Meiling chensignals & systems1 Lecture #04 Fourier representation for continuous-time signals.
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meiling chen signals & systems 1 Lecture #04 Fourier representation for continuous-time signals
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Transcript of Meiling chensignals & systems1 Lecture #04 Fourier representation for continuous-time signals.
meiling chen signals & systems 1
Lecture #04
Fourier representation
for continuous-time signals
meiling chen signals & systems 2
Fourier representations
• Fourier Series (FS) : for periodic signals
• Fourier-Transform (FT) : for nonperiodic signals
• Discrete-time Fourier series (DTFS): for discrete-time periodic signals
• Discrete-time Fourier transform : for discrete-time nonperiodic signals
meiling chen signals & systems 3
Orthogonal function：
A set of function
Is called orthogonal in the interval ),( 21 tt
if
niti ,,2,1,0,)(
jik
jidttt
ij
t
t
i ,
,0)()( *
2
1
where is the complex conjugate of )(* tj )(tj
),( 21 ttif 1ik then )(ti in is orthonormal
Continuous-time signals
meiling chen signals & systems 4
0
)()(i
ii tCtf For any function
We choose a orthogonal function set to be the basis
2
1
2
1
2
1
2
1
2
1
2
1
2
1
)()(1
)()(
)()(
)()()()()()()()(
)()()()(
)()(
*
*
*
**11
*00
*
*
0
*
0
t
t
jj
t
t
jj
t
t
j
j
t
t
jjj
t
t
j
t
t
j
t
t
j
ji
iij
iii
dtttfk
tt
dtttf
C
ttCttCttCdtttf
ttCttf
tCtf
Euler-Fourier formula
The question is how to find Ci
meiling chen signals & systems 5
Generalized Fourier series ：
Fourier series of function f(t)2
1
)()(1 *t
t
ii
i dtttfk
C
0
)()(i
ii tCtf
2
1
)(1t
t
tjnn dtetfT
C
0
0)(n
tjnneCtf
meiling chen signals & systems 6
example of orthogonal function :
,2,1,0, ne tjn o in the interval ),(0
200
tt
proof
0]12)sin(2)[cos()(
]1[)()(
,
1,
)(
0
)(
2)(
0
)(
0
)()(
20)(
)(
*
00
00
02
0
0
002
0
0
0
0
02
0
0
02
0
0
0
02
0
0
0
02
0
0
0
0
02
0
0
00
02
0
0
0
mnjmnmn
e
emn
e
mn
edtemn
dtdtedtemn
dte
dteedtee
tmnj
mnjtmnj
t
t
tmnjt
t
tmnj
t
t
t
t
tj
t
t
tmnj
t
t
tmnj
tjm
t
t
tjntjm
t
t
tjn
meiling chen signals & systems 7
For any function f(t) in the interval ),( 00 Ttt
n
tjnn TletTttteCtf
0
0 200 ,,)(
)()2sin2(cos
)(
)()(
)(1
)(1
00
002
0
0
0
0
0
02
0
0
0
0
2)(2
2
tfeCnjneC
eeCeCtf
Ttftfif
dtetfT
dtetfC
tjnn
tjnn
jntjnn
tjn
n
Tt
t
tjn
t
t
tjnn
meiling chen signals & systems 8
If f(t) is real function
]sin)((cos)[(
)]sin(cos)sin(cos[
)(
,,2,1,0,1,2,,)(
)(1
001
0
00001
0
10
*
00
0
0
0
0
tnCCjtnCCC
tnjtnCtnjtnCC
eCeCC
neCtf
CdtetfT
C
nnn
nn
nn
n
tjnn
n
tjnn
tjnn
n
Tt
t
tjn
n
nnn
nnn
jC
jC
letnnn
nnn
CC
CC
2
2let
nnnn
nnnn
CCb
CCa
2
2
meiling chen signals & systems 9
Tt
t
Tt
t
nmnnnn
Tt
t
nennnn
dttfT
Ca
tdtntfT
CICCb
tdtntfT
CRCCa
0
0
0
0
0
0
)(2
2
sin)(2
][2)(2
cos)(2
][22
00
0
0
,3,2,1,sin)(2
,2,1,0,cos)(2
]sincos[2
1)(
0
0
0
0
0
0
1000
ntdtntfT
b
ntdtntfT
a
tnbtnaatf
Tt
t
n
Tt
t
n
nnn
Fourier series：
meiling chen signals & systems 10
A periodic signal satisfying he following conditions can be extended into an infinite sum of sine and cosine functions.
1.The single-valued function f(t) is bounded, and hence absolutely integrable over the finite period T; that is
2.The function has a finite number of maxima and minima over the period T.
3. The function has a finite number of discountinuity points over the period T.
dttfT
T
2
2
)(
meiling chen signals & systems 11MIT signals & systems
meiling chen signals & systems 12
Example:
2E
2E 2
T T
]sin2sin2sin2[
][sin][sin][sin
coscoscos
cos2
2cos)
2(
2cos
2
2
cos)(2
043
0400
00
00
000
00
0
0
00
0
0
0
0
43
43
4
4
43
43
4
4
43
43
4
4
TnnnTn
E
tnTn
Etn
Tn
Etn
Tn
E
tdtnT
Etdtn
T
Etdtn
T
E
tdtnE
Ttdtn
E
Ttdtn
E
T
tdtntfT
a
TT
T
T
T
T
n
T
T
T
T
T
T
T
T
T
T
T
T
meiling chen signals & systems 13
t0cos
t03cos
]7cos7
15cos
5
13cos
3
1[cos
2)( 0000 tttt
Etf
]2sin22
3sin2
2
1sin2[
2
]sin2sin2sin2[
]sin2sin2sin2[
2432
42
2
043
0400
nnnn
E
TnnnTn
E
TnnnTn
Ea
TT
TT
TT
TTn
meiling chen signals & systems 14
0 02 03 04 05
E2
32E
52E
0 02 03 04 05
nC
n
o0
Frequency spectrum
meiling chen signals & systems 15
Fourier transform
n
tjnn TletTttteCtf
0
0 200 ,,)(
dtetfT
CTt
t
tjnn
0
0
0)(1
f(t) is not periodic function if T∞
00 ,2
ndT
dedtetf
edtetfd
eCtf
tjtj
tjtjtjnn
])([2
1
])(2[)( 0
)( jF
meiling chen signals & systems 16
dejFjFFtf
dtetftfFjF
tj
tj
)(2
1)]([)(
)()]([)(
1
Fourier transform of f(t)
Inverse Fourier transform
Comparing with Laplace transform
jr
jr
st
st
dsesFj
sFLtf
dtetftfLsF
)(2
1)]([)(
)()]([)(
1
0
js
meiling chen signals & systems 17
The properties of Fourier transform
(i) Linearity
(ii) Reversal
(iii) Scaling in time
)()()()(
)()(),()(
2121
2211
jFjFtftf
jFtfjFtf
),()( jFtf
)(1
)(a
jF
aatf
meiling chen signals & systems 18
(iv) Delay
(v) Frequency shifting modulation
(vi) Frequency differentiation
)()( 00 jFettf tj
)]([)( 00 jFetf tj
)()()(,)(
)(
jF
d
djtft
d
jdFjttf
n
nnn
(vii) Convolution
)()()()( 2121 jFjFtftf
meiling chen signals & systems 19
dejFjF
ddefejF
ddeejFf
ddejFftftf
dejFtf
jFjFtftf
tj
jtj
jtj
tj
tj
)()(2
1
})({)(2
1
)()(2
1
)(2
1)()()(
)(2
1)(
)()()()(
12
12
21
)(2121
)(22
2121
meiling chen signals & systems 20
(viii) multiplication
(ix) Derivative
(x) Integration
)]()([2
1)()( 2121
jFjFtftf
)()()( jFj
dt
tfd nn
n
)()0()(1
)(
jFjFj
dft
)(1
])([0
sFs
dfLt
meiling chen signals & systems
example
0,)(
1
)()(
)(
1
)(
0,0
0,0,)(
0
0)(
0
)(
aja
ja
e
ja
e
eja
dtedteeteF
t
taete
tt
tja
tjatjat
at
0,1
)]([
aja
tueF at
)()( jFtf
t
meiling chen signals & systems 22
2
4
24
meiling chen signals & systems 23
example
22
0
0
2
)(
1
)(
1
)()(
0,
0,)(
a
a
jaja
dteedtee
dtetfjF
te
tete
tjattjat
tj
at
at
meiling chen signals & systems 24
example
2sin2
2
)(2
)(1
)(1
)(
,0
,1)(
22
2222
2
2
2
2
j
ee
eej
eej
dtejF
t
ttf
jj
jjjjtj
meiling chen signals & systems 25
Cardinal sine function
x
xxc
sin)(sin
meiling chen signals & systems 26
Parseval’s theorem (時域頻域能量守恒 )
dttftfdttfE
)()()(2
dttftfdttfE
)()()(2If f(t) is real function
djF
djFjF
djFdtetf
dtdejFtf
tj
tj
2)(
2
1
)()(2
1
)(])([2
1
])(2
1[)(
djFdttf22)(
2
1)(
meiling chen signals & systems 27
Example
1)()}({
1)(
)()(
0
jtj edtettF
dtt
ttf
meiling chen signals & systems 28
)(tf
)( jF
meiling chen signals & systems 29
Example: Fourier series
2T
2T
TT
)(tf
)(tf
)(tf
dtetfT
C
eCtf
T
T
tjnn
n
tjnn
2
2
0
0
)(1
)(
n
tjnn
n
tjnn
n
tjnn
eCjntf
eCjntf
eCtf
ttt
At
t
Att
t
Atf
0
0
0
20
0
111
11
)()(
)(
)(
)()(2
)()(
1t1t
A
meiling chen signals & systems 30
]2cos2[
]2[
]2[
])()(2)([
)]()(2
)([1
)(1
)(
)()(
)()(2
)()(
101
1
0)(
1
111
111
11
20
20
111
11
1010
10010
2
2
000
2
2
0
2
2
0
0
tnTt
A
eeTt
A
eeeTt
A
dtettetettTt
A
dtettt
At
t
Att
t
A
T
dtetfT
Cn
eCntf
ttt
At
t
Att
t
Atf
tjn
tjn
T
T
T
T
T
T
tjn
jntjn
tjntjntjn
tjn
tjnn
n
tjnn
]1[cos)(
210
12
0
tnTtn
ACn
meiling chen signals & systems 31
2
)(tf
)(tf
)(tf
2
Example : Fourier transform A
meiling chen signals & systems 32
]cos22cos2[][
])2()()()2([
])2()()()2([
)]2()()()2([
)()()(
)()(2
1)(
)()(2
1)(
)(2
1)(
)2()()()2()(
2)()2(
2
2
Aeeee
A
etetetetA
dtetetetetA
dtetA
tA
tA
tA
dtetfjFj
dejFjtf
dejFjtf
dejFtf
tA
tA
tA
tA
tf
jnjnjnjn
tjntjntjntjn
tjntjntjntjn
tjn
tjn
tj
tj
tj
