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Hilbert Pair of Orthogonal Wavelet Bases: Revisiting the Condition

It is now well known that in order to have wavelet bases that form a Hilbert transform pair, the corresponding low-pass conjugate quadrature filters (CQF) should ideally be related through a half sampled delay, i.e., e-jomega/2. In this correspondence we revisit this condition and examine some subtleties associated with this condition that were overlooked in previous work. We show that there is a more general condition where the delay can be any "even+half samples, i.e., e-j(2N+1/2)omega. More importantly we examine the implications in formulating design strategies for Hilbert pairs and its implementation.