Discrete-time systems and the Z-transform. Part I

The Z-transform does to discrete-time signals and systems what the Laplace transform does to a continuous-time signals and systems.

The Z-transform

The Z-transform of the discrete-time signal $u[n]$, $n\in \mathbb{N}$, is the function of the complex variable $z$ defined by

$$U[z]=\mathcal{Z}\{u[n]\}=\sum _{n=0}^{\mathrm{\infty }}u[n]z^{-n}.$$

As with the Laplace transform, $u[n]$ is assumed to be zero for negative $n$, that is $u[n]=0$, $n<0$.

Exponential discrete-time signals are of great importance and the calculation

$$\mathcal{Z}\{a^n\}=\sum _{n=0}^{\mathrm{\infty }}{a}^n{z}^{-n}=\frac{1}{1–a{z}^{-1}},|z|>|a|.$$

reveals many of the inner workings of the Z-transform. The Z-transform of the exponential has a domain of convergence which is the exterior of the disc of radius $|a|$. As with the Laplace transform, the Z-transform has no trouble handling unbounded signals, e.g. $|a|>1$, by adapting its region of convergence.

A sufficient condition for a discrete-time signal to admit a Z-transform is for it to be of exponential order, that is if there exists $a>0$, and $M>0$ such that

$$|u[n]|\le M{a}^n,n\in \mathbb{N}.$$

Indeed, with $z=r{e}^{j\theta }$

$$|U[z]|\le \sum _{n=0}^{\mathrm{\infty }}|u[n]||z^{-n}|\le M\sum _{n=0}^{\mathrm{\infty }}{a}^n{r}^{-n}=\frac{1}{1+a{r}^{-1}}$$

is bounded for all $r>a$. The region of convergence of a typical Z-transform is the exterior of a disc centered at the origin, whereas the region of convergence of a Laplace transform were a right half-plane.

The Z-transform has many properties that are analogous to the properties of the Laplace transform. For example, one useful property is the delay property

$$\mathcal{Z}\{u[n-k]\}=z^{-k}U[z],k\ge 0,$$

obtained by assuming $u[n]=0$, $n<0$, and changing indices as in

$$\mathcal{Z}\{u[n-k]\}=\sum _{n=0}^{\mathrm{\infty }}u[n-k]z^{-n}=\sum _{m=-k}^{\mathrm{\infty }}u[m]z^{-m-k}=z^{-k}U[z].$$

As discussed in this post, the inverse Z-transform is given by the integral formula

$$u[n]=\frac{1}{2\pi j}{\oint }_{C}U[z]z^{n-1}dz,$$

a formula that is rarely used in practice, but of great theoretical importance.

We shall discuss other properties of the Z-transform in more detail later in this and on future posts.

Discrete-time linear systems, impulse response and convolution

As with continuous-time systems, discrete-time systems are completely described by one signal: the impulse response $g[n]$. The key ingredient is once again the definition of a suitable impulse

$$\delta [n]=\{\begin{array}{ll}1,& n=0\\ 0,& n\ne 0\end{array}$$

Note how a discrete-time impulse is much better behaved than its continuous-time counterpart: the discrete-time impulse is actually a true signal!

As discussed here and in Chapter 3 in the case of continuous-time systems, with the help of the impulse one can represent signals as a combination of delayed impulses

$$u[n]=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}u[k]\delta [n–k]$$

and the response of an arbitrary discrete-time linear time-invariant system by the convolution

$$y[n]=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}g[n–k]u[k]=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}g[k]u[n–k].$$

With causality, in which case $u[n]$ and $g[n]$ are zero when $n<0$, these reduce to

$$y[n]=\sum _{k=0}^{n}g[n–k]u[k]=\sum _{k=0}^{n}g[k]u[n–k]$$

as for causal continuous-time systems.

One of the most useful properties of the Z-transform is the ability to transform a convolution into a product as in

$$\begin{array}{rl}Y[z]=\mathcal{Z}\{y[n]\}& =\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}g[k]u[n-k]z^{-n}\\ & =\sum _{k=0}^{\mathrm{\infty }}g[k]\sum _{n=0}^{\mathrm{\infty }}u[n-k]z^{-n}\\ & =\sum _{k=0}^{\mathrm{\infty }}g[k]z^{-k}U[z]\\ & =G[z]U[z]\end{array}$$

after using the delay property of the Z-transform. One can recognize in the above formula the discrete-time transfer-function $G[z]=\mathcal{Z}\{g[n]\}$.

The continuous- versus discrete-time analogies go well beyond those discussed so far. We will leave some of those for future posts.