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17
Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
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Cited by 90 (16 self)
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The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
Design of orthonormal and overcomplete wavelet transforms based on rational sampling factors
 In Proc. Fifth SPIE Conference on Wavelet Applications in Industrial Processing
, 2007
"... Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for timescale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter ba ..."
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Cited by 3 (2 self)
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Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for timescale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter banks includes filter design methods and frequency domain implementations. We present several specific examples of Daubechiestype filters for a discrete orthonormal rational wavelet transform (FIR filters having a maximum number of vanishing moments) obtained using Gröbner bases. We also present the design of overcomplete rational wavelet transforms (tight frames) with FIR filters obtained using polynomial matrix spectral factorization.
Dispersive and PitchSynchronous Processing of Sounds
"... The aim of this paper is to present results on digital processing of sounds by means of both dispersive delay lines and pitchsynchronous transforms in a unified framework. The background on frequency warping is detailed and applications of this technique are pointed out with reference to the exi ..."
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Cited by 3 (2 self)
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The aim of this paper is to present results on digital processing of sounds by means of both dispersive delay lines and pitchsynchronous transforms in a unified framework. The background on frequency warping is detailed and applications of this technique are pointed out with reference to the existing literature. These include transient extraction, pitch shifting, harmonic detuning and auditory modeling.
S.Cavaliere, “Arbitrary Bandwidth Wavelet Sets
 Proc. ICASSP'98
, 1998
"... In this paper we consider an extension of the wavelet transform leading to the construction of wavelets with arbitrary bandwidth. The new wavelets are complete, orthonormal and dyadic; nevertheless their bandwidth is not constrained to be one octave, rather it may be designed by selecting a set of ..."
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Cited by 3 (3 self)
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In this paper we consider an extension of the wavelet transform leading to the construction of wavelets with arbitrary bandwidth. The new wavelets are complete, orthonormal and dyadic; nevertheless their bandwidth is not constrained to be one octave, rather it may be designed by selecting a set of parameters. The construction of the new bases starts in the discretetime domain, exploiting properties of the Laguerre transform. Furthermore, we provide a procedure to define continuoustime warped wavelets. Flexibility of the bandwidth allocation allows for more and improved applications of the wavelet transform, such as signal coding, the design of auditory model based filterbanks and transient detection in pseudoperiodic signals, pointed out in the paper. 1.
Harmonicband wavelet coefficient modeling for pseudoperiodic sounds processing
 DAFx00 Proceedings
, 2000
"... In previous papers [1], [2] we introduced a model for pseudoperiodic sounds based on Wornell results [3] concerning the synthesis of 1/f noise by means of the Wavelet transform (WT). This method provided a good model for representing not only the harmonic part of reallife sounds but also the stocha ..."
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Cited by 2 (1 self)
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In previous papers [1], [2] we introduced a model for pseudoperiodic sounds based on Wornell results [3] concerning the synthesis of 1/f noise by means of the Wavelet transform (WT). This method provided a good model for representing not only the harmonic part of reallife sounds but also the stochastic components. The latter are of fundamental importance from a perceptual point of view since they contain all the information related to the natural dynamic of musical timbres. In this paper we introduce a refinement of the method, making the spectralmodel technique more flexible and the resynthesis coefficient model more accurate. In this way we obtain a powerful tool for sound processing and crosssynthesis. 1.
TimeVarying Frequency Warping: Results and Experiments
"... Dispersive tapped delay lines are attractive structures for altering the frequency content of a signal. In previous papers we showed that in the case of a homogeneous line with first order allpass sections the signal formed by the output samples of the chain of delays at a given time is equivalent ..."
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Cited by 2 (1 self)
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Dispersive tapped delay lines are attractive structures for altering the frequency content of a signal. In previous papers we showed that in the case of a homogeneous line with first order allpass sections the signal formed by the output samples of the chain of delays at a given time is equivalent to compute the Laguerre transform of the input signal. However, most musical signals require a timevarying frequency modification in order to be properly processed. Vibrato in musical instruments or voice intonation in the case of vocal sounds may be modeled as small and slow pitch variations. Simulations of these effects require techniques for timevarying pitch and/or brightness modification that are very useful for sound processing. In our experiments the basis for timevarying frequency warping is a timevarying version of the Laguerre transformation. The corresponding implementation structure is obtained as a dispersive tapped delay line, where each of the frequency dependent delay elem...
RealTime TimeVarying Frequency Warping via ShortTime Laguerre Transform
 In Proceedings of COST G6 Conference on Digital Audio Effects (DAFX00
, 2000
"... In this paper we address the problem of the realtime implementation of timevarying frequency warping. Frequency warping based on a oneparameter family of onetoone warping maps can be realized by means of the Laguerre transform and implemented in a noncausal structure. This structure is not dir ..."
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Cited by 1 (0 self)
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In this paper we address the problem of the realtime implementation of timevarying frequency warping. Frequency warping based on a oneparameter family of onetoone warping maps can be realized by means of the Laguerre transform and implemented in a noncausal structure. This structure is not directly suited for realtime implementation since each output sample is formed by combining all of the input samples. Similarly, the recently proposed timevarying Laguerre transform has the same drawback. Furthermore, long frequency dependent delays destroy the time organization or macrostructure of the sound event. Recently, the author has introduced the ShortTime Laguerre Transform for the approximate realtime implementation of frequency warping. In this transform the shorttime spectrum rather than the overall frequency spectrum is frequency warped. The input is subdivided into frames that are tapered by a suitably selected window. By careful design, the output frames correspond to warped versions of the input frames modulated by a stretched version of the window. It is then possible to overlapadd these frames without introducing audible distortion. The overlapadd technique can be generalized to timevarying warping. However, several issues concerning the design of the window and the selection of the overlap parameters need to be addressed. In this paper we discuss solutions for the overlap of the frames when the Laguerre parameter is kept constant but distinct in each frame and solutions for the computation of full timevarying frequency warping when the Laguerre parameter is changing within each frame. 1.
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 ..."
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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.
EURASIP Journal on Applied Signal Processing 2001:1, 27–35 © 2001 Hindawi Publishing Corporation Audio Effects Based on Biorthogonal TimeVarying Frequency Warping
, 2001
"... We illustrate the mathematical background and musical use of a class of audio effects based on frequency warping. These effects alter the frequency content of a signal via spectral mapping. They can be implemented in dispersive tapped delay lines based on a chain of allpass filters. In a homogeneou ..."
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We illustrate the mathematical background and musical use of a class of audio effects based on frequency warping. These effects alter the frequency content of a signal via spectral mapping. They can be implemented in dispersive tapped delay lines based on a chain of allpass filters. In a homogeneous line with firstorder allpass sections, the signal formed by the output samples at a given time is related to the input via the Laguerre transform. However, most musical signals require a timevarying frequency modification in order to be properly processed. Vibrato in musical instruments or voice intonation in the case of vocal sounds may be modeled as small and slow pitch variations. Simulation of these effects requires techniques for timevarying pitch and/or brightness modification that are very useful for sound processing. The basis for timevarying frequency warping is a timevarying version of the Laguerre transformation. The corresponding implementation structure is obtained as a dispersive tapped delay line, where each of the frequency dependent delay element has its own phase response. Thus, timevarying warping results in a spacevarying, inhomogeneous, propagation structure. We show that timevarying frequency warping is associated to an expansion over biorthogonal sets generalizing the discrete Laguerre basis. Slow timevarying characteristics lead to slowly varying parameter sequences. The corresponding sound transformation does not suffer from discontinuities typical of delay lines based on unit delays.
PitchSynchronous Multiresolution Analysis of Music Signals
, 2007
"... ii In this thesis a novel multiresolution approach for note detection in a polyphonic mix is proposed. The idea is to use a set of wavelets whose lengths are adapted to the theoretical fundamental period of musical notes. Using the typical wavelet dyadic decomposition we can generate a set of wavele ..."
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ii In this thesis a novel multiresolution approach for note detection in a polyphonic mix is proposed. The idea is to use a set of wavelets whose lengths are adapted to the theoretical fundamental period of musical notes. Using the typical wavelet dyadic decomposition we can generate a set of wavelets that match the fundamental frequency (F0) of a given note in every octave. Therefore, using a set of 12 different wavelets, one per each semitone, we can represent the fundamental frequency of every note in every octave using one wavelet scale per each octave. The magnitude and phase continuity of wavelet coefficients across temporal frames is exploited to draw a special kind of spectrogram, namely PitchSynchronous Wavelet Spectrogram (PSWS). When the corresponding F0 and harmonics of a note are present in the signal, a special DC pattern appears in the PSWS, due to the aforementioned continuity. Any other harmonic signal or noise produces pseudoperiodic or random AC patterns. This way, by