written 6.5 years ago by |
Step-1: Identify the specification of filter
N=7∝=3ωc=−1<ω<1
Window Type: Hamming
Step-2: Calculate the Inverse Fourier Transform of H(ω)
hd(n)=12π∫−ωc(ωcHd(ω)ejωndω
=12π∫1−1e−j3ωejωnd
=12π∫1−1ej(n−3)ωdω
=12π[ej(n−3)ω)(j(n−3))]1−1
=1(π(n−3))[(ej(n−3)−e−j(n−3)))2j]
hd(n)=sin(n−3)(π(n−3))
hd(0)=0.014=hd(6) {by linear phase property}
hd(1)=0.144=hd(5) {by linear phase property}
hd(2)=0.267=hd(4) {by linear phase property}
By L-Hospital’s Rule
hd(3)=0.318
Step-3: Calculation of window response W(n)
W(n)=0.54−0.46cos(2πn(N−1))
W(0)=0.08=ω(6)
W(1)=0.31=ω(5)
W(2)=0.77=ω(4)
W(3)=1
Step-4: Calculate impulse Response of filter
h(n)=hd(n)∗W(n)
h(0)=0.0012=h(6)
h(1)=0.044=h(5)
h(2)=0.205=h(4)
h(3)=0.318
Step-5: Calculation of filter Transfer function
H(z)=∑N−1n=0h(n)z−n
H(z)=0.00112+0.044z−1+0.205z−2+0.318z−3+0.205z−4+0.044z−5+0.00112z−6
Step-6: Realization Structure