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
• Let s=jw
x ( t )  Ae
jw t
jw t
• 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
x ( t )  cos( w t ) 
jw t
1
2
y ( t )  AH ( jw ) e
LTI
e
jw t
e
 jw t

y (t ) 
1
2
H ( j w ) e
jw t
jw t
 H (  jw ) e
 jw t
• For real systems (h(t) is real): H ( j w )  H (  j w )
y ( t )  Aw cos( w t   )
• where Aw  H ( j w ) and    H ( j w )
• 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
Leo Lam © 2010-2012
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Quick note
e
est
estu(t)
Leo Lam © 2010-2012
st
 e u (t )
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
 T  5e e
jt
2 t
NOT LTI
cos(3 t )  T  cos(3 t )
NOT LTI
cos(3 t )  T  sin(3 t )
cos(3 t )  T  0
cos(3 t )  T  e
Leo Lam © 2010-2012
2 t
cos(3 t )
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 R  RI
• Capacitor
EC 
I 
Q
C
dQ
R
L
dt
• Inductor
Leo Lam © 2010-2012
EL  L
dI
dt
E(t) = E 0 sin wt
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Circuit example
• Kirchhoff’s Voltage Law (KVL)
C
L
dI
 RI 
dt
d I
dt
2
R
2
L
d I
dt
 E 0 sin w t
C
2
L
Q
2
R
dI
EC 

1 dQ
dt
C dt
dI
1
dt

C
R
 E 0 w cos wEt
I  E 0 w cos w t
R
C
EL  L
 RI
E(t) = E 0 sin wt
input
output
Leo Lam © 2010-2012
Q
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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
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Differential Equation: Linearity
• Define:
• Can we show that:
• What do we need to prove?
y ( t )  a  k 1 x1 ( t )  k 2 x 2 ( t )   b
Leo Lam © 2010-2012
d ( y1 ( t )  y 2 ( t ))
dt
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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
dt
 for t (time shift the input)
y ( ) 
Leo Lam © 2010-2012
y (t  t 0 ) 
dx ( t  t 0 )
dx ( )
d
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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)
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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