FIR filter design - Continuous and Discrete Time Signals and Systems - page 712

15.20 Using M ATLAB , determine the impulse response of the optimal FIR
filter for the specifications provided in Problem 15.4. You may use the
Kaiser window to determine the length of the optimal FIR filter. Sketch
the magnitude response of the optimal FIR filter and compare its fre-
quency characteristics with those of the FIR filters plotted in Problem
15.18.

15.21 Show that the alternation theorem is satisfied for the magnitude response
of the optimal FIR filter designed in Problem 15.20.

15.22 Using the fir1 function in M ATLAB , design a 41-tap lowpass filter
with a normalized cut-off frequency of Ω n = 0 . 55 using (i) rectangular;
(ii) Hamming; (iii) Blackman; and (iv) Kaiser (with β = 4) windows.
Plot the amplitude-frequency characteristics for the four filters. For each
plot, determine (i) the maximum pass-band ripple; (ii) the peak side lobe
gain; and (iii) the transition bandwidth. Assume that the transition band
is a band where the filter gain drops from -2 dB to -20 dB.

15.23 Using the fir1 function in M ATLAB , design a 45-tap linear-phase
bandpass FIR filter with pass-band edge frequencies of 0.45 π and 0.65 π ,
stop-band edge frequencies of 0.15 π and 0.9 π , maximum pass-band
attenuation of 0.1 dB, and minimum stop-band attenuation of 40 dB.
Use the Kaiser window for your design and sketch the frequency char-
acteristics of the resulting filter.

15.24 The fir2 function in M ATLAB is used to design FIR filters with
arbitrary frequency characteristics. Using fir2 , design a 95-tap FIR
filter with the following frequency characteristics:

0 . 85 0 ≤ Ω n ≤ 0 . 15

0 . 55 0 . 20 ≤ Ω n ≤ 0 . 45
1 0 . 55 ≤ Ω n ≤ 0 . 75
0 . 5 . 78 ≤ Ω n ≤ 1 ,
where Ω n is the normalized DT frequency. Use M ATLAB to confirm
that the designed FIR filter satisfies the given specifications.

H ( Ω ) =