Wavelet-based Lowpass/Bandpass Interpolation

F9 l is described by, x9 l (t) = P n x(n)9l(t): Let Wl = Span f9l(t 0 n)g. Then, the interpolation with 9l(t) is easily seen to equivalent to the Wavelet-Galerkin approximation to the shift operator on L 2 (IR) restricted to Wl. Moreover, in the multiplicity M case, we have an ecient M-adic interpolation scheme, similar to the dyadic interpolation scheme discussed in Section. 2.. In this case, the if XJ (z) denotes the Z-transform of x8 0 (M 0J n), then, XJ (z) = XJ01(z M)H0(z) (33) where H0(z) is the autocorrelation of h0. Similar results hold in the case of all M 0 1 bandpass schemes too. The only change is that X1(z) = X(z M)Hl(z) in this case, where interpolation is being done with respect to 8l(t). The in-terpolatory classes are precisely described in the wavelet transform domain. F9 l = fx(t)jx(t) = W f (l; 0;) for some f 2 Wlg (34) where the multiplicity M DSWT is dened by, f (t)M j=2 l(M j t 0) dt (35) In the multiplicity M case too, the Fourier Transform of the scaling function ^ 90(!), satises ^ 90(0) = 1, and therefore from Eqn. 4, g(t) = ^ 90(2t) is also an interpolatory function. Fig. 4. shows the M-adic lowpass interpolation lter for M = 4, and length N = 16 scaling vector h0. The corresponding function 90(t) is shown in Fig. 4.