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The following transfer function characterizes an FIR filter (N = 9). Determine the magnitude response and show that the phase and group delays are constant.H(z)=N1n=0h(n)zn
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Solution:

The transfer function of the filter is given by,

H(z)=N1n0h(n)zn=h(0)+h(1)z1+h(2)z2+h(3)z3+h(4)z4+h(5)z5+h(6)z6+h(7)z7+h(8)z8α=N12=912=4.

H(z)=z4[h(0)z4+h(1)z3+h(2)z2+h(3)z1+h(4)z0+h(5)z1+h(6)z2+h(7)z3+h(8)z4]

Since h(n)=h(N1n)

H(z)=z4[h(0)(z4+z4)+h(1)(z3+z3)+h(2)(z2+z2)+h(3)(z+z1)+h(4)]

H(ω)=ej4ω[h(0)[ej4ω+ej4ω]+h(1)[ej3ω+ej3ω]+h(2)[ej2ω+ej2ω]+h(3)[ejω+ejω]+h(4)]=ej4ω[h(4)+23n=0h(n)cos(4n)ω]=ej4ω|H(ω)|

where |H(ω)| is the magnitude response and θ(ω)=5ω is the phase response. The phase delay τp and group delay τg are given by

τp=θ(ω)ω=5 and τg=d(θ(ω))dω=d(5ω)dω=5

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