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.

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© 2000 Cuthbert A. Nyack.