Design of Hilbert transform pairs of orthonormal wavelet bases with improved analyticity

This paper proposes a class of Hilbert transform pairs of orthonormal wavelet bases with improved analyticity. To improve the analyticity of complex wavelet, a different allpass filter is used for the half-sample delay approximation. We present a design method for allpass filters with the specified degree of flatness at ω = 0 and equiripple phase response in the approximation band. Remez exchange algorithm is applied in the approximation band, and then a set of filter coefficients can be obtained easily by solving the eigen-value problem. Therefore, the equiripple phase response is attained through a few iterations. Furthermore, the corresponding filter banks are constructed from the designed allpass filters by using the method proposed in. The resulting orthonormal wavelet bases possess the maximum number of vanishing moments. Finally, one example is presented to demonstrate the improvement of the analyticity.