Let $\delta (n)$ be a digital impulse signal, that is, $\delta (0)=1$ and is zero everywhere else. By scaling, shfting and summing $\delta (n)$ functions, we can construct any input signal. LTI filters preserve scaling, shifting and summing, thus the behaviour of such a filter on $\delta (n)$ determines the behaviour of the filter on any input signal.

We can easily compute the transfer function of a filter by looking at the output of the filter given $\delta (n)$.

Given a function $x(n)$ defined for all $n\in \mathbb{Z}$, define the $z$-transform of $x$ by

Let $h(n)$ be the output of an LTI filter applied to $\delta (n)$. Define the transfer function $H(z)$ to be the $z$-transform of $h(n)$.

Then by linearity and time-invariance, for any input signal $x(n)$ and corresponding output signal $y(n)$, we have $Y(z)=H(z)X(z)$ where $X(z),Y(z)$ are the $z$-transforms of $x(n),y(n)$ respectively. Note also $y(n)=h(n)*x(n)$, where $*$ denotes convolution.