Abstract
In designing the Hilbert transform pairs of biorthogonal wavelet bases, it has been shown that the requirements of the equalmagnitude responses and the half-sample phase offset on the lowpass filters are the necessary and sufficient condition. In this paper, the relationship between the phase offset and the vanishing moment difference of biorthogonal scaling filters is derived, which implies a simple way to choose the vanishing moments so that the phase response requirement can be satisfied structurally. The magnitude response requirement is approximately achieved by a constrained optimization procedure, where the objective function and constraints are all expressed in terms of the auxiliary filters of scaling filters rather than the scaling filters directly. Generally, the calculation burden in the design implementation will be less than that of the current schemes. The integral of magnitude response difference between the primal and dual scaling filters has been chosen as the objective function, which expresses the magnitude response requirements in the whole frequency range. Two design examples illustrate that the biorthogonal wavelet bases designed by the proposed scheme are very close to Hilbert transform pairs.
Highlights
The dual-tree complex wavelet transform (DTCWT) had become an attractive signal processing tool since it was proposed by Kingsbury [1, 2]
The design is achieved by a constrained optimization procedure, in which the objective function and constraints are all expressed in terms of the auxiliary filters rather than the low-pass filters directly
This paper describes a new scheme for the design of Hilbert transform pairs of biorthogonal wavelets in detail
Summary
The dual-tree complex wavelet transform (DTCWT) had become an attractive signal processing tool since it was proposed by Kingsbury [1, 2]. A pair of filter banks is used in DTCWT, with which the wavelet bases associated form (approximate) Hilbert transform pairs. Yu and Ozkaramanli [7] had further proven that the requirements of half-sample offset are the necessary and sufficient condition for the design of Hilbert transform pairs of biorthogonal wavelet bases. In [8], Selesnick proposed a new design scheme and designed two filter banks that the corresponding wavelets forming approximate Hilbert transform pairs. The design is achieved by a constrained optimization procedure, in which the objective function and constraints are all expressed in terms of the auxiliary filters (ph, ph, pg , and pg ) rather than the low-pass filters directly. A better approximation quality to Hilbert transform pair is possible
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