## EE 451

Homework Assignment 10
Due Nov. 8, 1999

1. A discrete-time band-stop filter is required to meet the following specifications:
Passband: f < 5 kHz, f > 13 kHz
Stopband: 8 kHz < f < 10 kHz
Passband ripple: 1 dB
Stopband attenuation: 40 dB
Sampling frequency: 48 kHz

Not that you can use MATLAB to help with all parts of the following calculations, but do not just use the MATLAB butter function do all the work -- go through the steps as discussed in the handout. Be sure to turn in your MATLAB m-file.

1. Find the specifications for an continuous time low-pass filter which can be used as a prototype for the digital filter.
2. Design the continuous-time prototype. I.e., find the transfer function for the continuous-time filter. Plot the pole-zero diagram and gain of the continuous-time filter. (Use the MATLAB function freqs to find the gain.)
3. Use the bilinear transformation to get the discrete-time filter. Write down the transfer function. Plot the pole-zero diagram and gain of the discrete-time filter.

2. Using the specifications of Problem 1, do the following:
1. Design the butterworth filter using the MATLAB buttord and butter functions. Plot the pole-zero diagram and the gain of the filter.
2. Repeat Part (a) for a Chebyshev Type 1 filter.
3. Repeat Part (a) for a elliptical filter.

3. Problem 7.5. Add the following parts:
1. Plot the actual impulse response h[n] gotten by multiplying hd[n] times the Kaiser window function.
2. Plot the frequency response of the filter. Show that it meets the specs.

4. Do the design specified in Problem 7.5 using a Hamming window.
1. Determine the minimum length (M+1) of the impulse response for a filter that meets the specifications.
2. What is the delay of the filter?
3. Detemine the ideal impulse response hd[n] to which the Hamming window should be applied.
4. Plot the actual impulse response h[n] gotten by multiplying hd[n] times the Hamming window function.
5. Plot the frequency response of the filter. Show that it meets the specs.

5. Probelm 7.15

Bill Rison, <rison@ee.nmt.edu >