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Homework 2, ECE438, Fall 2011, Prof. Boutin
Due Wednesday September 6, 2011 (in class)
Question 1
Pick a signal x(t) representing a note of the middle scale of the piano (but not the middle C we did in class) and obtain its CTFT $ X(f) $. Then pick a sampling period $ T_1 $ for which no aliasing occurs and obtain the DTFT of the sampling $ x_1[n]=x(n T_1) $. More precisely, write a mathematical expression for $ X_1(\omega) $ and sketch its graph. Finally, pick a sampling frequency $ T_2 $ for which aliasing occurs and obtain the DTFT of the sampling $ x_2[n]=x(n T_2) $ (i.e., write a mathematical expression for $ X_2(f) $ and sketch its graph.) Note the difference and similarities between $ X(f) $ and $ X_1(\omega) $. Note the differences and similarities between $ X_1(\omega) $ and $ X_2(\omega) $.
Question 2
Pick five different DT signals and compute their z-transform. Then take the five z-transforms you obtained and compute their inverse z-transform.
Question 3
If you have not done so already, answer all the "collectively solved practice problems" posted on the course wiki and share your answers on the corresponding pages. Contact your instructor if you would like to be given an "anonymous" login (to be used in place of your career account login).
Discussion
- When posting your answers on the collective problem solving pages, please do not post a "scan" of your paper or a word document. Instead, use the markup language Wikitext to type your answers. A good resource to begin learning Wikitext is this help page describing how to add stuff to existing Rhea pagess. Your TA Trey also kindly created a great summary on how to enter math in Rhea with an emphasis on the math expressions used in ECE438. More help can be found on the Help page on "how to type math equations". If you really, absolutely, do not want to use the native markup language of Rhea, you may click on the "rich editor" button while in "edit" mode in order to get a wyzywyg editor. But I do not recommend this. After all, you are all aspiring engineers. -pm
- In class you listed a few things that were different between signals in CTFT and DTFT. One of these was the fact that they have different amplitudes, but is this not due to the fact that we converted into $ f $ in the CTFT and into $ \omega $ in the DTFT. This would account for the $ 2 \pi $ factor change in amplitude.
- Instructor's comment: You are very clever! Yes, the change from $ \omega $ to f does change the amplitude. However, there is something else going on which we will investigate further in class. The idea is to first look at the rescaling in frequency induced by the discretization. If you write this rescaling explicitly, you will see that the amplitude of the delta's is actually changed in a way that depends on the sampling period. -pm
- I was just asked whether you need to hand in a hard copy of your answers for Question 3. The answer is NO, you do not need to hand in a hard copy. We will check your answer directly on the wiki. -pm
