Signals and Systems - Electrical Engineering

704 C H A P T E R 11: Introduction to the Design of Discrete Filters

11.12. Butterworth versus Chebyshev filtering—MATLAB

If we wish to preserve low-frequency components of the input, a low-pass Butterworth filter could perform

better than a Chebyshev filter. MATLAB provides a second Chebyshev filter functioncheby2that has a

flat response in the passband and a rippled one in the stopband. Let the signal to be filtered be the first

100 samples from MATLAB’s “train” signal. To this signal add some Gaussian noise to be generated by

randn, multiply it by0.1, and add it to the 100 samples of the train signal. Design three discrete filters,

each of order 20, and a half frequency (for Butterworthbutter) and passband frequency (for the Chebyshev

filters) ofωn=0.5. For the design withcheby1let the maximum passband attenuation be0.01dB, and

for the design withcheby2let the minimum stopband attenuation be 60 dB. Obtain the three filters and

use them to filter the noisy “train” signal.

Using MATLAB plot the following for each of the three filters:

(a) Using thefftfunction compute the DFT of the original signal, the noisy signal, and the noise, and plot

their magnitudes. Is the cut-off frequency of the filters adequate to get rid of the noise? Explain.

(b) Compute and plot the magnitude and the unwrapped phase and the poles and the zeros for each of

the three filters. Comment on the differences in the magnitude responses.

(c) Use thefilterfunction to obtain the output of each of the filters, and plot the original noiseless signal

and the filtered signals. Compare them.

11.13. Butterworth, Chebyshev, and elliptic filters—MATLAB

The gain specifications of a filter are:

−0.1≤20 log 10 |H(ejω)|≤ 0 (dB) 0 ≤ω≤0.2π

20 log 10 |H(ejω)|≤− 60 (dB) 0.3π≤ω≤π

(a) Find the loss specifications for this filter.

(b) Design using MATLAB a Butterworth, a Chebyshev (usingcheby1), and an elliptic filter. Plot in one

plot the magnitude response of the three filters, and compare them and indicate which gives the

lowest order.

11.14. Notch and all-pass filters—MATLAB

Notch filters are a family of filters that include the all-pass filter. For the filter

H(z)=K

( 1 −α 1 z−^1 )( 1 +α 2 z−^1 )

( 1 −0.5z−^1 )( 1 +0.5z−^1 )

(a) Determine the values ofα 1 ,α 2 , andKthat would makeH(z)an all-pass filter of unit magnitude. Use

MATLAB to compute and plot the magnitude response ofH(z)using the obtained values forαand

K. Plot the poles and the zeros of this filter.

(b) If we would like the filterH(z)to be a notch filter of unit gain atω=π/ 2 rad, determine the values of

αandKto achieve this and then determine where the notch(es) are. Use MATLAB functions to verify

that the filter is a notch filter, and to plot the poles and the zeros.

(c) Place the zeros ofH(z)at positions between the zeros for the all-pass and the notch filters, and use

MATLAB to plot the magnitude responses. Each of these filters must have unit gain atω=π/ 2 rad.

Explain the connection between the all-pass and the notch filters.

(d) Suppose we use the transformationz−^1 =jZ−^1 to obtain a filterH(Z). Repeat the above part of the

problem forH(Z). Where are the notches of this new filter. What would be the difference between the

all-pass filtersH(z)andH(Z)?