Ve216LectureNotesChapter3Part1

Ve216LectureNotesChapter3Part1 - Ve216 Lecture Notes...

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Ve216 Lecture Notes Dianguang Ma Spring 2009
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Chapter 3 Fourier Representations of Signals and LTI Systems
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3.1 Introduction In this chapter, we represent a signal as a weighted superposition of complex sinusoids. If such a signal is applied to an LTI system, then the system output is a weighted superposition of the system response to each complex sinusoid. The study of signals and systems using sinusoidal representations is termed Fourier analysis .
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3.2 Complex Sinusoids and Frequency Response of LTI Systems Consider the output of a discrete-time LTI system with impulse response h[n] and unit amplitude complex sinusoidal input x[n]. ( ) [ ] [ ] [ ]* [ ] [ ] [ ] [ ] [ ] ( ) ( ): the frequency response of the system j n k j n k j n j k k k j n j j x n e y n h n x n h k x n k h k e e h k e e H e H e =-∞ - - Ω =-∞ =-∞ = = = - = = =
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3.2 Complex Sinusoids and Frequency Response of LTI Systems The output of an LTI system is a complex sinusoid of the same frequency as the input, multiplied by the frequency response of the system.
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3.2 Complex Sinusoids and Frequency Response of LTI Systems We say that the complex sinusoid e jΩn is an eigenfunction of the LTI system H associated with the eigenvalue H(e ). ( ) j n j j n e H e e
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3.2 Complex Sinusoids and Frequency Response of LTI Systems We write the complex-valued frequency response in polar form. { } { } arg ( ) ( ) [ ] ( ) ( ) ( ) : the magnitude response of the system arg ( ) : the phase response of the system j j j k k j H e j j j j H e h k e H e H e e H e H e - Ω =-∞ = =
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Response of LTI Systems The system thus modifies the amplitude of the input by the magnitude response |H(e )| and and the phase by the phase response arg{H(e )}. ( arg{
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This note was uploaded on 10/26/2009 for the course EECS EECS 216 taught by Professor Dianguangma during the Spring '09 term at University of Michigan-Dearborn.

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Ve216LectureNotesChapter3Part1 - Ve216 Lecture Notes...

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