Post on 12-Jan-2016
1
Review of Continuous-Time Fourier Series
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Example 3.5
T/2T1-T/2 -T1
This periodic signal x(t) repeats every T seconds.x(t)=1, for |t|<T1 , and x(t)=0, for T1 <|t|< T/2
Fundamental period= T,Fundamental frequency Choosing the period of integration to be between-T/2 and +T/2. Use eqn 3.39 to get at Fourier Series Coefficients.
3
Example 3.5 continued
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Continuous-Time Fourier Transform
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Continuous-Time Fourier Transform
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Continuous-Time Fourier Transform Pair Equation
13
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TransformFourierTimeContinuous
14
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4.1 Example
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22 ajX
ajXa
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An exponential time function and its Fourier Transform
16
Bode Plotrepresentationof the FourierTransform of
an exponentialtime function
17
18
19
20
21
Properties of Fourier Transform
• Linearity• Time Shifting• Conjugation• Differentiation in the time-domain• Integration in the time-domain• Time and Frequency Scaling.• Duality• Parseval’s Relation• Convolution• Multiplication
22
FT Property associated with Linearity
.)()()()(
)()(
)()(
2121
22
11
jbXjaXtbxtax
jXtx
jXtx
FT
FT
FT
23
FT Properties associated with Time shifting & Conjugation
).()(then
-:nConjugatio
).()(then
-:Shifting Time
),()(
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00
jXtx
jXettx
jXtx
FT
tjFT
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24
FT Properties associated with Differentiation & Integration in Time Domian
).()0()(1
d)(then
-:domain Time in then Integratio
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then
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t
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25
FT Properties associated with Time & Frequency Scaling.
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).(||
1)(then
-:ScalingFrequency and Time
).()(
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a
jX
aatx
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26
FT Properties associated with Duality
.||,1)(sin
)(
.sin2
)(||,1)(
-:Duality
).()(
22
1111
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jXtx
FT
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27
FT Properties –Duality continued.
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d
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tdx
jXtx
FT
FT
FT
28
FT Properties – Duality continued.
}.{))(()(
).()(}{
-:shiftingfrequency &shifting timeofDuality
).()(
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shiftingFrequencyjXtxe
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jXtx
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tjFT
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29
FT Properties-Duality continued
domian}.frequency in on {integrati
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x
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FT
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30
FT Properties- Duality continued
.d|)(|2
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-:Relation sParseval'
).()(
-
2
-
2
jXttx
jXtx FT
31
FT Property associated with Convolution in time domain
domain) (freqtion multiplicadomain) (timen convolutio
).()()(*)()(
)()(
)()(
FT
FT
FT
FT
jHjXthtxty
jHth
jXtx
32
FT Property associated with Multiplication in time domain
domain.equency in time/fr
cation n/multipliconvolutio ofDuality
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