advertisement

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. 1