Chapter 4-Fourier Series
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Transcript of Chapter 4-Fourier Series
CHAPTER 4 : FOURIER SERIES
4.0 Introduction
A periodic signal x(t) is periodic if x(t + T) = x (t) where T is the period and .
Sinusoid of frequency nf0 where n is positive integer is said to be the nth harmonic. If:
n = odd (odd harmonic)
n = even (even harmonic)
4.1 Trigonometric Fourier series
If x(t) is a periodic function with period T, then x(t) can be expressed as trigonometric
Fourier series:
x(t) is expressed as the sum of sinusoidal components having different frequencies
where:
and - the Fourier coefficients
- the dc value of x(t)
4.1.1 Orthogonal Fuctions for Sine and Cosine Functions
Consider a function and . According to orthogonal function properties
Where m and n are integers, T is the period and rn is some value.
Let and , then
1)
If n = 0, then
2)
Let and , then
3)
Let and , then
4)
If n = 0, then
5)
4.1.2 Determination of Trigonometric Fourier Coefficients
4.1.2.1 Determination of Fourier coefficient, :
Based on the trigonometric Fourier series expression
(4.1.2)
When Equation (4.1.2) is integrated both sides for one complete cycle, then
Using Orthogonal functions relation.
Thus,
4.1.2.2 Determination of Fourier coefficient, :
When Equation (4.1.2) is multiplied both sides with and then
integrated for one complete cycle, then
Using orthogonal function relation,
and
then,
Since m = n, then
4.1.2.3 Determination of Fourier coefficient, :
When Equation (4.1.2) is multiplied both sides with and then
integrated for one complete cycle, then
Using Orthogonal functions relation,
And
then,
Since m = n, then
Graph of sin nωt
Sin 2nπ n = even 0 0
n = odd 0Sin nπ n = even 0 0
n = odd 0
Sin
n = even 0 0n = odd 1,-1, 1,-1, 1
n = both 1, 0,-1, 1, 0,
where n = 2n-1
Graph of cos nωt
Cos 2nπ n = even 1 1n = odd 1
Cos nπ n = even 1n = odd -1
Cos
n = even -1, 1, -1, 1,
n = odd 0 0n = both 0,-1, 0, 1, 0, Cos(2n - 1)π = -1
where n = 2n-1
Example 4.1.1 Express the signal x(t) shown in Figure 4.1.1 as trigonometric
Fourier series.
Figure 4.1.1
SOLUTION:
The average value, is determined as follows:
or
= [Area under the curve for one complete cycle]
The Fourier coefficient, is determined as follows:
Since: sin nπ = 0 and sin 2nπ = 0;
The Fourier coefficient, bn is determined as follows:
Since: cos nπ = (-1)n and cos 2nπ = 1;
Thus,
or
Example 4.1.2 Express signal the x(t) shown in Figure 4.1.2 as trigonometric Fourier
series
Figure 4.1.2
SOLUTION:
The average value, is determined as follows:
or
= [Area under the curve for one complete cycle]
The Fourier coefficient, is determined as follows:
Since: sin nπ = 0 and sin 2nπ = 0;
The Fourier coefficient, bn is determined as follows:
Thus,
4.2 Symmetry Properties
i) Even Symmetry
ii) Odd Symmetry
iii) Half-Wave Symmetry
iv) Even And Half-Wave Symmetry (Half-Wave Even Symmetry)
v) Odd And Half-Wave Symmetry (Half-Wave Odd Symmetry)
vi) Hidden symmetry
4.2.1 Even Symmetry
Example 4.2.1 Consider a half-cycle signal x(t) shown in Figure 4.2.1(a) where T = 2
sec.
Figure 4.2.1(a)
The signal x(t) is said to be even symmetry if x(t) = x(-t). This-property is shown in
Figure 4.2.1(b)
Figure 4.2.1(b)
The even-symmetry signal x(t) for 3 complete cycles is shown in Figure 4.2.1(c).
Figure 4.2.1(c)
4.2.2 Odd Symmetry
Example 4.2.2 Consider a half-cycle signal x(t) shown in Figure 4.2.2(a) where T = 2
sec.
Figure 4.2.2(a)
The signal x(t) is said to be odd symmetry if x(t) = -x(-t). This property is shown in
Figure 4.2.2(b)
Figure 4.2.2(b)
1 cycle 1 cycle 1 cycle
The odd-symmetry signal x(t) for 3 complete cycles is shown in Figure 4.2.2(c)
Figure 4.2.2(c)
4.2.3 Half-Wave Symmetry
Example 4.2.3 Consider a half-cycle signal x(t) shown in Figure 4.2.3(a) where T = 2
sec.
Figure 4.2.3(a)
The signal x(t) is said to be half-wave symmetry if x(t) = -x(t + T/2). This property is
shown in Figure 4.2.3(b)
1 cycle 1 cycle 1 cycle
Figure 4.2.3(b)
The half-wave symmetry signal x(t) for 3 complete cycles is shown in Figure 4.2.3(c).
Figure 4.2.3(c)
4.2.4 Even and Half-Wave Symmetry (Half-Wave Even Symmetry)
Example 4.2.4 Consider a half-cycle signal x(t) shown in Figure 4.2.4(a) where T =
4 sec.
Figure 4.2.4(a)
For one complete cycle/the shape of signal x(t) is the same for both properties. This is
shown in Figure 4.2.4(b).
1 cycle 1 cycle 1 cycle
Figure 4.2.4(b)
Thus, the half-wave even symmetry signal x(t) for 3 complete cycles is shown in Figure
4.2.4(c).
Figure 4.2.4(c)
4.2.4 Odd and Half-Wave Symmetry (Half-Wave Odd Symmetry)
Example 4.2.5 Consider a half-cycle signal x(t) shown in Figure 4.2.5(a) where T
= 4 sec.
Figure 4.2.5(a)
1 cycle 1 cycle 1 cycle
For one complete cycle, the shape of signal x(t) is the same for both properties.This is
shown in Figure 4.2.5(b).
Figure 4.2.5(b)
Thus, the half-wave even symmetry signal x(t) for 3 complete cycles is shown in Figure
4.2.5 (c).
Figure 4.2.5 (c)
Example 4.2.6 The first half-cycle of a periodic signal y(t) is shown in Figure
4.2.6(a) and the period sec. Sketch y(t) clearly for.3
complete cycles if:
i) y(t) is an even-symmetric signal
ii) y(t) is an odd-symmetric signal
iii) y(t) is a half-wave symmetric signal
1 cycle 1 cycle 1 cycle
Figure 4.2.6(a)
SOLUTION:
The signal y(t) for all cases are given in Figure 4.2.6(b).
Figure 4.2.6(b)
4.3 Effects of Symmetry
i) If x(t) is an even symmetric signal, then its trigonometric Fourier series
expression is as follows:
Its Fourier series consists of a constant and cosine terms only where:
ii) If x(t) is an odd symmetric signal, then its trigonometric Fourier series
expression is as follows:
Its Fourier series consists of sine terms only where:
and
iii) If x(t) is a half-wave symmetric signal, then its trigonometric Fourier expression
is as follows:
Its Fourier series consists of odd harmonic of cosine and sine terms only where:
, and
iv) If x(t) is an even symmetric and also half-wave symmetric signal (half-wave
even symmetric signal), then its trigonometric Fourier series expression is as
follows:
Its Fourier series consists of a constant and cosine terms only where:
, and
v) If x(t) is an odd symmetric and also half-wave symmetric signal (half-wave odd
symmetric), then its trigonometric Fourier series expression is as follows:
Its Fourier series consists of odd harmonic of sine terms only where:
, and
The trigonometric Fourier series expressions of each symmetric signal are summarized
in Table 4.1.
Table 4.1 Fourier series Simplified Flow Techniques
Signal function
TFS Coefficients
Coefficients TFS expressions
Generala0
an
bn
Even symmetry
a0
an
bn=0
Odd symmetry
bn
a0 = an = 0
Half-wave symmetry
a0 =0
an (even) =0
bn (even) =0
Even and half-wave symmetry
a0 =0
bn =0
an (even) =0
Odd and half-wave symmetry
a0 =0
an =0
bn (even) =0
Example 4.3.1 Express signal x(t) shown in Figure 4.3.1 as trigonometric Fourier
series using symmetry property.
Figure 4.3.1
SOLUTION:
The signal x(t) is odd-symmetry and half- wave symmetry signal (half-wave odd
symmetry).
, and
The Fourier coefficient is determined as follows:
Thus,
Example 4.3.2 Express signal x(t) shown in Figure 4.3.2 as trigonometric Fourier
series using Symmetry property.
Figure 4.3.2
SOLUTION:
x(t) is even-symmetry and half- wave symmetry signal (half-wave even symmetry).
, and
The Fourier coefficient is determined as follows:
Thus,
4.4 Hidden Symmetry
Example 4.4.1 Express signal x(t) as trigonometric Fourier series.
Figure 4.4.1(a)
SOLUTION:
Signal x(t) does not posses any symmetry properties. The evaluation can be further
simplified by shifting the dc value of signal x(t). Signal g(t) is obtain from signal x(t)
where x(t) = 0.5A + g(t) and signal g(t) is shown in Figure 4.4.1 (b).
Figure 4.4.1 (b)
Signal g(t) posses odd-symmetry property. Thus,
Fourier series of x(t) = 0.5A + Fourier series of g(t)
For odd Symmetric signal, then
and
The Fourier coefficient, bn is determined as follows:
Thus,
Fourier series of x(t) = 0.5A + Fourier series of g(t)
Example 4.4.2 Express the signal x (t) shown in Figure 4.4.2(a) as trigonometric
Fourier series.
Figure 4.4.2(a)
SOLUTION:
Signal x(t) posses even-symmetry property. The evaluation can be further simplified by
shifting the dc value of signal x(t). Signal g(t) is obtain from signal x(t) where signal
and the signal g(t) is shown in Figure 4.4.2(b).
Figure 4.4.2(b)
Signal g(t) posses half-wave even symmetry property. Thus,
Fourier series of x(t) = + Fourier series of g(t)
Fourier series of g(t) is obtain as follows:
, and
The Fourier coefficient, an (n=odd) is determined as follows:
Fourier series of x(t) = + Fourier series of g(t)
4.5 Exponential Fourier series
The trigonometric Fourier series of signal x(t) is given as:
EULER'S IDENTITY:
The expression can be expressed as follows:
Let , and
Then,
The term can also be represented as follows:
Then,
Where:
EULER'S IDENTITY:
Then, where n = ±1, ±2, ….
Example 4.5.1 Express the signal x(t) shown in Figure 4.5.1 as an exponential
Fourier series.
Figure 4.5.1
SOLUTION:
EULER IDENTITY:
Then,
and
Thus,
Example 4.5.2 Express the signal x (t) shown in Figure 4.5.2 as an exponential
Fourier series.
Figure 4.5.2
SOLUTION:
EULER IDENTITY:
Then,
For n = 0, Cn has no meaning. Thus,
Thus,
4.6 Frequency Spectrum
Frequency spectrum consists of amplitude and phase spectrums. Amplitude spectrum
is the plot of |Cn| versus and phase spectrum is the plot of versus .
ASIDE:
To determine the amplitude and phase spectrums of Frequency spectrum, the
magnitude and phase of X are shown in Figure 4.6 and summarized in Table 4.6:
Figure 4.6
Table 4.6
Magnitude Amplitude
spectrums
Phase spectrums
1 X = a + jb |X| =
2 X = jb |X| = b
3 X = -jb |X| = b
4 X = a |X| = a
5 X = -a |X| = a
Example 4.6.1 Plot the frequency spectrum of signal x(t) shown in Figure 4.6.1(a).
Figure 4.6.1(a)
SOLUTION:
;
Im
Re
(a+jb)
a
b
X
|X|
And
Amplitude spectrum:
Since , or , it is satisfied magnitude no 3 in Table 4.6.
So the amplitude spectrum is:
and
Phase spectrum:
Since , it is satisfied magnitude no 2 and no 3 in Table 4.6.
And plot the frequency spectrum for n= 0, ±1, ±2, ±3, ±4, ±5.
The plotted amplitude and phase spectrums of signal x (t) are shown in Figure 4.6.1(b)
and in Figure 4.6.1(c).
Figure 4.6.1(b) Amplitude spectrums
n -5 -4 -3 -2 -1 1 2 3 4 5
0 0 0 0 0
0 0 0 0 0
Figure 4.6.1(c) Phase spectrums
Example 4.6.2 Plot the frequency spectrum of signal x (t) shown in Figure 4.6.2(a).
Figure 4.6.2(a)
SOLUTION:
;
And
Amplitude spectrum:
Since , or , it is satisfied magnitude no 2 in Table 4.6.
So the amplitude spectrum is:
and
Phase spectrum:
Since , it is satisfied magnitude no 2 and no 3 in Table 4.6.
And plot the frequency spectrum for n= 0, ±1, ±2, ±3, ±4, ±5.
The plotted amplitude and phase spectrums of signal x (t) are shown in Figure 4.6.2(b)
and in Figure 4.6.2(c).
Figure 4.6.2(b) Amplitude spectrums
n -5 -4 -3 -2 -1 1 2 3 4 5
Figure 4.6.2(c) Phase spectrums
4.7 Trigonometric Fourier Coefficients and Complex Fourier
Coefficients Relationship
, and
Example 4.7.1 Convert the trigonometric Fourier coefficients of signal x (t) of Figure
4.7.1 to complex Fourier coefficient.
Figure 4.7.1
; ; and
SOLUTION:
Thus,
. This is true for n= 0, ±1, ±2, ±3, …
Thus,
Example 4.7.2 Convert the trigonometric Fourier coefficients of signal x (t) of' Figure
4.7.2 to complex Fourier coefficient.
Figure 4.7.2
; ; and
SOLUTION:
Thus,
. This is true for n= 0, ±1, ±2, ±3…
Thus,