Hilbert transform pairs of wavelet bases

This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury (1999), that the dual-tree DWT is (nearly) shift-invariant when the scaling filters satisfy the same offset.