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Lecture 2: Rubber Sheet Geometry

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Schedule :: Intro :: Inverse :: Properties :: Solution :: Transfer Functions :: System Realization :: Bilateral :: Frequency Response :: Rubber Sheet Geometry :: FIR Filters

Let us write $H(z)$ in terms of its poles and zeros:
\begin{displaymath}H(z) = b_n \frac{(z-z_1)(z-z_2)\cdots
(z-z_n)}{(z-\gamma_1)(z-\gamma_2) \cdots (z-\gamma_n)}
\end{displaymath}

Consider evaluating this at a point $z = e^{j\Omega}$ , which is on the unit circle. We find

\begin{displaymath}\vert H(e^{j\Omega})\vert = \vert b_n\vert \frac{\vert e^{j\O...
...ga} - \gamma_1\vert \cdots
\vert e^{j\Omega} - \gamma_1\vert}
\end{displaymath}

Let us write $e^{j\Omega} - z_i = r_i e^{j\phi_i}$ (polar form for the line segment connecting them), and $e^{j\Omega} - z_i = d_i
e^{j\theta_i}$ (polar form). Then

\begin{displaymath}\vert H(e^{j\Omega})\vert = \vert b_n\vert \frac{r_1 r_2 \cdo...
...}$}}
{\text{product of distances from poles to $e^{j\Omega}$}}
\end{displaymath}

Similarly,
\begin{displaymath}\arg H(e^{j\Omega}) = (\phi_1 + \cdots \phi_n) - (\theta_1 + ...
... $e^{j\Omega}$} - \text{sum
of pole angles to $e^{j\Omega}$}.
\end{displaymath}

Discuss filter design by pole placement, and the rubber sheet idea: poles increase the gain, zeros decrease it. Notch filter. Overhead.

\begin{example}
Design using \lq\lq trial and error'' techniques a digital bandpass
...
...{z^2- \sqrt{2}\gamma z + \gamma^2}.
\end{displaymath}\vspace*{5in}
\end{example}

\begin{example}
We want to design a second-order notch filter to have zero
tra...
...o
\begin{displaymath}K = (1+a^2)/2.
\end{displaymath}\vspace*{3in}
\end{example}
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 18). Lecture 2: Rubber Sheet Geometry. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/node9.html. This work is licensed under a Creative Commons License Creative Commons License