M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 1
Fourier Seriesand
Fourier Transform• Complex exponentials• Complex version of Fourier Series• Time Shifting, Magnitude, Phase• Fourier Transform
Copyright © 2007 by M.H. PerrottAll rights reserved.
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 2
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 ejwt to have positive frequency
e jωt
I
Q
cos(ωt)
sin(ωt)
ωt
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 3
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 ejwt
e-jωt
I
Q
cos(ωt)
-sin(ωt)
−ωt
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 4
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 positive frequency
e-jωt
I
Q
ejωt
2cos(ωt)
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 5
Subtract Positive and Negative Frequencies
Note:
• 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
-e-jωt
I
Q
ejωt
2sin(ωt)
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 6
Fourier Series
• The Fourier Series is compactly defined using complex exponentials
• Where:
t
T
x(t)
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 7
From The Previous Lecture
• The Fourier Series can also be written in terms of cosines and sines:
t
T
x(t)
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 8
Compare Fourier Definitions• Let us assume the following:
• Then:
• So:
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 9
Square Wave Example
t
T
T/2
x(t)
A
-A
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 10
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:
2Aπ
2A
3π
-2Aπ
-2A
3π
nn
BnAn
1 3 5 7 9
-9 -7 -5 -3 -1
1 3 5 7 9
-9 -7 -5 -3 -1
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 11
2Aπ
2A
3π
-2Aπ
-2A
3π
ff
BfAf
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
• A given Fourier coefficient, ,represents the weight corresponding to frequency nwo
• It is often convenient to index in frequency (Hz)
Indexing in Frequency
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 12
The Impact of a Time (Phase) Shift
• Consider shifting a signal x(t) in time by Td
t
T/4
x(t)
T
T/4
A
-A
t
T
T/2
x(t)
A
-A
• Define:
• Which leads to:
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 13
Square Wave Example of Time Shift
• To simplify, note that except for odd n
t
T/4
x(t)
T
T/4
A
-A
t
T
T/2
x(t)
A
-A
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 14
2Aπ
f
f
Bf
Af
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
2Aπ
2A
3π
-2Aπ
-2A
3π
f
f
Bf
Af
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
Graphical View of Fourier Series
t
T/4
x(t)
T
T/4
A
-A
t
T
T/2
x(t)
A
-A
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 15
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:
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 16
Graphical View of Magnitude and Phase
t
T/4
x(t)
T
T/4
A
-A
t
T
T/2
x(t)
A
-A
f-3
T
-5
T
-1
T
-7
T
-9
T
2Aπ
2A
3π
f
f
Xf
Φf
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
2Aπ
2A
3π
Φf
2Aπ
2A
3π
f
Xf
1
T
-3
T
-5
T
-1
T
3
T
5
T
7
T
9
T
-7
T
-9
T
2Aπ
2A
3π
-π/2
π/2
1
T
3
T
5
T
7
T
9
T
π
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 17
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
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 18
Parseval’s Theorem• The squared magnitude of the Fourier Series
coefficients indicates power at corresponding frequencies– Power is defined as:
Note:* meanscomplexconjugate
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 19
The Fourier Transform• The Fourier Series deals with periodic signals
• The Fourier Transform deals with non-periodicsignals
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 20
Fourier Transform Example
• Note that x(t) is not periodic• Calculation of Fourier Transform:
t
x(t)
T
A
-T
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 21
Graphical View of Fourier Transform
t
x(t)
T
A
-T
This is calleda sinc function
X(f)2TA
2T1
2T-1
f
M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 22
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 …