Ten lectures on wavelets
Complex wavelet transforms with allpass filters
Signal Processing - Special section: Hans Wilhelm Schüßler celebrates his 75th birthday
Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions
IEEE Transactions on Signal Processing
On the phase condition and its solution for Hilbert transform pairs of wavelet bases
IEEE Transactions on Signal Processing
The design of approximate Hilbert transform pairs of wavelet bases
IEEE Transactions on Signal Processing
Hilbert transform pairs of biorthogonal wavelet bases
IEEE Transactions on Signal Processing - Part I
IEEE Transactions on Signal Processing
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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.