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Continuous-Time Fourier Transform
主講者：虞台文
ContentIntroductionFourier IntegralFourier TransformProperties of Fourier TransformConvolutionParseval’s Theorem
Continuous-Time Fourier Transform
Introduction
The Topic
FourierSeries
FourierSeries
DiscreteFourier
Transform
DiscreteFourier
Transform
ContinuousFourier
Transform
ContinuousFourier
Transform
FourierTransform
FourierTransform
ContinuousTime
DiscreteTime
Per
iodi
cA
peri
odic
Review of Fourier Series
Deal with continuous-time periodic signals. Discrete frequency spectra.
A Periodic SignalA Periodic Signal
T 2T 3T
t
f(t)
Two Forms for Fourier Series
T
ntb
T
nta
atf
nn
nn
2sin
2cos
2)(
11
0
T
ntb
T
nta
atf
nn
nn
2sin
2cos
2)(
11
0SinusoidalForm
ComplexForm:
n
tjnnectf 0)(
n
tjnnectf 0)( dtetf
Tc
T
T
tjnn
2/
2/
0)(1
dttfT
aT
T2/
2/0 )(2
tdtntfT
aT
Tn 0
2/
2/cos)(
2
tdtntfT
bT
Tn 0
2/
2/sin)(
2
How to Deal with Aperiodic Signal?
A Periodic SignalA Periodic Signal
T
t
f(t)
If T, what happens?
Continuous-Time Fourier Transform
Fourier Integral
tjn
nnT ectf 0)(
Fourier Integral
n
tjnT
T
jnT edef
T00
2/
2/)(
1
dtetfT
cT
T
tjnTn
2/
2/
0)(1
T
20
2
1 0
T
n
tjnT
T
jnT edef 00
0
2/
2/)(
2
1
LetT
20
0 dT
n
tjnT
T
jnT edef 00
2/
2/)(
2
1
dedef tjjT )(
2
1
dedeftf tjj)(2
1)(
Fourier Integral
F(j)
dtetfjF tj
)()(
dejFtf tj)(2
1)( Synthesis
Analysis
Fourier Series vs. Fourier Integral
n
tjnnectf 0)(
n
tjnnectf 0)(
FourierSeries:
FourierIntegral:
dtetfT
cT
T
tjnTn
2/
2/
0)(1 dtetf
Tc
T
T
tjnTn
2/
2/
0)(1
dtetfjF tj
)()( dtetfjF tj
)()(
dejFtf tj)(2
1)(
dejFtf tj)(2
1)(
Period Function
Discrete Spectra
Non-PeriodFunction
Continuous Spectra
Continuous-Time Fourier Transform
Fourier Transform
Fourier Transform Pair
dtetfjF tj
)()( dtetfjF tj
)()(
dejFtf tj)(2
1)(
dejFtf tj)(2
1)( Synthesis
Analysis
Fourier Transform:
Inverse Fourier Transform:
Existence of the Fourier Transform
dttf |)(|
dttf |)(|
Sufficient Condition:
f(t) is absolutely integrable, i.e.,
dtetfjF tj
)()(
Continuous Spectra
)()()( jjFjFjF IR
)(|)(| jejF FR(j)
FI(j)
|F(j)|
()
MagnitudePhase
Example
1-1
1
t
f(t)
dtetfjF tj
)()( dte tj
1
1
1
1
1
tje
j
)(
jj eej
sin2
Example
1-1
1
t
f(t)
dtetfjF tj
)()( dte tj
1
1
1
1
1
tje
j
)(
jj eej
sin2
-10 -5 0 5 10-1
0
1
2
3
F(
)-10 -5 0 5 10
0
1
2
3
|F(
)|
-10 -5 0 5 100
2
4ar
g[F
()]
-10 -5 0 5 10-1
0
1
2
3
F(
)-10 -5 0 5 10
0
1
2
3
|F(
)|
-10 -5 0 5 100
2
4ar
g[F
()]
Example
dtetfjF tj
)()( dtee tjt
0
t
f(t)
et
dte tj
0
)(
j
1
Example
dtetfjF tj
)()( dtee tjt
0
t
f(t)
et
dte tj
0
)(
j
1
-10 -5 0 5 100
0.5
1
|F(j
)|
-10 -5 0 5 10-2
0
2
arg[
F(j
)]
=2
-10 -5 0 5 100
0.5
1
|F(j
)|
-10 -5 0 5 10-2
0
2
arg[
F(j
)]
=2
Continuous-Time Fourier Transform
Properties of
Fourier Transform
Notation
)()( jFtf F )()( jFtf F
)()]([ jFtfF )()]([ jFtfF
)()]([1 tfjF- F )()]([1 tfjF- F
Linearity
)()()()( 22112211 jFajFatfatfa F )()()()( 22112211 jFajFatfatfa F
Proved by yourselves
Proved by yourselves
Time Scaling
a
jFa
atf||
1)( F
a
jFa
atf||
1)( F
Proved by yourselves
Proved by yourselves
Time Reversal
jFtf F)( jFtf F)(
Pf) dtetftf tj
)()]([F dtetft
t
tj
)(
)()( tdetft
t
tj
)()( tdetft
t
tj
dtetft
t
tj
)( dtetft
t
tj
)(
dtetf tj
)( )( jF
Time Shifting
0)( 0tjejFttf F 0)( 0
tjejFttf F
Pf) dtettfttf tj
)()]([ 00F dtettft
t
tj
)( 0
)()( 0)(0
0
0 ttdetftt
tt
ttj
dtetfet
t
tjtj
)(0
dtetfe tjtj
)(0 tjejF 0)(
Frequency Shifting (Modulation)
)()( 00 jFetf j F )()( 0
0 jFetf j F
Pf)dteetfetf tjtjtj
00 )(])([F
dtetf tj
)( 0)(
)( 0 jF
Symmetry Property
)(2)]([ fjtFF )(2)]([ fjtFF
Pf)
dejFtf tj)()(2
dejFtf tj)()(2
dtejtFf tj
)()(2
Interchange symbols and t
)]([ jtFF
Fourier Transform for Real Functions
If f(t) is a real function, and F(j) = FR(j) + jFI(j)
F(j) = F*(j)
dtetfjF tj
)()(
dtetfjF tj
)()(* )( jF
Fourier Transform for Real FunctionsFourier Transform for Real Functions
If f(t) is a real function, and F(j) = FR(j) + jFI(j)
F(j) = F*(j)
FR(j) is even, and FI(j) is odd.
F R( j) = F R(j) F I( j) = F I(j)
Magnitude spectrum |F(j)| is even, and phase spectrum () is odd.
Fourier Transform for Real FunctionsFourier Transform for Real Functions
If f(t) is real and even
F(j) is real
If f(t) is real and odd
F(j) is pure imaginary
Pf))()( tftf
Pf)Even
)()( jFjF
)(*)( jFjFReal
)(*)( jFjF
)()( tftf Odd
)()( jFjF
)(*)( jFjFReal
)(*)( jFjF
Example:
)()]([ jFtfF ?]cos)([ 0 ttfF
Sol)
))((2
1cos)( 00
0tjtj eetfttf
])([2
1])([
2
1]cos)([ 00
0tjtj etfetfttf FFF
)]([2
1)]([
2
100 jFjF
Example:
d/2d/2
1
t
wd(t)
d/2d/2t
f(t)=wd(t)cos0t
2sin
2)]([)(
2/
2/
ddtetwjW
d
d
tjdd F
]cos)([)( 0ttwjF d F0
0
0
0 )(2
sin)(2
sin
dd
Example:
d/2d/2
1
t
wd(t)
d/2d/2t
f(t)=wd(t)cos0t
2sin
2)]([)(
2/
2/
ddtetwjW
d
d
tjdd F
]cos)([)( 0ttwjF d F0
0
0
0 )(2
sin)(2
sin
dd
-60 -40 -20 0 20 40 60-0.5
0
0.5
1
1.5
F(j
)
d=2
0=5
-60 -40 -20 0 20 40 60-0.5
0
0.5
1
1.5
F(j
)
d=2
0=5
Example:
t
attf
sin)( ?)( jF
Sol)
d/2d/2
1
t
wd(t)
2sin
2)(
djWd
)(22
sin2
)]([
dd w
td
tjtW FF
)(sin
)]([ 2
aw
t
attf FF
||1
||0
a
a
Fourier Transform of f’(t)
0)(lim and )(
tfjFtft
F 0)(lim and )(
tfjFtft
F
Pf) dtetftf tj
)(')]('[F
dtetfjetf tjtj )()(
)()(' jFjtf F )()(' jFjtf F
)( jFj
Fourier Transform of f (n)(t)
0)(lim and )(
tfjFtft
F 0)(lim and )(
tfjFtft
F
)()()()( jFjtf nn F )()()()( jFjtf nn F
Proved by yourselves
Proved by yourselves
Fourier Transform of f (n)(t)
0)(lim and )(
tfjFtft
F 0)(lim and )(
tfjFtft
F
)()()()( jFjtf nn F )()()()( jFjtf nn F
Proved by yourselves
Proved by yourselves
Fourier Transform of Integral
00)( and )(
FdttfjFtf F 00)( and )(
FdttfjFtf F
jFj
dxxft 1
)(F
jFj
dxxft 1
)(F
Let dxxftt
)()( 0)(lim
t
t
)()()]([)]('[ jjjFtft FF
)(1
)(
jFj
j
The Derivative of Fourier Transform
d
jdFtjtf FF )]([
d
jdFtjtf FF )]([
Pf)dtetfjF tj
)()(
dtetfd
d
d
jdF tj
)()(
dtetf tj
)(
dtetjtf tj
)]([ )]([ tjtfF
Continuous-Time Fourier Transform
Convolution
Basic Concept
Linear SystemLinear Systemfi(t) fo(t)=L[fi(t)]
fi(t)=a1 fi1(t) + a2 fi2(t) fo(t)=L[a1 fi1(t) + a2 fi2(t)]
A linear system satisfies fo(t) = a1L[fi1(t)] + a2L[fi2(t)]
= a1fo1(t) + a2fo2(t)
Basic Concept
Time InvariantSystem
Time InvariantSystem
fi(t) fo(t)
fi(t +t0) fo(t + t0)fi(t t0) fo(t t0)
tfi(t)
tfo(t)
tfi(t+t0)
tfo(t+t0)
tfi(tt0)
tfo(tt0)
Basic Concept
CausalSystemCausalSystem
fi(t) fo(t)
A causal system satisfies
fi(t) = 0 for t < t0 fo(t) = 0 for t < t0
Basic Concept
CausalSystemCausalSystem
fi(t) fo(t)
tfi(t)
tfo(t)
t0t0
t0
tfi(t)
tfo(t)
t0fi(t)
t tfo(t)
t0t0
Which of the following systems are causal?
Which of the following systems are causal?
Unit Impulse Response
LTISystem
LTISystem
(t) h(t)=L[(t)]
f(t) L[f(t)]=?
Facts: )()()()()( tfdtfdtf
dtfLtfL )()()]([
dtLf )]([)(
dthf )()( Convolution
Unit Impulse Response
LTISystem
LTISystem
(t) h(t)=L[(t)]
f(t) L[f(t)]=?
Facts: )()()()()( tfdtfdtf
dtfLtfL )()()]([
dtLf )]([)(
dthf )()( Convolution
)(*)()]([ thtftfL )(*)()]([ thtftfL
Unit Impulse Response
Impulse ResponseLTI System
h(t)
Impulse ResponseLTI System
h(t)f(t) f(t)*h(t)
Convolution Definition
dtfftf )()()( 21
The convolution of two functions f1(t) and f2(t) is defined as:
)(*)( 21 tftf
Properties of Convolution
)(*)()(*)( 1221 tftftftf )(*)()(*)( 1221 tftftftf
dtfftftf )()()(*)( 2121
dtff )()( 21
)(])([)( 21
tdttftf
t
t
dftf )()( 21
dftf )()( 21 )(*)( 12 tftf
Impulse ResponseLTI System
h(t)
Impulse ResponseLTI System
h(t)
f(t) f(t)*h(t)
Properties of Convolution
)(*)()(*)( 1221 tftftftf )(*)()(*)( 1221 tftftftf
Impulse ResponseLTI System
f(t)
Impulse ResponseLTI System
f(t)
h(t) h(t)*f(t)
Properties of Convolution
)](*)([*)()(*)](*)([ 321321 tftftftftftf )](*)([*)()(*)](*)([ 321321 tftftftftftf
Prove by yourselves
Prove by yourselves
Properties of Convolution
)](*)([*)()(*)](*)([ 321321 tftftftftftf )](*)([*)()(*)](*)([ 321321 tftftftftftf
h1(t)h1(t) h2(t)h2(t) h3(t)h3(t)
h2(t)h2(t) h3(t)h3(t) h1(t)h1(t)
The following two systems are identical
The following two systems are identical
Properties of Convolution
)()(*)( tfttf )()(*)( tfttf (t)(t)f(t) f(t)
dtfttf )()()(*)(
dtf )()(
)(tf
Properties of Convolution
)()(*)( tfttf )()(*)( tfttf (t)(t)f(t) f(t)
)()(*)( TtfTttf )()(*)( TtfTttf
dTtfTttf )()()(*)(
dTtf )()(
)( Ttf
Properties of Convolution
f(t) f(t T)
)()(*)( TtfTttf )()(*)( TtfTttf
0 T
(tT)
tf (t)
0t
f (t)
0 T
Properties of Convolution
f(t) f(t T)
)()(*)( TtfTttf )()(*)( TtfTttf
0 T
(tT)
tf (t)
0t
f (t)
0 T
System function (tT) serves as an ideal delay or a copier.System function (tT) serves as an ideal delay or a copier.
Properties of Convolution
)()()(*)( 2121 jFjFtftf F )()()(*)( 2121 jFjFtftf F
dtedtfftftfF tj
)()()](*)([ 2121
ddtetff tj)()( 21
dejFf j)()( 21
defjF j)()( 12 )()( 21 jFjF
Properties of Convolution
)()()(*)( 2121 jFjFtftf F )()()(*)( 2121 jFjFtftf F
dtedtfftftfF tj
)()()](*)([ 2121
ddtetff tj)()( 21
dejFf j)()( 21
defjF j)()( 12 )()( 21 jFjF
Time Domain Frequency Domain
convolution multiplication
Properties of Convolution
)()()(*)( 2121 jFjFtftf F )()()(*)( 2121 jFjFtftf F
Time Domain Frequency Domain
convolution multiplication
Impulse ResponseLTI System
h(t)
Impulse ResponseLTI System
h(t)f(t) f(t)*h(t)
Impulse ResponseLTI System
H(j)
Impulse ResponseLTI System
H(j)F(j) F(j)H(j)
Properties of Convolution
)()()(*)( 2121 jFjFtftf F )()()(*)( 2121 jFjFtftf F
Time Domain Frequency Domain
convolution multiplication
F(j)
F(j)H1(j)
H2(j)H2(j)H1(j)H1(j) H3(j)H3(j)
F(j)H1(j)H2(j)
F(j)H1(j)H2(j)H3(j)
Properties of Convolution
)()()(*)( 2121 jFjFtftf F )()()(*)( 2121 jFjFtftf F
An Ideal Low-Pass Filter
0
Fi(j)
0
Fo(j)
0
H(j)
pp
1
Properties of Convolution
)()()(*)( 2121 jFjFtftf F )()()(*)( 2121 jFjFtftf F
An Ideal High-Pass Filter
0
Fi(j)
0
H(j)
pp
1
0
Fo(j)
Properties of Convolution
)(*)(2
1)()( 2121
jFjFtftf F )(*)(
2
1)()( 2121
jFjFtftf F
djFjFtftf )]([)(
2
1)()( 2121
F
djFjFtftf )]([)(
2
1)()( 2121
F
Prove by yourselvesProve by yourselves
Properties of Convolution
)(*)(2
1)()( 2121
jFjFtftf F )(*)(
2
1)()( 2121
jFjFtftf F
djFjFtftf )]([)(
2
1)()( 2121
F
djFjFtftf )]([)(
2
1)()( 2121
F
Prove by yourselvesProve by yourselves
Time Domain Frequency Domain
multiplication convolution
Continuous-Time Fourier Transform
Parseval’s Theorem
Properties of Convolution
djFjFdttftf ][)(
2
1)]()([ 2121
djFjFdttftf ][)(
2
1)]()([ 2121
djFjFtftf )]([)(
2
1)()( 2121
F
djFjFtftf )]([)(
2
1)()( 2121
F
djFjFdtetftf tj )]([)(
2
1)]()([ 2121
djFjFdttftf )]([)(
2
1)]()([ 2121=0=0
Properties of Convolution
djFjFdttftf ][)(
2
1)]()([ 2121
djFjFdttftf ][)(
2
1)]()([ 2121
If f1(t) and f2(t) are real functions,
djFjFdttftf ][)(
2
1)]()([ *
2121
djFjFdttftf ][)(
2
1)]()([ *
2121
f2(t) real ][][ *22 jFjF
Parseval’s Theorem:Energy Preserving
djFdttf 22 |)(|
2
1|)(|
djFdttf 22 |)(|
2
1|)(|
dtetftf tj
)(*)](*[F
djFjF )]([*)(
2
1
dttftfdttf
)(*)(|)(| 2
*
)(
dtetf tj )(* jF
djF 2|)(|
2
1