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Design linear phase FIR low pass filter of length T and cut of frequency 1 rad/sec using hamming window.
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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)=12πωc(ωcHd(ω)ejωndω

=12π11ej3ωejωnd

=12π11ej(n3)ωdω

=12π[ej(n3)ω)(j(n3))]11

=1(π(n3))[(ej(n3)ej(n3)))2j]

hd(n)=sin(n3)(π(n3))

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)=0.540.46cos(2πn(N1))

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)=N1n=0h(n)zn

H(z)=0.00112+0.044z1+0.205z2+0.318z3+0.205z4+0.044z5+0.00112z6

Step-6: Realization Structure

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