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Design FIR filter using frequency sampling technique for the following specification:

H(ejω)=ej3ω;0|ω|π2 H(ejω)=0;otherwise

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We know that

H(k)=H(ω)(ω=2πkN) ………(1)

H(ejω)=ej3ω

Here ∝=3

And∝=(N1)2

∴N=7

Now, By using equation (1)

H(k)=e(j6πk)7;02πk7π2

=0;π22πk7π

∴H(k)=e^{\frac{(-j6πk)}{7}} ;0≤k≤\frac{7π}{4π}

=0 ; \frac{7π}{4π}≤k≤\frac{7π}{2π}

∴H(k)=e^{\frac{(-j6πk)}{7}} ;0≤k≤2

=0 ;2≤k≤4

Now , Here K is vary from 0 to 2

When k=0

∴H(0)=1

When k=1

∴H(1)=e^{\frac{(-j6π)}{7}}

When K=2

∴H(2)=e^{\frac{(-j12π)}{7}}

According to transfer function equation of Frequency Sampling Realization:

H(z)=[\frac{(1-z^{-7})}{7}][\frac{H(0)}{(1-z^{-1}}+\frac{H(1)}{(1-e^{\frac{j2π}{7}} z^{-1}} +(\frac{H(2))}{(1-e^{\frac{j6π}{7}} z^{-1} )}]

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