eleg 3124 systems and signals ch. 4 fourier transform...find the fourier transform of x(t) cos(z 0...
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Department of Electrical EngineeringUniversity of Arkansas
ELEG 3124 SYSTEMS AND SIGNALS
Ch. 4 Fourier Transform
Dr. Jingxian Wu
2
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
3
INTRODUCTION: MOTIVATION
• Motivation:
– Fourier series: periodic signals can be decomposed as the
summation of orthogonal complex exponential signals
tjnctxn
n 0exp)( +
−=
=
• each harmonic contains a unique frequency: n/T
=T
n dttjntxT
c0
0exp)(1
How about aperiodic signals ?
( )=T• time domain ➔ frequency domain
t
x(t)
Time domain Frequency domain
4
INTRODUCTION: TRANSFER FUNCTION
• System transfer function
• System with periodic inputs
tje
)(th)( He tj
+
−= dttjthH exp)()(
tjne 0
)(th)( 0
0 nHe
tjn
tjn
n
nec 0+
−= )(th)( 0
0 nHec
tjn
n
n+
−=
)(tx)(th
)( 00
nHectjn
n
n+
−=
5
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
6
FOURIER TRANSFORM
• Inverse Fourier Transform
• Fourier Transform
– given x(t), we can find its Fourier transform
– given , we can find the time domain signal x(t)
– signal is decomposed into the “weighted summation” of complex exponential functions. (integration is the extreme case of summation)
+
−
−= dtetxX tj )()(
+
−=
deXtx tj)(2
1)(
)(X
)(X
➔)(tx )(X
7
FOURIER TRANSFORM
• Example
– Find the Fourier transform of )/()( trecttx =
t
x(t)
t
x(t)
8
FOURIER TRANSFORM
• Example
– Find the Fourier transform of |)|exp()( tatx −= 0a
9
FOURIER TRANSFORM
• Example
– Find the Fourier transform of )()exp()( tuattx −= 0a
10
FOURIER TRANSFORM
• Example
– Find the Fourier transform of )()( attx −=
11
FOURIER TRANSFORM: TABLE
12
FOURIER TRANSFORM
+
−dttx |)(|
)()exp()( tuttx =
• Example
–
• The existence of Fourier transform
– Not all signals have Fourier transform
– If a signal have Fourier transform, it must satisfy the following two
conditions
• 1. x(t) is absolutely integrable
• 2. x(t) is well behaved
– The signal has finite number of discontinuities, minima,
and maxima within any finite interval of time.
13
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
14
PROPERTIES: LINEARITY
• Linearity
– If
– then
)()( 11 Xtx )()( 22 Xtx
)()()()( 2121 bXaXtbxtax ++
• Example
– Find the Fourier transform of )(4)()2exp(3)/(2)( ttuttrecttx +−+=
15
PROPERTY: TIME-SHIFT
• Time shift
– If
– Then
)()( Xtx
]exp[)()( 00 tjXttx −−
• Time shift
– If
– Then
• Review: complex number
jbacjcecc j +=+== )sin(||)cos(||||
cos|| ca = sin|| cb =
22|| bac += )/tan( aba=
phase shift
time shift in time domain ➔ frequency shift in frequency domain
– Phase shift of a complex number c by : 0 )(exp||)exp( 00 += jcjc
16
PROPERTY: TIME SHIFT
• Example:
– Find the Fourier transform of 2)( −= trecttx
17
PROPERTY: TIME SCALING
• Time scaling
– If
– Then
• Example
– Let , find the Fourier transform of
)()( Xtx
aX
aatx
||
1)(
( ) 2/1)( −= rectX )42( +− tx
18
PROPERTY: SYMMETRY
• Symmetry
– If , and is a real-valued time signal
– Then
)()( Xtx )(tx
)()( * XX =−
19
PROPERTY: DIFFERENTIATION
• Differentiation
– If
– Then
)()( Xtx
)()(
Xjdt
tdx ( ) )(
)( Xj
dt
txd n
n
n
• Example
– Let , find the Fourier transform of ( ) 2/1)( −= rectXdt
tdx )(
20
PROPERTY: DIFFERENTIATION
• Example
– Find the Fourier transform of
(Hint: )
)sgn()( ttx =
)()sgn(2
1tt
dt
d=
21
PROPERTY: CONVOLUTION
• Convolution
– If ,
– Then
)()( Xtx )()( Hth
)()()()( HXthtx
)(tx
)(th)()( thtx )(X
)(H)()( HX
time domain frequency domain
22
PROPERTY: CONVOLUTION
• Example
– An LTI system has impulse response
If the input is
Find the output
( ) )(exp)( tuatth −=
( ) )()exp()()(exp)()( tuctactubtbatx −−+−−=
)0,0,0( cba
23
PROPERTY: MULTIPLICATION
• Multiplication
– If ,
– Then
)()( Xtx )()( Mtm
)()(2
1)()(
MXtmtx
24
PROPERTY: DUALITY
• Duality
– If
– Then
)()( Gtg
)(2)( − gtG
25
PROPERTY: DUALITY
• Example
– Find the Fourier transform of
(recall: )
=
2)(
tSath
2sinc )/(rect t
26
PROPERTY: DUALITY
• Example
– Find the Fourier transform of 1)( =tx
tjetx 0)(
=– Find the Fourier transform of
27
PROPERTY: SUMMARY
28
PROPERTY: EXAMPLES
• Examples
– 1. Find the Fourier transform of )cos()( 0ttx =
– 2. Find the Fourier transform of )()( tutx =
1)sgn(2
1)( += ttu
jt
2)sgn(
29
PROPERTY: EXAMPLES
• Examples
– 3. A LTI system with impulse response
Find the output when input is
)(exp)( tuatth −=
)()( tutx =
– 4. If , find the Fourier transform of
(Hint: )
)()( Xtx −
t
dx )(
)()()( tutxdxt
= −
30
PROPERTY: EXAMPLES
• Example
– 5. (Modulation) If ,
Find the Fourier transform of
)()( Xtx )cos()( 0ttm =
)()( tmtx
– 6. If , find x(t)
ja
X+
=1
)(
31
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN
• Differentiation in frequency domain
– If:
– Then:
)()( Xtx
n
nn
d
Xdtxjt
)()()( =−
PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN
32
),()exp( tuatt − 0a
• Example
– Find the Fourier transform of
33
PROPERTY: FREQUENCY SHIFT
• Frequency shift
– If:
– Then:
)()( Xtx
)()exp()( 00 − Xtjtx
• Example
– If , find the Fourier transform ( ) 2/1)( −= rectX )2exp()( tjtx −
34
PROPERTY: PARSAVAL’S THEOREM
• Review: signal energy
+
−= dttxE 2|)(|
• Parsaval’s theorem
+
−
+
−=
dXdttx 22 |)(|
2
1|)(|
35
PROPERTY: PARSAVAL’S THEOREM
• Example:
– Find the energy of the signal )()2exp()( tuttx −=
36
PROPERTY: PERIODIC SIGNAL
• Fourier transform of periodic signal
– Periodic signal can be written as Fourier series
tjnctxn
n 0exp)( +
−=
=
– Perform Fourier transform on both sides
)(2)( 0 ncXn
n −= +
−=
37
OUTLINE
• Introduction
• Fourier Transform
• Properties of Fourier Transform
• Applications of Fourier Transform
38
APPLICATIONS: FILTERING
• Filtering
– Filtering is the process by which the essential and useful part of a
signal is separated from undesirable components.
• Passing a signal through a filter (system).
• At the output of the filter, some undesired part of the signal
(e.g. noise) is removed.
– Based on the convolution property, we can design filter that only
allow signal within a certain frequency range to pass through.
)(tx
)(th)()( thtx )(X
)(H)()( HX
time domain frequency domain
filter filter
39
APPLICATIONS: FILTERING
• Classifications of filters
Low pass filter
Band pass filterBand stop (Notch) filter
PassbandStop
band PassbandStop
band
High pass filter
Passband Stop
band
Stop
band
Stop
bandPassband Passband
40
APPLICATION: FILTERING
• A filtering example
– A demo of a notch filter
)(X
)(H
)()( HX
Corrupted sound Filter Filtered sound
41
APPLICATIONS: FILTERING
• Example
– Find out the frequency response of the RC circuit
– What kind of filters it is?
RC circuit
42
APPLICATION: SAMPLING THEOREM
• Sampling theorem: time domain
– Sampling: convert the continuous-time signal to discrete-time signal.
+
−=
−=n
nTttp )()(
sampling period
)()()( tptxtxs =
)(tx
Sampled signal
43
APPLICATION: SAMPLING THEOREM
• Sampling theorem: frequency domain
– Fourier transform of the impulse train
• impulse train is periodic
+
−=
+
−=
=−=n
tjn
sn
sse
TnTttp
11
)()(
• Find Fourier transform on both sides
+
−=
−=n
s
s
nT
P )(2
)(
• Time domain multiplication ➔ Frequency domain convolution
)()(2
1)()(
PXtptx
+
−=
−n
s
s
nXT
tptx )(1
)()(
s
sT
2=
Fourier series
44
APPLICATION: SAMPLING THEOREM
• Sampling theorem: frequency domain
– Sampling in time domain ➔ Repetition in frequency domain
Time domain Frequency domain
45
APPLICATION: SAMPLING THEOREM
• Sampling theorem
– If the sampling rate is twice of the bandwidth, then the original
signal can be perfectly reconstructed from the samples.
Bs 2
Bs 2
Bs 2=
Bs 2
Frequency domain
46
APPLICATION: AMPLITUDE MODULATION
• What is modulation?
– The process by which some characteristic of a carrier wave is
varied in accordance with an information-bearing signal
modulationInformation
bearing signal
Carrier wave
Modulated signal
• Three signals:
– Information bearing signal (modulating signal)
• Usually at low frequency (baseband)
• E.g. speech signal: 20Hz – 20KHz
– Carrier wave
• Usually a high frequency sinusoidal (passband)
• E.g. AM radio station (1050KHz) FM radio station
(100.1MHz), 2.4GHz, etc.
– Modulated signal: passband signal
47
APPLICATION: AMPLITUDE MODULATION
• Amplitude Modulation (AM)
)2cos()()( tftmAts cc =
– A direct product between message signal and carrier signal
Mixer
Local
Oscillator
)(tm
)2cos( tfA cc
)(ts
Amplitude modulation
48
APPLICATION: AMPLITUDE MODULATION
• Amplitude Modulation (AM)
)()(2
)( ccc ffMffM
AfS ++−=
Amplitude modulation