For the FIR filtec wa impulse response of nn 2 deltan 2delt

For the FIR filtec w/a impulse response of n[n] = 2 delta[n] -2delta[n-1] + delta[n-2] the system function is H(z) = 2 -2 z^-1 + z^-2 Show mathematically how 2- 2z^-1 + z^-2 = 2Z^2 - z + 1/2/z^2 = 2[z-1/2(1+j)][z-1/2(1-j)]/z^2

Solution

The system function is H(z)= 2 - 2z^-1 + z^-2  = 2 - 2/z + 1/z²

= (2z² - 2z + 1) / z²

= 2(z² - z + 1/2) / z²

if we consider the numerator ie. z² - z + 1/2 , the roots of the equation is [-b + (b² - 4ac)^1/2] / 2a

z = [1 + (1- 2)^1/2] / 2

= [1 + (-1)^1/2] / 2

= [1 + j ] /2 , since j = (-1)^1/2

= (1 + j)/2 and (1-j)/2

the roots of the equation z² - z + 1/2 is [ z- (1 + j)/2 ] [ z- (1 - j)/2]

thus the transfer function H(Z) = 2(z² - z + 1/2) / z² = 2[ [ z- (1 + j)/2 ] [ z- (1 - j)/2]] / z²

= 2[ [ z- 1/2(1 + j) ] [ z- 1/2(1 - j)] / z²