L03_fourier_series_transform

L03_fourier_series_transform - Fourier Series and Fourier...

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1 6.02 Spring 2008 Fourier Series and Fourier Transform, Slide 1 Fourier Series and Fourier Transform Complex exponentials Complex version of Fourier Series Time Shifting, Magnitude, Phase Fourier Transform Copyright © 2007 by M.H. Perrott & C. G. Sodini All rights reserved. 6.02 Spring 2008 Fourier Series and Fourier Transform, Slide 2 From The Previous Lecture The Fourier Series can also be written in terms of cosines and sines: t T x(t)
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2 6.02 Spring 2008 Fourier Series and Fourier Transform, Slide 3 The Complex Exponential as a Vector Euler’s Identity: Note: Consider I and Q as the real and imaginary parts As explained later, in communication systems, stands for in-phase and for quadrature As t increases, vector rotates counterclockwise We consider e jwt to have positive frequency 6.02 Spring 2008 Fourier Series and Fourier Transform, Slide 4 The Concept of Negative Frequency Note: As t increases, vector rotates clockwise We consider -jwt to have negative frequency Note: A-jB is the complex conjugate of A+jB So, is the complex conjugate of
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3 6.02 Spring 2008 Fourier Series and Fourier Transform, Slide 5 Add Positive and Negative Frequencies Note: As t increases, the addition of positive and negative frequency complex exponentials leads to a cosine wave Note that the resulting cosine wave is purely real and considered to have a frequency e -j ω t I Q e j ω t 2cos( ω t) 6.02 Spring 2008
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This note was uploaded on 08/23/2009 for the course EECS 6.02 taught by Professor Terman during the Spring '08 term at MIT.

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L03_fourier_series_transform - Fourier Series and Fourier...

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