SDP Approximation of a Fractional Delay and the Design of Dual-Tree Complex Wavelet Transform

We show that an H_infin optimization problem related to fractional delay approximation can be formulated as a semideflnite programming (SDP) problem and thus solved reliably. Particularly, given the finite-impulse-response (FIR) filter H(z), we find the FIR filter G(z) of given degree such that ||G(z) - z^-1/2H(z)||_infin is minimum. This half-sample delay approximation problem is used in the design of filters generating orthogonal dual-tree complex wavelet transforms. Since the solution does not conform to the orthogonality constraints exactly, we propose their enforcement in a second stage of optimization, in which an analyticity criterion is optimized. The proposed designs compare favorably with previous ones.