On the phase condition and its solution for Hilbert transform pairs of wavelets bases

In this work, the phase condition on the scaling filters of two wavelet bases which ensures that the corresponding wavelets are Hilbert transform pairs of each other is studied. An alternative and equivalent phase condition is derived. With the equivalent condition and Fourier series expansions, it is shown that the solution for which the corresponding scaling filters are offset from one another by a half sample is the only solution satisfying the phase condition which results in Hilbert transform pairs of wavelet bases.