# Transfer function to impulse response

## Homework Statement

Given the transfer function of a linear, time-invariant system
$$H(z)=\frac{z^2+5z}{z^2+5}$$
compute the impulse response.

## Homework Equations

We are supposed to compute the inverse z-transform with partial fraction decomposition but the problem here is the irreducible quadratic function $$z^2+5$$.

## The Attempt at a Solution

In our table of inverse z-transforms they are only functions of the the type
$$\frac{z^{m+1}}{(z-a)^{m+1}}$$
I tried this.
$$H(z)=\frac{z^2+5z}{z^2+5}=\frac{z^2}{z^2+5}+5\frac{z}{z^2+5}$$
I can compute the inverse z-transform of $$\frac{z^2}{z^2+5}$$ just fine, but how do I compute the inverse z-transform of $$\frac{z}{z^2+5}$$

Thanks.

It's not irreducible. $z^2+5=(z-\sqrt{5}i)(z+\sqrt{5}i)$. Complex numbers are an important part of z transforms.