Results 1  10
of
45
Simple Regularity Criteria For Subdivision Schemes. II. The Rational Case
 SIAM J. Math. Anal
, 1997
"... We study regularity properties of special functions obtained as limits of "p/qadic subdivision schemes." Such "rational" schemes generalize  in a flexible way  binary (or dyadic) subdivision schemes, used in computeraided geometric design and in functional analysis to const ..."
Abstract

Cited by 99 (0 self)
 Add to MetaCart
We study regularity properties of special functions obtained as limits of "p/qadic subdivision schemes." Such "rational" schemes generalize  in a flexible way  binary (or dyadic) subdivision schemes, used in computeraided geometric design and in functional analysis to construct compactly supported wavelets. This finds natural applications in the signal processing area, where it may be desirable to decompose a signal into compactly supported wavelets over fractions of an octave. This results in a finer decomposition than in the dyadic case, which corresponds to an octave by octave decomposition. The main difficulty here, as compared to the dyadic case, is the lack of shift invariance of the limit functions. In this case, a direct extension of Daubechies and Lagarias ideas concerning regularity order estimation becomes impossible, because what they call "twoscale difference equations" cannot be obtained. Using another, "discrete approach", originally proposed in an earlier work for the d...
Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain
 IEEE Transactions on Signal Processing
, 1993
"... AbstractA maximally decimated filter bank system (with possibly unequal decimation ratios in the subbands) can be regarded as a generalization of the shorttime Fourier transformer. In fact, it is known that such a “filter bank transformer” is closely related to the wavelet transformation. A natura ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
AbstractA maximally decimated filter bank system (with possibly unequal decimation ratios in the subbands) can be regarded as a generalization of the shorttime Fourier transformer. In fact, it is known that such a “filter bank transformer” is closely related to the wavelet transformation. A natural question that arises when we conceptually pass from the traditional Fourier transformer to the filter bank transformer is: what happens to the convolution theorem, i.e., is there an analog of the convolution theorem in the world of “filter bank transforms”? In this paper we address the question first for uniform decimation and then generalize it to the nonuniform case. The result takes a particularly simple and useful form for paraunitary or orthonormal filter banks. It shows how we can convolve two signals x(n) and g(n) by directly convolving the subband signals of a paraunitary filter bank and adding the results. The advantage of the method is that we can quantize in the subbands based on the signal variance and other perceptual considerations, as in traditional subband coding. As a result, for a fixed bit rate, the result of convolution is much more accurate than direct convolution. That is, we obtain a coding gain over direct convolution. We will derive expressions for optimal bit allocation and optimal coding gain for such paraunitary convolvers. As a special case, if we take one of the two signals to be the delta function (e.g., g(n) = 6(n)), we can recover the wellknown bit allocation and coding gain formulas of traditional subband coding. The derivations also show that these formulas are valid regardless of the filter quality, as long as orthonormality is not violated. A special case similar to orthogonal transform coding is also considered and good convolutional coding gains for speech are demonstrated, with the use of the DCT matrix. Finally, the result is extended to the case of nonuniform biorthonormal filter banks; with the incorporation of an additional trick, the convolution theorems in this case become as simple as for the orthonormal case. I.
ResonanceBased Signal Decomposition: A New SparsityEnabled Signal Analysis Method
"... Numerous signals arising from physiological and physical processes, in addition to being nonstationary, are moreover a mixture of sustained oscillations and nonoscillatory transients that are difficult to disentangle by linear methods. Examples of such signals include speech, biomedical, and geoph ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
(Show Context)
Numerous signals arising from physiological and physical processes, in addition to being nonstationary, are moreover a mixture of sustained oscillations and nonoscillatory transients that are difficult to disentangle by linear methods. Examples of such signals include speech, biomedical, and geophysical signals. Therefore, this paper describes a new nonlinear signal analysis method based on signal resonance, rather than on frequency or scale, as provided by the Fourier and wavelet transforms. This method expresses a signal as the sum of a ‘highresonance ’ and a ‘lowresonance ’ component — a highresonance component being a signal consisting of multiple simultaneous sustained oscillations; a lowresonance component being a signal consisting of nonoscillatory transients of unspecified shape and duration. The resonancebased signal decomposition algorithm presented in this paper utilizes sparse signal representations, morphological component analysis, and constantQ (wavelet) transforms with adjustable Qfactor. Keywords: sparse signal representation, constantQ transform, wavelet transform, morphological component analysis 1.
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 ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
(Show Context)
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 Upsampling, Downsampling And Rational Sampling Rate Filter Banks
 COMPUTATIONAL MATHEMATICS LABORATORY, RICE UNIVERSITY
, 1992
"... Recently, solutions to the problem of design of rational sampling rate filter banks in one dimension has been proposed. The ability to interchange the operations of upsampling, downsampling, and filtering plays an important role in these solutions. This paper develops a complete theory for the analy ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Recently, solutions to the problem of design of rational sampling rate filter banks in one dimension has been proposed. The ability to interchange the operations of upsampling, downsampling, and filtering plays an important role in these solutions. This paper develops a complete theory for the analysis of arbitrary combinations of upsamplers, downsamplers and filters in multiple dimensions. Though some of the simpler results are well known, the more difficult results concerning swapping upsamplers and downsamplers and variations thereof are new. As an application of this theory, we obtain algebraic reductions of the general multidimensional rational sampling rate problem to a multidimensional uniform filter bank problem. However, issues concerning the design of the filters themselves are not addressed. In multiple dimensions, upsampling and downsampling operators are determined by integer matrices (as opposed to scalars in one dimension), and the noncommutativity of matrices, makes th...
Frequencydomain design of overcomplete rationaldilation wavelet transforms
 IEEE Trans. on Signal Processing
, 2009
"... The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Qfactor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible fami ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Qfactor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher Qfactors (desirable for processing oscillatory signals) or the same low Qfactor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible ‘constantQ’ discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for L2(R). The wavelet can be made to resemble a Gabor function and can hence have good concentration in the timefrequency plane. The construction of the new wavelet transform depends on the judicious use of both the transform’s redundancy and the flexibility allowed by frequencydomain filter design. I.
General multirate building structures with application to nonuniform filter banks,”
 Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on,
, 1998
"... ..."
(Show Context)
Overcomplete Discrete Wavelet Transforms with Rational Dilation Factors
, 2008
"... This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discretetime signals. The proposed overcomplete rational DWT is implemented using selfinverting FIR filter banks, is approximately shiftinvariant, and can provide a dense sampling of the t ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discretetime signals. The proposed overcomplete rational DWT is implemented using selfinverting FIR filter banks, is approximately shiftinvariant, and can provide a dense sampling of the timefrequency plane. A straightforward algorithm is described for the construction of minimallength perfect reconstruction filters with a specified number of vanishing moments; whereas, in the nonredundant rational case, no such algorithm is available. The algorithm is based on matrix spectral factorization. The analysis/synthesis functions (discretetime wavelets) can be very smooth and can be designed to closely approximate the derivatives of the Gaussian function.
Stable filtering schemes with rational dilations
 J. Fourier Analysis and Applications
, 2007
"... Abstract. The relationship between multiresolution analysis and filtering schemes is a wellknown facet of wavelet theory. However, in the case of rational dilation factors, the wavelet literature is somewhat lacking in its treatment of this relationship. This work seeks to establish a means for the ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The relationship between multiresolution analysis and filtering schemes is a wellknown facet of wavelet theory. However, in the case of rational dilation factors, the wavelet literature is somewhat lacking in its treatment of this relationship. This work seeks to establish a means for the construction of stable filtering schemes with rational dilations through the theory of shiftinvariant spaces. In particular, principal shiftinvariant spaces will be shown to offer frame wavelet decompositions for rational dilations even when the associated scaling function is not refinable. Moreover, it will be shown that such decompositions give rise to stable filtering schemes with finitely supported filters, reminiscent of those studied by Kovačevic ́ and Vetterli. 1.
Fractional Biorthogonal Partners in Channel Equalization and Signal Interpolation
, 2002
"... The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers. Hence the name fractional biorthogonal partners. The cond ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers. Hence the name fractional biorthogonal partners. The conditions for the existence of stable and of FIR fractional biorthogonal partners are derived. It is also shown that the FIR solutions (when they exist) are not unique. This property is further explored in one of the applications of fractional biorthogonal partners, namely the fractionally spaced equalization in digital communications. The goal is to construct zeroforcing equalizers that also combat the channel noise. The performance of these equalizers is assessed through computer simulations. Another application considered is the allFIR interpolation technique with the minimum amount of oversampling required in the input signal. We also consider the extension of the least squares approximation problem to the setting of fractional biorthogonal partners.