If we wish to preserve low frequencies components of the input, a low-pass Butterworth filter could perform better than a Chebyshev filter. MATLAB provides a second Chebyshev filter function cheby2 that has a flat response in the pass-band 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 r and n, multiply it by 0.1 and add it to the 100 samples of the train signal. Design three discrete filters, each of order 20, and half-frequency (for Butterworth butter) and the passband frequency (for the Chebyshev filters) of ω_n = 0.5. For the design with cheby1 let the maximum passband attenuation be 0.01 dB and for the design with cheby2 let 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:

• Using the fft function compute the DFT of the original signal, the noisy signal, and the noise, and plot their magnitudes. Is the cutoff frequency of the filters adequate to get rid of the noise? Explain.

• Compute and plot the magnitude and the unwrapped phase, as well as the poles and zeros for each of the three filters. Comment on the differences in the magnitude responses.

• Use the function filter to obtain the output of each of the filters, and plot the original noiseless signal and the filtered signals. Compare them.