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Continuous-Time Fourier Transform
Content
Introduction
Fourier Integral
Fourier Transform
Properties of Fourier Transform
Convolution
Parseval’s Theorem
Continuous-Time Fourier Transform
Introduction
The Topic
Fourier
Series
Discrete
Fourier
Transform
Continuous
Fourier
Transform
Fourier
Transform
Continuous
Time
Discrete
Time
Per
iodic
Aper
iod
ic
Review of Fourier Series
Deal with continuous-time periodic signals.
Discrete frequency spectra.
A Periodic Signal
T 2T 3T
t
f(t)
Two Forms for Fourier Series
T
ntb
T
nta
atf
n
n
n
n
2sin
2cos
2)(
11
0Sinusoidal
Form
Complex
Form:
n
tjn
nectf 0)( dtetfT
cT
T
tjn
n
2/
2/
0)(1
dttfT
aT
T2/
2/0 )(
2tdtntf
Ta
T
Tn 0
2/
2/cos)(
2
tdtntfT
bT
Tn 0
2/
2/sin)(
2
How to Deal with Aperiodic Signal?
A Periodic Signal
T
t
f(t)
If T, what happens?
Continuous-Time Fourier Transform
Fourier Integral
tjn
n
nT ectf 0)(
Fourier Integral
n
tjnT
T
jn
T edefT
002/
2/)(
1
dtetfT
cT
T
tjn
Tn
2/
2/
0)(1
T
20
2
1 0
T
n
tjnT
T
jn
T edef 00
0
2/
2/)(
2
1
LetT
20
0 dT
n
tjnT
T
jn
T edef 002/
2/)(
2
1
dedef tjj
T )(2
1
dedeftf tjj)(2
1)(
Fourier Integral
F(j)
dtetfjF tj
)()(
dejFtf tj)(2
1)( Synthesis
Analysis
Fourier Series vs. Fourier Integral
n
tjn
nectf 0)(Fourier
Series:
Fourier
Integral:
dtetfT
cT
T
tjn
Tn
2/
2/
0)(1
dtetfjF tj
)()(
dejFtf tj)(2
1)(
Period Function
Discrete Spectra
Non-Period
Function
Continuous Spectra
Continuous-Time Fourier Transform
Fourier Transform
Fourier Transform Pair
dtetfjF tj
)()(
dejFtf tj)(2
1)( Synthesis
Analysis
Fourier Transform:
Inverse Fourier Transform:
Existence of the Fourier Transform
dttf |)(|
Sufficient Condition:
f(t) is absolutely integrable, i.e.,
dtetfjF tj
)()(
Continuous Spectra
)()()( jjFjFjF IR
)(|)(| jejF FR(j)
FI(j)
()
MagnitudePhase
Example
1-1
1
t
f(t)
dtetfjF tj
)()( dte tj
1
1
1
1
1
tje
j
)(
jj eej 2sin
2sin 2c f
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
4arg
[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
Continuous-Time Fourier Transform
Properties of
Fourier Transform
Notation
)()( jFtf F
)()]([ jFtfF
)()]([1 tfjF- F
Linearity
)()()()( 22112211 jFajFatfatfa F
Time Scaling
ajF
aatf
||
1)( F
Time Reversal
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)( 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
Pf)dteetfetf tjtjtj
00 )(])([F
dtetftj
)( 0)(
)( 0 jF
Symmetry Property
)(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.
FR(j) = FR(j) FI(j) = FI(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
0
tjtjeetfttf
])([2
1])([
2
1]cos)([ 00
0
tjtjetfetfttf
FFF
)]([2
1)]([
2
100 jFjF
Example:
d/2d/2
1
t
wd(t)
d/2d/2t
f(t)=wd(t)cos0t
/2
/2
2 1( ) [ ( )] sin sin sin
2
dj t
d dd
dW j w t e dt fd d c fd
f
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
tj
dd 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
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
Pf)dtetftf tj
)(')]('[F
dtetfjetf tjtj )()(
)()(' jFjtf F
)( jFj
Fourier Transform of f (n)(t)
0)(lim and )(
tfjFtft
F
)()()()( jFjtf nn F
Fourier Transform of f (n)(t)
0)(lim and )(
tfjFtft
F
)()()()( jFjtf nn F
Fourier Transform of Integral
00)( and )(
FdttfjFtf F
jFj
dxxft 1
)(F
Let dxxftt
)()( 0)(lim
t
t
)()()]([)]('[ jjjFtft FF
)(1
)(
jFj
j
The Derivative of Fourier Transform
d
jdFtjtf FF )]([
Pf)
dtetfjF tj
)()(
dtetfd
d
d
jdF tj
)(
)(dtetf tj
)(
dtetjtf tj
)]([ )]([ tjtfF
Continuous-Time Fourier Transform
Convolution
Basic Concept
Linear 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 Invariant
System
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
Causal
System
fi(t) fo(t)
A causal system satisfies
fi(t) = 0 for t < t0 fo(t) = 0 for t < t0
Basic Concept
Causal
System
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?
Unit Impulse Response
LTI
System
(t) h(t)=L[(t)]
f(t) L[f(t)]=?
Facts: )()()()()( tfdtfdtf
dtfLtfL )()()]([
dtLf )]([)(
dthf )()( Convolution
Unit Impulse Response
LTI
System
(t) h(t)=L[(t)]
f(t) L[f(t)]=?
Facts: )()()()()( tfdtfdtf
dtfLtfL )()()]([
dtLf )]([)(
dthf )()( Convolution
)(*)()]([ thtftfL
Unit Impulse Response
Impulse Response
LTI 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
dtfftftf )()()(*)( 2121
dtff )()( 21
)(])([)( 21
tdttftf
t
t
dftf )()( 21
dftf )()( 21
)(*)( 12 tftf
Impulse Response
LTI System
h(t)
f(t) f(t)*h(t)
Properties of Convolution
)(*)()(*)( 1221 tftftftf
Impulse Response
LTI System
f(t)
h(t) h(t)*f(t)
Properties of Convolution
)](*)([*)()(*)](*)([ 321321 tftftftftftf
Properties of Convolution
)](*)([*)()(*)](*)([ 321321 tftftftftftf
h1(t) h2(t) h3(t)
h2(t) h3(t) h1(t)
The following two
systems are identical
Properties of Convolution
)()(*)( tfttf (t)f(t) f(t)
dtfttf )()()(*)(
dtf )()(
)(tf
Properties of Convolution
)()(*)( tfttf (t)f(t) f(t)
)()(*)( TtfTttf
dTtfTttf )()()(*)(
dTtf )()(
)( Ttf
Properties of Convolution
f(t) f(t T)
)()(*)( TtfTttf
0 T
(tT)
tf (t)
0t
f (t)
0 T
Properties of Convolution
f(t) f(t T)
)()(*)( TtfTttf
0 T
(tT)
tf (t)
0t
f (t)
0 T
System function (tT) serves as an
ideal delay or a copier.
Properties of Convolution
)()()(*)( 2121 jFjFtftf F
dtedtfftftfF tj
)()()](*)([ 2121
ddtetff tj)()( 21
dejFf j)()( 21
defjF j)()( 12)()( 21 jFjF
Properties of Convolution
)()()(*)( 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
Time Domain Frequency Domain
convolution multiplication
Impulse Response
LTI System
h(t)f(t) f(t)*h(t)
Impulse Response
LTI System
H(j)F(j) F(j)H(j)
Properties of Convolution
)()()(*)( 2121 jFjFtftf F
Time Domain Frequency Domain
convolution multiplication
F(j)
F(j)H1(j)
H2(j)H1(j) H3(j)
F(j)H1(j)H2(j)
F(j)H1(j)H2(j)H3(j)
Properties of Convolution
)()()(*)( 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
An Ideal High-Pass Filter
0
Fi(j)
0
H(j)
pp
1
0
Fo(j)
Properties of Convolution
)(*)(2
1)()( 2121
jFjFtftf F
djFjFtftf )]([)(
2
1)()( 2121
F
Properties of Convolution
)(*)(2
1)()( 2121
jFjFtftf F
djFjFtftf )]([)(
2
1)()( 2121
F
Time Domain Frequency Domain
multiplication convolution
Continuous-Time Fourier Transform
Parseval’s Theorem
Properties of Convolution
djFjFdttftf ][)(
2
1)]()([ 2121
djFjFtftf )]([)(
2
1)()( 2121
F
djFjFdtetftf tj )]([)(
2
1)]()([ 2121
djFjFdttftf )]([)(
2
1)]()([ 2121=0
Properties of Convolution
djFjFdttftf ][)(
2
1)]()([ 2121
If f1(t) and f2(t) are real functions,
djFjFdttftf ][)(
2
1)]()([ *
2121
f2(t) real ][][ *
22 jFjF
Parseval’s Theorem:Energy Preserving
djFdttf 22 |)(|
2
1|)(|
dtetftf tj
)(*)](*[F
djFjF )]([*)(
2
1
dttftfdttf
)(*)(|)(| 2
*
)(
dtetf tj )(* jF
djF 2|)(|
2
1
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