Fourier Series Representation of Periodic Signals

1
Fourier Series Representation of Periodic Signals
2
The Response of LTI Systems to Complex
Exponentials

• The response of an LTI systems to a complex
  exponential input is the same complex exponential
  with only a change in amplitude

3
The Response of LTI Systems to Complex
Exponentials(2)

• A signal for which the system output is a
  constant times the input is referred to as an
  eigenfunction of the system , and the amplitude
  factor is referred to as the systems eigenvalue.

4
The Response of LTI Systems to Complex
Exponentials(3)
5
The Response of LTI Systems to Complex
Exponentials(4)
6
The Response of LTI Systems to Complex
Exponentials(5)
7
Example 3.1-1
8
Example 3.1-2
9
Fourier Series Representation of Continuous-Time
Periodic Signals

• A signal is periodic if

10
Example 3.2
11
Example 3.2(2)
12
F.S. Representation of Continuous-Time Periodic
Signals(2)

• Suppose that x(t) is real and can be represented
  in
• then

13
F.S. Representation of Continuous-Time Periodic
Signals(3)
14
F.S. Representation of Continuous-Time Periodic
Signals(4)
15
F.S. Representation of Continuous-Time Periodic
Signals(5)
16
F.S. Representation of Continuous-Time Periodic
Signals(6)
17
Example 3.3
18
Example 3.4
19
Example 3.4(2)
20
Example 3.5
21
Example 3.5(2)
22
Example 3.5(3)
23
Convergence of the Fourier Series

• Periodic signal x(t) by a linear combination of
  a finite number

24
Convergence of the Fourier Series(2)

• The energy in the error over one period
• One class of periodic signals are represent
• -able through the F.S. those signals which
  have finite energy over a single period

25
Convergence of the Fourier Series(3)

• Most of the periodic signals we consider do have
  finite energy over a single period, they have
  F.S. representation.

26
Dirichlet conditions 1

• x(t) must be absolutely integrable

27
Dirichlet conditions 1(2)

• A periodic signal that violates the first
  Dirichlet conditions

28
Dirichlet conditions 2

• In any finite interval of time, x(t) is of
  bounded variation that is, there are no more
  than a finite number of maxima and minima during
  any single period of the signal

29
Dirichlet conditions 2(2)

• An example of a function that meets condition 1
  but not condition 2

30
Dirichlet conditions 3

• In any finite interval of time, there are only a
  finite number of discontinuities
• An example of a function that violates Dirichlet
  conditions 3

31
Gibbs phenomenon

• xN(t) of a discontinuous signal x(t) will in
  general exhibit high-frequency ripples and
  overshoot x(t) near the discontinuous

32
Gibbs phenomenon(2)
33
Properties of Continuous-Time Fourier Series
34
Time Shifting
35
Time Reversal
36
Time Scaling

• The Fourier coefficients have not changed, the
  Fourier series representation has changed because
  of the change in the fundamental frequency

37
Multiplication
38
Conjugation and Parsevals Relation

• Conjugation
• Parsevals relation

39
Example 3.6
40
Example 3.6(2)
41
Example 3.7
42
Example 3.8
43
Example 3.8(2)
44
FS Representation of Discrete-Time Periodic Signal

• A discrete-time signal xn with period N
• The set of all discrete-time complex exponential
  signals with period N

45
FS Representation of Discrete-Time Periodic
Signal(2)

• To consider the representation of more general
  periodic sequences

46
FS Representation of Discrete-Time Periodic
Signal(3)
47
Example 3.10
48
Example 3.11
49
Example 3.11(2)
50
Example 3.12
51
Example 3.12(2)
52
Example 3.12(2)
53
Signal and system
54
3.7 Properties of discrete-time fourier series

• 3.7.1 Multiplication
• 3.7.2 First difference
• 3.7.3 Parsevals relation for discrete-time
  periodic signals
• 3.7.4 Example

55
3.7.1

• Multiplication
• Periodic convolution
• Aperiodic convolution
• Ranges form

56
3.7.2

• First difference
• Defined as

57
3.7.3

• Parsevals relation
• The average power in a periodic signal equals the
  sum of the average powers in all of its harmonic
  components

58
3.7.4

• Example 3.13

59
3.7.4
60
3.7.4

• Example 3.14

61
3.7.4
62
3.8 Fourier series and LTI systems

• Continuous-time LTI system
• Discrete-time LTI system

63
3.8

• Continuous-time
• Special case
• Discrete-time
• Focus in

64
3.8

• Continuous-time
• Example 3.16

65
3.8
66
3.8

• Discrete-time
• Example 3.17

67
3.8
68
3.9 Filtering

• 3.9.1 Frequency-shaping filters
• 3.9.2 Frequency-selective filters

69
3.9.1

• Frequency-shaping filters
• Linear time-invariant systems that change the
  shape of the spectrum
• Example

70
3.9.1
71
3.9.2

• Frequency-selective filters
• Systems that are designed to pass some
  frequencies essentially undistorted and
  significantly attenuate or eliminate others
• Example

72
3.9.2
73
3.10 Examples of continuous-time filters

• 3.10.1 A simple RC lowpass filter
• 3.10.2 A simple RC highpass filter

74
3.10.1
75
3.10.1
76
3.10.2
77
3.11 Example of discrete-time filters

• 3.11.1 First-order recursive discrete-time
  filters
• 3.11.2 Nonrecursive discrete-time filters

78
3.11.1
79
3.11.2
80
3.11.2