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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, I stands for in-phase and Q 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 e -jwt to have negative frequency Note: A-jB is the complex conjugate of A+jB So, e -jwt is the complex conjugate of e jwt