The document discusses Fourier series and their application to periodic functions. It introduces periodic functions and their properties. Fourier series can be used to decompose any periodic function into a sum of sines and cosines. This allows periodic waveforms to be analyzed and approximated as the sum of their harmonic components. Examples are provided to demonstrate how to calculate the Fourier series of simple periodic functions like square waves.

1. Admission in India 2015
By:
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2. Fourier Series
主講者：虞台文
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3. 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
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4. Fourier Series
Periodic Functions
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5. 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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6. Example:
f (t) = cos t + t Find its period.
4
cos
3
cos t + t = t +T + t +T
( ) cos 1
3
f (t) = f (t +T) ( )
4
cos 1
4
cos
3
Fact: cosq = cos(q + 2mp)
T = 2mp
3
T = 2np
4
T = 6mp
T = 8np
T = 24p smallest T
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7. Example:
f t t t 1 2 ( ) = cosw + cosw Find its period.
f (t) = f (t +T) cos cos cos ( ) cos ( ) 1 2 1 2 w t + w t = w t +T + w t +T
w T = 2mp 1
w T = 2np 2
= m
n
w
1
w
2
w must be a
1
w
2
rational number
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8. Example:
f (t) = cos10t + cos(10 + p)t
Is this function a periodic one?
10
1 not a rational
+ p
=
w
w
10
2
number
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9. Fourier Series
Fourier Series
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10. Introduction
 Decompose a periodic input signal into
primitive periodic components.
AA p peerriiooddiicc s seeqquueennccee
t
T 2T 3T
f(t)
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11. Synthesis
b nt
= +å p +å p ¥
T
f t a a nt
T
n
n
n
n
=
¥
=
cos 2 sin 2
2
( )
1 1
0
DC Part Even Part Odd Part
T is a period of all the above signals
f t a0 a n t b n t
n = +å w +å w ¥
( ) 0
cos( ) sin( )
2
1
0
1
n
n
n
=
¥
=
Let w0=2p/T.
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12. Orthogonal Functions
Call a set of functions {fk
} orthogonal
on an interval a < t < b if it satisfies
î í ì
m ¹
n
ò f t f t dt
= r m =
n
n
b
a m n
0
( ) ( )
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13. Orthogonal set of Sinusoidal
Functions
Define w0=2p/T.
m t dt m T
T
cos( ) 0, 0 / 2
/ 2 0 ò w = ¹ -
m t dt m T
T
sin( ) 0, 0 / 2
/ 2 0 ò w = ¹ -
î í ì
m ¹
n
0
ò w w = - T m =
n
T
/ 2
cos( m t ) cos( n t ) dt T / 2 0 0
/ 2
î í ì
m ¹
n
0
ò w w = - T m =
n
T
/ 2
sin( m t )sin( n t ) dt T / 2 0 0
/ 2
We now prove this one
m t n t dt m n T
T
sin( ) cos( ) 0, for all and / 2
/ 2 0 0 ò w w = -
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14. Proof
cosacosb = 1 a +b + a -b
òT
/ 2
cos( m w t ) cos( n w t )
dt -
T / 2 0 0 cos[( ) ] 1
T
T ò ò- -
1
= + w + - w / 2
/ 2
/ 2 0 cos[( ) ]
1 T
sin[( ) ] 1
2sin[( ) ] 1
= m n
0
[cos( ) cos( )]
2
m n t dt m n t dt T
T
/ 2 0
2
2
/ 2
0 / 2
0
/ 2
0 / 2
0
sin[( ) ]
1
( )
2
1
( )
2
T
T
T m n t
m n
m n t
+ w +
- w
m + n w
- - w
- =
m ¹ n
2sin[( ) ]
1
( )
2
1
( )
1
2
0 0
- p
+ w
+ p +
+ w
m n
m n
m n
0 0 =admission.edhole.com

15. Proof
cosacosb = 1 a +b + a -b
cos2 a = 1 + a
òT
/ 2
cos( m w t ) cos( n w t )
dt -
T / 2 0 0 1
1 T
t
- -
0
[cos( ) cos( )]
2
m t dt T
T ò-
= w / 2
cos2 ( )
/ 2 0
/ 2
/ 2
0
0
/ 2
/ 2
sin 2 ]
4
1
2
T
T
T
m t
m
w
w
= +
m = n
= T
2
[1 cos 2 ]
2
m t dt T
T ò-
= + w / 2
/ 2 0 [1 cos 2 ]
2
î í ì
m ¹
n
0
ò w w = - T m =
n
T
/ 2
cos( m t ) cos( n t ) dt T / 2
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16. Orthogonal set of Sinusoidal
Functions
Define w0=2p/T.
1,
ì
m t dt m T
T
cos( ) 0, 0 / 2
/ 2 0 ò w = ¹ -
ü
m t dt m T
T
sin( ) 0, 0 / 2
/ 2 0 ò w = ¹ -
t t t
w w w
cos ,cos 2 ,cos3 ,
î í ì
m ¹
n
0
ò w w = - T m n
0 0 0
t t =
t
T
/ 2
cos( m t ) cos( n t ) dt T / 2 0 0
/ 2
î í ì
m ¹
n
0
ò w w = - T m =
n
T
/ 2
sin( m t )sin( n t ) dt T / 2 0 0
/ 2
m t n t dt m n T
T
sin( ) cos( ) 0, for all and / 2
/ 2 0 0 ò w w = -
ïþ
ïý
ïî
ïí
w w w


sin ,sin 2 ,sin 3 ,
0 0 0
aann oorrtthhooggoonnaall sseett..
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17. Decomposition
f t a0 a n t b n t
n = +å w +å w ¥
( ) 0
2 ( )
f t dt
a t T
t ò + = 0
T
0
0
( ) cos 1,2, 2
a t T
n t
= ò 0
+ f t n w tdt n
= T
0
0
( )sin 1,2, 2
b t T
n t
= ò 0
+ f t n w tdt n
= T
0
0
cos( ) sin( )
2
1
0
1
n
n
n
=
¥
=
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18. Proof
Use the following facts:
m t dt m T
T
cos( ) 0, 0 / 2
/ 2 0 ò w = ¹ -
m t dt m T
T
sin( ) 0, 0 / 2
/ 2 0 ò w = ¹ -
î í ì
m ¹
n
0
ò w w = - T m =
n
T
/ 2
cos( m t ) cos( n t ) dt T / 2 0 0
/ 2
î í ì
m ¹
n
0
ò w w = - T m =
n
T
/ 2
sin( m t )sin( n t ) dt T / 2 0 0
/ 2
m t n t dt m n T
T
sin( ) cos( ) 0, for all and / 2
/ 2 0 0 ò w w = admi-ssion.edhole.com

19. Example (Square Wave)
f(t)
1
-6p -5p -4p -3p -2p -p p 2p 3p 4p 5p
= òp a dt
0 0 =
1 1
2
2
p
sin 0 1,2, cos 1
2
= p p ò nt n
2
0 0
= =
p
=
p
n
a ntdt n
=
n n
2 / 1,3,5,
p p
= ò = - = - - = 
0 2,4,6,
sin 1 cos 1 (cos 1)
2
2
0 0 î í ì
=

n
n
n
nt
n
b ntdt n
p
p
p p p
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20. sin 3 1
3
ö çè
( ) 1
Example (Square Wave)
-6p -5p -4p -3p -2p -p p 2p 3p 4p 5p
= òp a dt
0 0 =
1 1
2
2
p
sin 0 1,2, cos 1
2
= p p ò nt n
2
0 0
= =
p
=
p
n
a ntdt n
n n
p =
2 / 1,3,5,
= p p ò 
0 2,4,6,
sin 1 cos 1 (cos 1)
1
2
0 0 î í ì
=
p - =
p
= -
p
= -
p

n
n
n
nt
n
b ntdt n
f(t)
1
÷ø
æ + + +
p
f t = + t t sin 5t 
5
2 sin 1
2
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21. sin 3 1
3
ö çè
( ) 1
Example (Square Wave)
-6p -5p -4p -3p -2p -p p 2p 3p 4p 5p
= òp a dt
0 0 =
1 1
2
2
p
sin 0 1,2, cos 1
2
1.5
1
= p p ò nt n
2
0 0
= =
p
=
p
n
a ntdt n
n n
p =
2 / 1,3,5,
0.5
0
= p p ò 
0 2,4,6,
sin 1 cos 1 (cos 1)
1
2
0 0 î í ì
=
p - =
p
= -
p
= -
p

n
n
n
nt
n
b ntdt n
f(t)
1
-0.5
÷ø
æ + + +
p
f t = + t t sin 5t 
5
2 sin 1
2
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22. Harmonics
b nt
= +å p +å p ¥
T
f t a a nt
T
n
n
n
n
=
¥
=
cos 2 sin 2
2
( )
1 1
0
f t a0 a n t b n t
n = +å w +å w ¥
( ) 0
cos( ) sin( )
DC Part Even Part Odd Part
T is a period of all the above signals
2
1
0
1
n
n
n
=
¥
=
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23. Harmonics
w = 2pf = 2p Define 0 0 , called the fundamental angular frequency.
f t a a n t b n t
( ) = +å w +å w ¥
0 cos sin
2
n
n
n
n 0
1
0
1
=
¥
=
T
0 w = nw Define n , called the n-th harmonic of the periodic function.
f t a a t b t n
= +å w n n +å ¥
w n
n
n
=
¥
=
cos sin
2
( )
1 1
0
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24. Harmonics
f t a a t b t n
= +å w n n +å ¥
w n
n
n
=
¥
=
cos sin
2
( )
1 1
0
a0 a t b t
n w + w + = å¥
( cos sin )
2 1
n n n
n
=
ö
æ
t b
a a b a
= ÷ ÷
å¥
ø
ç ç
0 2 2 cos sin
2 n
n n t
è
w
n
+
w +
n
+
= + +
1
2 2 2 2
n
n n
n
n n
a b
a b
n n n n n na a b t t
( ) å¥
=
0 2 2 cos cos sin sin
2 n
= + + q w + q w
1
n nt C C q - w + = å¥
cos( )
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0 n
1
n

25. Amplitudes and Phase Angles
n nt C C t f q - w + = å¥ =
( ) cos( )
0 n
1
n
harmonic amplitude phase angle
0
C = a
2
0
2 2
n n n C = a + b
ö
÷ ÷ø
æ
tan 1 b
ç çè
q = -
n
n
n a
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26. Fourier Series
Complex Form of the
Fourier Series
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27. Complex Exponentials
e jn t n t j n t
0 0 w0 = cos w + sin w
e jn t n t j n t
0 0 - w0 = cos w - sin w
n t (e jn 0t e jn 0t )
cos 1 0
w = w + - w
2
(e jn t e jn t ) j (e jn t e jn t )
j
sin 1 0
w = w - - w = - w - - w
n t 0 0 0 0
2 2
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28. Complex Form of the Fourier Series
f t a a n t b n t
( ) = +å w +å w ¥
0 cos sin
2
n
n
n
n 0
1
0
1
=
¥
=
( ) ( jn t jn t )
n a a e e j b e e 0 0 0 0
= + å + - å -
n
n
jn t jn t
n
1 1
0
1
2 2
2
w - w
¥
=
w - w
¥
=
n n a a jb e a jb e
å¥
( ) 1
2
0 0 ( ) 0
=
w - w
ù
úû
êëé + + - + =
1
2
1
2 n
jn t
n n
jn t
[ - w
] -
å¥
=
= + w +
1
0
0 0
n
jn t
n
jn t
n c c e c e
c a
0
2
1
=
c = a -
jb
( )
2
n n n
c = 1
a +
jb
( )
2
0
n n n
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29. Complex Form of the Fourier Series
[ - w
å¥
] =
( ) 0 0
-
= + w +
1
0
n
jn t
n
jn t
n f t c c e c e
å å-
=-¥
w
¥
= + w +
=
1
1
0
0 0
n
jn t
n
n
jn t
n c c e c e
å¥
= w
=-¥
n
jn t
nc e 0
c a
0
2
1
=
c = a -
jb
( )
2
n n n
c = 1
a +
jb
( )
2
0
n n n
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30. Complex Form of the Fourier Series
= = 1 / 2
( )
0 ò-
/ 2
0
2
T
T
f t dt
T
c a
1
n n n c = a - jb
( )
2
úû ù
/ 2
/ 2 0 1 ( ) cos T ( )sin
T
êë é
/ 2
/ 2 0
T
T
= ò w - ò w - -
f t n tdt j f t n tdt
T
ò-
/ 2 0 0 1 T ( )(cos sin )
T
= w - w / 2
f t n t j n t dt
T
ò-
1 ( ) 0 T
= - w / 2
/ 2
T
f t e jn tdt
T
( ) 1 ( ) 0
2
c a
0
2
1
=
c = a -
jb
( )
2
n n n
c = 1
a +
jb
c a jb ( )
ò-
w
- = + = / 2
/ 2
1 T
T
jn t
n n n f t e dt
T
2
0
n n n
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31. Complex Form of the Fourier Series
å¥
= w
=-¥
n
jn t
nf (t) c e 0
1 ( ) 0
= - w / 2
f t e dt
c T
T
T
jn t
n ò-
/ 2
c a
0
2
1
=
c = a -
jb
( )
2
n n n
c = 1
a +
jb
( )
2
0
n n n
-
If f(t) is real,
*
n n c = c -
n j n
=| | f , = * =| |
n n n
j
c c e c c c e- f
n n -
| | | | 1 n n n n c = c = a + b -
2 2
2
ö
c = 1 a admission.edhole.com
÷ ÷ø
æ
tan 1 b
ç çè
f = - -
n
n
n a
n = ±1,±2,±3,
0 0 2

32. Complex Frequency Spectra
n j n
=| | f , = * =| |
n n n
j
c c e c c c e- f
n n -
| | | | 1 n n n n c = c = a + b -
2 2
2
ö
tan 1 b n = ±1,±2,±3,
÷ ÷ø
æ
ç çè
f = - -
n
n
n a
c = 1 a
0 0 2
|cn|
amplitude
spectrum
w
fn
phase
spectrum
w
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33. Example
-T - T T
T
2
2
d
2
t
f(t)
A
- d
2
= - w / 2
e dt
c A d
T
d
jn t
n ò-
/ 2
0
/ 2
0 1 d
0 d
/ 2
e jn t
A
T jn
-
- w
- w
=
ö
÷ ÷ø
æ
ç çè
0 0 1 jn d 1 e jn d
A
= - w w / 2
- w
-
- w
0
/ 2
0
jn
e
T jn
A - w
- w
1 ( 2 sin / 2)
0
0
j n d
T jn
=
A w
1 sin / 2
n d
T 1 n
w
0
2 0
=
n d
ö çè
n d
ö çè
÷ø
æ p
÷ø
æ p
=
T
T
Ad
T
sin
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34. n d
ö çè
n d
ö çè
÷ø
æ p
÷ø
æ p
c =
Ad n
T
T
T
sin
A/5
d T d
, 1
20
= = =
2 8
1
5
T
,
4
1
0 w = p = p
T
Example
-120p -80p -40p 0 40p 80p 120p
5w0 10w0 15w0 -5w0 -10w0 -15w0
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35. n d
ö çè
n d
ö çè
÷ø
æ p
÷ø
æ p
c =
Ad n
T
T
T
sin
A/10
d T d
, 1
20
= = =
2 4
1
5
T
,
2
1
0 w = p = p
T
Example
-120p -80p -40p 0 40p 80p 120p
10w0 20w0 30w0 -10w0 -20w0 -30w0
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36. Example
c A d jn t
n ò = - w
-T d T
e dt
T
0
0
d
0 1 - w
- w
e jn t
A
T jn
0 0
=
ö
÷ ÷ø
æ
A jn d
ç çè
1 1 0
jn
- w
-
= - w
- w
e
0 0
T jn
A - - w
1 (1 ) 0
0
e jn d
T jn
w
=
A - w w - - w
0 / 2
sin
ö çè
Ad - w
e jn d
n d
n d
ö T
çè
T
T
÷ø
æ p
÷ø
æ p
=
t
f(t)
A
0
1 / 2 ( / 2 / 2 )
0
e jn 0d e jn 0d e jn 0d
T jn
w
=
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37. Fourier Series
Impulse Train
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38. Dirac Delta Function
î í ì
¹
t
0 0
t and ò d( ) =1 ¥
¥ =
d =
0
( )
t
-¥
t dt
0 t
Also called unit impulse function.
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39. Property
ò ¥
d( t )f( t ) dt
= f(0) -¥
f(t): Test Function
ò ¥
d( t )f( t ) dt = ò ¥
d( t )f(0) dt = f(0)ò ¥
d( t ) dt
= f(0) -¥
-¥
-¥
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40. Impulse Train
-3T -2T -T 0 T 2T 3T t
å¥
T (t) (t nT)
d = d -
=-¥
n
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41. Fourier Series of the Impulse Train
å¥
T (t) (t nT)
d = d -
=-¥
n
2 / 2 ( ) 2
T
= ò d t dt
= 0 -
/ 2 a T
T
T T
2 / 2 ( ) cos( ) 2
/ 2 0 = ò d w = -
2 / 2 ( )sin( ) 0
T
t n t dt
a T
n T T
T
b T
n T T
= ò d t n w t dt
= -
/ 2 0 T
t 0 ( ) 1 2 cos
å¥
T n t
d = + w
=-¥
n
T T
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42. Complex Form
Fourier Series of the Impulse Train
1 ( ) 1
T
= = ò d t dt
= 0 -
c a T
T
T T
2
/ 2
/ 2
0
1 / 2 ( ) 1
T
= ò d t e 0 dt
= -
c T
T
T
jn t
n T
/ 2
- w
å¥
T (t) (t nT)
d = d -
=-¥
n
t 0 ( ) 1
å¥
d = w
=-¥
n
jn t
T e
T
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43. Fourier Series
Analysis of
Periodic Waveforms
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44. Waveform Symmetry
Even Functions
f (t) = f (-t)
Odd Functions
f (t) = - f (-t)
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45. Decomposition
Any function f(t) can be expressed as the
sum of an even function fe(t) and an odd
function fo(t).
f (t) f (t) f (t) e o = +
( ) [ ( ) ( )] 2
f t 1 f t f t e = + -
( ) [ ( ) ( )] 2
f t 1 f t f t o = - -
Even Part
Odd Part
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46. Example
î í ì
>
<
=
-
0 0
0
( )
e t
t
f t
t
Even Part
Odd Part
î í ì <
>
=
-
0
0
( )
1
2
1
e t
2
e t
f t t
t
e
î í ì
>
t
1
e t
- <
=
-
0
0
( )
1
2
2
e t
f t o
t
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47. Half-Wave Symmetry
f (t) = f (t +T) and f (t) = - f (t +T / 2)
-T/2 T/2 T
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48. Quarter-Wave Symmetry
Even Quarter-Wave Symmetry
-T/2 T/2 T
Odd Quarter-Wave Symmetry
T
-T/2 T/2
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49. Hidden Symmetry
 The following is a asymmetry periodic function:
A
-T T
 Adding a constant to get symmetry property.
A/2
-T T
-A/2
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50. 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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51. Fourier Coefficients of Even Functions
f (t) = f (-t)
f ( t a å¥
) = + a n w t
0 cos
2
n
n 0
=
1
0 0 4 T ( ) cos( )
= ò w / 2
n f t n t dt
T
a
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52. Fourier Coefficients of Even Functions
f (t) = - f (-t)
sin ) ( w =å¥
f t b n t
n
n 0
=
1
= ò / 2
w 0 0 4 T ( )sin( )
n f t n t dt
T
b
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53. Fourier Coefficients for Half-Wave Symmetry
f (t) = f (t +T) and f (t) = - f (t +T / 2)
-T/2 T/2 T
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54. Fourier Coefficients for Half-Wave Symmetry
f (t) = f (t +T) and f (t) = - f (t +T / 2)
0 0 å¥
=
f t = a n w t + b n w
t
n n ( ) ( cos sin )
1
n
n
0 for even
ïî
ïí ì
= w 4 ò ( ) cos( ) for odd
a T
n
/ 2
0 0 f t n t dt n
T
n
0 for even
ïî
ïí ì
= w 4 ò ( )sin( ) for odd
b T
n admission.edhole.com
/ 2
0 0 f t n t dt n
T

55. Fourier Coefficients for
Even Quarter-Wave Symmetry
-T/2 T/2 T
n w - =å¥
2 1 f t a n t
( ) cos[(2 1) ] 0
1
n
=
-
8 T / 4
( ) cos[(2 1) ]
2 1 0 0 = ò - w -
n f t n t dt
T
a
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56. Fourier Coefficients for
Odd Quarter-Wave Symmetry
n w - =å¥
2 1 f t b n t
( ) sin[(2 1) ] 0
1
n
=
-
8 T / 4
( )sin[(2 1) ]
2 1 0 0 = ò - w -
n f t n t dt
T
b
T
-T/2 T/2
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57. Example
Even Quarter-Wave Symmetry
1
-T -T/4 T/4
2 1 0 0 8 T ( ) cos[(2 1) ]
a = n -
ò / 4
f t n - w t dt
= ò / 4
- w T
0 0 8 T cos[(2 1) ]n t dt
T
/ 4
8 T
0
0
0
sin[(2 1) ]
(2 1)
n t
n T
- w
- w
=
( 1) 1 4
- p
= - -
n
(2 1)
n
T
-T/2 T/2
-1
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58. cos3 1
3
ö çè
Example
Even Quarter-Wave Symmetry
1
-T -T/4 T/4
2 1 0 0 8 T ( ) cos[(2 1) ]
a = n -
ò / 4
f t n - w t dt
= ò / 4
- w T
0 0 8 T cos[(2 1) ]n t dt
T
/ 4
8 T
0
0
0
sin[(2 1) ]
(2 1)
n t
n T
- w
- w
=
( 1) 1 4
- p
= - -
n
(2 1)
n
T
-T/2 T/2
-1
÷ø
æ w - w + w +
p
f t = t t t  0 0 0 cos5
5
( ) 4 cos 1
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59. Example
T
1
Odd Quarter-Wave Symmetry
-T/2 T/2
-T -T/4 T/4
-1
2 1 0 0 8 T ( )sin[(2 1) ]
b = n -
ò / 4
f t n - w t dt
= ò / 4
- w T
0 0 8 T sin[(2 1) ]n t dt
T
/ 4
8 T
0
0
0
cos[(2 1) ]
= -
(2 1)
n t
n T
- w
- w
4
n
=
(2 1)
- p
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60. sin 3 1
3
ö çè
Example
T
1
Odd Quarter-Wave Symmetry
-T/2 T/2
-T -T/4 T/4
-1
2 1 0 0 8 T ( )sin[(2 1) ]
b = n -
ò / 4
f t n - w t dt
= ò / 4
- w T
0 0 8 T sin[(2 1) ]n t dt
T
/ 4
8 T
0
0
0
cos[(2 1) ]
= -
(2 1)
n t
n T
- w
- w
4
n
=
(2 1)
- p
÷ø
æ w + w + w +
p
f t = t t t  0 0 0 sin 5
5
( ) 4 sin 1
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61. Fourier Series
Half-Range
Expansions
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62. Non-Periodic Function Representation
t
 A non-periodic function f(t) defined over (0, t)
can be expanded into a Fourier series which is
defined only in the interval (0, t).
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63. Without Considering Symmetry
t T
 A non-periodic function f(t) defined over (0, t)
can be expanded into a Fourier series which is
defined only in the interval (0, t).
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64. Expansion Into Even Symmetry
t T=2t
 A non-periodic function f(t) defined over (0, t)
can be expanded into a Fourier series which is
defined only in the interval (0, t).
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65. Expansion Into Odd Symmetry
t
T=2t
 A non-periodic function f(t) defined over (0, t)
can be expanded into a Fourier series which is
defined only in the interval (0, t).
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66. Expansion Into Half-Wave Symmetry
t T=2t
 A non-periodic function f(t) defined over (0, t)
can be expanded into a Fourier series which is
defined only in the interval (0, t).
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67. Expansion Into
Even Quarter-Wave Symmetry
t
T/2=2t
T=4t
 A non-periodic function f(t) defined over (0, t)
can be expanded into a Fourier series which is
defined only in the interval (0, t).
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68. Expansion Into
Odd Quarter-Wave Symmetry
t
T/2=2t T=4t
 A non-periodic function f(t) defined over (0, t)
can be expanded into a Fourier series which is
defined only in the interval (0, t).
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69. Fourier Series
Least Mean-Square
Error Approximation
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70. Approximation a function
k
S t a a n t b n t
k n n Use = 0 + ( cos w + sin
w
) å=
2
n
1
0 0
( )
to represent f(t) on interval -T/2 < t < T/2.
Define (t) f (t) S (t) k k e = -
1 T [ ( )]2
E = / 2
e Mean-Square
ò-
t dt
k T / 2
k T
Error
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71. Approximation a function
Show that using Sk(t) to represent f(t) has
least mean-square property.
1 T [ ( )]2
= e / 2
ò-
t dt
k T / 2
k T
E
n n f t a a n t b n t dt
= é - - w + w / 2
ò å( ) -
0 cos sin
2
=
ù
úû
êë
/ 2
2
1
0 0
1 T ( )
T
k
n
T
Proven by setting ¶Ek/¶ai = 0 and ¶Ek/¶bi = 0. admission.edhole.com

72. Approximation a function
1 T [ ( )]2
= e / 2
ò-
t dt
k T / 2
k T
E
n n f t a a n t b n t dt
ò å( ) -
0 cos sin
2
=
ù
úû
êë= é - - w + w / 2
/ 2
2
1
0 0
1 T ( )
T
k
n
T
1 ( ) 0
E 2 / 2 ( ) cos 0
¶ ò-
2
/ 2
/ 2
0
0
= - =
¶
T
T
k f t dt
T
a
a
E
¶ ò-
k f t n tdt
/ 2 0 = - w =
¶
T
n T
n
T
a
a
E
¶ ò-
2 / 2 ( )sin 0
k f t n tdt
/ 2 0 = - w =
¶
T
n T
n
T
b
b
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73. Mean-Square Error
1 T [ ( )]2
= e / 2
ò-
t dt
k T / 2
k T
E
n n f t a a n t b n t dt
ò å( ) -
0 cos sin
2
T
=
ò å -
ù
úû
êë= é - - w + w / 2
/ 2
2
1
0 0
1 T ( )
T
k
n
= / 2
f t dt - a - 1
a + b
k n n =
/ 2
1
2 2
2
2 0 ( )
2
4
1 T [ ( )]
T
k
n
T
E
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74. Mean-Square Error
1 T [ ( )]2
= e / 2
ò-
t dt
k T / 2
k T
E
n n f t a a n t b n t dt
ò å( ) -
0 cos sin
2
T
=
ò å -
ù
úû
êë= é - - w + w / 2
/ 2
2
1
0 0
1 T ( )
T
k
n
f t dt a 1
a b
n n ³ + + / 2
/ 2
=
1
2 2
2
2 0 ( )
2
4
1 T [ ( )]
T
k
n
T
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75. Mean-Square Error
1 T [ ( )]2
= e / 2
ò-
t dt
k T / 2
k T
E
n n f t a a n t b n t dt
ò å( ) -
0 cos sin
2
T
=
ò å -
ù
úû
êë= é - - w + w / 2
/ 2
2
1
0 0
1 T ( )
T
k
n
f t dt a 1
¥
a b
n n = + + / 2
/ 2
=
1
2 2
2
2 0 ( )
2
4
1 T [ ( )]
T
n
T