Overview and Introduction to the z-Transform (Polynomial & Rational Functions)

[John Santiago]

Here's the first of a series on the z-Transform. Comments are welcomed.

We introduce the z-transform bringing polynomials and rational functions to help analyze linear discrete-time systems. The discrete-time convolution (or FIR convolution) is equivalent to polynomial multiplication and algebraic operations in the z transform domain can be translated as combining or decomposing linear time-invariant (LTI) systems. The most common z-transforms are rational functions, that is, the numerator polynomial divided by the denominator polynomial.

We consider three representations of signals and systems. The first one, the time-domain or n-domain, involves sequences, impulse responses and differences. The next representation is the frequency or w-domain (omega-domain). Here, we consider frequency responses and spectrum descriptions. Finally and most important when analyzing discrete-time systems is the z-domain. This consist of z transforms, operators, and poles and zeros.


One application of the z-transform is the use of the discrete-time convolution mentioned earlier. Here, the operation in the z-domain or z-transform domain involves multiplication between two polynomials. We'll see its the multiplication between the z transform of the input signal and the z-transform of the system or filter.

The above application shows the value of having three different domain representation. A difficult analysis in one domain (discrete-time convolution) is simpler to analyze in the other domain (in this case the z-transform domain involving polynomial multiplication).

Therefore, having increased understanding will result form developing skills for moving from one representation to another. The z transform domain exists primarily for its mathematical convenience in analyzing and synthesizing discrete-time signals and systems.