]> Sound and music - The z-Transform

## The z-Transform

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

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

$X\left(z\right)=\sum _{n=-\infty }^{\infty }x\left(n\right){z}^{-n}$

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

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