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Homework 6, ECE438, Fall 2010, Prof. Boutin
Due in class, Friday October 15, 2010.
The discussion page for this homework is here. Feel free to share your answers/thoughts/questions on that page.
Question 1
Consider the signal
$ x[n]=\cos \left( \omega_1 n \right)+ k \cos \left( \omega_2 n \right) $
where k is a real-valued constant.
a) Write a program that will
- Plot x[n].
- Compute the N point DFT X[k]. (Yes, you may use FFT routines.)
- Plot the magnitude of X[k].
Turn in a print out of your code.
b) Run your program and generate outputs for the cases shown below.
| Case | N | $ \omega_1 $ | k | $ \omega_2 $ |
|---|---|---|---|---|
| 1 | 20 | 0.62831853 | ||
| 2 | 200 | 0.62831853 | 0 | N/A |
| 3 | 20 | 0.64402649 | 0 | N/A |
| 4 | 200 | 0.64402649 | 0 | N/A |
| 5 | 200 | 0.64402649 | 0.2 | 1.27234502 |
| 6 | 200 | 0.64402649 | 0.2 | 0.79168135 |
Question 2
Draw a complete flow diagram for a decimation-in-time FFT algorithm for an 8 point FFT. How many complex operations does your algorithm takes? How many operations would this DFT computation take if you were using the summation formula (i.e., the definition of the DFT) instead?
Question 3
a) Draw a complete flow diagram for a decimation-in-time FFT algorithm for an 6 point FFT beginning with two three-point DFTs. How many complex operations does your algorithm takes? How many operations would this DFT computation take if you were using the summation formula (i.e., the definition of the DFT) instead?
b) Draw a complete flow diagram for a decimation-in-time FFT algorithm for an 6 point FFT beginning with three two-point DFTs. How many complex operations does your algorithm takes? Compare with your answers in part a).
