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LECTURE 11: FOURIER TRANSFORM PROPERTIES. Objectives: Linearity Time Shift and Time Reversal Multiplication Integration Convolution Parseval’s Theorem Duality - PowerPoint PPT Presentation
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ECE 8443 – Pattern RecognitionECE 3163 – Signals and Systems
• Objectives:LinearityTime Shift and Time ReversalMultiplicationIntegrationConvolutionParseval’s TheoremDuality
• Resources:BEvans: Fourier Transform PropertiesMIT 6.003: Lecture 8 DSPGuide: Fourier Transform PropertiesWiki: Audio Timescale ModificationISIP: Spectrum Analysis
LECTURE 11: FOURIER TRANSFORM PROPERTIES
Audio:URL:
ECE 3163: Lecture 11, Slide 2
• Recall our expressions for the Fourier Transform and its inverse:
• The property of linearity:
Proof:
Linearity
dtetxT
XjX
dejXtx
tj
tj
)(1
)()(
)(2
1)( (synthesis)
(analysis)
)()()()()()()()( jbYjaXtbytaxjbYjaXtbytax F
)()(
)(1
)(1
)(1
)(1
)()(1
)()(
jbYjaX
dtetyT
bdtetxT
a
dtetbyT
dtetaxT
dtetbytaxT
tbytax
tjtj
tjtj
tj
F
ECE 3163: Lecture 11, Slide 3
0
0
0
)(
)(1
)(1
)(
implieswhich, :variablesofchangeamake
)(1
)(
)(0
00
00
tj
tjj
tj
tj
ejX
edexT
dexT
ttx
tλtt-tλ
dtettxT
ttx
F
F
• Time Shift:
Proof:
• Note that this means time delay is equivalent to a linear phase shift in the frequency domain (the phase shift is proportional to frequency).
• We refer to a system as an all-pass filter if:
• Phase shift is an important concept in the development of surround sound.
Time Shift
0)()( 0cjejXttx
0)(1)( jXjX
ECE 3163: Lecture 11, Slide 4
• Time Scaling:
Proof:
• Generalization for a < 0 , the negative value is offset by the change in the limits of integration.
• What is the implication of a < 1 on the time-domain waveform? On the frequency response? What about a > 1?
• Any real-world applications of this property? Hint: sampled signals.78
Time Scaling
)(1
)(a
jX
aatx
)()1
(
)(1
)1
(
)1
()(1
)(
)/1(and,/implieswhich, :variablesofchangeamake ,0 assume
)(1
)(
)/(
)(
a
jX
a
dexTa
da
exT
tax
dadtatatλa
dteatxT
atx
aj
aj
tj
F
F
ECE 3163: Lecture 11, Slide 5
• Time Reversal:
Proof:
We can also note that for real-valued signals:
• Time reversal is equivalent to conjugation in the frequency domain.
• Can we time reverse a signal? If not, why is this property useful?
Time Reversal
)()( jXtx
)()(1
)(1
jX
a
jX
atx
a
F
conjugate)(complex)()()(
)()()(*
jXjXjX
jXjXjX
ECE 3163: Lecture 11, Slide 6
)()()(1
)()(1
)()(
:by multiply
)()(1)(
: respect to with atedifferenti
)(1
)(
txtdtetxtT
dtetxjtT
jd
jdXj
j
dtetxjtTd
jdX
dtetxT
jX
tj
tj
tj
tj
F
• Multiplication by a power of t:
Proof:
• We can repeat the process for higher powers of t.
Multiplication by a Power of t
)()()(
jXd
djtxt
n
nnn
ECE 3163: Lecture 11, Slide 7
))((
)(1
)(1
0
)( 0
00
ωjX
dtetxT
dteetxT
x(t)e
tωj
tjtjωtjω
F
• Multiplication by a complex exponential:
Proof:
• Why is this property useful?
• First, another property:
• This produces a translation in the frequencydomain. How might this be useful in acommunication system?
Multiplication by a Complex Exponential (Modulation)
00 ))(()( numberrealanyforjXetx tj
))](())(([2
1]
2)[()cos()( 000
00
jXjXee
txttxtjtj
ECE 3163: Lecture 11, Slide 8
• Differentiation in the Time Domain:
• Integration in the Time Domain:
• What are the implications of time-domain differentiation in the frequency domain?
• Why might this be a problem? Hint: additive noise.
• How can we apply these properties? Hint: unit impulse, unit step, …
Differentiation / Integration
)()()( jXjtxdt
d nn
n
)()0()(1
)(
XjXj
dxt
ECE 3163: Lecture 11, Slide 9
• Convolution in the time domain:
• Proof:
Convolution in the Time Domain
)()()()( jHjXthtx
)()(
)()(
)()(
:variablesofchange
)()(
)()()()(
)()()()(
)(
jHjX
dehdex
ddehx
dtdt
ddtethx
dtedthxthtx
dthxthtx
jj
j
tj
tj
F
ECE 3163: Lecture 11, Slide 10
• Multiplication in the time domain:
• Parseval’s Theorem:
• Duality:
• Note: please read the textbook carefully for the derivations and interpretation of these results.
Other Important Properties
dYXjYjXtytx )()(2
1)]()([
2
1)()(
djXdttx22 )(
2
1)(
)(2)( xtX
ECE 3163: Lecture 11, Slide 12
Example: Cosine Function
ECE 3163: Lecture 11, Slide 13
Example: Periodic Pulse Train
ECE 3163: Lecture 11, Slide 14
Example: Gaussian Pulse
