How to find out transient response of z-transform (discrete)

Question

Given z-transform transfer function $H(z) = \frac{Y(z)}{X(z)}$, with the corresponding linear ODE, how does one find out transient response of such a transfer function given a certain input?

Answer 1

Let's assume that the system described by the transfer function $H(z)$ is causal and stable, i.e. all its poles are inside the unit circle of the complex $z$-plane. You can compute the $\mathcal{Z}$-transform of the output signal

$$Y(z)=H(z)X(z)$$

where $X(z)$ is the $\mathcal{Z}$-transform of the input signal. Transforming $Y(z)$ back to the time domain gives the output signal $y[n]$. The part of $y[n]$ that decays to zero with increasing $n$ is the transient response, the remaining part is the steady-state response.