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462 Chapter 8 IIR FILTER DESIGN
P8.2 A digital resonator is to be designed with ω 0 = π/4 that has 2 zeros at z =1and z = −1.
1. Compute and plot the frequency response of this resonator for r =0.8, 0.9, and 0.99.
2. For each case in part 1 determine the 3 dB bandwidth and the resonant frequency ω r
from your magnitude plots.
3. Compare your results in part 2 with (8.48) and (8.47 ), respectively.
P8.3 We want to design a digital resonator with the following requirements: a 3 dB bandwidth of
0.05 rad, a resonant frequency of 0.375 cycles/sam, and zeros at z =1and z = −1. Using
trial-and-error approach, determine the difference equation of the resonator.
P8.4 A notch filter is to be designed with a null at the frequency ω 0 = π/2.
1. Compute and plot the frequency response of this notch filter for r =0.7, 0.9, and 0.99.
2. For each case in part 1, determine the 3 dB bandwidth from your magnitude plots.
3. By trial-and-error approach, determine the value of r if we want the 3 dB bandwidth to
be 0.04 radians at the null frequency ω 0 = π/2.
P8.5 Repeat Problem P8.4 for a null at ω 0 = π/6.
P8.6 Aspeech signal with bandwidth of 4 kHz is sampled at 8 kHz. The signal is corrupted by
sinusoids with frequencies 1 kH, 2 kHz, and 3 kHz.
1. Design an IIR filter using notch filter components that eliminates these sinusoidal
signals.
2. Choose the gain of the filter so that the maximum gain is equal to 1, and plot the
log-magnitude response of your filter.
3. Load the handel sound file in MATLAB, and add the preceding three sinusoidal signals
to create a corrupted sound signal. Now filter the corrupted sound signal using your
filter and comment on its performance.
P8.7 Consider the system function of an IIR lowpass filter
H(z) =K
1+z−1
(8.72)
1 − 0.9z −1
where K is a constant that can be adjusted to make the maximum gain response equal to 1.
We obtain the system function of an Lth-order comb filter H L(z) using H L(z) =H ( z L) .
1. Determine the value of K for the system function in (8.72).
2. Using the K value from part 1, determine and plot the log-magnitude response of the
comb filter for L =6.
3. Describe the shape of your plot in part 2.
P8.8 Consider the system function of an IIR highpass filter
H(z) =K
1 − z−1
(8.73)
1 − 0.9z −1
where K is a constant that can be adjusted to make the maximum gain response equal to 1.
We obtain the system function of an Lth-order comb filter H L(z) using H L(z) =H ( z L) .
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