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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/q-adic subdivision schemes." Such "rational" schemes generalize -- in a flexible way -- binary (or dyadic) subdivision schemes, used in computer-aided geometric design and in functional analysis to const ..."
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Cited by 99 (0 self)
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We study regularity properties of special functions obtained as limits of "p/q-adic subdivision schemes." Such "rational" schemes generalize -- in a flexible way -- binary (or dyadic) subdivision schemes, used in computer-aided 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 "two-scale 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
"... Abstract-A maximally decimated filter bank system (with possibly unequal decimation ratios in the subbands) can be regarded as a generalization of the short-time Fourier transformer. In fact, it is known that such a “filter bank transformer” is closely related to the wavelet transformation. A natura ..."
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Cited by 19 (4 self)
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Abstract-A maximally decimated filter bank system (with possibly unequal decimation ratios in the subbands) can be regarded as a generalization of the short-time 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 well-known 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.
Resonance-Based Signal Decomposition: A New Sparsity-Enabled Signal Analysis Method
"... Numerous signals arising from physiological and physical processes, in addition to being non-stationary, are moreover a mixture of sustained oscillations and non-oscillatory transients that are difficult to disentangle by linear methods. Examples of such signals include speech, biomedical, and geoph ..."
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Cited by 16 (8 self)
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Numerous signals arising from physiological and physical processes, in addition to being non-stationary, are moreover a mixture of sustained oscillations and non-oscillatory 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 ‘high-resonance ’ and a ‘low-resonance ’ component — a high-resonance component being a signal consisting of multiple simultaneous sustained oscillations; a low-resonance component being a signal consisting of non-oscillatory transients of unspecified shape and duration. The resonance-based signal decomposition algorithm presented in this paper utilizes sparse signal representations, morphological component analysis, and constant-Q (wavelet) transforms with adjustable Q-factor. Keywords: sparse signal representation, constant-Q transform, wavelet transform, morphological component analysis 1.
A New Design Algorithm for Two-Band 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 two-band 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 13 (0 self)
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In this paper, we present a new algorithm for the design of orthonormal two-band 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 ..."
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Cited by 11 (2 self)
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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 non-commutativity of matrices, makes th...
Frequency-domain design of overcomplete rational-dilation 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 Q-factor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible fami ..."
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Cited by 10 (5 self)
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The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Q-factor) 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 Q-factors (desirable for processing oscillatory signals) or the same low Q-factor 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 ‘constant-Q’ 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 frequency-domain 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
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Overcomplete Discrete Wavelet Transforms with Rational Dilation Factors
, 2008
"... This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discrete-time signals. The proposed overcomplete rational DWT is implemented using self-inverting FIR filter banks, is approximately shift-invariant, and can provide a dense sampling of the t ..."
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Cited by 9 (5 self)
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This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discrete-time signals. The proposed overcomplete rational DWT is implemented using self-inverting FIR filter banks, is approximately shift-invariant, and can provide a dense sampling of the time-frequency plane. A straightforward algorithm is described for the construction of minimal-length perfect reconstruction filters with a specified number of vanishing moments; whereas, in the non-redundant rational case, no such algorithm is available. The algorithm is based on matrix spectral factorization. The analysis/synthesis functions (discrete-time 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 well-known 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 ..."
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Cited by 7 (0 self)
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Abstract. The relationship between multiresolution analysis and filtering schemes is a well-known 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 shift-invariant spaces. In particular, principal shift-invariant 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 decom-positions 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 ..."
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Cited by 5 (3 self)
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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 all-FIR 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.
