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Fourier Series
1
Content
Periodic Functions
Fourier Series
Complex Form of the Fourier Series
Impulse Train
Analysis of Periodic Waveforms
Half-Range Expansion
Least Mean-Square Error Approximation
2
Periodic Functions
Fourier Series
3
The Mathematic Formulation
Any function that satisfies
( ) ( )f t f t m mT
where T is a constant and is called the period
of the function.
4
Example:
Find its period.4
cos3
cos)(tt
tf
)()( Ttftf )(4
1cos)(
3
1cos
4cos
3cos TtTt
tt
Fact: )2cos(cos m
mT
23
nT
24
mT 6
nT 8
24T smallest T
5
Example:
Find its period.tttf 21 coscos)(
)()( Ttftf )(cos)(coscoscos 2121 TtTttt
mT 21
nT 22
n
m
2
1
2
1
must be a
rational number
6
Example:
Is this function a periodic one?
tttf )10cos(10cos)(
10
10
2
1 not a rational
number
7
Fourier Series
Fourier Series
8
Introduction
Decompose a periodic input signal into
primitive periodic components.
A periodic sequence
T 2T 3T
t
f(t)
9
Synthesis
T
ntb
T
nta
atf
n
n
n
n
2sin
2cos
2)(
11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 0
1
0
1
0 tnbtnaa
tfn
n
n
n
Let 0=2/T.
10
Orthogonal Functions
Call a set of functions {k} orthogonal
on an interval a < t < b if it satisfies
*
,
0( ) ( )
( ), ( )
b
m na
m n m n
m nt t dt or
N m n
t t N
11
Orthogonal set of Sinusoidal Functions
Define 0=2/T.
0 ,0)cos(2/
2/0 mdttm
T
T
/2
0/2
sin( ) 0, T
Tm t dt m
/2
0 0 ,/2
0cos( )cos( )
/ 2 2
T
m nT
m n Tm t n t dt
T m n
/2
0 0 ,/2
sin( )sin( )2
T
m nT
Tm t n t dt
nmdttntmT
T and allfor ,0)cos()sin(
2/
2/00
We now prove this one
12
Proof
dttntmT
T 2/
2/00 )cos()cos(
0
)]cos()[cos(2
1coscos
dttnmdttnmT
T
T
T
2/
2/0
2/
2/0 ])cos[(
2
1])cos[(
2
1
2/
2/0
0
2/
2/0
0
])sin[()(
1
2
1])sin[(
)(
1
2
1 T
T
T
Ttnm
nmtnm
nm
m n
0 0
1 1 1 12sin[( ) ] 2sin[( ) ]
2 ( ) 2 ( )m n m n
m n m n
00
13
Proof
dttntmT
T 2/
2/00 )cos()cos(
0
)]cos()[cos(2
1coscos
dttmT
T 2/
2/0
2 )(cos
2/
2/
0
0
2/
2/
]2sin4
1
2
1T
T
T
T
tmm
t
m = n
2
T
]2cos1[2
1cos2
dttmT
T 2/
2/0 ]2cos1[
2
1
nmT
nmdttntm
T
T 2/
0)cos()cos(
2/
2/00
14
Orthogonal set of Sinusoidal Functions
Define 0=2/T.
0 ,0)cos(2/
2/0 mdttm
T
T0 ,0)sin(
2/
2/0 mdttm
T
T
nmT
nmdttntm
T
T 2/
0)cos()cos(
2/
2/00
nmT
nmdttntm
T
T 2/
0)sin()sin(
2/
2/00
nmdttntmT
T and allfor ,0)cos()sin(
2/
2/00
,3sin,2sin,sin
,3cos,2cos,cos
,1
000
000
ttt
ttt
an orthogonal set.15
Decomposition
dttfT
aTt
t
0
0
)(2
0
,2,1 cos)(2
0
0
0
ntdtntfT
aTt
tn
,2,1 sin)(2
0
0
0
ntdtntfT
bTt
tn
)sin()cos(2
)( 0
1
0
1
0 tnbtnaa
tfn
n
n
n
16
Convergence
Let picewise continuous periodic function
If continuous at the limit is
2- if continuous in and picewise
continuous function within interval, CVU to
00 0
1 1
( ) cos( ) sin( )2
11 lim ( ) ( ) ( )
2
N N
N n n
n n
NN
af t a n t b n t
t f t f t f t
( ),f t T
( )f t 0t 0( )f t
( )f t '( )f t(a,b) ( )Nf t ( )f t
17
(Suite)
3- if exists except on a finite number
points on closed interval, then in a point where
exists then
''( )f t
''
0( )f t
' 'lim ( ) ( )NN
f t f t
18
Convergence and existence
1- absolutely integrable over any period
2- has a finite number of extremum within any finite
interval of t
3- has a finite number of discontinuities within any
interval of t and each of these discontinuities is finite
….sufficient but not necessary conditions
19
0
( )T
f t dt
𝑓(𝑡
𝑓(𝑡
Proof
Use the following facts:
0 ,0)cos(2/
2/0 mdttm
T
T0 ,0)sin(
2/
2/0 mdttm
T
T
nmT
nmdttntm
T
T 2/
0)cos()cos(
2/
2/00
nmT
nmdttntm
T
T 2/
0)sin()sin(
2/
2/00
nmdttntmT
T and allfor ,0)cos()sin(
2/
2/00 20
Example (Square Wave)
00
21 1 : 2
2a dt T
,2,1 0sin1
cos2
200
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(
1 cos
1sin
2
200
n
nnn
nnt
nntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
21
112
2
00
dta
,2,1 0sin1
cos2
200
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(
1 cos
1sin
2
100
n
nnn
nnt
nntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
Example (Square Wave)
ttttf 5sin
5
13sin
3
1sin
2
2
1)(
22
112
2
00
dta
,2,1 0sin1
cos2
200
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(
1 cos
1sin
2
100
n
nnn
nnt
nntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
Example (Square Wave)
-0.5
0
0.5
1
1.5
ttttf 5sin
5
13sin
3
1sin
2
2
1)(
23
Harmonics
T
ntb
T
nta
atf
n
n
n
n
2sin
2cos
2)(
11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 0
1
0
1
0 tnbtnaa
tfn
n
n
n
24
Harmonics
tnbtnaa
tfn
n
n
n 0
1
0
1
0 sincos2
)(
Tf
22 00
Define , called the fundamental angular frequency.
0 nnDefine , called the n-th harmonic of the periodic function.
tbtaa
tf n
n
nn
n
n
sincos2
)(11
0
25
Harmonics
tbtaa
tf n
n
nn
n
n
sincos2
)(11
0
)sincos(2 1
0 tbtaa
nnn
n
n
12222
220 sincos2 n
n
nn
nn
nn
nnn t
ba
bt
ba
aba
a
1
220 sinsincoscos2 n
nnnnnn ttbaa
)cos(1
0 n
n
nn tCC
26
Amplitudes and Phase Angles
)cos()(1
0 n
n
nn tCCtf
2
00
aC
22
nnn baC
n
nn
a
b1tan
harmonic amplitude phase angle
27
Complex Form of the Fourier Series
Fourier Series
28
Complex Exponentials
tnjtnetjn
00 sincos0
tjntjneetn 00
2
1cos 0
tnjtnetjn
00 sincos0
tjntjntjntjnee
jee
jtn 0000
22
1sin 0
29
Complex Form of the Fourier Series
tnbtnaa
tfn
n
n
n 0
1
0
1
0 sincos2
)(
tjntjn
n
n
tjntjn
n
n eebj
eeaa
0000
11
0
22
1
2
1
0 00 )(2
1)(
2
1
2 n
tjn
nn
tjn
nn ejbaejbaa
1
000
n
tjn
n
tjn
n ececc
)(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
30
Complex Form of the Fourier Series
1
000)(
n
tjn
n
tjn
n ececctf
1
1
000
n
tjn
n
n
tjn
n ececc
n
tjn
nec 0
)(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
31
Complex Form of the Fourier Series
2/
2/
00 )(
1
2
T
Tdttf
T
ac
)(2
1nnn jbac
2/
2/0
2/
2/0 sin)(cos)(
1 T
T
T
Ttdtntfjtdtntf
T
2/
2/00 )sin)(cos(
1 T
Tdttnjtntf
T
2/
2/
0)(1 T
T
tjndtetf
T
2/
2/
0)(1
)(2
1 T
T
tjn
nnn dtetfT
jbac )(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
32
Complex Form of the Fourier Series
n
tjn
nectf 0)(
dtetfT
cT
T
tjn
n
2/
2/
0)(1
)(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
If f(t) is real,
*
n nc c
hermitian symmetry
nn j
nnn
j
nn ecccecc
|| ,|| *
22
2
1|||| nnnn bacc
n
nn
a
b1tan
,3,2,1 n
002
1ac
33
Complex Frequency Spectra
nn j
nnn
j
nn ecccecc
|| ,|| *
22
2
1|||| nnnn bacc
n
nn
a
b1tan ,3,2,1 n
002
1ac
|cn|
amplitude
spectrum
n
phase
spectrum
34
Example
2
T
2
T TT
2
d
t
f(t)
A
2
d
dteT
Ac
d
d
tjn
n
2/
2/
0
2/
2/0
01
d
d
tjne
jnT
A
2/
0
2/
0
0011 djndjn
ejn
ejnT
A
)2/sin2(1
0
0
dnjjnT
A
2/sin1
0
021
dnnT
A
sin
sin
n d
Ad Ad n dTc
n dT T T
T
35
T
dn
T
dn
T
Adcn
sin
82
5
1
T ,
4
1 ,
20
1
0
T
dTd
Example
40 80 120-40 0-120 -80
A/5
50 100 150-50-100-150
36
T
dn
T
dn
T
Adcn
sin
42
5
1
T ,
2
1 ,
20
1
0
T
dTd
Example
40 80 120-40 0-120 -80
A/10
100 200 300-100-200-300
37
Example
dteT
Ac
dtjn
n
0
0
d
tjne
jnT
A
00
01
00
110
jne
jnT
A djn
)1(1
0
0
djne
jnT
A
2/0
sindjn
e
T
dn
T
dn
T
Ad
TT d
t
f(t)
A
0
)(1 2/2/2/
0
000 djndjndjneee
jnT
A
38
Impulse Train
Fourier Series
39
Dirac Delta “Function”
0
00)(
t
tt and 1)(
dtt
0t
Also called unit impulse function.
( ) 1 0t dt
40
Generalized function
But an ordinary function which is everywhere 0
except at a single point must have integral 0 (in
the Riemann integral sense).
– Thus cannot be an ordinary function and
mathematically it is defined by
Where is an regular function continuous at
( ) ( ) (0)t t dt
0t ( )t41
Generalized function (suite)
An alternative definition of is given by
It is why, is often called a generalized
function and a testing function
(0) 0
( ) ( ) 0 0 0
0 0
b
a
a b
t t dt a b or a b
undefined a or b
42
0 0( ) ( ) ( )t t t dt t
0( )t t is defined by • Also
Some properties of ( )t
43
Some properties
0 0 0
1( ) ( ) ( ) ( )
( ) ( ) (0) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )* ( )
at t t ta
x t t x t x t t t x t t t
x t x t d x t x t t
44
Generalized derivatives
( ) ( )
'
( ) ( ) ( 1) ( ) ( )
( ( ) int )
'( ) ( ) (0)
( )( )
n n ng t t dt g t t dt
and because t vanished out side some fixed erval
t t dt
du texample t
dt
45
Property
)0()()(
dttt
)0()()0()0()()()(
dttdttdttt
(t): Test Function
46
Impulse Train
0t
T 2T 3TT2T3T
n
T nTtt )()(
47
Fourier Series of the Impulse Train
n
T nTtt )()(T
dttT
aT
TT
2)(
2 2/
2/0
Tdttnt
Ta
T
TTn
2)cos()(
2 2/
2/0
0)sin()(2 2/
2/0 dttnt
Tb
T
TTn
n
T tnTT
t 0cos21
)(
48
Complex FormFourier Series of the Impulse Train
Tdtt
T
ac
T
TT
1)(
1
2
2/
2/
00
Tdtet
Tc
T
T
tjn
Tn
1)(
1 2/
2/
0
n
tjn
T eT
t 01
)(
n
T nTtt )()(
49
Analysis ofPeriodic Waveforms
Fourier Series
50
Waveform Symmetry
Even Functions
Odd Functions
)()( tftf
)()( tftf
51
Decomposition
Any function f(t) can be expressed as the
sum of an even function fe(t) and an odd
function fo(t).
)()()( tftftf oe
)]()([)(21 tftftfe
)]()([)(21 tftftfo
Even Part
Odd Part
52
Example
00
0)(
t
tetf
t
Even Part
Odd Part
0
0)(
21
21
te
tetf
t
t
e
0
0)(
21
21
te
tetf
t
t
o
53
Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
TT/2T/2
54
Quarter-Wave Symmetry
Even Quarter-Wave Symmetry
TT/2T/2
Odd Quarter-Wave Symmetry
T
T/2T/2
55
Hidden Symmetry
The following is a asymmetry periodic function:
Adding a constant to get symmetry property.
A
TT
A/2
A/2
TT
56
Fourier Coefficients of Symmetrical Waveforms
The use of symmetry properties simplifies the
calculation of Fourier coefficients.
– Even Functions
– Odd Functions
– Half-Wave
– Even Quarter-Wave
– Odd Quarter-Wave
– Hidden
57
Fourier Coefficients of Even Functions
)()( tftf
tnaa
tfn
n 0
1
0 cos2
)(
2/
00 )cos()(
4 T
n dttntfT
a
58
Fourier Coefficients of Even Functions
)()( tftf
tnbtfn
n 0
1
sin)(
2/
00 )sin()(
4 T
n dttntfT
b
59
Fourier Coefficients for Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
TT/2T/2
The Fourier series contains only odd harmonics.60
Fourier Coefficients for Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
)sincos()(1
00
n
nn tnbtnatf
odd for )cos()(4
even for 02/
00 ndttntf
T
na T
n
odd for )sin()(4
even for 02/
00 ndttntf
T
nb T
n
61
Fourier Coefficients forEven Quarter-Wave Symmetry
TT/2T/2
])12cos[()( 0
1
12 tnatfn
n
4/
0012 ])12cos[()(
8 T
n dttntfT
a
62
Fourier Coefficients forOdd Quarter-Wave Symmetry
])12sin[()( 0
1
12 tnbtfn
n
4/
0012 ])12sin[()(
8 T
n dttntfT
b
T
T/2T/2
63
ExampleEven Quarter-Wave Symmetry
4/
0012 ])12cos[()(
8 T
n dttntfT
a 4/
00 ])12cos[(
8 T
dttnT
4/
0
0
0
])12sin[()12(
8T
tnTn
)12(
4)1( 1
n
n
T
T/2T/2
1
1
T T/4T/4
64
ExampleEven Quarter-Wave Symmetry
4/
0012 ])12cos[()(
8 T
n dttntfT
a 4/
00 ])12cos[(
8 T
dttnT
4/
0
0
0
])12sin[()12(
8T
tnTn
)12(
4)1( 1
n
n
T
T/2T/2
1
1
T T/4T/4
ttttf 000 5cos
5
13cos
3
1cos
4)(
65
Example
T
T/2T/2
1
1
T T/4T/4
Odd Quarter-Wave Symmetry
4/
0012 ])12sin[()(
8 T
n dttntfT
b 4/
00 ])12sin[(
8 T
dttnT
4/
0
0
0
])12cos[()12(
8T
tnTn
)12(
4
n
66
Example
T
T/2T/2
1
1
T T/4T/4
Odd Quarter-Wave Symmetry
4/
0012 ])12sin[()(
8 T
n dttntfT
b 4/
00 ])12sin[(
8 T
dttnT
4/
0
0
0
])12cos[()12(
8T
tnTn
)12(
4
n
ttttf 000 5sin
5
13sin
3
1sin
4)(
67
Half-Range Expansions
Fourier Series
68
Non-Periodic Function Representation
A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
69
Without Considering Symmetry
A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
T
70
Expansion Into Even Symmetry
A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
T=2
71
Expansion Into Odd Symmetry
A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
T=2
72
Expansion Into Half-Wave Symmetry
A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
T=2
73
Expansion Into Even Quarter-Wave Symmetry
A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
T/2=2
T=4
74
Expansion Into Odd Quarter-Wave Symmetry
A non-periodic function f(t) defined over (0, )
can be expanded into a Fourier series which is
defined only in the interval (0, ).
T/2=2 T=4
75
Least Mean-Square
Error Approximation
Fourier Series
76
Approximation a function
Use 00 0
1
( ) cos sin2
N
N n n
n
aS t a n t b n t
to represent f(t) on interval T/2 < t < T/2.
Define ( ) ( ) ( )N Ne t f t S t
/22 2
/2
1[ ( )]
T
N NT
e t dtT
Mean-Square
Error
77
Approximation a function
Show that using SN(t) to represent f(t) has
least mean-square property
/22 2
/2
1[ ( )]
T
N NT
e t dtT
2
/20
0 0/2
1
1( ) cos sin
2
NT
n nT
n
af t a n t b n t dt
T
Proven by setting /ai = 0 and /bi = 0.78
Approximation a function
/22 2
/2
1[ ( )]
T
N NT
e t dtT
2
/20
0 0/2
1
1( ) cos sin
2
NT
n nT
n
af t a n t b n t dt
T
2/2
0
/20
1( ) 0
2
TN
T
af t dt
a T
2
/2
0/2
2( )cos 0
TN
nT
n
a f t n tdta T
2/2
0/2
2( )sin 0
TN
nT
n
b f t n tdtb T
79
Mean-Square Error
/22 2
/2
1[ ( )]
T
N NT
e t dtT
2
/20
0 0/2
1
1( ) cos sin
2
NT
n nT
n
af t a n t b n t dt
T
2/2
2 2 2 20
/21
1 1[ ( )] ( )
4 2
NT
N n nT
n
af t dt a b
T
80
Mean-Square Error
/22 2
/2
1[ ( )]
T
N NT
e t dtT
2
/20
0 0/2
1
1( ) cos sin
2
NT
n nT
n
af t a n t b n t dt
T
2/2
2 2 20
/21
1 1[ ( )] ( )
4 2
NT
n nT
n
af t dt a b
T
81
Mean-Square Error
/22 2
/2
1[ ( )]
T
N NT
e t dtT
2
/20
0 0/2
1
1( ) cos sin
2
NT
n nT
n
af t a n t b n t dt
T
2/
2/1
222
02 )(2
1
4)]([
1 T
Tn
nn baa
dttfT
22lim lim ( ) 0N NN N
e t
82
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