Winter 2002/2003 Homework MATH 640
DUE: February 20, 2003
1. Let Cdenote the moving average £lter. Write out the operator ↓2Cby indicating its action on a vector. Let D
be the moving difference £lter. Write out the action of ↓2Don a vector.
Verify in all details the basic identity:
(C↓2)T(C↓2)T+(D↓2)T(D↓2)T+I.
2. Find the lowpass £lter C(ω)for the max¤at £lter with p=5, using the spectral factorization technique.
3. Find H1(z),F0(z), and F1(z)for the biorthogonal £lter bank with
P0(z) = 1
16 ³−1+9z−2+16z−3+9z−4−z−6´,H0(z) = µ1+z−1
2¶3
.
4. Let φ(t)and w(t)be the Haar scaling and wavelet functions. Let Vjand Wjbe the spaces generated by φj,k(t) =
2j/2φ(2jt−k)and wj,k(t) = 2j/2w(2jt−k), for any integer k. Let f(t)be de£ned on 0 ≤t<1 be given as
follows: it is -1 for 0 ≤t<1/4, 4 for 1/4≤t<1/2, 2, for 1/2≤t<3/4, and £nally -3 for 3/4≤t<1.
(a) Express fin terms of the basis for V2.
(b) Decompose finto its component parts in W1,W0, and V0. That is, £nd the Haar wavelet decomposition for
f.
(c) Sketch each of these decompositions.
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