The Continuous-Time Fourier
Series
Representing a Signal
8/2/13
M. J. Roberts - All Rights Reserved
2
Linearity and Superposition
If an excitation can be expressed as a linear combination of complex sinusoids, the response of an LTI system can be expressed as a linear combination of responses to complex sinusoids.
8/2/13
M. J. Roberts - All Rights Reserved
3
Real and Complex Sinusoids
8/2/13
M. J. Roberts - All Rights Reserved
4
Real and Complex Sinusoids
8/2/13
M. J. Roberts - All Rights Reserved
5
Jean Baptiste Joseph Fourier
3/21/1768 - 5/16/1830
8/2/13
M. J. Roberts - All Rights Reserved
6
Conceptual Overview
The Fourier series represents a signal as a sum of sinusoids.
The best approximation to the dashed-line signal below using
only a constant is the solid
line. (A constant is a cosine of zero frequency.)
8/2/13
M. J. Roberts - All Rights Reserved
7
Conceptual Overview
The best approximation to the dashed-line signal using a constant plus one real sinusoid of the same fundamental frequency as the dashed-line signal is the solid line. 8/2/13
M. J. Roberts - All Rights Reserved
8
Conceptual Overview
The best approximation to the dashed-line signal using a constant plus one sinusoid of the same fundamental frequency as the dashed-line signal plus another sinusoid of twice the fundamental
frequency of the dashed-line signal is the solid line. The frequency
of this second sinusoid is the second harmonic of the fundamental
frequency. 8/2/13
M. J. Roberts - All Rights Reserved
9
Conceptual Overview
The best approximation to the dashed-line signal using a constant plus three harmonics is the solid line. In this case (but not in
general), the third harmonic has zero amplitude. This means that no
sinusoid of three times the fundamental frequency improves the
approximation. 8/2/13
M. J. Roberts - All Rights Reserved
10
Conceptual Overview
The best approximation to the dashed-line signal using a constant plus four harmonics is the solid line. This is a good approximation
that gets better with the addition of more harmonics.
8/2/13
M. J. Roberts - All Rights Reserved
11
Continuous-Time Fourier Series
Definition
8/2/13
M. J. Roberts - All Rights Reserved
12
Orthogonality
8/2/13
M. J. Roberts - All Rights Reserved
13
Orthogonality
8/2/13
M. J. Roberts - All Rights Reserved
14
Orthogonality
8/2/13
M. J. Roberts - All Rights Reserved
15
Orthogonality
8/2/13
M. J. Roberts - All Rights Reserved
16
Orthogonality
8/2/13
M. J. Roberts - All Rights Reserved
17
Continuous-Time Fourier Series
Definition
8/2/13
M. J. Roberts - All Rights Reserved
18
CTFS of a Real Function
8/2/13
M. J. Roberts - All Rights Reserved
19
The Trigonometric CTFS
8/2/13
M. J. Roberts - All Rights Reserved
20
The Trigonometric CTFS
8/2/13
M. J. Roberts - All Rights Reserved
21
CTFS Example #1
8/2/13
M. J. Roberts - All Rights Reserved
22
CTFS Example #1
8/2/13
M. J. Roberts - All Rights Reserved
23
CTFS Example #2
8/2/13
M. J. Roberts - All Rights Reserved
24
CTFS Example #2
8/2/13
M. J. Roberts - All Rights Reserved
25
CTFS Example #3
8/2/13
M. J. Roberts - All Rights Reserved
26
CTFS Example #3
8/2/13
M. J. Roberts - All Rights Reserved
27
CTFS Example #3
8/2/13
M. J. Roberts - All Rights Reserved
28
CTFS Example #3
8/2/13
M. J. Roberts - All Rights Reserved
29
Linearity of the CTFS
These relations hold only if the harmonic functions of all
the component functions are based on the same
representation time T.
8/2/13
M. J. Roberts - All Rights Reserved
30
CTFS Example #4
8/2/13
M. J. Roberts - All Rights Reserved
31
CTFS Example #4
8/2/13
M. J. Roberts - All Rights Reserved
32
CTFS Example #4
8/2/13
M. J. Roberts - All Rights Reserved
33
CTFS Example #4
8/2/13
M. J. Roberts - All Rights Reserved
34
CTFS Example #4
A graph of the magnitude and phase of the harmonic function as a function of harmonic number is a good way of illustrating it.
Notice that the magnitude is an even function of k and the phase is an odd function of k.
8/2/13
M. J. Roberts - All Rights Reserved
35
The Sinc Function
8/2/13
M. J. Roberts - All Rights Reserved
36
CTFS Example #5
8/2/13
M. J. Roberts - All Rights Reserved
37
CTFS Example #5
8/2/13
M. J. Roberts - All Rights Reserved
38
CTFS Example #5
8/2/13
M. J. Roberts - All Rights Reserved
39
CTFS Example #5
The CTFS representation of this cosine is the signal
below, which is an odd function, and the discontinuities
make the representation have significant higher harmonic
content. Although correct in the time interval from zero
to 7.5 ms, this is a very inelegant representation.
8/2/13
M. J. Roberts - All Rights Reserved
40
CTFS of Even and Odd Functions
8/2/13
M. J. Roberts - All Rights Reserved
41
Convergence of the CTFS
For continuous signals, convergence is exact at every point.
A Continuous Signal
8/2/13
M. J. Roberts - All Rights Reserved
42
Convergence of the CTFS
Partial CTFS Sums
For discontinuous signals, convergence is exact at every point of continuity.
Discontinuous Signal
8/2/13
M. J. Roberts - All Rights Reserved
43
Convergence of the CTFS
At points of discontinuity
the Fourier series
representation converges
to the mid-point of the
discontinuity.
8/2/13
M. J. Roberts - All Rights Reserved
44
Numerical Computation of the CTFS
How could we find the CTFS of a signal that has no
known functional description?
Numerically.
Unknown
8/2/13
M. J. Roberts - All Rights Reserved
45
Numerical Computation of the CTFS
Samples from x(t)
8/2/13
M. J. Roberts - All Rights Reserved
46
Numerical Computation of the CTFS
8/2/13
M. J. Roberts - All Rights Reserved
47
Numerical Computation of the CTFS
8/2/13
M. J. Roberts - All Rights Reserved
48
CTFS Properties
Linearity
8/2/13
M. J. Roberts - All Rights Reserved
49
CTFS Properties
Time Shifting
8/2/13
M. J. Roberts - All Rights Reserved
50
CTFS Properties
Frequency Shifting (Harmonic Number Shifting)
A shift in frequency (harmonic number) corresponds to multiplication of the time function by a complex exponential.
Time Reversal
8/2/13
M. J. Roberts - All Rights Reserved
51
CTFS Properties
8/2/13
M. J. Roberts - All Rights Reserved
52
CTFS Properties
Time Scaling (continued)
8/2/13
M. J. Roberts - All Rights Reserved
53
CTFS Properties
8/2/13
M. J. Roberts - All Rights Reserved
54
CTFS Properties
Change of Representation Time
8/2/13
M. J. Roberts - All Rights Reserved
55
CTFS Properties
Time Differentiation
8/2/13
M. J. Roberts - All Rights Reserved
56
CTFS Properties
Time Integration
Case 1
Case 2
M. J. Roberts - All Rights Reserved
57
is not periodic
8/2/13
CTFS Properties
8/2/13
M. J. Roberts - All Rights Reserved
58
CTFS Properties
8/2/13
M. J. Roberts - All Rights Reserved
59
CTFS Properties
8/2/13
M. J. Roberts - All Rights Reserved
60
Some Common CTFS Pairs
8/2/13
M. J. Roberts - All Rights Reserved
61
CTFS Examples
8/2/13
M. J. Roberts - All Rights Reserved
62
CTFS Examples
8/2/13
M. J. Roberts - All Rights Reserved
63
CTFS Examples
8/2/13
M. J. Roberts - All Rights Reserved
64
LTI Systems with Periodic Excitation
8/2/13
M. J. Roberts - All Rights Reserved
65
LTI Systems with Periodic Excitation
8/2/13
M. J. Roberts - All Rights Reserved
66
LTI Systems with Periodic Excitation
8/2/13
M. J. Roberts - All Rights Reserved
67
LTI Systems with Periodic Excitation
8/2/13
M. J. Roberts - All Rights Reserved
68
The Continuous-Time Fourier
Transform
Extending the CTFS
• The CTFS is a good analysis tool for systems with
periodic excitation but the CTFS cannot represent
an aperiodic signal for all time
• The continuous-time Fourier transform (CTFT)
can represent an aperiodic (and also a periodic)
signal for all time 8/2/13
M. J. Roberts - All Rights Reserved
70
CTFS-to-CTFT Transition
8/2/13
M. J. Roberts - All Rights Reserved
71
CTFS-to-CTFT Transition
8/2/13
M. J. Roberts - All Rights Reserved
72
CTFS-to-CTFT Transition
8/2/13
M. J. Roberts - All Rights Reserved
73
CTFS-to-CTFT Transition
8/2/13
M. J. Roberts - All Rights Reserved
74
CTFS-to-CTFT Transition
8/2/13
M. J. Roberts - All Rights Reserved
75
Definition of the CTFT
8/2/13
M. J. Roberts - All Rights Reserved
76
Definition of the CTFT
8/2/13
M. J. Roberts - All Rights Reserved
77
Definition of the CTFT
8/2/13
M. J. Roberts - All Rights Reserved
78
Some Remarkable Implications
of the Fourier Transform
The CTFT can express a finite-amplitude, real-valued, aperiodic signal, which can also, in general, be time-limited, as a summation (an integral) of an infinite continuum of weighted, infinitesimal-
amplitude, complex-valued sinusoids, each of which is unlimited in
time. (Time limited means “having non-zero values only for a finite time.”)
8/2/13
M. J. Roberts - All Rights Reserved
79
Frequency Content
8/2/13
M. J. Roberts - All Rights Reserved
80
Some CTFT Pairs
8/2/13
M. J. Roberts - All Rights Reserved
81
Convergence and the
Generalized Fourier Transform
8/2/13
M. J. Roberts - All Rights Reserved
82
Convergence and the Generalized
Fourier Transform
8/2/13
M. J. Roberts - All Rights Reserved
83
Convergence and the
Generalized Fourier Transform
8/2/13
M. J. Roberts - All Rights Reserved
84
Convergence and the
Generalized Fourier Transform
8/2/13
M. J. Roberts - All Rights Reserved
85
Convergence and the Generalized
Fourier Transform
8/2/13
M. J. Roberts - All Rights Reserved
86
More CTFT Pairs
8/2/13
M. J. Roberts - All Rights Reserved
87
Negative Frequency
This signal is obviously a sinusoid. How is it described
mathematically?
8/2/13
M. J. Roberts - All Rights Reserved
88
Negative Frequency
8/2/13
M. J. Roberts - All Rights Reserved
89
Negative Frequency
8/2/13
M. J. Roberts - All Rights Reserved
90
CTFT Properties
Linearity
8/2/13
M. J. Roberts - All Rights Reserved
91
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
92
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
93
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
94
The “Uncertainty” Principle
The time and frequency scaling properties indicate that if a signal is expanded in one domain it is compressed in the other domain.
This is called the “uncertainty principle” of Fourier analysis.
8/2/13
M. J. Roberts - All Rights Reserved
95
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
96
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
97
CTFT Properties
In the frequency domain, the cascade connection multiplies
the frequency responses instead of convolving the impulse
responses.
8/2/13
M. J. Roberts - All Rights Reserved
98
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
99
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
100
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
101
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
102
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
103
CTFT Properties
8/2/13
M. J. Roberts - All Rights Reserved
104
Numerical Computation of the CTFT
8/2/13
M. J. Roberts - All Rights Reserved
105
8/2/13
M. J. Roberts - All Rights Reserved
106
8/2/13
M. J. Roberts - All Rights Reserved
107
8/2/13
M. J. Roberts - All Rights Reserved
108
8/2/13
M. J. Roberts - All Rights Reserved
109
8/2/13
M. J. Roberts - All Rights Reserved
110
8/2/13
M. J. Roberts - All Rights Reserved
111