Design of Hilbert transform pairs of orthonormal wavelet bases using IIR filters

Conventionally, FIR filters have been often used to design the dual tree complex wavelet transforms (DTCWTs), where two real orthonormal wavelet bases form a Hilbert transform pair, whereas IIR filters are seldom used, although they require a lower computational complexity than FIR filters. In this paper, a new class of Hilbert transform pairs of orthonormal wavelet bases is proposed by using general IIR filters. To obtain the maximum number of vanishing moments, the conventional design methods located as many zeros as possible at z = -1. This paper proposes a new design method for DTCWTs by locating a specified number of zeros at z = -1 and minimizing the stopband error. The proposed method uses the well-known Remez exchange algorithm to approximate an equiripple magnitude response in the stopband. Therefore, a set of filter coefficients can be easily obtained by solving the eigenvalue problem. Furthermore, the optimal solution is attained through a few iterations. The advantage of the proposed method is that the number of zeros at z = -1 can be specified arbitrarily and an improved frequency selectivity can be obtained.