$H(e^{jω} )=e^{-j3ω} ; 0≤|ω|≤\frac{3π}{4}$

=0 ; otherwise

Step-1: Identify the specification of filter

$N=7 ∝=3 ω_c=\frac{-3π}{4}≤ω≤\frac{3π}{4}$

Window Type: Hamming

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

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

$=\frac{1}{2π} ∫_{\frac{(-3π)}{4}}^{\frac{3π}{4}}e^{-j3ω} e^{jωn} dω$

$=\frac{1}{2π} ∫_{\frac{(-3π)}{4}}^{\frac{3π}{4}}e^{j(n-3)ω} dω$

$=\frac{1}{2π} [(e^{j(n-3)\frac{3π}{4}}-e^{\frac{-j(n-3)\frac{3π}{4})}{(j(n-3)}}]$

$=\frac{1}{π(n-3)} [(e^{j(n-3)\frac{3π}{4}}-e^{\frac{-j(n-3)\frac{3π}{4}))}{2j}}]$

$h_d (n)=\frac{sin⁡\frac{3π}{4}(n-3)}{(π(n-3)}$

$h_d (0)=0.075=h_d (6 )$ {by linear phase property}

$h_d (1)=-0.159=h_d (5 …