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
|ĥ(ω)|2 + |ĥ(ω + π)| = 2,
and
√
ĥ(0) = 2.
2) To help yourself believe Theorem 7.4 ...
Suppose ψ̂(0) = ψ̂ 0 (0) = 0. Prove that ĥ(π) = ĥ0 (π) = 0.
(Hint: you’ll need to remember something about ϕ̂(0).)
3) Prove that g[n] = (−1)1−n h[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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