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382 Chapter 7 FIR FILTER DESIGN
P7.31 Using the Parks-McClellan algorithm, design a 25-tap FIR differentiator with slope equal to
1 sample/cycle.
1. Choose the frequency band of interest between 0.1π and 0.9π. Plot the impulse response
and the amplitude response.
2. Generate 100 samples of the sinusoid
x(n) =3sin(0.25πn), n =0, ..., 100
and process through the preceding FIR differentiator. Compare the result with the
theoretical “derivative” of x(n). Note: Don’t forget to take the 12-sample delay of the
FIR filter into account.
P7.32 Design a lowest-order equiripple linear-phase FIR filter to satisfy the specifications given in
Figure P7.2. Provide a plot of the amplitude response and a plot of the impulse response.
P7.33 A digital signal x(n) contains a sinusoid of frequency π/2 and a Gaussian noise w(n) ofzero
mean and unit variance; i.e.,
x(n) =2cos πn
2 + w(n)
We want to filter out the noise component using a 50th-order causal and linear-phase FIR
filter.
1. Using the Parks-McClellan algorithm, design a narrow bandpass filter with passband
width of no more than 0.02π and stopband attenuation of at least 30 dB. Note that no
other parameters are given and that you have to choose the remaining parameters for
the firpm function to satisfy the requirements. Provide a plot of the log-magnitude
response in dB of the designed filter.
2. Generate 200 samples of the sequence x(n) and processed through the preceding filter to
obtain the output y(n). Provide subplots of x(n) and y(n) for 100 ≤ n ≤ 200 on one plot
and comment on your results.
1.00
0.90
Amplitude Response
0.45
0.35
0.05
0.00
0 0.4 0.5 0.7 0.8 1
Frequency ω in π Units
FIGURE P7.2 Filter Specifications for Problem P7.32
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