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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

$N=7 ∝=3 ω_c=-1\ltω\lt1$

Window Type: Hamming

Step-2: Calculate the Inverse Fourier Transform of H(ω)

$h_d (n)=\frac{1}{2π} ∫_{-ω_c}{(ω_c}H_d (ω) e^{jωn} dω$

$=\frac{1}{2π} ∫_{-1}^1e^{-j3ω} e^{jωn }d$

$=\frac{1}{2π} ∫_{-1}^1e^{j(n-3)ω} dω$

$=\frac{1}{2π} [e^{\frac{j(n-3)ω)}{(j(n-3))}}]_{-1}^1$

$=\frac{1}{(π(n-3))} [(e^{j(n-3)}-e^{\frac{-j(n-3)))}{2j}}]$

$h_d (n)=\frac{sin⁡(n-3)}{(π(n-3))}$

$h_d (0)=0.014=h_d (6)$ {by linear phase property}

$h_d (1)=0.144=h_d (5)$ {by linear phase property}

$h_d (2)=0.267=h_d (4)$ {by linear phase property}

By L-Hospital’s Rule

$h_d (3)=0.318$

Step-3: Calculation of window response W(n)

$W(n)=0.54-0.46cos⁡(\frac{2πn}{(N-1))}$

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

$h(n)=h_d (n)*W(n)$

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

$H(z)= ∑_{n=0}^{N-1}h(n) z^{-n}$

$H(z)=0.00112+0.044z^{-1} +0.205z^{-2}+0.318z^{-3}+0.205z^{-4}+0.044z^{-5}+0.00112z^{-6}$

Step-6: Realization Structure