Fourier Series
and
Fourier Transform
•
•
•
•
6.082 Spring 2007
Complex exponentials
Complex version of Fourier Series
Time Shifting, Magnitude, Phase
Fourier Transform
Copyright © 2007 by M.H. Perrott
All rights reserved.
Fourier Series and Fourier Transform, Slide 1
The Complex Exponential as a Vector
Q
Note:
e jωt
sin(ωt)
ωt
cos(ωt)
I
• Euler’s Identity:
• 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 ejwt to have positive frequency
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 2
The Concept of Negative Frequency
Q
Note:
cos(ωt)
I
−ωt
-sin(ωt)
e-jωt
• 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 ejwt
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 3
Add Positive and Negative Frequencies
Q
Note:
ejωt
2cos(ωt)
I
e-jωt
• 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 positive frequency
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 4
Subtract Positive and Negative Frequencies
Q
Note:
2sin(ωt)
-e-jωt
ejωt
I
• As t increases, the subtraction of positive and
negative frequency complex exponentials leads to a
sine wave
– Note that the resulting sine wave is purely imaginary and
considered to have a positive frequency
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 5
Fourier Series
x(t)
t
T
• The Fourier Series is compactly defined using
complex exponentials
• Where:
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 6
From The Previous Lecture
x(t)
t
T
• The Fourier Series can also be written in terms of
cosines and sines:
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 7
Compare Fourier Definitions
• Let us assume the following:
• Then:
• So:
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 8
Square Wave Example
A
x(t)
T/2
t
-A
T
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 9
Graphical View of Fourier Series
• As in previous lecture, we can plot Fourier Series
coefficients
– Note that we now have positive and negative values of n
• Square wave example:
An
2A
π
Bn
2A
3π
1 3 5 7 9
-9 -7 -5 -3 -1
n
1 3 5 7 9
-9 -7 -5 -3 -1
n
-2A
3π
6.082 Spring 2007
-2A
π
Fourier Series and Fourier Transform, Slide 10
Indexing in Frequency
• A given Fourier coefficient,
,represents the
weight corresponding to frequency nwo
• It is often convenient to index in frequency (Hz)
Af
1 3 5 7 9
T T T T T
-9 -7 -5 -3 -1
T T T T T
6.082 Spring 2007
2A
π
2A
3π
f
Bf
1 3 5 7 9
T T T T T
-9 -7 -5 -3 -1
T T T T T
f
-2A
3π
-2A
π
Fourier Series and Fourier Transform, Slide 11
The Impact of a Time (Phase) Shift
x(t)
x(t)
A
A
T/2
T
t
-A
t
-A
T
T/4 T/4
• Consider shifting a signal x(t) in time by Td
• Define:
• Which leads to:
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 12
Square Wave Example of Time Shift
x(t)
x(t)
A
A
T/2
T
t
-A
t
-A
T
• To simplify, note that
6.082 Spring 2007
T/4 T/4
except for odd n
Fourier Series and Fourier Transform, Slide 13
Graphical View of Fourier Series
x(t)
x(t)
A
A
T/2
T
t
-A
t
-A
T/4 T/4
T
Af
1 3 5 7 9
T T T T T
-9 -7 -5 -3 -1
T T T T T
2A
π
2A
3π
-9 -7 -5 -3 -1
T T T T T
6.082 Spring 2007
Af
2A
π
f
-7
T
-9
T
-3
T
-5
T
3
T
-1 1
T T
7
T
5
T
9
T
Bf
Bf
1 3 5 7 9
T T T T T
1 3 5 7 9
T T T T T
-2A
3π
-2A
π
f
-9 -7 -5 -3 -1
T T T T T
f
f
Fourier Series and Fourier Transform, Slide 14
Magnitude and Phase
• We often want to ignore the issue of time (phase)
shifts when using Fourier analysis
– Unfortunately, we have seen that the An and Bn
coefficients are very sensitive to time (phase) shifts
• The Fourier coefficients can also be represented in
term of magnitude and phase
• where:
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 15
Graphical View of Magnitude and Phase
x(t)
x(t)
A
A
T/2
T
t
-A
t
-A
T/4 T/4
T
Xf
2A
3π
2A
π
2A
π
Xf
2A
3π
-9 -7 -5 -3 -1 1 3 5 7 9
T T T T T T T T T T
2A
3π
f
Φf
1 3 5 7 9
T T T T T
π/2
-9 -7 -5 -3 -1
T T T T T
6.082 Spring 2007
2A
π
2A
π
2A
3π
f
-9 -7 -5 -3 -1 1 3 5 7 9
T T T T T T T T T T
Φf
f
-π/2
π
-9 -7 -5 -3 -1 1 3 5 7 9
T T T T T T T T T T
f
Fourier Series and Fourier Transform, Slide 16
Does Time Shifting Impact Magnitude?
• Consider a waveform x(t) along with its Fourier
Series
• We showed that the impact of time (phase)
shifting x(t) on its Fourier Series is
• We therefore see that time (phase) shifting does
not impact the Fourier Series magnitude
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 17
Parseval’s Theorem
• The squared magnitude of the Fourier Series
coefficients indicates power at corresponding
frequencies
– Power is defined as:
Note:
* means
complex
conjugate
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 18
The Fourier Transform
• The Fourier Series deals with periodic signals
• The Fourier Transform deals with non-periodic
signals
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 19
Fourier Transform Example
x(t)
A
-T
T
t
• Note that x(t) is not periodic
• Calculation of Fourier Transform:
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 20
Graphical View of Fourier Transform
x(t)
A
-T
2TA
T
X(j2πf)
t
This is called
a sinc function
f
6.082 Spring 2007
-1
2T
1
2T
Fourier Series and Fourier Transform, Slide 21
Summary
• The Fourier Series can be formulated in terms of
complex exponentials
– Allows convenient mathematical form
– Introduces concept of positive and negative frequencies
• The Fourier Series coefficients can be expressed in
terms of magnitude and phase
– Magnitude is independent of time (phase) shifts of x(t)
– The magnitude squared of a given Fourier Series coefficient
corresponds to the power present at the corresponding
frequency
• The Fourier Transform was briefly introduced
– Will be used to explain modulation and filtering in the
upcoming lectures
– We will provide an intuitive comparison of Fourier Series
and Fourier Transform in a few weeks …
6.082 Spring 2007
Fourier Series and Fourier Transform, Slide 22