Chapter-6-Geometric-evaluation-of-the-Fourier-Transform-from-the-pole-zero-plot-Student

# Chapter-6-Geometric-evaluation-of-the-Fourier-Transform-from-the-pole-zero-plot-Student

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ELEC 215: Tim Woo Spring 2009/10 Chapter 6 - 1 Chapter 6: Geometric evaluation of the Fourier Transform from the pole-zero plot Spring 2009/10 Lecture: Tim Woo

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ELEC 215: Tim Woo Spring 2009/10 Chapter 6 - 2 Laplace Transform Where we are Differential equations State-space model CTFT Hardware Implementation System Characteristics System Responses Closed-loop Systems Continuous-time z-Transform Difference equations State-space model DTFT Hardware Implementation System Characteristics System Responses Closed-loop Systems Discrete-time Mapping Done in 211 To be covered In progress Done Will be covered if available Open-loop Systems Open-loop Systems
ELEC 215: Tim Woo Spring 2009/10 Chapter 6 - 3 Expected Outcome • In this chapter, you will be able to – Evaluate the frequency response of rational system function from the pole-zero plot geometrically.

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ELEC 215: Tim Woo Spring 2009/10 Chapter 6 - 4 Outline • Textbook – Section 9.4 Geometric evaluation of the Fourier Transform from the pole-zero plot – Section 10.4 Geometric evaluation of the Fourier Transform from the pole-zero plot • Reference book – A. V. Oppenheim, et. al., Discrete-time Signal Processing , 2nd edition, Prentice-Hall, 1999 – Section 5.5 All-pass systems
ELEC 215: Tim Woo Spring 2009/10 Chapter 6 - 5 Introduction As we saw in Section 9.1 (or Section 10.1), the CTFT (or DTFT) of a signal is the Laplace transform (or z-transform) evaluated on the j ω - axis (or unit circle). We discuss an alternative way, a procedure for geometrically evaluating the Fourier Transform from the pole-zero plot associated with a rational system function.

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ELEC 215: Tim Woo Spring 2009/10 Chapter 6 - 6 From the figure, the value of X ( s 1 ) has – A magnitude that is the length of vector – A phase that is the angle of the vector relative to the real axis. 9.4 Geometric evaluation of the Fourier Transform from the pole-zero plot To develop the procedure, let first consider a Laplace transform with a single zero [i.e. X ( s ) = s a ], which we evaluate at a specific value of s , says, s = s 1 .
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Chapter-6-Geometric-evaluation-of-the-Fourier-Transform-from-the-pole-zero-plot-Student

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