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31
Hilbert Transform Pairs of Wavelet Bases
, 2001
"... This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an a ..."
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Cited by 51 (6 self)
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This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury, that the dualtree DWT is (nearly) shiftinvariant when the scaling filters satisfy the same offset.
The Design of Approximate Hilbert Transform Pairs of Wavelet Bases
, 2002
"... Several authors have demonstrated that significant improvements can be obtained in waveletbased signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of p ..."
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Cited by 34 (7 self)
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Several authors have demonstrated that significant improvements can be obtained in waveletbased signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.
The DoubleDensity DualTree DWT
, 2004
"... This paper introduces the doubledensity dualtree discrete wavelet transform (DWT), which is a DWT that combines the doubledensity DWT and the dualtree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on ..."
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Cited by 17 (0 self)
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This paper introduces the doubledensity dualtree discrete wavelet transform (DWT), which is a DWT that combines the doubledensity DWT and the dualtree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. One pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, one pair of wavelets are designed to be approximate Hilbert transforms of the other pair of wavelets so that two complex (approximately analytic) wavelets can be formed. Therefore, they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractionaldelay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shiftinvariant.
Elimination Of Transients In TimeVarying Allpass Fractional Delay Filters With Application To Digital Waveguide Modeling
 in Proc. Int. Computer Music Conf
, 1995
"... : This paper considers discretetime allpass filters that implement a timevarying fractional delay. These recursive filters are desirable from the point of view of implementational efficiency and flat magnitude response, but they are prone to transient effects when their parameters are changed. A s ..."
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Cited by 14 (4 self)
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: This paper considers discretetime allpass filters that implement a timevarying fractional delay. These recursive filters are desirable from the point of view of implementational efficiency and flat magnitude response, but they are prone to transient effects when their parameters are changed. A state variable update approach to this problem is reviewed, the basic idea of which is to modify also the state of the filter when coefficients are changed so that the filter enters a new state smoothly without transient attacks. This method is adapted to give a new and simple practical method for eliminating the transients. The effectiveness of the new technique is verified by applying it to a digital waveguide model of a vibrating string. 1. INTRODUCTION A fractional delay (FD) filter is a device for implementing a noninteger delay, a process equivalent to bandlimited interpolation between samples. FD filters are of fundamental importance in digital waveguide models of musical instruments....
The phaselet transform  an integral redundancy nearly shiftinvariant wavelet transform
 IEEE Trans. on Signal Proc
, 2003
"... This paper introduces an approximately shift invariant redundant dyadic wavelet transform the phaselet transform that includes the popular dualtree complex wavelet transform of Kingsbury [1] as a special case. The main idea is to use a finite set of wavelets that are related to each other in a sp ..."
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Cited by 11 (1 self)
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This paper introduces an approximately shift invariant redundant dyadic wavelet transform the phaselet transform that includes the popular dualtree complex wavelet transform of Kingsbury [1] as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way and hence called phaseletsto achieve approximate shiftredundancy; bigger the set better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are fractional shifts of each other. Algorithms for the design of phaselets with a fixed number vanishing moments is presented building upon the work of Selesnick [2] for the design of wavelet pairs for Kingsbury’s dualtree complex wavelet transform. Construction of 2dimensional directional bases from tensor products of 1d phaselets is also described. Phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader independently of the approximate shift invariance property that this paper argues they possess. 1
Generalized Digital Butterworth Filter Design
 IN PROC. IEEE INT. CONF. ACOUST., SPEECH, SIGNAL PROCESSING (ICASSP
, 1995
"... This paper presents a formulabased method for the design of IIR filters having more zeros than (nontrivial) poles. The filters are designed so that their square magnitude frequency responses are maximallyflat at ! = 0 and at ! = ß and are thereby generalizations of classical digital Butterworth fi ..."
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Cited by 8 (4 self)
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This paper presents a formulabased method for the design of IIR filters having more zeros than (nontrivial) poles. The filters are designed so that their square magnitude frequency responses are maximallyflat at ! = 0 and at ! = ß and are thereby generalizations of classical digital Butterworth filters. A main result of the paper is that, for a specified halfmagnitude frequency and a specified number of zeros, there is only one valid way in which to split the zeros between z = \Gamma1 and the passband. Moreover, for a specified number of zeros and a specified halfmagnitude frequency, the method directly determines the appropriate way to split the zeros between z = \Gamma1 and the passband. IIR filters having more zeros than poles are of interest, because often, to obtain a good tradeoff between performance and the expense of implementation, just a few poles are best.
Phaselets of Framelets
 IEEE Trans. SP, submitted
, 2002
"... Phaselets are a set of dyadic wavelets that are related in a particular way such that the associated redundant wavelet transform is nearly shiftinvariant [1]. Framelets are a set of functions that generalize the notion of a single dyadic wavelet in the sense that dyadic dilates and translates of th ..."
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Cited by 6 (1 self)
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Phaselets are a set of dyadic wavelets that are related in a particular way such that the associated redundant wavelet transform is nearly shiftinvariant [1]. Framelets are a set of functions that generalize the notion of a single dyadic wavelet in the sense that dyadic dilates and translates of the framelets form a frame in L²(R) [2]. This paper generalizes the notion of phaselets to framelets. Sets of framelets that only differ in their Fourier transform phase are constructed such that the resulting redundant framelet transform is approximately shiftinvariant. Explicit constructions of phaselets are given for frames with two and three framelet generators. The results in this paper generalize the construction of Hilbert transform pairs of framelets [3].