniform band splitting is extended to allow for nonuniform 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...
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 20 (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 13 (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.
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. In pa ..."
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Cited by 10 (0 self)
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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...
Structures for MChannel PerfectReconstruction FIR QMF Banks Which Yield LinearPhase Analysis Filters
 IEEE Trans. on ASSP
, 1990
"... In this paper, we develop structures for FIR perfectrec. nstruction QMF banks which cover a subclass of systems that yield linearphase 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 ..."
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Cited by 9 (2 self)
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In this paper, we develop structures for FIR perfectrec. nstruction QMF banks which cover a subclass of systems that yield linearphase 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 linearphase and satisfy the perfectreconstruction 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 perfectreconstruction and some do not. For a given M, we develop a formula for the number of combinatiuns for a subclass of linearphase perfectreconstruction structures. As an example, we elaborate on a perfectreconstruction linearphase 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 perfectreconstruction 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.