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Transcript of Fourier Series Content
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
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
Introduction
Decompose a periodic input signal into
primitive periodic components.
A periodic sequence
T 2T 3T
t
f(t)
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.
Orthogonal Functions
Call a set of functions {k} orthogonal on an interval a < t < b if it satisfies
nmr
nmdttt
n
b
anm
0)()(
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
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/
2/0
0
2/
2/0
0
])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
0
0
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
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.
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
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
Example (Square Wave)
112
2
00
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
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)(
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)(
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
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
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
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
Complex Exponentials
tnjtnetjn
00 sincos0
tjntjneetn 00
2
1cos 0
tnjtnetjn
00 sincos0
tjntjntjntjnee
jee
jtn 0000
22
1sin 0
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
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
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
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,
*
nn cc
nn j
nnn
j
nn ecccecc
|| ,|| *
22
2
1|||| nnnn bacc
n
nn
a
b1tan
,3,2,1 n
002
1ac
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
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
T
dn
T
dn
T
Adsin
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
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
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
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
)(
Complex Form Fourier 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 )()(
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
Hidden Symmetry
The following is a asymmetry periodic function:
Adding a constant to get symmetry property.
A
T T
A/2
A/2
T T
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 0
1
0 cos2
)(
2/
00 )cos()(
4 T
n dttntfT
a
Fourier Coefficients for Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
T T/2 T/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()(4
even for 02/
00 ndttntf
T
na T
n
odd for )sin()(4
even for 02/
00 ndttntf
T
nb T
n
Fourier Coefficients for Even Quarter-Wave Symmetry
T T/2 T/2
])12cos[()( 0
1
12 tnatfn
n
4/
0012 ])12cos[()(
8 T
n dttntfT
a
Fourier Coefficients for Odd Quarter-Wave Symmetry
])12sin[()( 0
1
12 tnbtfn
n
4/
0012 ])12sin[()(
8 T
n dttntfT
b
T
T/2 T/2
Example Even 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/2 T/2
1
1
T T/4 T/4
Example Even 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/2 T/2
1
1
T T/4 T/4
ttttf 000 5cos
5
13cos
3
1cos
4)(
Example
T
T/2 T/2
1
1
T T/4 T/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
Example
T
T/2 T/2
1
1
T T/4 T/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)(
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
Approximation a function
Use
k
n
nnk tnbtnaa
tS1
000 sincos
2)(
to represent f(t) on interval T/2 < t < T/2.
Define )()()( tStft kk
2/
2/
2)]([1 T
Tkk dtt
TE
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
Tkk dtt
TE
2/
2/
2
1
000 sincos
2)(
1 T
T
k
n
nn dttnbtnaa
tfT
Proven by setting Ek/ai = 0 and Ek/bi = 0.
Approximation a function
2/
2/
2)]([1 T
Tkk dtt
TE
2/
2/
2
1
000 sincos
2)(
1 T
T
k
n
nn dttnbtnaa
tfT
0)(1
2
2/
2/
0
0
T
T
k dttfT
a
a
E0cos)(
2 2/
2/0
T
Tn
n
k tdtntfT
aa
E
0sin)(2 2/
2/0
T
Tn
n
k tdtntfT
bb
E
Mean-Square Error
2/
2/
2)]([1 T
Tkk dtt
TE
2/
2/
2
1
000 sincos
2)(
1 T
T
k
n
nn dttnbtnaa
tfT
2/
2/1
222
02 )(2
1
4)]([
1 T
T
k
n
nnk baa
dttfT
E
Mean-Square Error
2/
2/
2)]([1 T
Tkk dtt
TE
2/
2/
2
1
000 sincos
2)(
1 T
T
k
n
nn dttnbtnaa
tfT
2/
2/1
222
02 )(2
1
4)]([
1 T
T
k
n
nn baa
dttfT