Need help with z-inverse transform

Question

So the question is

**Use partial reaction method to determine the inverse z-transform for F(z) = $$\frac{z^3 -z^2 +z-1/16}{z^3 -5/4z^2 +1/2z -1/16}$$ **

Your help would be highly appreciated thanks!

Answer 1

If I have understood correctly, we need

$$F(z)=\dfrac{16z^3-16z^2+16z-1}{16z^3-20z^2+8z-1}$$

As $$16z^3-20z^2+8z-1=(2z-1)^2(4z-1)$$ and the coefficients of the terms containing higher power are same in the numerator & the denominator

let

$$F(z)=1+\dfrac A{4z-1}+\dfrac B{2z-1}+\dfrac C{(2z-1)^2}$$

where $A,B,C$ are arbitrary constants which can be determined comparing the coefficients of the different powers of $z$