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