Download 20120720-EE235-Lecture18

Signals and Systems
EE235
Lecture 18
Leo Lam © 2010-2012
Today’s scary menu
• Transfer Functions
• LCCDE!
Leo Lam © 2010-2012
LTI system transfer function
est
H(s)est
LTI

H ( s) 
 h( )e
 s
d

• s is complex
• H(s): two-sided Laplace Transform of h(t)
Leo Lam © 2010-2012
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LTI system transfer function
est
LTI
H(s)est
LTI
y(t )  AH ( jw)e jwt
• Let s=jw
x(t )  Ae jwt
• LTI systems preserve frequency
• Complex exponential output has same
frequency as the complex exponential input
Leo Lam © 2010-2012
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LTI system transfer function
• Example:
x(t )  Ae jwt

LTI
1 jwt
x(t )  cos(wt )  e  e  jwt
2

y(t )  AH ( jw)e jwt

1
y (t )  H ( jw )e jwt  H ( jw )e  jwt
2
• For real systems (h(t) is real): H ( jw )  H ( jw )
y(t )  Aw cos(wt   )
• where Aw  H ( jw) and   H ( jw )
• LTI systems preserve frequency
Leo Lam © 2010-2012
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
Importance of exponentials
• Makes life easier
• Convolving with est is the same as
multiplication
• Because est are eigenfunctions of LTI systems
• cos(wt) and sin(wt) are real
• Linked to est
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Quick note
e  e u(t )
st
est
estu(t)
Leo Lam © 2010-2012
st
LTI
LTI
H(s)est
H(s)estu(t)
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Which systems are not LTI?
e
e
2 t
2 t
 T  5e
2 t
jt 2 t
 T  5e e
NOT LTI
cos(3t )  T  cos(3 t )
NOT LTI
cos(3t )  T  sin(3t )
cos(3t )  T  0
cos(3t )  T  e
Leo Lam © 2010-2012
2 t
cos(3t )
NOT LTI
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Summary
• Eigenfunctions/values of LTI System
Leo Lam © 2010-2012
LCCDE, what will we do
• Why do we care?
• Because it is everything!
• Represents LTI systems
• Solve it: Homogeneous Solution + Particular
Solution
• Test for system stability (via characteristic
equation)
• Relationship between HS (Natural Response)
and Impulse response
• Using exponentials est
Leo Lam © 2010-2012
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Circuit example
• Want to know the current i(t) around the circuit
C
• Resistor
E  RI
R
• Capacitor
• Inductor
Leo Lam © 2010-2012
Q
EC 
C
dQ
I
dt
dI
EL  L
dt
R
L
E(t) = E0 sin wt
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Circuit example
• Kirchhoff’s Voltage Law (KVL)
C
dI
Q
L  RI   E0 sin wt
dt
C
EC 
d 2I
dI 1 dQ
L 2 R 
 E0w cos wEt
dt C dt
dt
R
R
2
d I
dI 1
L 2  R  I  E0w cos wt
dt
dt C
EL  L
 RI
E(t) = E0 sin wt
input
output
Leo Lam © 2010-2012
Q
C
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dI
dt
L
Differential Eq as LTI system
x(t)
T
y(t)
• Inputs and outputs to system T have a
relationship defined by the LTI system:
• Let “D” mean d()/dt
(a2D2+a1D+a0)y(t)=(b2D2+b1D+b0)x(t)
Defining
Q(D)
Leo Lam © 2010-2012
Defining
P(D)
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Differential Eq as LTI system (example)
x(t)
T
y(t)
• Inputs and outputs to system T have a
relationship defined by the LTI system:
• Let “D” mean d()/dt
Leo Lam © 2010-2012
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Differential Equation: Linearity
• Define:
• Can we show that:
• What do we need to prove?
d ( y1 (t )  y2 (t ))
y (t )  ak1 x1 (t )  k 2 x2 (t )   b
dt
Leo Lam © 2010-2012
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Differential Equation: Time Invariance
• System works the same whenever you use it
• Shift input/output – Proof
• Example:
dx(t )
y (t ) 
dt
• Time shifted system:
• Time invariance?
• Yes: substitute
dx(t  t0 )
y (t  t0 ) 
dt
 for t (time shift the input)
dx ( )
y ( ) 
d
Leo Lam © 2010-2012
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Differential Equation: Time Invariance
• Any pure differential equation is a timeinvariant system:
• Are these linear/time-invariant?
Linear, time-invariant
Linear, not TI
Non-Linear, TI
Linear, time-invariant
Linear, time-invariant
Linear, not TI
Leo Lam © 2010-2012
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LTI System response
• A little conceptual thinking
• Time: t=0
Unknown past
T
Initial condition
zero-input response (t)
Input x(t)
T
zero-state output (t)
• Linear system: Zero-input response and Zero-state
output do not affect each other
Total response(t)=Zero-input response (t)+Zero-state output(t)
Leo Lam © 2010-2012
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Zero input response
• General nth-order differential equation
• Zero-input response: x(t)=0
Homogeneous Equation
• Solution of the Homogeneous Equation is the
natural/general response/solution or complementary
function
Leo Lam © 2010-2012
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Zero input response (example)
• Using the first example:
• Zero-input response: x(t)=0
• Need to solve:
• Solve (challenge)
Leo Lam © 2010-2012
n for “natural response”
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Zero input response (example)
• Solve
• Guess solution:
• Substitute:
• One term must be 0:
Characteristic Equation
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Zero input response (example)
• Solve
• Guess solution:
• Substitute:
• We found:
• Solution:
Unknown constants:
Need initial conditions
Leo Lam © 2010-2012
Characteristic roots =
natural frequencies/
eigenvalues
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Summary
• Differential equation as LTI system
• Complete example tomorrow
Leo Lam © 2010-2012