Homework 3
Due: Thursday, April 16
The following exercises are for both 5750 and 6880.
1) Using the scaling function associated with the Haar wavelet, verify
|ˆ
h(ω)|2+|ˆ
h(ω+π)|= 2,and
ˆ
h(0) = √2.
2) To help yourself believe Theorem 7.4 ...
Suppose ˆ
ψ(0) = ˆ
ψ0(0) = 0. Prove that ˆ
h(π) = ˆ
h0(π) = 0.
(Hint: you’ll need to remember something about ˆϕ(0).)
3) Prove that g[n]=(−1)1−nh[1 −n].
4) Prove equations (7.102) and (7.104) from the text. (I did (7.103) in class
and (7.102) is essentially in the text. Refer to these only as a last resort!)
5) Set up the wavelet matrix for the Haar wavelet and send the signal
s= [1 1 2 3 4 5 4 4]
through the filter twice. Then invert the transform to recover s.
The following exercise is for students registered for 6880.
G6) Use Lemma A and Lemma 7.1 (from class notes) to prove Theorem 7.3.
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