Suppose the impulse response of an FIR filter of order m 5 5 is as follows where the X terms are to be determined. h 5 f2, 4, 3, X, X, Xg (a) Assuming H(z) is a linear-phase filter, find the complete impulse response. If there are multiple solutions, find each of them.

Complete Question:

Suppose the impulse response of an FIR filter of order m =5 is as follows where the X terms are to be determined.

h = [2,4,3,X,X,X]

a) Assuming H(z) is a linear- phase filter, find the complete impulse response. If there are multiple solutions, find each of them

b) For each solution in part (a), indicate the linear-phase FIR filter type

c) For each solution in part (a), find the phase offset, α, and the group delay, D(f)

a1) For symmetric impulse response, h(n) = [2,4,3,3,4,2]

a2)For asymmetric impulse response h(n) = [2,4,3,-3,-4,-2]

b1)  For symmetric impulse response,the FIR filter is a low pass filter

b2)  For symmetric impulse response,, the FIR filter is a differentiator

c1) Phase offset = -2.5w

c2) D(f) = π/2 - 2.5w

Step-by-step explanation:

a) The impulse response, h can either be symmetric or anti-symmetric about α = m/2

i) When the impulse response, h is symmetric about  α = m/2, h becomes:

h(n) = [2,4,3,3,4,2]

ii) When the impulse response, h is anti-symmetric about   α = m/2, h becomes:

h(n) = [2,4,3,-3,-4,-2]

b) Relationship for converting h(n) into the frequency domain using Fourier transform: , n= 1,2,3,4,5

i) For h(n) = [2,4,3,3,4,2]

When ,

When w = π,

This is the property of a low pass filter, hence the FIR filter is a low pass filter

ii) For h(n) = [2,4,3,-3,-4,-2]

When

When

This is the property of a differentiator, hence the FIR filter is a differentiator.

c1) Phase offset = -wα

α = 5/2 = 2.5

Phase offset = -2.5w

c2) Group delay, D(f) = π/2 - αw

D(f) = π/2 - 2.5w

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step-by-step explanation:

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