Direct Design of Phase Factor in the Common-Factor Technique for Hilbert-Pairs

Two filter banks whose corresponding wavelets are Hilbert transforms of each other constitute a Hilbert-Pair. The common-factor technique is a simple technique proposed by Selesnick [1] for its design. The technique first requires the design of a phase factor which approximates the half-sample-delay condition. The technique then requires the computation of a factor that is common to both filter banks so that the perfect reconstruction condition is satisfied. The design of the phase factor by Selesnick [1] is based on all-pass filters which does not take the magnitude response into account. This paper proposes a direct approach to designing the phase factor that takes both the magnitude and phase into account. The new approach is more flexible and can yield better filters and wavelets.