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Chapter 2. Fourier Representation of Signals and Systems. Overview. Fourier transform Frequency content of a given signal Signals and systems Linear time-invariant system. Concept – Dirac Delta Function. Unit impulse function Unit step function. 0. 1. 2. 3. 0. 1. 2. 3. - PowerPoint PPT Presentation
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Chapter 2. Fourier Representation of Signals
and Systems
Overview
• Fourier transform– Frequency content of a given signal
• Signals and systems– Linear time-invariant system
Concept – Dirac Delta Function
• Unit impulse function
• Unit step function
1. 0
2. 0, 0
3. 1
4.
t t
t dt
t t
0 ,0
0,2
1
0 ,1
)(
t
t
t
tu
tdtttu )()(
Concept – Impulse Response
• The response of the system to a unit impulse – A function of time
Impulse function: t
0 1 2 3 0 1 2 3
Impulseresponseh[t]
Inputx [t]
Outputy[t]
0.70.5
Impulse response: h t
1
0 0, 1 0.7,
2 0.5, 3 0
h h
h h
tt
Concept – Linear Time Invariant System
• A common model for many engineering systems– Linearity
– Time invariance
1 2 1 2 x t x t y t y t
1 0 1 0 x t t y t t
0 1 2 3
0 1 2 3
0 1 2 3
0 1 2 3
0.70.5
1
0.7 0.490.35
Concept – Convolution
• Computes the output for an arbitrary input– LTI system
y t x h t y t x t h t x h t d
Impulseresponseh[t]
Inputx [t]
Outputy[t]
Concept – Euler's formula
*
exp cos sin ,
exp cos sin cos sin
exp
exp exp exp
exp exp exp exp exp
exp exp exp
j
a b a b
e j j
j j j
j
a jb a jb
a jb a jb a jb
a jb a jb
d d d
Concept – Fourier Transform
• A mathematical operation that decomposes a signal into its constituent frequencies
2.1 The Fourier Transform
• Definitions– Fourier transform of the signal g(t) : analysis equation
– Inverse Fourier transform : synthesis equation
• Notations– –
)1.2()2exp()()(
dtftjtgfG
)2.2()2exp()()( dfftjfGtg
]/[2 sradfw )3.2()]([F)( tgfG
)4.2()]([F)( 1 fGtg )5.2()()( fGtg
2.1 The Fourier Transform
• Dirichlet’s conditions1. The function g(t) is single-valued, with a finite number of
maxima and minima in any finite time interval. 2. The function g(t) has a finite number of discontinuities in
any finite time interval.3. The function g(t) is absolutely integrable
– For physical realizability of a signal g(t), the energy of the signal defined by
must satisfy the condition
– Such a signal is referred to as an energy signal. – All energy signals are Fourier transformable.
dttg )(
dttg
2)(
dttg
2)(
2.1 The Fourier Transform
• Continuous Spectrum– A pulse signal g(t) of finite energy is expressed as a
continuous sum of exponential functions with frequencies in the interval -∞ to ∞.
– We may express the function g(t) in terms of the continuous sum infinitesimal components,
– The signal in terms of its time-domain representation by specifying the function g(t) at each instant of time t.
– The signal is uniquely defined by either representation.– The Fourier transform G(f) is a complex function of
frequency f,
dfftjfGtg )2exp()()(
)6.2()](exp[)()( fjfGfG
g(t) of spectrum amplitude continuous : )( fG
g(t) of spectrum phase continuous : )( f
2.1 The Fourier Transform
– The spectrum of a real-valued signal• : complex conjugate• : even function• : odd function
)()( * fGfG )()( fGfG )()( ff
2.2 Properties of the Fourier Transfrom
1. Linearity (Superposition)
2. Dilation
3. Conjugation Rule
4. Duality
5. Time Shifting6. Frequency Shifting
7. Area Under g(t)8. Area Under G(f)
)14.2()()()()( 22112211 fGcfGctgctgc
)20.2(1
)(
a
fG
aatg
)22.2()()( ** fGtg
)24.2()()( fgtG
)26.2()2exp()()( 00 ftjfGttg )27.2()()()2exp( cc ffGtgtfj
)31.2()0()( Gdttg
)32.2()()0( dffGg
2.2 Properties of the Fourier Transfrom
9. Differentiation in the Time Domain
10. Integration in the Time Domain
11. Modulation Theorem12. Convolution Theorem13. Correlation Theorem
14. Rayleigh’s Energy Theorem
)33.2()(2)( ffGjtgdt
d
)41.2()(2
1)( tG
fjdg
t
)49.2()()()()( 2121 dfGGtgtg
)51.2()()()()( 2121 fGfGdtgg
)53.2()()()()( *
21
*
21 fGfGdttgtg
)55.2()()(22
dffGdttg
2.2 Properties of the Fourier Transfrom
• Property 1 : Linearity (Superposition)
then for all constants c1 and c2,
• Property 2 : Dilation
(proof) If a>0,
: reflection property
)14.2()()()()( 22112211 fGcfGctgctgc
)20.2(1
)(
a
fG
aatg
)()( and )()(Let 2211 fGtgfGtg
dtftjatgatgF
)2exp()()]([
)21.2()()( fGtg
2.2 Properties of the Fourier Transfrom
• Property 3 : Conjugation Rule
• Property 4 : Duality
)22.2()()( ** fGtg
dfftjfGtg
)2exp()()(
dfftjfGtg
)2exp()()( **
dfftjfG
dfftjfGtg
)2exp()(
)2exp()()(
*
**
)23.2()()( ** fGtg
)24.2()()( fgtG
dfftjfGtg
)2exp()()(
dtftjtGfg
)2exp()()(
2.2 Properties of the Fourier Transfrom
• Property 5 : Time Shifting
• Property 6 : Frequency Shifting
)26.2()2exp()()( 00 ftjfGttg
)()2exp(
)2exp()()2exp()]([
0
00
fGftj
djgftjttgF
)27.2()()()2exp( cc ffGtgtfj
)(
])(2exp[)()]()2[exp(
c
cc
ffG
dtfftjtgtgtfjF
2.2 Properties of the Fourier Transfrom
• Property 7 : Area Under g(t)
• Property 8 : Area Under G(t)
)31.2()0()( Gdttg
)32.2()()0( dffGg
2.2 Properties of the Fourier Transfrom
• Property 9 : Differentiation in the Time Domain
• Property 10 : Integration in the Time Domain– Assuming G(0)=0,
)33.2()(2)( ffGjtgdt
d
)34.2()()2()( fGfjtgdt
d n
n
n
)41.2()(2
1)( tG
fjdg
t
t
dgdt
dtg )()(
tdgFfjfG )()2()(
2.2 Properties of the Fourier Transfrom
• Property 11 : Modulation Theorem
– The multiplication of two signals in the time domain is transformed into the convolution of their individual Fourier transforms in the frequency domain.
)49.2()()()()( 2121 dfGGtgtg
)()()( 1221 fGtgtg
dtftjtgtgfG )2exp()()()( 2112
'''
22 )2exp()()( dftfjfGtg
dtdftffjfGtgfG '''
2112 ])(2exp[)()()(
ddttjtgfGfG
)2exp()()()( 1212
)50.2()()()()( 2121 fGfGtgtg
2.2 Properties of the Fourier Transfrom
• Convolution
– f(t)*g(t) = g(t)*f(t) : signal = system
2.2 Properties of the Fourier Transfrom
• Property 12 : Convolution Theorem
• Property 13 : Correlation Theorem
)51.2()()()()( 2121 fGfGdtgg
)52.2()()()()( 2121 fGfGtgtg
)53.2()()()()( *
21
*
21 fGfGdttgtg
)54.2()()()()( 2121 fGfGdttgtg
2.2 Properties of the Fourier Transfrom
• Property 14 : Rayleigh’s Energy Theorem
– Total energy of a Fourier-transformable signal equals the total area under the curve of squared amplitude spectrum of this signal.
)55.2()()(22
dffGdttg
2** )()()()()( fGfGfGdttgtg
)56.2()2exp()()()(2
* dffjfGdttgtg
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