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# Conformal Mapping between S-Plane to Z-Plane

The s-plane and the z-plane are related by a conformal mapping specified by the analytic complex function

where

The mapping is continuous, i.e., neighboring points in s-plane are mapped to neighboring points in z-plane and vice versa. Consider the mapping of these specific features:
• The origin of s-plane is mapped to on the real axis in z-plane.
• Each vertical line in s-plane is mapped to a circle centered about the origin in z-plane. In particular,
• Leftmost vertical line is mapped as the origin
• Imaginary axis is mapped as the unit circle
• Rightmost vertical line is mapped as a circle of infinite radius .
• Each horizontal line in s-plane is mapped to , a ray from the origin in z-plane of angle with respect to the positive horizontal direction.
• A right angle formed by a pair vertical and horizontal lines in s-plane is conserved by the mapping, as the corresponding circle and ray in z-plane also form a right angle. (In fact any angle is conserved, an important property of the conformal mapping.)
The infinite range for frequency along a vertical line in s-plane is mapped repeatedly to a finite range around a circle in z-plane, corresponding to the conversion of a continuous signal with non-periodic spectrum for to a discrete signal with periodic spectrum for .

Next: Region of Convergence and Up: Z_Transform Previous: From Discrete-Time Fourier Transform
Ruye Wang 2014-10-28