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378 Chapter 7 FIR FILTER DESIGN
h(n) isofalinear-phase type. Verify your function on sequences given here.
h I(n) =(0.9) |n−5| cos[π(n − 5)/12] [u(n) − u(n − 11)]
h II(n) =(0.9) |n−4.5| cos[π(n − 4.5)/11] [u(n) − u(n − 10)]
h III(n) =(0.9) |n−5| sin[π(n − 5)/12] [u(n) − u(n − 11)]
h IV(n) =(0.9) |n−4.5| sin[π(n − 4.5)/11] [u(n) − u(n − 10)]
h(n) =(0.9) n cos[π(n − 5)/12] [u(n) − u(n − 11)]
P7.7 Prove the following properties of linear-phase FIR filters.
1. If H(z) has four zeros at z 1 = re jθ , z 2 = 1 r e−jθ , z 3 = re −jθ , and z 4 = 1 r e−jθ then H(z)
represents a linear-phase FIR filter.
2. If H(z) has two zeros at z 1 = e jθ and z 2 = e −jθ then H(z) represents a linear-phase FIR
filter.
3. If H(z) has two zeros at z 1 = r and z 2 = 1 then H(z) represents a linear-phase FIR
r
filter.
4. If H(z) has a zero at z 1 =1or a zero at z 1 = −1 then H(z) represents a linear-phase
FIR filter.
5. For each of the sequences given in Problem P7.6, plot the locations of zeros. Determine
which sequences imply linear-phase FIR filters.
P7.8 A notch filter is an LTI system, which is used to eliminate an arbitrary frequency ω = ω 0.
The ideal linear-phase notch filter frequency response is given by
H d
( e
jω ) =
{
0, |ω| = ω0;
1 · e −jαω , otherwise.
(α is a delay in samples)
1. Determine the ideal impulse response, h d (n), of the ideal notch filter.
2. Using h d (n), design a linear-phase FIR notch filter using a length 51 rectangular window
to eliminate the frequency ω 0 = π/2 rad/sample. Plot amplitude the response of the
resulting filter.
3. Repeat part 2 using a length 51 Hamming window. Compare your results.
P7.9 Design a linear-phase bandpass filter using the Hann window design technique. The
specifications are
lower stopband edge: 0.2π
As =40dB
upper stopband edge: 0.75π
lower passband edge: 0.35π
upper passband edge: 0.55π R p =0.25 dB
Plot the impulse response and the magnitude response (in dB) of the designed filter. Do not
use the fir1 function.
P7.10 Design a bandstop filter using the Hamming window design technique. The specifications are
lower stopband edge: 0.4π
As =50dB
upper stopband edge: 0.6π
lower passband edge: 0.3π
upper passband edge: 0.7π R p =0.2 dB
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