Z-transform for D.T. System Analysis

The Z-transform is the discrete time equivalent of Laplace Transform for C.T. System Analysis.

• The Laplace transform converts integro-differential equations into algebraic equations.
• The Z-transform changes difference equations into algebraic equations, thereby simplifying the analysis of discrete-time systems.

Definition

We define $X[z]$, the direct Z-transform of $x[n]$, as

$$X[z]=\sum _{n=-\infty }^{\infty }x[n]z^{-n}$$

where $z$ is a complex variable.

The inverse Z-transform is:

$$x[n]=\frac{1}{2\pi j}\oint X[z]z^{n-1}dz$$

In comparison to the Laplace transform, we have:

$$\begin{array}{rl}z& =e^s\\ & =e^{\sigma +j\omega }=e^{\sigma }{e}^{j\omega }\end{array}$$

such that $e^{sn}=z^n$.

Application

Let’s say we want to find the response of an LTI system $h[n]$ to the complex exponential $z^n$.

$$\begin{array}{rl}y[n]& =h[n]\ast x[n]\\ & =h[n]\ast z^n\\ & =\sum _{m=-\infty }^{\infty }h[m]z^{n-m}\\ & =z^n\sum _{m=-\infty }^{\infty }h[m]z^{-m}\end{array}$$

Then, we have:

$$y[n]=z^{n}H[z]$$

where $H[z]={\sum }_{m=-\infty }^{\infty }h[m]z^{-m}$ is just the Z-transform of $h[n]$!

Thus, the input passes through (eigenfunction), and we have a transfer function:

$$H[z]=\frac{y(t)}{x(t)}{|}_{x(t)=z^n}$$