**Homework Assignment 10 **

Due Nov. 8, 1999

- 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 kHzNot 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.- Find the specifications for an continuous time low-pass filter which can be used as a prototype for the digital filter.
- 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.) - 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.

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

- Design the butterworth filter using the MATLAB
- Problem 7.5. Add the following parts:
- Plot the actual impulse response h[n] gotten by multiplying
h
_{d}[n] times the Kaiser window function. - Plot the frequency response of the filter. Show that it meets the specs.

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

- Probelm 7.15