Chapter 2 : Fourier Series
1.0 Introduction
• Fourier Series : representation of periodic signals as weighted sums of
harmonically related frequencies.
• If a signal x(t) is periodic signal , then x(t) can be represented in terms of Fourier
Series either in :
a) Trigonometric Form
b) Exponential (Complex) Form
Fourier
Series
Saw tooth waveform
Infinite number of sine waves (harmonics)
Chapter 3 : Fourier Series
1.1 Trigonometric Fourier Series
• General equation :
• x(t) is expressed as the sum of sinusoidal components having different frequencies
where :
* an and bn : the Fourier coefficients
* a0 : DC Value
1
Chapter 3 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as Trigonometric Fourier
Series:
x(t)
1
-0.5T
-T
0
t
0.5T
T
-1
Chapter 3 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as Trigonometric Fourier
Series:
x(t)
1
x(t) =
-2
-
t /
0
;0<t<
;  < t < 2
t
0

2
3
-1
2
Chapter 3 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as Trigonometric Fourier
Series:
x(t)
1
-1
-2
t
0
1
2
3
-1
Chapter 3 : Fourier Series
1.2 Symmetry Properties
• The symmetry properties can be classified into 5 types :
a)
Even Symmetry
b)
Odd Symmetry
c)
Half Wave Symmetry
d)
Even And Half Wave Symmetry
e)
Odd And Half Wave Symmetry
3
Chapter 3 : Fourier Series
1.2 Symmetry Properties
a)
Even Symmetry
The signal x(t) is said to be even symmetry if x(t) = x(-t)
x(t)
x(t) = x(-t)
1
1
t
t
1
1
x(t)
x(t) = x(-t)
t
t
Chapter 3 : Fourier Series
1.2 Symmetry Properties
b)
Odd Symmetry
The signal x(t) is said to be even symmetry if x(t) = -x(-t)
x(t) = -x(-t)
x(t)
1
1
-1
t
1
t
1
-1
x(t)
x(t) = x(-t)
t
4
Chapter 3 : Fourier Series
1.2 Symmetry Properties
c)
Half Wave Symmetry
The signal x(t) is said to be half wave symmetry if x(t) = -x(t + T/2)
x(t) = -x(t+T/2)
x(t)
1
1
-1
t
t
T/2
1
-1
x(t)
x(t) = -x(t+T/2)
T/2
T
t
T/2
T/2
T
t
Chapter 3 : Fourier Series
Example
The first half-cycle of a periodic signal is shown in figure below and the period is
T = 2 sec. Sketch y(t) clearly for 3 complete cycles if :
i)
y(t) is an even-symmetrical signal
ii)
y(t) is an odd-symmetrical signal
iii)
y(t) is a half-wave symmetrical signal
x(t)
1
t
1
5
Chapter 3 : Fourier Series
Answer
i)
even-symmetrical signal
iii) odd-symmetrical signal
x(-t)
-x(-t)
1
1
t
t
-3
-2
-1
0
1
3
2
-3
-2
0
-1
1
2
3
-1
Chapter 3 : Fourier Series
Answer
iii)
half-wave symmetrical signal
x-(t+1)
1
t
-3
-2
-1
0
1
2
3
-1
6
Chapter 3 : Fourier Series
Example
The first half-cycle of a periodic signal is shown in figure below and the period is
T = 4 sec. Sketch y(t) clearly for 3 complete cycles if :
i)
y(t) is an even-symmetrical signal
ii)
y(t) is an odd-symmetrical signal
iii)
y(t) is a half-wave symmetrical signal
y(t)
3
1
t
2
1
Chapter 3 : Fourier Series
Answer
i)
-5 -4
even-symmetrical signal
-3
-2
iii) odd-symmetrical signal
x(-t)
-x(-t)
3
3
-1
1
2
3 4
5
t
-5 -4
-3 -2
-1
1
2
3 4
5
t
-1
-3
7
Chapter 3 : Fourier Series
Answer
iii)
half-wave symmetrical signal
x-(t+1)
3
-5 -4 -3 -2
-1
2
1
3 4
t
5
-3
Chapter 3 : Fourier Series
1.2 Symmetry Properties
d)
Even And Half-Wave Symmetry
Consider a half cycle signal x(t) shown in figure below where T = 4 sec
x(t)
1
1
t
-1
For one complete cycle , the shape of signal x(t) is the same properties .
-x(t+T/2)
x(-t)
1
1
-1
1
-1
t
t
-1
1
-1
8
Chapter 3 : Fourier Series
1.2 Symmetry Properties
d)
Even And Half-Wave Symmetry
Thus , the half-wave even symmetry signal x(t) for 3 complete cycles is
shown below :
x(t) = half-wave even
1
-6
-5
-4
-3
-2
-1
t
0
1
2
3
4
5
6
-1
Chapter 3 : Fourier Series
1.2 Symmetry Properties
e)
Odd And Half-Wave Symmetry
Consider a half cycle signal x(t) shown in figure below where T = 4 sec
x(t)
1
t
1
2
-1
For one complete cycle , the shape of signal x(t) is the same properties .
-2
-x(-t)
-x(t+T/2)
1
1
-1
1
-1
2
t
-2
-1
1
2
t
-1
9
Chapter 3 : Fourier Series
1.2 Symmetry Properties
e)
Odd And Half-Wave Symmetry
Thus , the half-wave even symmetry signal x(t) for 3 complete cycles is
shown below :
x(t) = half-wave odd
1
-6
-5
-4
-3
-2
-1
0
t
1
2
3
4
5
6
-1
Chapter 3 : Fourier Series
1.3 Effects of Symmetry Properties
n = odd
10
Chapter 3 : Fourier Series
1.3 Effects of Symmetry Properties
Chapter 3 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as Trigonometric Fourier
Series using symmetry property.
x(t)
1
-T
-0.5T
0
t
0.5T
T
-1
11
Chapter 3 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as Trigonometric Fourier
Series using symmetry property.
x(t)
1
-T
t
-0.5T
0.5T
T
-1
Chapter 3 : Fourier Series
1.4 Hidden Symmetry
Example
Express signal x(t) as TFS
x(t)
A
-2T
-T
t
T
2T
Solution
x(t) does not posses any symmetry properties.
12
Chapter 3 : Fourier Series
1.4 Hidden Symmetry
Solution
However applying the hidden symmetry by shifting 0.5A of the DC value for signal
x(t) will result signal below :
g(t)
A
0.5A
-2T
-T
t
T
2T
t
Now , signal g(t) posses odd symmetry , thus FS of x(t) :
x(t) = 0.5 A + FS of signal g(t)
Chapter 3 : Fourier Series
1.4 Hidden Symmetry
Example
Express signal x(t) as TFS
x(t)
π
-3π
-2π
π
0
π
2π
3π
t
13
Chapter 3 : Fourier Series
1.4 Hidden Symmetry
Solution
Signal x(t) posses even-symmetry property . Thus FS of x(t) is :
x(t) = π/2 + FS of signal g(t)
x(t)
π
t
-3π
-2π
π
0
π
2π
3π
t
Chapter 3 : Fourier Series
1.5 Trigonometric Fourier Series
The trigonometric FS of signal x(t) is given as :
Euler’s Identity
The expression (
) can be expressed as follows :
14
Chapter 3 : Fourier Series
1.5 Complex Fourier Series
Thus x(t) :
Let :
Then x(t) :
Chapter 3 : Fourier Series
1.5 Complex Fourier Series
The term
can also be represented as follows :
=
=
Then ,
x(t) =
+
+
15
Chapter 2 : Fourier Series
1.5 Complex Fourier Series
Where :
where :
(Euler’s Identity)
Chapter 3 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as an Exponential Fourier
Series.
x(t)
1
-T
-0.5T
0
t
0.5T
T
-1
16
Chapter 2 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as an Exponential Fourier
Series.
x(t)
1
-2T
t
-T
T
2T
3T
Chapter 2 : Fourier Series
Example
The following figure below shows a half cycle of a periodic signal x(t) . If x(t) is an
odd symmetry signal , determine its EFS :
x(t)
2
t
2
17
Chapter 2 : Fourier Series
Solution
x(t) is an odd symmetry signal
T
ω0
x(t)
2
=4
= 2π / 4
=π/2
Odd symmetry
-4
-2
2
t
4
a0 = 0 (C0)
a n= 0
bn = ????
-2
Chapter 2 : Fourier Series
1.6 Frequency Spectrum
• Frequency spectrum consist of :
a) Amplitude spectrum
the plot of І Cn І against nω0
b) Phase spectrum
the plot of Cn against nω0
Cn
І Cn І
nω0
nω0
18
Chapter 2 : Fourier Series
1.6 Frequency Spectrum
• For complex number
Im (jω)
Magnitude , І Cn І =
Phase
a + jb
Cn
=
І Cn І
Cn
Real (σ)
Chapter 2 : Fourier Series
1.6 Frequency Spectrum
• Conditions
i) If b = 0 , Cn = a
jω
ii) If a = 0 , Cn = jb
jω
jb
π/2
σ
-a
a
І Cn І = a
Cn
-jb
σ
= 0 if a > 0 (positive)
= π or – π if a < 0 (negative)
-π/2
І Cn І = b
Cn
= π/2 if b > 0 (positive)
= -π/2 if b < 0 (negative)
19
Chapter 2 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as an Exponential Fourier
Series. Plot the frequency spectrum for signal x(t).
x(t)
1
-T
-0.5T
t
0
0.5T
T
-1
Chapter 2 : Fourier Series
Example
Express the following signal x(t) as shown in figure below as an Exponential Fourier
Series. Plot the frequency spectrum of signal x(t).
x(t)
1
-2T
-T
t
T
2T
3T
20
Chapter 2 : Fourier Series
Example
The Fourier Series of signal x(t) is given as :
Sketch the frequency spectrum of x(t) for n = ±5 , ±4 , ±3 , ±2 , ±1 and 0
Chapter 2 : Fourier Series
1.7 TFS Coefficients and EFS Coefficients Relationship
or
21
Chapter 2 : Fourier Series
Example
Convert the TFS coefficients of signal x(t) of figure below to EFS coefficients.
x(t)
From previous TFS
1
a0 = 0
-T
-0.5T
0
t
0.5T
T
an = 0
0
; n = even
bn =
-1
4 / nπ ; n = odd
Chapter 2 : Fourier Series
Example
Convert the TFS coefficients of signal x(t) of figure below to EFS coefficients.
From previous TFS
x(t)
a0 = π / 2
π
bn = 0
-3π
-2π
π
0
π
2π
3π
t
0
; n = even
an =
-4/n2π ; n = odd
22
Chapter 2 : Fourier Series
Example
Consider the signal y(t) as shown below :
i) Determine the TFS of y(t)
ii) From the TFS of y(t) , write the EFS of y(t)
x(t)
2
1
-4
-3
-2
-1
t
0
1
2
3
4
-1
-2
23