Z Transform and Laplace Transform.
The Laplace Transform of a sampled signal can be
If the following substitution is made in the
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
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.
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© 2007 Cuthbert A. Nyack.