Rectangular Function Impulse Function Continuous Time Systems 2.4 &2.6.
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Transcript of Rectangular Function Impulse Function Continuous Time Systems 2.4 &2.6.
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Rectangular FunctionImpulse Function
Continuous Time Systems
2.4 &2.6
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How do you represent a unit rectangular function mathematically?
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Fundamental(1)
u(t)
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Fundamental(2)
u(t-0.5)
u(-t-0.5)
u(t+0.5)
u(-t+0.5)
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Block Function (window)• rect(t/T)
• Can be expressed as u(T/2-t)-u(-T/2-t) – Draw u(t+T/2) first; then reverse it!
• Can be expressed as u(t+T/2)-u(t-T/2)
• Can be expressed as u(t+T/2)u(T/2-t)
-T/2 T/2
1
-T/2 T/2
1
-T/2 T/2
1
-T/2 T/2
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Application
• The rectangular pulse can be used to extract part of a signal
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c03f10A Simple Cell Phone Charger Circuit
(R1 is necessary)
Another Application:Signal strength indicator
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Mathematical Modeling
Modify the unit rectangular pulse:1. Shift to the right by To/42. The period is To/2
V1(t) V1(t-To) V1(t-2To)
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Application of Impulse Function
The unit impulse function is used to model sampling operation, i.e. the selection of a value of function at a particular time instant using analog to digital converter.
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Generation of an Impulse Function
Ramp function
epsilon approaches 0
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Shifted Impulse Function
0
0 to
d(t)
d(t-to)
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The Impulse Function
We use a vertical arrow to represent 1/ε because g(t) Increases dramatically as ε approaches 0.
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Another Definition of the Impulse Function
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Mathematica Connection
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Property
f[t]
f[t-2]
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Property
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Property
Shifted Unit Step Function
Slope is sharp at t=2
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Property
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Property
g(at), a>1, e.g. 2
area: 1/ ε ε /2=1/21/ ε 2ε=2
to/2 to/2+ ε/2
1/ ε
g(2t),
11/2
δ(t)
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Property
g(at), a<1, e.g. 1/2
area: 1/ ε 2ε=2
2to 2to+ 2ε
1/ ε
g(2t),
1
δ(t) 2
2to
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Property
d(t)
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Example
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• A system is an operation for which cause-and-effect relationship exists– Can be described by block diagrams – Denoted using transformation T[.]
• System behavior described by mathematical model
Continuous-Time Systems
T [.]X(t) y(t)
(meat grinder)
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Inverting Amplifier
Vout=-(R1/R2)
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Inverting Summer Example
Vout=-RF(V1/R1+V2/R2)If RF/R1=1, RF/R2=1Vout=-(V1+V2)
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Multiplier
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Parallel Connection
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Cascade Connection
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