Signal Processing Chapter 2

1. FOURIERSERIES
CHAPTER 2
BELT 4093
SIGNALS & NETWORKS

2. SUBTOPICS
Trigonometric
Fourier Series
Exponential
Fourier
Series
Symmetry
considerations
in Fourier
Series
Amplitude &
phase
spectra in
Fourier
Series
Applications
1 2 3 4 5

3. Courseoutcome
#2
Analyze electrical problems & passive
filters using circuit laws, Fourier
and/or Laplace technique
At the end oh this chapter, student
should be able to:
• Determine the trigonometry FS with a
variety of periodic function
• Plot the amplitude and phase spectra
of trigonometry FS
• Determine the Exponential FS
coefficients with a variety of periodic
function
• Plot the double sided spectrum of
Exponential FS for magnitude and
phase
• Determine the output signal from
Low-pass filter
3
Learning Outcomes Chapter 2

4. WhatisFourier
Series
Fourier Series  periodic signals
Fourier Transform  aperiodic signals
 Fourier Series is a technique for
expressing a periodic function in
terms of sinusoid.
 Fourier analysis provides a set of
mathematical tools which enable
the engineer to break down a
wave into its various frequency
components.
 The really cool thing about
Fourier series is that first, almost
any type of a wave can be
approximated.
 Second, when Fourier series
converge, they converge very
speed.
4

5. Isn'titCOOL?!
Compression:
Everyone's favorite MP3 format uses this for
audio squeezing.
You take a sound, extend its Fourier series. It'll
most same to be an infinite series but it
converges so fast that taking the first few terms
is enough to reproduce the original sound.
The rest of the terms can be avoided because
they add so little that a human ear can likely tell
no difference.
5
Telephones use an ridiculous number
of DSP tricks to keep an acceptable
voice quality, including deconvolution,
noise reduction, compression, echo
cancellation, equalization, etc. All this
done by codec DSPS or FPGA that
calculate Fourier millions of Fourier
coefficients per second.
All of Internet uses Spatial-domain
Fourier transforms to handle pictures
and videos. All Photoshop tricks and
video morphing are based on Fourier
transforms. Without Fourier, Pictures
would be still edited in photographic
labs, with scissors and ink pen, and
videos would look like those funny
psychodelic effects
If Fourier wasn’t used, phone calls
would sound like pre-WW2 radios or
even worse: people would have to talk
louder the farther away they called
from, calls would be interrupted by
noise every time people walked behind
a tree or around cars because of
reflections.
Without Fourier, we could only encode
data like old FM radios, and we’d have
to wait 1 minute to load this page, or
leave the laptop all night downloading
a single Tube video.
https://hackaday.com/2018/12/21/explaining-
fourier-again/
https://www.youtube.com/watch?v=lfwoPpsh9PI

6. 2.1
Trigonometric
FourierSeries
Learning Outcome:
• Student should know how to
determine the trigonometry FS with a
variety of periodic function
• Student should be able to plot the
amplitude and phase spectra of
trigonometry FS

7. 2.1TrigonometricFourierSeries
If f(t) is a periodic function with period T, the Fourier series of is given by
Where the Fourier coefficients are
Where;
ω0 = 2π / T
n = 1, 2, 3, …

8. AnalternativeformofFourierSeries
8
The amplitude-phase form:
Phasor form:
2
1
2
1
2
1
2
1
2
1
2
1










r
r
z
z
r
r
z
z
Multiplication:
Division:

9. ThingstoRemember
9
j
j
x
j
x
e
x
x
x
x
x
x
x
x
jx
















1
sin
cos
)
180
sin(
sin
)
180
cos(
cos
)
90
cos(
sin
)
90
sin(
cos
0
0
0
0

 
  2
1
2
1
2
sin
1
2
cos
)
1
(
cos
)
tan(
)
tan(
)
cos(
)
cos(
)
sin(
)
sin(















n
n
n
n
n
n
x
x
x
x
x
x


 ,n integer
,n even integer
,n odd integer
sin 𝑛𝜋 = 0
𝑗 = −1, 𝑗2
=-1

10. ReviewofIntegration
10

11. Example17.1
2

T
T



 


2
2
2
0
T
𝑎0 =
1
𝑇
0
𝑇
𝑓 𝑡 𝑑𝑡 =
1
2
0
1
1𝑑𝑡 +
1
2
0𝑑𝑡
=
1
2
1 =
1
2
𝑎𝑛 =
2
𝑇
0
𝑇
𝑓 𝑡 cos 𝑛𝜔0𝑡 𝑑𝑡
=
2
2
0
1
1 cos 𝑛𝜋𝑡 𝑑𝑡 +
1
2
0 cos 𝑛𝜋𝑡 𝑑𝑡
= 1
1
𝑛𝜋
sin 𝑛𝜋𝑡 0
1
= 0

12. 12
𝑏𝑛 =
2
𝑇
0
𝑇
𝑓 𝑡 sin 𝑛𝜔0𝑡 𝑑𝑡 =
2
2
0
1
1 sin 𝑛𝜋𝑡 𝑑𝑡 +
1
2
0 sin 𝑛𝜋𝑡 𝑑𝑡
= 1
1
𝑛𝜋
−cos 𝑛𝜋𝑡 0
1
=
1
𝑛𝜋
(− cos 𝑛𝜋 + cos 0)
=
1
𝑛𝜋
− −1 𝑛
+ 1 =
1
𝑛𝜋
1 − −1 𝑛
=
2
𝑛𝜋
, 𝑛 = 𝑜𝑑𝑑
0, 𝑛 = 𝑒𝑣𝑒𝑛
So we will get: 











odd
n
n
odd
n
n
t
n
n
t
n
n
t
f
1
1
sin
1
2
2
1
sin
2
2
1
)
( 



Change to amplitude-phase form: Phasor form:
0
90
2
)
( 



n
t
f
 







odd
n
n
t
n
n
t
f
1
0
90
cos
2
2
1
)
( 


13. 2.2Amplitude&PhaseSpectra–Example17.1
13
• Expand the answer:








 )
90
5
cos(
5
2
)
90
3
cos(
3
2
)
90
cos(
2
2
1
)
( 0
0
0
t
t
t
t
f 





Amplitude spectra Phase spectra
 







odd
n
n
t
n
n
t
f
1
0
90
cos
2
2
1
)
( 

Truncating the series at N=11














even
n
,
0
odd
n
,
90
even
n
,
0
odd
n
,
/
2
n
n
n
A



14. SummationofsignalinExample17.1
Fourier series with 11th
number of harmonics:

15. GibbsPhenomenon
• As more and more Fourier components are added, the sum gets closer and closer
to the original waveform
• At the neighborhood of the points of discontinuity (x = 0, 1, 2, ...), there is
overshoot and damped oscillation. In fact, an overshoot of about 9 percent of the
peak value is always present, regardless of the number of terms used to
approximate f(t).
Fourier Series Graph
Interactive:
https://www.intmath.c
om/fourier-
series/fourier-graph-
applet.php

16. 2.3Symmetry
Considerations

17. Symmetry
Considerations
3 Types of symmetry
• Even symmetry
• Odd symmetry
• Half-wave symmetry
The symmetry consideration is for
simplifying the calculation needed
to determine Fourier coefficients
)
(
)
( t
f
t
f 

)
(
)
( t
f
t
f 


)
(
2
t
f
T
t
f 








Even
Odd
Half-wave

18. 18
1. Even Symmetry : a function f(t) if its
plot is symmetrical about the vertical axis.
In this case,
2.1SymmetryConsiderations(1)
)
(
)
( t
f
t
f 

0
)
cos(
)
(
4
)
(
2
2
/
0
0
2
/
0
0





n
T
n
T
b
dt
t
n
t
f
T
a
dt
t
f
T
a

Typical examples of even periodic
function
Then, the Fourier series can be represented as
𝑓 𝑡 = 𝑎0 +
𝑛=
∞
𝑎𝑛 cos 𝑛𝜔𝑡

19. 19
2.Odd Symmetry : a function f(t) if its plot is
anti-symmetrical about the vertical axis.
In this case,
2.2SymmetryConsiderations(2)
)
(
)
( t
f
t
f 





2
/
0
0
0
)
sin(
)
(
4
0
T
n dt
t
n
t
f
T
b
a

Typical examples of odd periodic
function
𝑓 𝑡 =
𝑛=
∞
𝑏𝑛 sin 𝑛𝜔𝑡
𝑎𝑛 = 0
Then, the Fourier series can be represented as

20. 20
3.Half-wave Symmetry : a function f(t) if
2.2SymmetryConsiderations(3)
)
(
)
2
( t
f
T
t
f 

















even
an
for
,
0
odd
n
for
,
)
sin(
)
(
4
even
an
for
,
0
odd
n
for
,
)
cos(
)
(
4
0
2
/
0
0
2
/
0
0
0
T
n
T
n
dt
t
n
t
f
T
b
dt
t
n
t
f
T
a
a


Typical examples of half-wave odd periodic functions
In this case,
𝑓 𝑡 =
𝑛=
∞
(𝑎𝑛 cos 𝑛𝜔𝑡 + 𝑏𝑛sin 𝑛𝜔𝑡)
Then, the Fourier series can be represented as

21. 2.4Exponential
Fourierseries
Learning Outcome:
• Student should be able to determine
the Exponential FS coefficients with a
variety of periodic function
• Student should be able to plot the
double sided spectrum of
Exponential FS for magnitude and
phase

22. ExponentialFourierSeries
• Exponential form:
• Where: and:
• Relation with sin-cos form:
• Phasor form:







0
0
0
)
(
n
n
t
jn
ne
c
c
t
f 



T t
jn
n dt
e
t
f
T
c
0
0
)
(
1 



T
dt
t
f
T
a
c
0
0
0 )
(
1
2
n
n
n
jb
a
c


n
n
c 

use normal complex equation
]
[
2
1
)
sin(
]
[
2
1
)
cos(
t
jn
t
jn
o
t
jn
t
jn
o
o
o
o
o
e
e
j
t
n
e
e
t
n












Cos & sin in exponential form,
Euler's formula:
• A compact way to express a Fourier series function
• No more sin or cos and no amplitude-phase form

23. Example17.11
For dc component:
For ac component:
T
1

T 


 2
1
2
2
0 


T
t
t
f 
)
( 1
0
, 
 t










n
j
n
n
j
n
j
n
n
n
j
e
dt
te
dt
e
t
f
T
c
n
j
t
n
j
T t
jn
n
2
4
1
)
1
2
)(
2
sin
2
(cos
4
1
)
1
2
(
)
(
1
2
2
2
2
2
1
0
2
0
0



















  



T
tdt
dt
t
f
T
a
c
0
1
0
0
0
2
1
)
(
1

24. Example17.11
• Add both dc & ac components:
• To plot amplitude and phase spectra in exponential FS, consider both positive &
negative n:













0
2
2
2
1
)
(
n
n
t
n
j
e
n
j
t
f 

Positive n Negative n
0
2
2
90
2
1
0
)
(
2
1
0
2
1
)
(
2
1
0





































ve
ve
n
n
n
n
n
j
n
j
c
n
n
c






0
2
2
90
2
1
0
)
(
2
1
0
2
1
)
(
2
1
0




































ve
ve
n
n
n
n
n
j
n
j
c
n
n
c






𝑟 = 𝑧 = 𝑥2 + 𝑦2 𝜃 = 𝑡𝑎𝑛−1 𝑦
𝑥
𝑐𝑛∠𝜃 = 𝑥 + 𝑗𝑦
0+jy  𝜃=90
X+j0  𝜃=0
Complex number:

25. Example17.11
Amplitude spectrum Phase spectrum

26. Example1
Given:
Find the signal in
exponential form FS and,
indicate the exponential
Fourier coefficients.
n Cn C-n
0 10 -
1 1.5 1.5
2 2.530 2.5-30
3 2-90 -290
𝑣 𝑡 = 10 + 3 cos 𝜔0𝑡
+5 cos 2𝜔0𝑡 + 30°
+4 sin 3𝜔0𝑡

27. ,
,
Exponential Fourier Coefficients Table

28. Example2
Given the signal x(t) in Figure below.
(i) Find the expression for x(t) using Complex Exponential Fourier Series up
to the 10th harmonic.
(ii) Plot the double-sided spectrum of x(t) for magnitude and phase.
x(t)
5
t
2π
-2π

29. 29
Signal number 3 from Fourier Coefficient table:
Based on the signal in question ; 𝑋0 = 5, 𝜔0 =
2𝜋
2𝜋
= 1
𝑐0 =
𝑥
2
, 𝑐𝑛 =
−2𝑋0
(𝜋𝑛)2
, 𝑐𝑛 = 0 |𝑛 𝑒𝑣𝑒𝑛
Using 𝑥 𝑡 = 𝑛=−10
10
𝑐𝑛𝑒𝑗𝑛𝜔0𝑡
= 𝑐−9𝑒−9𝑗𝑡 + 𝑐−7𝑒−7𝑗𝑡 + 𝑐−5𝑒−5𝑗𝑡 + 𝑐−3𝑒−3𝑗𝑡 + 𝑐−1𝑒−1𝑗𝑡 +
5
2
+ 𝑐1𝑒𝑗𝑡 + 𝑐3𝑒3𝑗𝑡 + 𝑐5𝑒5𝑗𝑡 + 𝑐7𝑒7𝑗𝑡 + 𝑐9𝑒9𝑗𝑡

30. 30
Where the cn values are as follows:
𝑐−9 =
−10
81𝜋2
= 𝑐9
𝑐−7 =
−10
49𝜋2
= 𝑐7
𝑐−5 =
−10
25𝜋2
= 𝑐5
𝑐−3 =
−10
9𝜋2
= 𝑐3
𝑐−1 =
−10
𝜋2
= 𝑐1
The magnitude spectra of x(t);
The phase spectra of x(t);

31. Learning Outcome:
• Able to determine the output
signal from Low-pass filter
2.5Applications -
Filter

32. FilterAnimation
32
Original signal in
time-domain
Blue: Original signal
in frequency-domain
Red: Band-pass filter
Filtered signal

33. Filter
33
33
•Filter are an important component of electronics and
communications system.
•This filtering process cannot be accomplished without the Fourier
series expansion of the input signal.
•For example,
(a) Input and output
spectra of a lowpass filter,
(b) the lowpass filter
passes only the dc
component when c << 0

34. If the sawtooth waveform in Fig. (a) is applied to an ideal low-pass filter with the transfer
function shown in Fig. (b),
i) Find the trigonometry FS expansion
ii) Determine the filter output, y(t).
Example17.14
34

35. ii) Filter Output, y(t)
𝑥 𝑡 =
1
2
−
1
𝜋
sin 2𝜋𝑡 −
1
2𝜋
sin 4𝜋𝑡 −
1
3𝜋
sin 6𝜋𝑡 − ⋯
=
1
2
−
1
𝜋
sin 6.28𝑡 −
1
2𝜋
sin 12.57𝑡 −
1
3𝜋
sin 18.85𝑡 − ⋯
i) Fourier series expansion:
𝑇 = 1 𝜔𝑜 = 2𝜋𝑟𝑎𝑑/𝑠 𝜔𝑐 = 10𝑟𝑎𝑑/𝑠
𝑥 𝑡 =
1
2
−
1
𝜋
sin 𝜔𝑜𝑡 −
1
2𝜋
sin 2𝜔𝑜𝑡 −
1
3𝜋
sin 3𝜔𝑜𝑡 − ⋯
35
Only signal in between this
range, 𝑛𝜔𝑜 < 10 will be passed
2nd , 3rd and higher harmonics
will be rejected since their
frequency > 10
𝑦(𝑡) =
1
2
−
1
𝜋
sin 2𝜋𝑡

36. SummaryofFourierSeries
Trigonometric
Fourier Series
• Coefficients: a0,
an & bn
• Amplitude-phase
form
• Phasor form
• Gibbs
phenomenon
Exponential
Fourier
Series
• Coefficients: c0
& cn
• Phasor form
Symmetry
considerations
in FS
• Even
• Odd
• Half-wave
Amplitude &
phase
spectra
To plot frequency
spectrum, we
need:
• Amplitude
• Phase
Applications
• Filter
1 2 3 4 5

37. THANKYOU
Dr. Siti Nur Sakinah binti Jamaludin, FOE
+09-424 2715
sakinah@dhu.edu.my
37
Acknowledgment to Nurul Wahidah Arshad, FKEE, UMP for this notes.