CT LTI Systems

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Chapter 3

• CT LTI Systems
• Updated 9/16/13

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A Continuous-Time System

• How do we know the output?

X(t)
y(t)
System
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LTI Systems

• Time Invariant
• X(t)? y(t) x(t-to) ? y(t-to)
• Linearity
• a1x1(t) a2x2(t)? a1y1(t) a2y2(t)
• a1y1(t) a2y2(t) Ta1x1(t)a2x2(t)
• Meet the description of many physical systems
• They can be modeled systematically
• Non-LTI systems typically have no general
  mathematical procedure to obtain solution

What is the input-output relationship for LTI-CT
Systems?
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Convolution Integral

• An approach (available tool or operation) to
  describe the input-output relationship for LTI
  Systems
• In a LTI system
• d(t)? h(t)
• Remember h(t) is Td(t)
• Unit impulse function ? the impulse response
• It is possible to use h(t) to solve for any
  input-output relationship
• One way to do it is by using the Convolution
  Integral

X(t)d(t)
y(t)h(t)
LTI System
X(t)
y(t)
LTI System h(t)
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Convolution Integral

• Remember
• So what is the general solution for

X(t)Ad(t-kto)
y(t)Ah(t-kto)
LTI System
X(t)
y(t)
LTI System
?
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Convolution Integral

• Any input can be expressed using the unit impulse
  function

Proof
Sifting Property
to?t and integrate by dt
X(t)
y(t)
LTI System
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Convolution Integral

• Given
• We obtain Convolution Integral
• That is A system can be characterized using its
  impulse response
• y(t)x(t)h(t)

X(t)
y(t)
LTI System
X(t)
y(t)
LTI System h(t)
By definition
Do not confuse convolution with multiplication!
y(t)x(t)h(t)
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Convolution Integral
X(t)
y(t)
LTI System h(t)
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Convolution Integral - Properties

• Commutative
• Associative
• Distributive
• Thus, using commutative property

Next We draw the block diagram representation!
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Convolution Integral - Properties

• Commutative
• Associative
• Distributive

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Simple Example

• What if a step unit function is the input of a
  LTI system?
• S(t) is called the Step Response

u(t)
y(t)Su(t)s(t)
LTI System
Step response can be obtained by integrating the
impulse response!
Impulse response can be obtained by
differentiating the step response
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Example 1

• Consider a CT-LTI system. Assume the impulse
  response of the system is h(t)e(-at) for all
  agt0 and tgt0 and input
• x(t)u(t). Find the output.

u(t)
y(t)
h(t)e-at
Because tgt0
Draw x(t), h(t), h(t-t),etc. ? next slide
The fact that agt0 is not an issue!
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Example 1 Cont.
y(t) for a3
y(t)

t
t
U(-(t-t))
tgt0
U(-(t-t))
tlt0
Remember we are plotting it over t and t is the
variable
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Example 1 Cont. Graphical Representation
(similar)
a1 In this case!
http//www.wolframalpha.com/input/?iconvolutiono
ftwofunctionslk4num4lk4num4
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Example 1 Cont. Graphical Representation
(similar)
Note in our case We have u(t) rather than
rectangular function!
http//www.jhu.edu/signals/convolve/
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Example 2

• Consider a CT-LTI system. Assume the impulse
  response of the system is h(t)e-at for all agt0
  and tgt0 and input x(t) eat u(-t). Find the
  output.

x(t)
y(t)
h(t)e-at
Note that for tgt0 x(t) 0 so the integration
can only be valid up to t0
Draw x(t), h(t), h(t-t),etc. ? next slide
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Example Cont.

x(t) eat u(-t)
h(t)e-at u(t)
?
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Another Example
notes
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Properties of CT LTI Systems

• When is a CT LTI system memory-less (static)
• When does a CT LTI system have an inverse system
  (invertible)?
• When is a CT LTI system considered to be causal?
  Assuming the input is causal
• When is a CT LTI system considered to be Stable?

notes
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Example

• Is this an stable system?
• What about this?

notes
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Differential-Equations Models

• This is a linear first order differential
  equation with constant coefficients (assuming a
  and b are constants)
• The general nth order linear DE with constant
  equations is

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Is the First-Order DE Linear?

• Consider
• Does a1x1(t) a2x2(t)? a1y1(t) a2y2(t)?
• Is it time-invariant? Does input delay results in
  an output delay by the same amount?
• Is this a linear system?

notes
notes

Y(t)
e(t)
Sum
Integrator
X(t)
-
a
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Example

• Is this a time invariant linear system?

R
L
V(t)
Ldi(t)/dt Ri(t) v(t) a -R/L b1/L
y(t)i(t) x(t) V(t)
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Solution of DE

• A classical model for the solution of DE is
  called method of undermined coefficients
• yc(t) is called the complementary or natural
  solution
• yp(t) is called the particular or forced solution

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Solution of DE
Thus, for x(t) constant ? yp(t)P x(t) Ce-7t ?
yp(t) Pe-7t x(t) 2cos(3t) ? yp(t)P1cos(3t)P2s
in(3t)
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Example

• Solve
• Assume x(t) 2 and y(0) 4
• What happens if

notes
yc(t) Ce-2t yp(t) P P 1 y(t) Ce-2t
1 y(0) 4 ? C 3 ? y(t) 3e-2t 1
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Schaums Examples

• Chapter 2
• 2, 4-6, 8, 10, 11-14, 18, 19, 48, 52, 53,