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GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Summer 2008 Problem Set #9 Assigned: Week of 14-Jul-08 Due Date: 22/23-Jul-08 Please check the T-square “Announcements” daily. All official course announcements will be posted there. ALL of the STARRED problems should be turned in for grading. To give students in the Monday recitations some breathing room after Quiz 3, students with recitations on Monday may turn in this homework at the start of lab on Wednesday. Students in Tuesday recitations should turn it in at the start of their recitation as usual. PROBLEM 9.1 : Try parts (b) and (c) of Problem 11.6 on p. 342 of Signal Processing First . PROBLEM 9.2 : Try Problem 11-8 on p. 343 of Signal Processing First . PROBLEM 9.3 : Suppose a mysterious function f ( a ) is defined by f ( a ) = integraldisplay 10 0 t exp ( jat ) dt + integraldisplay 20 10 (10 t ) exp ( jat ) dt Sketch the function x ( t ) = 1 2 π integraldisplay -∞ f ( a ) exp ( jat ) da (I often put questions like this on test to make sure you recognize the form of the Fourier transform and the inverse Fourier transform.)
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PROBLEM 9.4 *: The homework problems that one gets in engineering and math classes often feel unrelated to anything in the real world. To try to make this material more “real,” the exposition in this problem and the next problem “dresses up” some standard ECE2025 questions with some background information about where these things actually show up in real life! (If you don’t care about the background, you need not pay close attention to it. You can just work the problems, but hopefully you’ll find the background interesting.) (a) The “ramp filter” is often used in medical imaging applications such as X-ray computer-aided tomography. It has a frequency response given by H ( ) = braceleftBigg for | ω | < ω 0 0 otherwise Find the impulse response, h ( t ), of this filter. (You may need to go back to your old calculus text if you’ve forgotten how to take derivatives of quotients.) Also make a plot of the impulse response (this is pretty complicated, so you should feel free to use MATLAB or one of those fancy graphing calculators to help you make the plot.) If you’ve ever had a CAT scan done, you’ve run across this filter in practice! A hint on making the plot: We’re interested in the overall shape of the curve; you can pick whatever ω 0 you find the most interesting. I found that the following bit of MATLAB code made a nice plot. You should feel free to use it: omega_0 = 2*pi; period = 2*pi/omega; t = -5*period:period/100:5*period; h = [fill in code that goes here] h([t == 0]) = [fill in value that goes here] plot(t,h); You may ask yourself what the h([t == 0]) = business is all about. Well, it turns out that h ( t ) is indeterminite at t = 0, i.e. we wind up dividing zero by zero. To find a meaningful value for h (0), use L’Hopital’s rule. Alternatively, you can probably guess what
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