Question 1. Suppose you wish to design a 25-tap FIR filter that approximates the following ideal frequency response

H_d(ω) = 1, for |ω|≤ Π/6

           0, for Π/6 < |ω|Π    

a. Make the design using the windowed method using a rectangular window:
i. As part of the design find an explicit equation for the impulse response of the designed filter
ii. Using matlab plot the impulse response of the filter
iii. Plot (use matlab) magnitude (in dB) and phase (unwrapped) of the frequency response against only positive frequencies

b. Make the design using the windowed method using a Hamming window:
i. Use MATLAB to compute and plot the impulse response of the filter
ii. Plot (use matlab) magnitude (in dB) and phase (unwrapped) of the frequency response against only positive frequencies

c. Make the design using the windowed method using a Bartlett window:
i. Use MATLAB to compute and plot the impulse response of the filter
ii. Plot (use matlab) magnitude (in dB) and phase (unwrapped) of the frequency response against only positive frequencies

Question 2. The continuous-time filter with frequency response shown below is to be used as the basis for designing a digital filter via the bilinear transformation method without pre- warping. If the sampling frequency to be used is Fs = 5 kHz, then find the digital frequency ω at which the frequency response of the digital filter has magnitude of 2.

Ω (rad / sec)

Question 3. Consider the pole-zero plot shown in fig.

a) Does it represent an FIR filter?
b) is it a linear-phase system?

Question 4. An IIR digital lowpass filter is required to meet the following specifications,

Passband ripple (or peak-to-peak ripple): ≤ 0.5 dB
Passband edge: 1.2 kHz
Stopband attenuation: ≥ 40 dB
Stopband edge: 2.0 kHz
Sample rate: 8.0 kHz

Use the design formulas in the book to determine the required filter order for
(a) A digital Butterworth filter
(b) A digital Chebyshev filter
(c) A digital elliptic filter

a. Answer the question posed in the book using the book's formulas for estimating the needed filter order (see Section 10.3.4 of the book)
b. Answer the question posed in the book using the MATLAB commands discussed in the notes for estimating the needed filter order
c. Design the three filters using the MATLAB commands discussed in the notes.
i. Plot the magnitude of the frequency response in dB vs the corresponding analog frequency in Hz
ii. Verify that it meets the specs.

Question 5. Use the Parks-McClellan method to design an FIR filter that meets the specs given in Question 4

a. Compare the resulting frequency response to that for the IIR designs found for Question 4 (i.e., #4 above).
b. Compare the computational complexity (multiplies and adds per output sample) for this FIR design compared to the IIR designs found in #4 above