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

$$U[z]=\mathcal{Z} \{ u[n] \} = \sum_{n=0}^{\infty} u[n] \, z^{-n}.$$

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

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

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

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

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

$$|U[z]|\leq \sum_{n=0}^{\infty} |u[n]| \, |z^{-n}| \leq M \sum_{n=0}^{\infty} a^n \, r^{-n} = \frac{1}{1 + a r^{-1}}$$

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

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

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

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

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

$$
\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \neq 0 \end{cases}
$$

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

$$
u[n] = \sum_{k=-\infty}^{\infty} u[k] \, \delta[n – k]
$$

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

$$
y[n] = \sum_{k=-\infty}^{\infty} g[n – k] \, u[k] = \sum_{k=-\infty}^{\infty} g[k] \, u[n – k].
$$

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

$$
\begin{aligned}
Y[z]=\mathcal{Z}\{y[n]\} &= \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} g[k] \, u[n-k] \, z^{-n} \\
&= \sum_{k=0}^{\infty} g[k] \sum_{n=0}^{\infty} u[n-k] \, z^{-n} \\
&= \sum_{k=0}^{\infty} g[k] z^{-k} U[z] \\
&= G[z] \, U[z]
\end{aligned}
$$

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.