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TRANSCRIPT
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Fourier Series
For more ppt’s, visit www.studentsaarthi.com
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ContentPeriodic FunctionsFourier SeriesComplex Form of the Fourier Series Impulse TrainAnalysis of Periodic WaveformsHalf-Range ExpansionLeast Mean-Square Error Approximation
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Fourier Series
Periodic Functions
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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.
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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
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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
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Example:
tttf )10cos(10cos)(
Is this function a periodic one?
10
10
2
1 not a rational number
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Fourier Series
Fourier Series
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Introduction
Decompose a periodic input signal into primitive periodic components.
A periodic sequenceA periodic sequence
T 2T 3T
t
f(t)
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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.
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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)()(
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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
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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
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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
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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.
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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
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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
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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
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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)(
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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)(
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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
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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
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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
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Amplitudes and Phase Angles
)cos()(1
0 nn
nn tCCtf
20
0
aC
22nnn baC
n
nn a
b1tan
harmonic amplitude phase angle
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Fourier Series
Complex Form of the Fourier Series
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Complex Exponentials
tnjtne tjn00 sincos0
tjntjn eetn 00
2
1cos 0
tnjtne tjn00 sincos0
tjntjntjntjn eej
eej
tn 0000
22
1sin 0
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Fourier Series
Impulse Train
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Dirac Delta Function
0
00)(
t
tt and 1)(
dtt
0 t
Also called unit impulse function.
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Property
)0()()(
dttt )0()()(
dttt
)0()()0()0()()()(
dttdttdttt
(t): Test Function
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Impulse Train
0 tT 2T 3TT2T3T
n
T nTtt )()(
n
T nTtt )()(
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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
)(
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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 )()(
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Fourier Series
Analysis ofPeriodic Waveforms
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Waveform Symmetry
Even Functions
Odd Functions
)()( tftf
)()( tftf
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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
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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
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Half-Wave Symmetry
)()( Ttftf and 2/)( Ttftf
TT/2T/2
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Quarter-Wave Symmetry
Even Quarter-Wave Symmetry
TT/2T/2
Odd Quarter-Wave Symmetry
T
T/2T/2
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Hidden Symmetry
The following is a asymmetry periodic function:
Adding a constant to get symmetry property.
A
TT
A/2
A/2
TT
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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
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Fourier Coefficients of Even Functions
)()( tftf
tnaa
tfn
n 01
0 cos2
)(
2/
0 0 )cos()(4 T
n dttntfT
a
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Fourier Coefficients of Even Functions
)()( tftf
tnbtfn
n 01
sin)(
2/
0 0 )sin()(4 T
n dttntfT
b
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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.
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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
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Fourier Coefficients forEven Quarter-Wave Symmetry
TT/2T/2
])12cos[()( 01
12 tnatfn
n
4/
0 012 ])12cos[()(8 T
n dttntfT
a
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Fourier Coefficients forOdd Quarter-Wave Symmetry
])12sin[()( 01
12 tnbtfn
n
4/
0 012 ])12sin[()(8 T
n dttntfT
b
T
T/2T/2
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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
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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)(
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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
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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)(
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Fourier Series
Half-Range Expansions
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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, ).
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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
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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
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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
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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
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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
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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
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Fourier Series
Least Mean-Square Error Approximation
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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
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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.
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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
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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
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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
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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