b Now suppose we have a wavelet ψ t which can be written as a superposition of

# B now suppose we have a wavelet ψ t which can be

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(b) Now suppose we have a wavelet ψ 0 ( t ), which can be written as a superposition of scaling functions at scale j = 1 using ψ 0 ( t ) = X n b [ n ] φ 1 ,n ( t ) . Show that if the discrete sequence b [ n ] has p vanishing moments (as in ( 1 )), then the continuous time wavelet ψ 0 ( t ) must also have p vanishing moments, meaning Z -∞ t q ψ 0 ( t ) d t = 0 for all q = 0 , . . . , p. Note that the φ 1 ,n ( t ) will not in general have vanishing moments — just make the following constant substitutions when you see the integrals below: C 0 = Z -∞ φ 1 , 0 ( t ) d t, C 1 = Z -∞ 1 , 0 ( t ) d t, · · · , C p = Z -∞ t p φ 1 , 0 ( t ) d t. (Hint: Start by showing this for p = 0, then p = 1, then generalize ... maybe by using the binomial theorem at some point.) 1 Last updated 0:38, October 4, 2019

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3. Implement the Haar wavelet transform and its inverse in MATLAB. Do this by writing two MATLAB functions, haar.m and ihaar.m that are called as w=haar(x,L) and x = ihaar(w,L) . Here, x is the original signal, and L is the number of levels in the transform. You may assume that the length of x is dyadic; that is, the length of the input is 2 J for some positive integer J .
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