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

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 November 23, 2009, 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.

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	Info Lecture 2: Rubber Sheet Geometry Document Actions Schedule :: Intro :: Inverse :: Properties :: Solution :: Transfer Functions :: System Realization :: Bilateral :: Frequency Response :: Rubber Sheet Geometry :: FIR Filters Let us write 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 November 23, 2009, 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.	Reuse Course Download this course

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Lecture 2: Rubber Sheet Geometry

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