7.3. Exercises


a. Calculate the coefficients of a digital IIR 4th order lowpass filter for a sampling frequency of 10 kHz and a cutoff frequency of 3 kHz.

b. Plot the amplitude and phase of its frequency response.

c. Plot the poles and zeros of the filter transfer function.

d. Plot the filter impulse response.

e. Repeat the exercise for a Chebyshev type I filter, having the same specifications and a maximum passband ripple of 0.5 dB. Then compare the two filters.


Design a lowpass filter with the following specifications: sampling frequency 2 Hz, passband upper edge 0.28 Hz, lower stopband edge 0.32 Hz, bandpass maximum ripple 0.1 dB and stopband minimum attenuation 30 dB.

a. Find out the order of the Butterworth, Chebychev type I and II, and elliptic filters, which meet these specifications.

b. Why is the elliptic filter order always the lowest for the same required specifications?


A linear sampled filter, with initial conditions equal to zero, is defined by the following recurrent relationship:

y[nT] = x[nT] + 2π[(n – 1)T] + 3x[[n – 2]T] + 4π[(n – 3)T]

+3x[(n – 4)T] + 2x[(n – 5)T]+ x[(n – 6)Y]

a. Calculate the filter transfer function and impulse response.

b. Plot the magnitude of the transfer function. Indicate the type of corresponding filter.

c. Find the zeros of the transfer function. What is their effect on the filter frequency response?

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