Objectives: Derivation Transform Pairs Response of LTI Systems Transforms of Periodic Signals...
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ECE 8443 – Pattern RecognitionECE 3163 – Signals and Systems
• Objectives:DerivationTransform PairsResponse of LTI SystemsTransforms of Periodic SignalsExamples
• Resources:Wiki: The Fourier TransformCNX: DerivationMIT 6.003: Lecture 8 Wikibooks: Fourier Transform TablesRBF: Image Transforms (Advanced)
• URL: .../publications/courses/ece_3163/lectures/current/lecture_10.ppt• MP3: .../publications/courses/ece_3163/lectures/current/lecture_10.mp3
LECTURE 10: THE FOURIER TRANSFORM
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ECE 3163: Lecture 10, Slide 2
Motivation• We have introduced a Fourier Series for analyzing periodic signals. What
about aperiodic signals? (e.g., a pulse instead of a pulse train)
• We can view an aperiodic signal as the limit of a periodic signal as T .
• The harmonic components are spaced apart.
• As T , 0 0, then k0 becomes continuous.
• The Fourier Series becomes the Fourier Transform.
T 2
0
1
01
0
110
sin2,,
)/(2sin2sin
TTc
kTfixedTTk
TTkkTk
c
k
k
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ECE 3163: Lecture 10, Slide 3
Derivation of Analysis Equation• Assume x(t) has a finite duration.
• Define as a periodic extensionof x(t):
• As
• Recall our Fourier series pair:
• Since x(t) and are identical over this interval:
• As
2
22)(
)(~ Ttperiodic
TtTtxtx
)(~ tx
)()(~, txtxT
k
tjkkectx 0)(~
2/
2/
0)(~1 T
T
tjkk dtetxT
c
2/
2/
2/
2/
00 )(1)(~1 T
T
tjkT
T
tjkk dtetx
Tdtetx
Tc
)(~ tx
0, kT
dtetxT
jX tj )(1)(
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ECE 3163: Lecture 10, Slide 4
Derivation of the Synthesis Equation
• Recall:
• We can substitute for ck the sampled value of :
• As
and we arrive at our Fourier Transform pair:
• Note the presence of the eigenfunction:
• Also note the symmetry of these equations (e.g., integrals over time and frequency, change in the sign of the exponential, difference in scale factors).
22)()(~ 0
TtTforectxtxk
tjkk
k
tjk
k
tjk
ejX
ejXT
txtx
0
0
)(21
))(1()()(~
00
0
)( jX
000 ,,0, kdT
k
dtetxT
jX
dejXtx
tj
tj
)(1)(
)(21)( (synthesis)
(analysis)
tj
js
st ee
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ECE 3163: Lecture 10, Slide 5
Frequency Response of a CT LTI System• Recall that the impulse response of
a CT system, h(t), defines the properties of that system.
• We apply the Fourier Transform toobtain the system’s frequency response:
except that now this is valid for finite duration (energy) signals as well as periodic signals!
• How does this relate to what you have learned in circuit theory?
CT LTI)()(jXtx
)()(jYty
)()( jHth
CT LTItje tjejH )()()( jHth
dtethT
jH
dejHth
tj
tj
)(1)(
)(21)(
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ECE 3163: Lecture 10, Slide 6
Existence of the Fourier Transform• Under what conditions does this transform exist?
x(t) can be infinite duration but must satisfy these conditions:
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ECE 3163: Lecture 10, Slide 7
Example: Impulse Function
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ECE 3163: Lecture 10, Slide 8
Example: Exponential Function
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ECE 3163: Lecture 10, Slide 9
Example: Square Pulse
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ECE 3163: Lecture 10, Slide 10
Example: Gaussian Pulse
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ECE 3163: Lecture 10, Slide 11
CT Fourier Transforms of Periodic Signals
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ECE 3163: Lecture 10, Slide 12
Example: Cosine Function
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ECE 3163: Lecture 10, Slide 13
Example: Periodic Pulse Train
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ECE 3163: Lecture 10, Slide 14
• Motivated the derivation of the CT Fourier Transform by starting with the Fourier Series and increasing the period: T
• Derived the analysis and synthesis equations (Fourier Transform pairs).
• Applied the Fourier Transform to CT LTI systems and showed that we can obtain the frequency response of an LTI system by taking the Fourier Transform of its impulse response.
• Discussed the conditions under which the Fourier Transform exists. Demonstrated that it can be applied to periodic signals and infinite duration signals as well as finite duration signals.
• Worked several examples of important finite duration signals.
• Introduced the Fourier Transform of a periodic signal.
• Applied this to a cosinewave and a pulse train.
Summary