1. Show that the DFT matrix is the matrix of eigenvectors of ANY circulant matrix. What would the eigenvalues be then?
Obviously you can find proofs in any book or on the web, but there is still benefit to doing it yourself once.

2. Signal Processing problems (some of you will recognize these!)

3. A practical problem from biology: Look at this image: imgw.tif. The biologist wants you to count the number of cells (the circular/elliptical shapes) in this image and the number of  chloroplasts (dark blue regions).  Not something that I expect you to do based on your current knowledge, but may want to try it by the middle/end of the class.