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-1X(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.