# Partial Fractions Decomposition and Inverse Z Transform of H(z) = (z^2) / (z^2 - 1)

1. Jul 19, 2009

### mdmman

1. The problem statement, all variables and given/known data
What is the inverse Z transform of
$$H(z)=\frac{z^2}{z^{2}-1}$$

2. Relevant equations
Z transform table can be found at http://en.wikipedia.org/wiki/Z-transform" [Broken]

3. The attempt at a solution

$$H(z)=\frac{z^2}{z^{2}-1} = 1+\frac{1}{z^{2}-1}$$

$$\frac{1}{z^{2}-1}=\frac{A}{z+1}+\frac{B}{z-1}$$

$$1=A(z-1)+B(z+1)=z(A+B)+(-A+B)$$

$$0=(A+B), 1=(-A+B)$$

$$A=-0.5, B=0.5$$

$$H(z)=\frac{z^2}{z^{2}-1} = 1+\frac{1}{z^{2}-1} = 1+\frac{-0.5}{z+1}+\frac{0.5}{z-1}$$

$$h(k)=Z^{-1}\{H(z)\} = Z^{-1}\{1\} + Z^{-1}\{\frac{-0.5}{z+1}\} + Z^{-1}\{\frac{0.5}{z-1}\} = \delta(n) + ? + ?$$

I don't know how to figure out what the inverse of $\frac{-0.5}{z+1}$ and $\frac{0.5}{z-1}$. Any help?

Last edited by a moderator: May 4, 2017