Dear all, I am trying to transform the discrete LTI sytem E(z) into its partial fraction expansion
E(z) = \frac{N(z)}{D(z)}, N(z) = Mz^2 - Mz D(z) = (1+K)z^3+(-2K-3)z^2+(K+3)z -1
M and K are const. D(z) is of 3rd order. E(z) has 3 poles, {1,1, \frac{1}{K+1} }
My approach was to find the coefficients A,B,C of the partial fraction terms: \frac{A}{ (z-1)^2 } + \frac{B}{z-1} + \frac{C}{ z-\frac{1}{K+1} }
However, all my attempts ended up with A = 0, B = \frac{M}{K} C = - \frac{M}{K(1+K)}
This is confusing. E(z) has 3 poles, while its partial fraction expansion counts only 2. Further, E(z) is not stable, while its partial fraction expansion is conditional stable, i.e. its impulse response will remain on a constant value.
What am I doing wrong here?
Best regards, Chris