Hilbert transform pairs of biorthogonal wavelet bases

The forming of Hilbert transform pairs of biorthogonal wavelet bases of two-band filter banks is studied in this paper. We first derive necessary and sufficient conditions on the scaling filters that render two Hilbert transform pairs: one decomposition pair and one reconstruction pair. We show that the Hilbert transform pairs are achieved if and only if the decomposition scaling filter of one filter bank is half-sample delayed from that of the other filter bank; and the reconstruction scaling filter of the former is half-sample advanced from that, of the latter. Hilbert transform pairs of wavelet bases are also characterized by equivalent relationships on the wavelet filters and the scaling functions associated with the two filter banks. An illustrative example is provided.