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LECTURE 07: CONVOLUTION FOR CT SYSTEMS. Objectives: Convolution Definition Graphical Convolution Examples Properties Resources: Wiki: Convolution MIT 6.003: Lecture 4 JHU: Convolution Tutorial ISIP: Java Applet. URL:. Representation of CT Signals (Review). - PowerPoint PPT Presentation
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ECE 8443 – Pattern RecognitionEE 3512 – Signals: Continuous and Discrete
• Objectives:Convolution DefinitionGraphical ConvolutionExamplesProperties
• Resources:Wiki: ConvolutionMIT 6.003: Lecture 4JHU: Convolution TutorialISIP: Java Applet
LECTURE 07: CONVOLUTION FOR CT SYSTEMS
URL:
EE 3512: Lecture 07, Slide 2
Representation of CT Signals (Review)• We approximate a CT signal
as a weighted pulse function.
• The signal can be written as a sum of these pulses:
k
ktkxtx )()()(ˆ
• In the limit, as :0
dtxtx )()()(
• Mathematical definition of an impulsefunction (the equivalent of the unit pulsefor DT signals and systems):
0
0
1)(
000
)(
dtt
tfortfor
t
• Unit pulses can be constructed from many functional shapes (e.g., triangular or Gaussian) as long as they have a vanishingly small width. The rectangular pulse is popular because it is easy to integrate
EE 3512: Lecture 07, Slide 3
• Denote the system impulse response, h(t), as the output produced when the input is a unit impulse function, (t).
• From time-invariance:
• From linearity:
• This is referred to as the convolution integral for CT signals and systems.
• Its computation is completely analogous to the DT version:
Response of a CT LTI System
CT LTI)(tx )(*)()( thtxty )(th
)()( tht
)(*)()()()()()()( thtxdthxtydtxtx
EE 3512: Lecture 07, Slide 4
Example: Unit Pulse Functions
• t < 0: y(t) = 0
• t > 2: y(t) = 0
• 0 t 1: y(t) = t
• 1 t 2: y(t) = 2-t
EE 3512: Lecture 07, Slide 5
Example: Negative Unit Pulse
• t < 0.5: y(t) = 0
• t > 2.5: y(t) = 0
• 0.5 t 1.5: y(t) = 0.5-t
• 1 t 2: y(t) = -2.5+t
EE 3512: Lecture 07, Slide 6
Example: Combination Pulse
• p(t) = 1 0 t 1
• x(t) = p(t) - p(t-1)
• y(t) = ???
EE 3512: Lecture 07, Slide 7
Example: Unit Ramp
• p(t) = 1 0 t 1
• x(t) = r(t) p(t)
• y(t) = ???
EE 3512: Lecture 07, Slide 8
Properties of Convolution• Sifting Property:
Proof:)()(*)( 00 ttxtttx
)()()()()()(*)( 0000
0
0
ttxdttxdttxtttxtt
tt
• Integration:
Proof:
t
dxtutx )()(*)(
t
tfortubecausedxdtuxtutx 0)()()()()(*)(
• Step Response (follows from the integration property):
Comments: Requires proof of the commutative property. In practice, measuring the step response of a system is much easier than
measuring the impulse response directly. How can we obtain the impulse response from the step response?
t
dhtuththtu )()(*)()(*)(
EE 3512: Lecture 07, Slide 9
Properties of Convolution (Cont.)• Commutative Property:
Proof:)(*)()(*)( txththtx
• Implications (from DT lecture):
)(*)()()())(()()(*)(
,,
)()()(*)(
txthdtxhdhtxthtx
ddandtortlet
dthxthtx
• Distributive Property:
Proof:)(*)()(*)()]()([*)( 2121 thtxthtxththtx
)(*)()(*)(
)()()()(
)]()()[()]()([*)(
21
21
2121
thtxthtx
dthxdthx
dththxththtx
EE 3512: Lecture 07, Slide 10
Properties of Convolution (Cont.)• Associative Property:
Proof:
)(*))(*)(()(*))(*)(()(*)(*)(
12
2121
ththtxththtxththtx
• Implications (from DT lecture):
)](*)([*)(
))(()()(
))(()()(
)()()(
)(])()([)(*)](*)([
)()()](*)([
21
21
21
21
2121
11
ththtx
ddthhx
ddthhx
ddandLet
ddthhx
dthdhxththtx
dthxthtx
EE 3512: Lecture 07, Slide 11
Useful Properties of CT LTI Systems• Causality: which implies:
This means y(t) only depends on x( < t).
• Stability:
Bounded Input ↔ Bounded Output
00 nth
)(th
Sufficient Condition:
dthxdthxty
xtx
max
max
)()(
)(for
Necessary Condition:
dhdhhhdhx
txththtx
th
00)()(0)(y(0)But
(bounded)1)(then,)(/)()(Let
if
*
*
dthxdthxt
)()(
EE 3512: Lecture 07, Slide 12
• We introduced CT convolution.
• We worked some analytic examples.
• We also demonstrated graphical convolution.
• We discussed some general properties of convolution.
• We also discussed constraints on the impulse response for bounded input / bounded output (stability).
Summary
