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Lecture 11:Introduction to Fourier Series
Sections 2.2.3, 2.3
Trigonometric Fourier Series
•Outline▫Introduction▫Visualization▫Theoretical Concepts▫Qualitative Analysis▫Example▫Class Exercise
Introduction• What is Fourier Series?
▫ Representation of a periodic function with a weighted, infinite sum of sinusoids.
• Why Fourier Series?▫ Any arbitrary periodic signal, can be approximated by
using some of the computed weights
▫ These weights are generally easier to manipulate and analyze than the original signal
Periodic Function
• What is a periodic Function?▫ A function which remains unchanged when time-shifted
by one period f(t) = f(t + To) for all values of t
• What is To
To To
Properties of a periodic function 1
• A periodic function must be everlasting▫ From –∞ to ∞
• Why?
• Periodic or Aperiodic?
Properties of a periodic function
• You only need one period of the signal to generate the entire signal▫ Why?
• A periodic signal cam be expressed as a sum of sinusoids of frequency F0 = 1/T0 and all its harmonics
VisualizationCan you represent this simple function using sinusoids?
Single sinusoid representation
Visualization
To obtain the exact signal, an infinite number of sinusoids are required
)cos( 01 ta amplitude Fundamental
frequency
)3cos( 03 ta
New amplitude 2nd Harmonic
)5cos( 05 ta amplitude 4th Harmonic
Theoretical Concepts
(6)
,...3,2,1....)sin()(2
,...3,2,1....)cos()(2
2
)sin()cos()(
01
1
01
1
00
00
00
01
01
0
ndttntfT
b
ndttntfT
a
T
tnbtnaatf
Tt
t
n
Tt
t
n
nn
nn
PeriodCosine terms
Sine terms
Theoretical Concepts
(6)
n
nn
nnn
nn
n
a
b
bac
ac
tncctf
1
22
00
01
0
tan
)cos()(
DC Offset
What is the difference between these two functions?
A
0 1 2-1-2
-A
A
0 1 2-1-2
Average Value = 0
Average Value ?
DC OffsetIf the function has a DC value:
01
1
)(1
)sin()cos(2
1)(
00
01
01
0
Tt
t
nn
nn
dttfT
a
tnbtnaatf
Qualitative Analysis• Is it possible to have an idea of what your solution
should be before actually computing it?
For Sure
Properties – DC Value• If the function has no DC value, then a0 = ?
-1 1 2
-A
A
DC?
A
0 1 2-1-2
DC?
Properties – Symmetry
A
A
0 π/2 π 3π/2
f(-t) = -f(t)
• Even function
• Odd function
0
-A
A
π/2 π 3π/2
f(-t) = f(t)
Properties – Symmetry• Note that the integral over a period of an odd function is?
,...3,2,1....)sin()(2 01
1
00
ndttntfT
bTt
tn
If f(t) is even:
EvenOddX = Odd
,...3,2,1....)cos()(2 01
1
00
ndttntfT
aTt
tn
EvenEvenX = Even
Properties – Symmetry• Note that the integral over a period of an odd function is
zero.
,...3,2,1....)cos()(2 01
1
00
ndttntfT
aTt
tn
If f(t) is odd:
OddEvenX = Odd
,...3,2,1....)sin()(2 01
1
00
ndttntfT
bTt
tn
OddOddX = Even
Properties – Symmetry•If the function has:
▫even symmetry: only the cosine and associated coefficients exist
▫odd symmetry: only the sine and associated coefficients exist
▫even and odd: both terms exist
Properties – Symmetry
• If the function is half-wave symmetric, then only odd harmonics exist
Half wave symmetry: f(t-T0/2) = -f(t)
-1 1 2
-A
A
Properties – Discontinuities•If the function has
▫ Discontinuities: the coefficients will be proportional to 1/n
▫ No discontinuities: the coefficients will be proportional to 1/n2
• Rationale:
-1 1 2
-A
A
A
0 1 2-1-2
Which is closer to a sinusoid?
Which function has discontinuities?
Example
• Without any calculations, predict the general form of the Fourier series of:
-1 1 2
-A
A
DC? No, a0 = 0;
Symmetry? Even, bn = 0;
Half wave symmetry?
Yes, only odd harmonics
Discontinuities?No, falls of as
1/n2Prediction an 1/n2 for n = 1, 3, 5, …;
Example• Now perform the calculation
2/
00
00
0
001
1
)cos()(4
)cos()(2
TTt
tn dttntf
Tdttntf
Ta
;20 T 2
20
...5,3,1...8
22 n
n
Aan
)cos(14
)cos(2222
1
0
nn
AdttnAtan
zero for n even
Example• Now compare your calculated answer with your
predicted form
DC? No, a0 = 0;
Symmetry?Even, bn = 0;
Half wave symmetry?
Yes, only odd harmonics
Discontinuities?No, falls of as
1/n2
Class exercise
• Discuss the general form of the solution of the function below and write it down
• Compute the Fourier series representation of the function
• With your partners, compare your calculations with your predictions and comment on your solution
A
0 1 2-1-2
Spectral Lines
,...3,2,1....)sin()(2
,...3,2,1....)cos()(2
2
)sin()cos()(
01
1
01
1
00
00
00
01
01
ndttntfT
b
ndttntfT
a
T
tnbtnatf
Tt
tn
Tt
tn
nn
nn
n
nn
nnn
nn
n
a
b
bac
ac
tncctf
1
22
00
01
0
tan
)cos()(
Spectral Lines
•Gives the frequency composition of the function▫Amplitude, phase of sinusoidal components
•Could provide information not found in time signal▫E.g. Pitch, noise components
•May help distinguish between signals ▫E.g speech/speaker recognition
Spectral Lines Example
-3 -2 -1 0 1 2 30.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t/pi
y(t
)
exp(-t/2)
QUESTIONS --DC Yes____ ao = ? No_____ ao = 0Symmetry
Even____ an = ? bn = 0Odd____ an = 0 bn = ?Nether even nor odd ____ an = ? bn = ?
Halfwave symmetryYes_____ only odd harmonicsNo______ all harmonics
DiscontinuitiesYes_____ proportional to1/nNo______ proportional to1/n2
Note ? means find that variable.Comment on the general form of the Fourier Series coefficients [an and/or bn.]
X
X
X
X
Spectral Lines Example
,3,2,1....0
)2sin(2/
,...3,2,1....01
1
)0sin()(
0
2
,3,2,1....0
)2cos(2/
,3,2,1....01
1
)0cos()(
0
2
22
0
20
5042.02/01
1
)(
00
0
11
ndttnt
e
n
Tt
tdttntf
Tnb
ndtntt
e
n
Tt
tdttntf
Tna
T
dtt
e
Tt
tdttf
Ta
-3 -2 -1 0 1 2 30.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t/pi
y(t
)
exp(-t/2)
Spectral Lines Example
,....3,2,1.....1
216
4.0342n
12
16
)2/
1(2/
82
12
16
)2/
44(2/
22
,...3,2,1.....1
216
2/4)2sin()2cos(4(
2/22
,...3,2,1.....1
216
008.1
12
16
)2/
1(2/
22
,...3,2,1.....1
216
2/)2cos()2sin(4(
2/22
nnn
ene
n
nene
nn
nennnenb
nnn
ee
nn
ennnena
Spectral Lines Example
-3 -2 -1 0 1 2 30
0.5
1
t/pi
y(t)
exp(-t/2)
0 1 2 3 4 5 6 7 8 9 100
0.5
1
a n
n
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
b n
n
0 1 2 3 4 5 6 7 8 9 100
0.5
1C
n
n
0 1 2 3 4 5 6 7 8 9 10-2
-1
0
n [ra
d]
n
n
nn
nnn
nn
n
a
b
bac
ac
tncctf
1
22
00
01
0
tan
)cos()(
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