fourier series
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Fourier Series. 主講者：虞台文. 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. Fourier Series. Periodic Functions. The Mathematic Formulation. - PowerPoint PPT PresentationTRANSCRIPT
Fourier Series
主講者：虞台文
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 )(41cos)(
31cos
4cos
3cos TtTttt
Fact: )2cos(cos m
mT 23
nT 24
mT 6
nT 824T smallest T
Example:
tttf 21 coscos)( Find its period.
)()( Ttftf )(cos)(coscoscos 2121 TtTttt
mT 21
nT 22
nm
2
1
2
1
must be a
rational number
Example:tttf )10cos(10cos)(
Is this function a periodic one?
1010
2
1 not a rational number
Fourier Series
Fourier Series
Introduction Decompose a periodic input signal into
primitive periodic components.
A periodic sequence
T 2T 3T
t
f(t)
Synthesis
Tntb
Tntaatf
nn
nn
2sin2cos2
)(11
0 DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 01
01
0 tnbtnaatfn
nn
n
Let 0=2/T.
Orthogonal FunctionsCall a set of functions {k} orthogonal
on an interval a < t < b if it satisfies
nmrnm
dtttn
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
nmTnm
dttntmT
T 2/0
)cos()cos(2/
2/ 00
nmTnm
dttntmT
T 2/0
)sin()sin(2/
2/ 00
nmdttntmT
T and allfor ,0)cos()sin(
2/
2/ 00
We now prove this one
Proofdttntm
T
T 2/
2/ 00 )cos()cos(
0
)]cos()[cos(21coscos
dttnmdttnmT
T
T
T
2/
2/ 0
2/
2/ 0 ])cos[(21])cos[(
21
2/
2/00
2/
2/00
])sin[()(
121])sin[(
)(1
21 T
T
T
Ttnm
nmtnm
nm
m n
])sin[(2)(
121])sin[(2
)(1
21
00
nmnm
nmnm
0
0
Proofdttntm
T
T 2/
2/ 00 )cos()cos(
0
)]cos()[cos(21coscos
dttmT
T 2/
2/ 02 )(cos
2/
2/0
0
2/
2/
]2sin4
121
T
T
T
T
tmm
t
m = n
2T
]2cos1[21cos2
dttmT
T 2/
2/ 0 ]2cos1[21
nmTnm
dttntmT
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
nmTnm
dttntmT
T 2/0
)cos()cos(2/
2/ 00
nmTnm
dttntmT
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
tttttt
an orthogonal set.
Decomposition
dttfT
aTt
t
0
0
)(20
,2,1 cos)(20
0
0
ntdtntfT
aTt
tn
,2,1 sin)(20
0
0
ntdtntfT
bTt
tn
)sin()cos(2
)( 01
01
0 tnbtnaatfn
nn
n
ProofUse the following facts:
0 ,0)cos(2/
2/ 0 mdttmT
T0 ,0)sin(
2/
2/ 0 mdttmT
T
nmTnm
dttntmT
T 2/0
)cos()cos(2/
2/ 00
nmTnm
dttntmT
T 2/0
)sin()sin(2/
2/ 00
nmdttntmT
T and allfor ,0)cos()sin(
2/
2/ 00
Example (Square Wave)
1122
00
dta
,2,1 0sin1cos22
00
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(1 cos1sin
22
00
nnn
nn
ntn
ntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
1122
00
dta
,2,1 0sin1cos22
00
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(1 cos1sin
21
00
nnn
nn
ntn
ntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
Example (Square Wave)
ttttf 5sin
513sin
31sin2
21)(
1122
00
dta
,2,1 0sin1cos22
00
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(1 cos1sin
21
00
nnn
nn
ntn
ntdtbn
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
513sin
31sin2
21)(
Harmonics
Tntb
Tntaatf
nn
nn
2sin2cos2
)(11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 01
01
0 tnbtnaatfn
nn
n
Harmonics
tnbtnaatfn
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.
tbtaatf nn
nnn
n
sincos2
)(11
0
Harmonicstbtaatf n
nnn
nn
sincos2
)(11
0
)sincos(2 1
0 tbtaannn
nn
12222
220 sincos2 n
n
nn
nn
nn
nnn t
ba
btba
abaa
1
220 sinsincoscos2 n
nnnnnn ttbaa
)cos(1
0 nn
nn tCC
Amplitudes and Phase Angles
)cos()(1
0 nn
nn tCCtf
20
0aC
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
21cos 0
tnjtne tjn00 sincos0
tjntjntjntjn eejeej
tn 0000
221sin 0
Complex Form of the Fourier Series
tnbtnaatfn
nn
n 01
01
0 sincos2
)(
tjntjn
nn
tjntjn
nn eebjeeaa
0000
11
0
221
2
1
0 00 )(21)(
21
2 n
tjnnn
tjnnn ejbaejbaa
1
000
n
tjnn
tjnn ececc
)(21
)(212
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
)(21
)(212
00
nnn
nnn
jbac
jbac
ac
Complex Form of the Fourier Series
2/
2/
00 )(1
2T
Tdttf
Tac
)(21
nnn 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)(
21 T
T
tjnnnn dtetf
Tjbac )(
21
)(212
00
nnn
nnn
jbac
jbac
ac
Complex Form of the Fourier Series
n
tjnnectf 0)(
dtetfT
cT
T
tjnn
2/
2/0)(1
)(21
)(212
00
nnn
nnn
jbac
jbac
ac
If f(t) is real,*nn cc
nn jnnn
jnn ecccecc
|| ,|| *
22
21|||| nnnn bacc
n
nn a
b1tan
,3,2,1 n
00 21 ac
Complex Frequency Spectrann j
nnnj
nn ecccecc
|| ,|| *
22
21|||| nnnn bacc
n
nn a
b1tan ,3,2,1 n
00 21 ac
|cn|
amplitudespectrum
n
phasespectrum
Example
2T
2T
TT2d
t
f(t)A
2d
dteTAc
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(10
0
dnjjnT
A
2/sin10
021
dnnT
A
TdnT
dn
TAd
sin
TdnT
dn
TAdcn
sin
82
51
T ,
41 ,
201
0
T
dTd
Example
40 80 120-40 0-120 -80
A/5
50 100 150-50-100-150
TdnT
dn
TAdcn
sin
42
51
T ,
21 ,
201
0
T
dTd
Example
40 80 120-40 0-120 -80
A/10
100 200 300-100-200-300
Example
dteTAc
d tjnn
00
dtjne
jnTA
00
01
00
110
jne
jnTA djn
)1(10
0
djnejnT
A
2/0
sindjne
TdnT
dn
TAd
TT d
t
f(t)A
0
)(1 2/2/2/
0
000 djndjndjn eeejnT
A
Fourier Series
Impulse Train
Dirac Delta Function
000
)(tt
t and 1)(
dtt
0 t
Also called unit impulse function.
Property)0()()(
dttt
)0()()0()0()()()(
dttdttdttt
(t): Test Function
Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()(
Fourier Series of the Impulse Train
n
T nTtt )()(T
dttT
aT
T T2)(2 2/
2/0
Tdttnt
Ta
T
T Tn2)cos()(2 2/
2/ 0 0)sin()(2 2/
2/ 0 dttntT
bT
T Tn
n
T tnTT
t 0cos21)(
Complex FormFourier Series of the Impulse Train
Tdtt
Tac
T
T T1)(1
22/
2/0
0
Tdtet
Tc
T
T
tjnTn
1)(1 2/
2/0
n
tjnT e
Tt 0
1)(
n
T nTtt )()(
Fourier Series
Analysis ofPeriodic Waveforms
Waveform SymmetryEven Functions
Odd Functions
)()( tftf
)()( tftf
DecompositionAny 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
000
)(tte
tft
Even Part
Odd Part
00
)(21
21
tete
tf t
t
e
00
)(21
21
tete
tf t
t
o
Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
TT/2T/2
Quarter-Wave SymmetryEven Quarter-Wave Symmetry
TT/2T/2
Odd Quarter-Wave Symmetry
TT/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
tnaatfn
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.
Fourier Coefficients for Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
)sincos()(1
00
n
nn tnbtnatf
odd for )cos()(4even for 0
2/
0 0 ndttntfT
na T
n
odd for )sin()(4even for 0
2/
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
TT/2T/2
ExampleEven Quarter-Wave Symmetry
4/
0 012 ])12cos[()(8 T
n dttntfT
a 4/
0 0 ])12cos[(8 Tdttn
T4/
00
0
])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 Tdttn
T4/
00
0
])12sin[()12(
8T
tnTn
)12(4)1( 1
nn
TT/2T/2
1
1T T/4T/4
ttttf 000 5cos
513cos
31cos4)(
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 Tdttn
T4/
00
0
])12cos[()12(8
T
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 Tdttn
T4/
00
0
])12cos[()12(8
T
tnTn
)12(4
n
ttttf 000 5sin
513sin
31sin4)(
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 tnbtnaatS
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 dttnbtnaatf
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 dttnbtnaatf
T
0)(12
2/
2/0
0
T
Tk dttf
Ta
aE 0cos)(2 2/
2/ 0
T
Tnn
k tdtntfT
aaE
0sin)(2 2/
2/ 0
T
Tnn
k tdtntfT
bbE
Mean-Square Error
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtnaatf
T
2/
2/1
22202 )(
21
4)]([1 T
T
k
nnnk baadttf
TE
Mean-Square Error
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtnaatf
T
2/
2/1
22202 )(
21
4)]([1 T
T
k
nnn baadttf
T
Mean-Square Error
2/
2/
2)]([1 T
T kk dttT
E
2/
2/
2
100
0 sincos2
)(1 T
T
k
nnn dttnbtnaatf
T
2/
2/1
22202 )(
21
4)]([1 T
Tn
nn baadttfT