The Laplace Transform of a sampled signal can be written as:-

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 image below show how the applet can appear when enabled.

© 2007 Cuthbert A. Nyack.