# Z Transform,
Step and Related Functions.

Cuthbert Nyack

The definition of the Z transform is
shown below.

The step function is defined as:-
and is shown graphically below.
A continuous step function shown above is plotted
in blue and the sampled step in red.
When a step function is sampled, each sample
has a constant value of 1. The Z Transform can be
written as a sum of terms as indicated below.
The expression for X(z) is a geometric
series which converges if |z| > 1 to:-

A Step function delayed by 1 sampling
interval is shown above and its Z transform
shown below.
This can be summed to give the Z transform of
the delayed step.
##
The Z transform of
x(k-1) can be written as z^{-1}X(z)
where X(z) is the Z transform of x(k).

For a kT interval delay of the step function
the Z transform is multiplied by z^{-k}

A Pulse of width 3 sample times and delayed
by 4 sample times is shown below.
Its Z transform is given by.
And this simplifies to:-
This Z transform has 3 zeros at -1, +j, -j and 7 poles at the origin.
A Z Transform like this one which includes only
a finite number of terms converges for all
values of z, its region of convergence covers the
z plane except at z = 0.

The applet below shows the magnitude, phase, real part and
imaginary part of the Z transform X(z) = 1 + ... + z^{-n}
for n from 2 to 10. Setting Fn = 4 shows the magnitude of the
Z transform and can be used to find the poles by varying zr and zi.
eg for n = 4, zeros are -0.809 ± j 0.588 and 0.309 ± j 0.951.
eg parameters (10.0, 0.0, 0.0, 2.5, 3.0, 1.5, 0, -124.0, 85.0, 1.5)
show the 10 zeros, 8 in dark blue and 2 in lighter blue. This transform
has an nth order pole at the origin which is not included in the plot
because it makes the zeros difficult to see.

*Return to main page*

*Return to page index*

© 2000 Cuthbert A. Nyack.