The Z-Transform

Introduction

A linear system can be represented in the complex frequency domain (s-domain where s = s + jw) using the LaPlace Transform.

Where the direct transform is:

And x(t) is assumed zero for t ≤ 0

The Inversion integral is a contour integral in the complex plane (seldom used, tables are used instead)

Where s is chosen such that the contour integral converges.

If we now assume that x(t) is ideally sampled as in:

Where:

and

Analyzing this equivalent system using standard analog tools will establish the z-Transform.

Sampling

Substituting the Sampled version of x(t) into the definition of the LaPlace Transform we get

But

(For x(t) = 0 when t < 0 )

Therefore

Now interchanging the order of integration and summation and using the sifting property of d-functions

(We are assuming that the first sample occurs at t = 0+)

if we now adjust our nomenclature by letting:

z = e^sT , x(n*Ts) = x_n , and

Which is the direct z-transform (one-sided; it assumes x_n = 0 for n < 0).

The inversion integral is:

(This is a contour integral in the complex z-plane)

(The use of this integral can be avoided as tables can be used to invert the transform.)

To prove that these form a transform pair we can substitute one into the other.

Now interchanging the order of summation and integration (valid if the contour followed stays in the region of convergence):

If “C” encloses the origin (that’s where the pole is), the Cauchy Integral theorem says:

And we get x_k = x_k Q.E.D

An Example

Find the z-transform of

This is the “Unit Pulse” at n = k (assume k > 0)

(Note: dividing by z is equivalent to a delay of one sample time)

A Short Table of z-Transforms

f(t) (sampled)	F(z)	Region of Convergence
U(t)		\|z\| > 1
d_n-k		\|z\| > 1
t		\|z\| > 1
t²		\|z\| > 1
e^at		\|z\| > e^at
sin(bt)		\|z\| > 1
cos(bt)		\|z\| > 1

Properties of the z-Transform

The z-transform has properties that are analogous to those of the LaPlace Transform. The following table has some of the more useful ones listed.

		Û	where C is a closed contour that includes z=0
	Signal		z-Transform
		Û
Superposition		Û
Time Shifting		Û
		Û
		Û
		Û
Time inversion		Û
Time Convolution		(convolution)
Frequency Differentiation		Û
Summation		Û

You should familiarize yourself with these as they will be used, along with the table of transforms to move between time series and the z-domain.

Finding the Inverse z-Transform

There are three common ways to find the time series, x_n when X(z) is given:

Infinite Series – done by dividing out the rational polynomial in z
Partial Fraction Expansion – Same as in LaPlace
The Inversion Integral – a contour integral in the complex z-plane

Example: , determine f_n

A. By Infinite Series

Now divide (long division) with the polynomials written in descending powers of z

2z^-2+8z^-3+22z^-4+52z^-5+114z^-6+…

Z³-4z²+5z-2|2z

2z-8+10z^-1-4z^-2

8-10z^-1+04z^-2

8-32z^-1+40z^-2-16z^-3

22z^-1-36z^-2+016z^-3

22z^-1-88z^-2+110z^-3-44z^-4

52z^-2-094z^-3+044z^-4

52z^-2-208z^-3+260z^-4-104z^-5

114z^-3-216z^-4+104z^-5

And the time sequence for f_n is

n	0	1	2	3	4	5	6	…
f_n	0	0	2	8	22	52	114	…

Note that this method does NOT give a closed form for the answer, but it is a good method for finding the first few sample values or to check out that the closed form given by another method at least starts out correctly.