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Evaluation of Fourier Transform from Pole-Zero Plot

Given the pole-zero plot of the transfer function $H(s)$, we can qualitatively learn the system's behavior as frequency $\omega$ changes from 0 to infinity (i.e., the Bode plot if in log scale). We first review the representation of complex values in the complex plane (e.g., the s-plane).

In general, the magnitude and phase angle of a rational expression $X(s)$ can be found as

\begin{displaymath}H(s)=\frac{\prod_{k=1}^M (s-s_{z_k})}{\prod_{k=1}^N (s-s_{z_k...
...}
\;exp[j(\sum_{k=1}^M \angle u_k-\sum_{k=1}^N \angle v_k)]
\end{displaymath}

where

\begin{displaymath}u_k\stackrel{\triangle}{=}s-s_{z_k},\;\;\;\;(k=1,\cdots,M),
...
...\;\;v_k\stackrel{\triangle}{=}s-s_{p_k},\;\;\;\;(k=1,\cdots,N)
\end{displaymath}

i.e.,

\begin{displaymath}\vert H(s)\vert=\frac{\prod_{k=1}^M \vert u_k\vert}{\prod_{k=...
...;
\angle H(s)=\sum_{k=1}^M \angle u_k-\sum_{k=1}^N \angle v_k \end{displaymath}

Given the pole-zero plot of a system's rational transfer function $H(s)$, the magnitude and phase angle of the corresponding Fourier transform $H(j\omega)$, i.e., its Bode plot, can be obtained qualitatively by observing the magnitude and angle changes of the zero and pole vectors, as shown below for the first and second order systems.



Subsections
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Next: First order system Up: Laplace_Transform Previous: LTI Systems Characterized by
Ruye Wang 2012-01-28