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Fourier Series
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ContentPeriodic FunctionsFourier SeriesComplex Form of the Fourier Series Impulse TrainAnalysis of Periodic WaveformsHalf-Range ExpansionLeast Mean-Square Error Approximation
Fourier Series
Periodic Functions
The Mathematic Formulation
Any function that satisfies
( ) ( )f t f t T
where T is a constant and is called the period of the function.
Example:
4cos
3cos)(
tttf Find its period.
)()( Ttftf )(4
1cos)(
3
1cos
4cos
3cos TtTt
tt
Fact: )2cos(cos m
mT
23
nT
24
mT 6
nT 8
24T smallest T
Example:
tttf 21 coscos)( Find its period.
)()( Ttftf )(cos)(coscoscos 2121 TtTttt
mT 21
nT 22
n
m
2
1
2
1
must be a
rational number
Example:
tttf )10cos(10cos)(
Is this function a periodic one?
10
10
2
1 not a rational number
Fourier Series
Fourier Series
Introduction
Decompose a periodic input signal into primitive periodic components.
A periodic sequenceA periodic sequence
T 2T 3T
t
f(t)
Synthesis
T
ntb
T
nta
atf
nn
nn
2sin
2cos
2)(
11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 01
01
0 tnbtnaa
tfn
nn
n
Let 0=2/T.
Orthogonal Functions
Call a set of functions {k} orthogonal on an interval a < t < b if it satisfies
nmr
nmdttt
n
b
a nm
0)()(
Orthogonal set of Sinusoidal Functions
Define 0=2/T.
0 ,0)cos(2/
2/ 0 mdttmT
T0 ,0)sin(
2/
2/ 0 mdttmT
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
We now prove this one
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/
2/00
2/
2/00
])sin[()(
1
2
1])sin[(
)(
1
2
1 T
T
T
Ttnm
nmtnm
nm
m n
])sin[(2)(
1
2
1])sin[(2
)(
1
2
1
00
nmnm
nmnm
00
Proof
dttntmT
T 2/
2/ 00 )cos()cos(
0
)]cos()[cos(2
1coscos
dttmT
T 2/
2/ 02 )(cos
2/
2/
00
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
Orthogonal set of Sinusoidal Functions
Define 0=2/T.
0 ,0)cos(2/
2/ 0 mdttmT
T0 ,0)sin(
2/
2/ 0 mdttmT
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
,3sin,2sin,sin
,3cos,2cos,cos
,1
000
000
ttt
ttt
an orthogonal set.an orthogonal set.
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
)( 01
01
0 tnbtnaa
tfn
nn
n
ProofUse the following facts:
0 ,0)cos(2/
2/ 0 mdttmT
T0 ,0)sin(
2/
2/ 0 mdttmT
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
Example (Square Wave)
112
200
dta
,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
112
200
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)(
ttttf 5sin
5
13sin
3
1sin
2
2
1)(
112
200
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)(
ttttf 5sin
5
13sin
3
1sin
2
2
1)(
Harmonics
T
ntb
T
nta
atf
nn
nn
2sin
2cos
2)(
11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 01
01
0 tnbtnaa
tfn
nn
n
Harmonics
tnbtnaa
tfn
nn
n 01
01
0 sincos2
)(
Tf
22 00Define , called the fundamental angular frequency.
0 nnDefine , called the n-th harmonic of the periodic function.
tbtaa
tf nn
nnn
n
sincos2
)(11
0
Harmonics
tbtaa
tf nn
nnn
n
sincos2
)(11
0
)sincos(2 1
0 tbtaa
nnnn
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 nn
nn tCC
Amplitudes and Phase Angles
)cos()(1
0 nn
nn tCCtf
20
0
aC
22nnn baC
n
nn a
b1tan
harmonic amplitude phase angle
Fourier Series
Complex Form of the Fourier Series
Complex Exponentials
tnjtne tjn00 sincos0
tjntjn eetn 00
2
1cos 0
tnjtne tjn00 sincos0
tjntjntjntjn eej
eej
tn 0000
22
1sin 0
Complex Form of the Fourier Series
tnbtnaa
tfn
nn
n 01
01
0 sincos2
)(
tjntjn
nn
tjntjn
nn eeb
jeea
a0000
11
0
22
1
2
1
0 00 )(2
1)(
2
1
2 n
tjnnn
tjnnn ejbaejba
a
1
000
n
tjnn
tjnn ececc
)(2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
)(
2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
Complex Form of the Fourier Series
1
000)(
n
tjnn
tjnn ececctf
1
10
00
n
tjnn
n
tjnn ececc
n
tjnnec 0
)(2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
)(
2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
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
tjn dtetfT
2/
2/
0)(1
)(2
1 T
T
tjnnnn dtetf
Tjbac )(
2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
)(
2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
Complex Form of the Fourier Series
n
tjnnectf 0)(
n
tjnnectf 0)(
dtetfT
cT
T
tjnn
2/
2/
0)(1 dtetf
Tc
T
T
tjnn
2/
2/
0)(1
)(2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
)(
2
1
)(2
12
00
nnn
nnn
jbac
jbac
ac
If f(t) is real,*nn cc
nn jnnn
jnn ecccecc
|| ,|| *
22
2
1|||| nnnn bacc
n
nn a
b1tan
,3,2,1 n
00 2
1ac
Complex Frequency Spectra
nn jnnn
jnn ecccecc
|| ,|| *
22
2
1|||| nnnn bacc
n
nn a
b1tan ,3,2,1 n
00 2
1ac
|cn|
amplitudespectrum
n
phasespectrum
Example
2
T
2
T TT
2
d
t
f(t)
A
2
d
dteT
Ac
d
d
tjnn
2/
2/
0
2/
2/0
01
d
d
tjnejnT
A
2/
0
2/
0
0011 djndjn ejn
ejnT
A
)2/sin2(1
00
dnjjnT
A
2/sin1
002
1dn
nT
A
TdnT
dn
T
Adsin
TdnT
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
TdnT
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
Example
dteT
Ac
d tjnn
0
0
d
tjnejnT
A
00
01
00
110
jne
jnT
A djn
)1(1
0
0
djnejnT
A
2/0
sindjne
TdnT
dn
T
Ad
TT d
t
f(t)
A
0
)(1 2/2/2/
0
000 djndjndjn eeejnT
A
Fourier Series
Impulse Train
Dirac Delta Function
0
00)(
t
tt and 1)(
dtt
0 t
Also called unit impulse function.
Property
)0()()(
dttt )0()()(
dttt
)0()()0()0()()()(
dttdttdttt
(t): Test Function
Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()(
n
T nTtt )()(
Fourier Series of the Impulse Train
n
T nTtt )()(
n
T nTtt )()(T
dttT
aT
T T
2)(
2 2/
2/0
Tdttnt
Ta
T
T Tn
2)cos()(
2 2/
2/ 0 0)sin()(
2 2/
2/ 0 dttntT
bT
T Tn
n
T tnTT
t 0cos21
)(
n
T tnTT
t 0cos21
)(
Complex FormFourier Series of the Impulse Train
Tdtt
T
ac
T
T T
1)(
1
2
2/
2/
00
Tdtet
Tc
T
T
tjnTn
1)(
1 2/
2/
0
n
tjnT e
Tt 0
1)(
n
tjnT e
Tt 0
1)(
n
T nTtt )()(
n
T nTtt )()(
Fourier Series
Analysis ofPeriodic Waveforms
Waveform Symmetry
Even Functions
Odd Functions
)()( tftf
)()( tftf
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
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
Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
TT/2T/2
Quarter-Wave Symmetry
Even Quarter-Wave Symmetry
TT/2T/2
Odd Quarter-Wave Symmetry
T
T/2T/2
Hidden Symmetry
The following is a asymmetry periodic function:
Adding a constant to get symmetry property.
A
TT
A/2
A/2
TT
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
Fourier Coefficients of Even Functions
)()( tftf
tnaa
tfn
n 01
0 cos2
)(
2/
0 0 )cos()(4 T
n dttntfT
a
Fourier Coefficients of Even Functions
)()( tftf
tnbtfn
n 01
sin)(
2/
0 0 )sin()(4 T
n dttntfT
b
Fourier Coefficients for Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
TT/2T/2
The Fourier series contains only odd harmonics.The Fourier series contains only odd harmonics.
Fourier Coefficients for Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
)sincos()(1
00
n
nn tnbtnatf
odd for )cos()(4
even for 02/
0 0 ndttntfT
na T
n
odd for )sin()(4
even for 02/
0 0 ndttntfT
nb T
n
Fourier Coefficients forEven Quarter-Wave Symmetry
TT/2T/2
])12cos[()( 01
12 tnatfn
n
4/
0 012 ])12cos[()(8 T
n dttntfT
a
Fourier Coefficients forOdd Quarter-Wave Symmetry
])12sin[()( 01
12 tnbtfn
n
4/
0 012 ])12sin[()(8 T
n dttntfT
b
T
T/2T/2
ExampleEven Quarter-Wave Symmetry
4/
0 012 ])12cos[()(8 T
n dttntfT
a 4/
0 0 ])12cos[(8 T
dttnT
4/
0
00
])12sin[()12(
8T
tnTn
)12(
4)1( 1
nn
TT/2T/2
1
1T T/4T/4
ExampleEven Quarter-Wave Symmetry
4/
0 012 ])12cos[()(8 T
n dttntfT
a 4/
0 0 ])12cos[(8 T
dttnT
4/
0
00
])12sin[()12(
8T
tnTn
)12(
4)1( 1
nn
TT/2T/2
1
1T T/4T/4
ttttf 000 5cos
5
13cos
3
1cos
4)(
ttttf 000 5cos
5
13cos
3
1cos
4)(
Example
TT/2T/2
1
1
T T/4T/4
Odd Quarter-Wave Symmetry
4/
0 012 ])12sin[()(8 T
n dttntfT
b 4/
0 0 ])12sin[(8 T
dttnT
4/
0
00
])12cos[()12(
8T
tnTn
)12(
4
n
Example
TT/2T/2
1
1
T T/4T/4
Odd Quarter-Wave Symmetry
4/
0 012 ])12sin[()(8 T
n dttntfT
b 4/
0 0 ])12sin[(8 T
dttnT
4/
0
00
])12cos[()12(
8T
tnTn
)12(
4
n
ttttf 000 5sin
5
13sin
3
1sin
4)(
ttttf 000 5sin
5
13sin
3
1sin
4)(
Fourier Series
Half-Range Expansions
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, ).
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
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
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
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
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
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
Fourier Series
Least Mean-Square Error Approximation
Approximation a function
Use
k
nnnk tnbtna
atS
100
0 sincos2
)(
to represent f(t) on interval T/2 < t < T/2.
Define )()()( tStft kk
2/
2/
2)]([1 T
T kk dttT
E Mean-Square Error
Approximation a function
Show that using Sk(t) to represent f(t) has least mean-square property.
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtna
atf
T
Proven by setting Ek/ai = 0 and Ek/bi = 0.
Approximation a function
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtna
atf
T
0)(1
2
2/
2/
0
0
T
T
k dttfT
a
a
E 0)(1
2
2/
2/
0
0
T
T
k dttfT
a
a
E0cos)(
2 2/
2/ 0
T
Tnn
k tdtntfT
aa
E 0cos)(2 2/
2/ 0
T
Tnn
k tdtntfT
aa
E
0sin)(2 2/
2/ 0
T
Tnn
k tdtntfT
bb
E 0sin)(2 2/
2/ 0
T
Tnn
k tdtntfT
bb
E
Mean-Square Error
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtna
atf
T
2/
2/1
22202 )(
2
1
4)]([
1 T
T
k
nnnk ba
adttf
TE
2/
2/1
22202 )(
2
1
4)]([
1 T
T
k
nnnk ba
adttf
TE
Mean-Square Error
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtna
atf
T
2/
2/1
22202 )(
2
1
4)]([
1 T
T
k
nnn ba
adttf
T
2/
2/1
22202 )(
2
1
4)]([
1 T
T
k
nnn ba
adttf
T
Mean-Square Error
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtna
atf
T
2/
2/1
22202 )(
2
1
4)]([
1 T
Tn
nn baa
dttfT
2/
2/1
22202 )(
2
1
4)]([
1 T
Tn
nn baa
dttfT
