Z-Transform vs. Inverse Z-Transform: Key Differences

This article explains the fundamental differences between the Z-Transform and its inverse.

Z-Transform

The Z-Transform is essentially the discrete-time equivalent of the Laplace Transform. Think of it as a way to analyze discrete signals in a different domain, making certain operations and analyses simpler.

Mathematically, the Z-transform of a discrete-time signal is defined as:

$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$

A crucial concept associated with the Z-Transform is the Region of Convergence (ROC). The ROC defines the range of ‘Z’ values for which the summation above converges to a finite value.

Properties of the ROC:

• The ROC of X(z) forms a ring in the Z-plane, centered around the origin.
• The ROC cannot contain any poles.

Properties of the Z-Transform:

The Z-Transform has several useful properties that simplify signal processing tasks.

Inverse Z-Transform

The Inverse Z-Transform, as the name suggests, is the process of converting a signal from the Z-domain back to the time domain. It essentially reverses the Z-Transform operation.

The equation for the Inverse Z-Transform is:

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

Where the integral represents a counter-clockwise contour integration around a circular path centered at the origin with radius ‘a’.

Here’s a table of common Z-Transform pairs for some frequently used functions: