[Problems 1 thru 4 due Feb 12 in class, rest due Feb 14].

1. Problem 9.53 (Textbook)
2. Prove properties 9.1 thru 9.9 in table 9.1 and the 16 Laplace transforms in table 9.2. Write all the necessary steps of the proofs (you do not need to prove a result more than once: just number it the first time you prove it and refer to it later).
3. Prove that if the transfer function is rational then causality is equivalent with right-sided ROC (statement in a box at the top of page 694 of your textbook).
4. Prove that if the impulse response of an LTI dynamical system is not absolutely integrable, then the dynamical system is not stable (cfr sec. 2.3.7 of your textbook). Start with a rigorous definition of stability.
5. Write a Matlab function that takes two images, calculates their Fourier transform, swaps the phase of the two Fourier transforms, inverts the transforms and returns two images. Does any of the new images look like the input ones? Does most `information' go with the phase or with the magnitude of the Fourier transform? Do you have any idea why? (Hint: calculate the FT for a number of images and observe the behavior of the magnitude vs the frequency -- do you observe any regularities?). Turn in your code (use fft2 in Matlab to compute the DFT of an image).  Use the two standard images dog.jpg and cat.jpg, then use two images of your choice. Use the Matlab functions imcrop, imresize, interp2 to make all of your images have a standard size (e.g. 256x256). Also: the function fftshift will be useful for displaying the images' FT in a coordinate system that you are familiar with. Turn in both your code and the images that you have produced along with your observations.
6. Do the same with voice data. Again: does the phase carry more information, or does the magnitude? Why? Turn in your code along with your observations.