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 dual-tree DWT is (nearly) shift-invariant when the scaling filters satisfy the same offset.

Several authors have demonstrated that significant improvements can be obtained in wavelet-based 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.

: This paper considers discrete-time allpass filters that implement a time-varying 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....

This paper introduces the double-density dual-tree discrete wavelet transform (DWT), which is a DWT that combines the double-density DWT and the dual-tree 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 fractional-delay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shift-invariant.

This paper introduces an approximately shift invariant redundant dyadic wavelet transform- the phaselet transform- that includes the popular dual-tree 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 phaselets-to achieve approximate shift-redundancy; bigger the set better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are frac-tional 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 dual-tree complex wavelet transform. Construction of 2-dimensional directional bases from tensor products of 1-d 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

This paper presents a formula-based 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 maximally-flat 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 half-magnitude 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 half-magnitude 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 trade-off between performance and the expense of implementation, just a few poles are best.

Phaselets are a set of dyadic wavelets that are related in a particular way such that the associated redundant wavelet transform is nearly shift-invariant [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 shift-invariant. 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].

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This paper describes a new class of maximally flat low-pass recursive digital filters. The filters are realizable as a parallel sum of two all-pass filters, a structure for which low-complexity low-noise implementations exist. Note that, with the classical Butterworth filter of degree x which is retrieved as a special case, it is not possible to adjust the delay (or phase linearity). However, with the more general class of filters described in this paper, the adjustment of the delay becomes possible, and the tradeoff between the delay and the phase linearity can be chosen. The construction of these low-pass filters depends upon a new maximally flat delay allpole filter, for which the degrees of flatness at 3 aHand 3 a % are not necessarily equal. For the coefficients of this flat delay filter, an explicit solution is introduced, which also specializes to a previously known result.