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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

\begin{displaymath}z=e^s=e^{\sigma+j\omega}=e^\sigma e^{j\omega}=r e^{j\omega} \end{displaymath}

where

\begin{displaymath}\left\{ \begin{array}{l} Re[s]=\sigma  Im[s]=j\omega \end{a...
...rt z\vert=r=e^\sigma  \angle{z}=\omega
\end{array} \right.
\end{displaymath}

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 infinite range $-\infty < \omega < \infty$ for frequency $\omega$ along a vertical line $Re[s]=\sigma_0$ in s-plane is mapped repeatedly to a finite range $0 \le \omega < 2\pi$ around a circle $\vert z\vert=e^{\sigma_0}$ in z-plane, corresponding to the conversion of a continuous signal $x(t)$ with non-periodic spectrum $X(j\omega)$ for $-\infty < \omega < \infty$ to a discrete signal $x[n]$ with periodic spectrum $X(e^{j\omega})$ for $0 \le \omega < 2\pi$.

conformal.gif


next up previous
Next: Region of Convergence and Up: Z_Transform Previous: From Discrete-Time Fourier Transform
Ruye Wang 2012-01-28