ELEC 0547 Project #5 Fall 1999

Pole-Zero Response and Filter Design

This project will give you some exposure to the effects of pole and zero placement on the frequency response of a digital filter.

Exercises

1. Consider a two-pole/two-zero system function

   

   Find the pole and zero locations of this system. (Hint: the functions roots and poly may be useful in this exercise.) Generate a pole-zero plot using zplane. Sketch the frequency response you expect to see by looking at the pole-zero plot. Display the frequency response with freqz.

2. Using trial-and-error, determine the pole and zero locations of a system so that it implements the best lowpass filter possible with cutoff using two poles and two zeros. ("Best" may be in the eye of the beholder to a degree.) Be sure to choose the poles and zeros so that the difference equation has real coefficients (complex-conjugate pole and zero locations).
3. Determine the difference equation defined by the filter you designed in the previous step. Generate a signal of length 100. Filter this signal with filter and examine the result. Explain this result in terms of the frequency response of your filter.
4. Generate a 50-point impulse response from the difference equation. Does this impulse response look like it would implement a lowpass filter? Explain.
5. Read in ('/opt/demo/audio/fanfare.au'); with auread. Filter this signal with the filter you designed and listen to the result with sound.

   If this doesn't work on your machine, you will need to save the file using wavwrite and then listen to it with the PC sound capability. (Double-click the file in Windows Explorer.)

   Describe the effect of the filter. The sampling frequency on this sound file is 8 kHz. What is the approximate cutoff of the equivalent analog lowpass filter?

Write a short report describing your findings. The report should contain a concise description of your results. Be sure to answer all questions.

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