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Unit Impulse Overview
Unit step
Switches
Unit impulse Relationships
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Unit Step for Switches
vs
Linear
Circuit
t=0
vsu(t)
Linear
Circuit
Linear
Circuit
t=0
is
isu(t)
Linear
Circuit
u(t) useful for representing the opening or closing of switches
We will often solve for or be given initial conditions at t = 0
We can then represent independent sources as though they wereimmediately applied at t = 0. More later.
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Continuous-Time Unit Impulse
t
ue(t)
t-e e -e e
t
u(t)
t
1 1
e(t)
(t)
e(t) due(t)
dt
As e 0 ,
ue(t) u(t)
e(t) for t = 0 becomes very large
e(t) for t = 0 becomes zero
(t) lime0 e(t)
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Continuous-Time Unit Impulse Continued
t
1
(t)
(t)
0 t = 0
t = 0
ee
(t) dt = 1 for any e > 0
Also known as the Dirac delta function
Is zero everywhere except zero
The impulse integral serves as a measure of the impulse amplitude
Drawn as an arrow with unit height
5(t) would be drawn as an arrow with height of 5
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Continuous-Time Unit Impulse Comments
(t) =
0 t = 0
t = 0
The impulse should be viewed as an idealization
Real systems with finite inertia do not respond instantaneously
The most important property of an impulse is its area
Most systems will respond nearly the same to sharp pulsesregardless of their shape - if
They have the same amplitude (area)
Their duration is much briefer than the systems response The idealized unit impulse is short enough for any system
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Unit Impulse Properties
(t)x(t) = (t)x(0)
(at) =1
|a|
(t)
(t) = (t)
(t) =du(t)
dt
u(t) =t
() d
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Unit Impulse Sampling Property+
x(t)(t) dt =
+
x(0)(t) dt
= x(0)
+
(t) dt
= x(0)
Similarly, +
x(t)(t t0) dt =
+
x(t0)(t t0) dt
= x(t0)+
(t t0) dt
= x(t0)
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Unit Impulse Sampling Property
x(t) =
+
x()( t) d
This integral does not appear to be useful It will turn out to be very useful
It states that x(t) can be written as a linear combination of scaledand shifted unit impulses
This will be a key concept when we discuss convolution
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Example 1: Continuous-Time Unit-Ramp
t
r(t)
1
r(t) 0 t 0t t 0
What is the first derivative?
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Example 2: Continuous-Time Unit-Ramp Integral
t
r(t)
1
What is the integral of the unit ramp?
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Basis Function Relationships
u(t) =
t
() d
du(t)dt
= (t)
t
u() d = r(t)
r(t) =
t
u() d
dr(t)dt
= u(t)
t
r() d = 12 r(t)2
If we can write a signal x(t) in terms of u(t) and r(t), it is easy tofind the derivative
Similarly, it is easy to integrate
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Basis Functions Translated
t
u(t-t0)
t
1 1
t
r(t-t0)
1
t0 t0t0
(t-t0)
Can write simple expressions for the functions translated in time
Can scale the amplitude
Any piecewise linear signal can be written in terms of basisfunctions
This makes it easy to calculate derivatives and integrals
Will not discusshow
this term Sufficient to know it can be done
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