# Z Transform and Laplace Transform.

Cuthbert Nyack
The Laplace Transform of a sampled signal can be written as:- If the following substitution is made in the Laplace Transform The definition of the z tranaform results. The relation between s and z can also be written:-  The mapping of the s plane to the z plane is illustrated by the above diagram and the following 2 relations. Lines of any given color in the s plane maps to lines of the same color in the z plane.  The above relations show the following:-
The imaginary axis of the s plane between minus half the sampling and plus half the sampling frequency maps onto the unit circle in the z plane.
The portion of the s plane to the left of the red line maps to the interior of the unit circle in the z plane.
The portion of the s plane to the right of the red line maps to the exterior of the unit circle in the z plane.
The green line(line of constant sigma) maps to a circle inside the unit circle in the z plane.
Lines of constant frequency in the s plane maps to radial lines in the z plane.
The origin of the s plane maps to z = 1 in the z plane.
The negative real axis in the s plane maps to the unit interval 0 to 1 in the z plane.

The s plane can be divided into horizontal strips of width equal to the sampling frequency. Each strip maps onto a different Riemann surface of the z "plane".
Mapping of different areas of the s plane onto the Z plane is shown below. The applet below shows the mapping from the s plane to the z plane.

The image below show how the applet can appear when enabled. 