### Characterization of Complex Wavelets

#### Abstract

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.

Hilbert Transform Pairs of Wavelet Bases IEEE Signal Processing Letters; Vol. 8, No. 6, pp. 170-173, June 2001.

The Charaterization and Design of Hilbert Transform Pairs of Wavelet Bases. Invited paper at the Conference on Information Sciences and Systems, The Johns Hopkins University, March 21-23, 2001.

#### Examples from the paper

The following examples were obtained using Gröbner basis methods.

Additional examples, also obtained using Gröbner methods

### Design of Complex Wavelets

#### Abstract

This paper describes a simple procedure, based on spectral factorization, for the design of a pair of orthonormal wavelet bases where the two wavelets form a Hilbert transform pair. The two scaling filters respectively have the numerator and denominator of the flat delay all-pass filter as factors. The design procedure allows for an arbitrary number of zero wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

The design of Hilbert transform pairs of wavelet bases via the flat delay filter. ICASSP 2001 (Int. Conf. on Acoustics, Speech, and Sig. Proc.)

The design of approximate Hilbert transform pairs of wavelet bases. IEEE Trans. on Signal Processing, 50(5):1144-1152, May 2002.