18.327  Wavelets and Filter Banks 
Gilbert Strang 

1.130  Wavelets and Multiscale Methods in 
Kevin Amaratunga 
Gilbert Strang  email: gs@math.mit.edu  Office: Room 2240. 
Kevin Amaratunga  email: kevina@mit.edu  Office: Room 1274. 
Schedule for Spring 2004: Monday and Wednesday 1:303:00 in Room 1390.
Text: WAVELETS AND FILTER BANKS by Strang and Nguyen, WellesleyCambridge Press, 2nd Ed. (1997.)
Syllabus: Please see the class schedule. Also, see the course announcement for general course overview.
To launch the graphical user interface for the Matlab wavelet toolbox, type "wavemenu" at the Matlab prompt. For starters, try selecting "Wavelet 1D", then go to "File  Load Signal". To locate the sample signals on Athena, type the following path into the "Filter" field and hit the return key.
/mit/matlab_v6.5.1/distrib/toolbox/wavelet/wavedemo/*.mat
You will then be able to choose a signal from the list of files. A good one to start with is noisdopp.mat. Once the signal has loaded, choose a wavelet and the number of decomposition levels. The default wavelet filter is "haar". To use the Daubechies 4tap filter, for example, you would choose "db2". The 2 means that the frequency response has 2 zeros at pi. Similarly, to use the linear phase 9/7 filters, you would choose "bior4.4". In this case, 4.4 means that the synthesis lowpass filter has 4 zeros at pi and the analysis lowpass filter has 4 zeros at pi (in that order.) Next, click "Analyze" to perform the wavelet analysis. You will then see graphs of the original signal (s) and the various subbands of the signal (an,dn,...,d1) where n is the number of levels. Note that the subband signals are of full length in this display. In other words, what is being displayed is not the actual coefficients produced in the output of the analysis bank, but the reconstructed version of each channel that is seen in the synthesis filter bank immediately prior to recombining the channels. Note that you can look at the statistics of the signal and even compress or denoise it.
The "Wavelet 2D" menu item should be straightforward to use once you have seen the 1D version. A good image to start with is woman.mat, which is the famous Barbara image.
Also interesting to beginners is the "Wavelet Display" menu item. This allows you to view the scaling function(s) and wavelet(s) and their associated filters.