ECE 802-601 Homework 5
due April 13, 2006
1. Suppose that h, ˜
hdefine a pair of perfect reconstruction filters in a biorthogonal system.
a) Show that H∗(ω)˜
H(ω) + H∗(ω+π)˜
H(ω+π) = 2.
b) Prove that hnew[n] = 1
2(h[n] + h[n−1]),˜
hnew =1
2(˜
h[n] + ˜
h[n−1]) defines a new pair
of perfect reconstruction filters.
c) The Deslauriers-Dubuc filters are H(ω) = 1 and ˜
H(ω) = 1
16 (−e−3jω + 9e−jω + 16 +
9ejω −e3jω ). Compute hnew and ˜
hnew as well as the corresponding biorthogonal wavelets
ψnew,˜
ψnew.
2. Consider the wavelet packet decomposition of the following signal.
a)Use the ’wpdec’ function in MATLAB using the Shannon entropy and log energy
criteria. Compare the results with each other and to the wavelet decomposition for
this signal.
b) Now add random white noise to this signal using randn(length(x)). Repeat the
same procedure as in part (a). Do the results change? Why or why not?
3. Find the equivalent lifting implementation of Daubechies 4 wavelets, i.e. length is equal
to 4. Sketch the block diagram of the one-step decomposition and reconstruction. Write
a MATLAB function that will implement this decomposition. Test your program with
the following signal, x= (32.0,10.0,20.0,38.0,37.0,28.0,38.0,34.0,18.0,24.0,
18.0,9.0,23.0,24.0,28.0,34.0).
4. Wavelets can be used to remove noise from signal. Let f(t) = sin(8πt) cos(3πt) + n(t),
with n(t) being the noise. Numerically, we can model n(t) by using a random number
generator, such as MATLAB’s rand. Take 1500 samples on [−2,3] of f and do an
analysis with N= 2,3 and 6 Daubechies wavelets. Experiment with different levels.
Which wavelet does the best job? Include a copy of the original and the denoised
signal, and explain your procedure.
5. 7.29 from the book. A smoothed image is given on the webpage.
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