Hilbert transform pairs of orthogonal wavelet bases: necessary and sufficient conditions

The condition on scaling filters of two orthogonal wavelet bases that render the corresponding wavelets as Hilbert transform pairs is re-examined in this note. Without making any pre-assumption on the relationship between the two scaling filters, the authors derive necessary and sufficient conditions for forming Hilbert transform pairs. They lead to new magnitude conditions and Selesnick's phase condition. Unique solutions to these conditions are concluded. It is shown that orthogonal wavelet bases form Hilbert transform pairs if and only if the two scaling filters are offset from one another by half a sample.