Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 18.
23
Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 18
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Transcript of Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 18.
Leo Lam © 2010-2012
Signals and Systems
EE235Lecture 18
Leo Lam © 2010-2012
Today’s scary menu
• Transfer Functions• LCCDE!
Leo Lam © 2010-2012
LTI system transfer function
3
LTIest H(s)est
( ) ( ) sH s h e d
• s is complex• H(s): two-sided Laplace Transform of h(t)
Leo Lam © 2010-2012
LTI system transfer function
4
• Let s=jw
• LTI systems preserve frequency• Complex exponential output has same
frequency as the complex exponential input
LTIest H(s)est
( ) j tx t Ae LTI ( ) ( ) j ty t AH j e
Leo Lam © 2010-2012
LTI system transfer function
5
• Example:
• For real systems (h(t) is real):
• where and• LTI systems preserve frequency
( ) j tx t Ae LTI ( ) ( ) j ty t AH j e
tjtj eettx 2
1)cos()( tjtj ejHejHty )()(
2
1)(
)()( jHjH
)cos()( tAty
)( jHA )( jH
Leo Lam © 2010-2012
Importance of exponentials
6
• 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
Quick note
7
LTIest H(s)est
( )st ste e u t
LTIestu(t) H(s)estu(t)
Leo Lam © 2010-2012
Which systems are not LTI?
8
2 2
2 2
2
5
5
cos(3 ) cos(3 )
cos(3 ) sin(3 )
cos(3 ) 0
cos(3 ) cos(3 )
t t
t jt t
t
e T e
e T e e
t T t
t T t
t T
t T e t
NOT LTI
NOT LTI
NOT LTI
Leo Lam © 2010-2012
Summary
• Eigenfunctions/values of LTI System
Leo Lam © 2010-2012
LCCDE, what will we do
10
• 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
Circuit example
11
• Want to know the current i(t) around the circuit• Resistor
• Capacitor
• Inductor
R L
C
E(t) = E 0 s in t
RIER
C
QEC
dt
dQI
dt
dILEL
Leo Lam © 2010-2012
Circuit example
12
• Kirchhoff’s Voltage Law (KVL)
R L
C
E(t) = E 0 s in t
RIER
C
QEC
dt
dILEL
tEC
QRI
dt
dIL sin0
tEdt
dQ
Cdt
dIR
dt
IdL cos1
02
2
tEICdt
dIR
dt
IdL cos1
02
2
output
input
Leo Lam © 2010-2012
Differential Eq as LTI system
13
• Inputs and outputs to system T have a relationship defined by the LTI system:
• Let “D” mean d()/dt
Tx(t) y(t)
(a2D2+a1D+a0)y(t)=(b2D2+b1D+b0)x(t)Defining
Q(D)Defining
P(D)
Leo Lam © 2010-2012
Differential Eq as LTI system (example)
14
• Inputs and outputs to system T have a relationship defined by the LTI system:
• Let “D” mean d()/dt
Tx(t) y(t)
Leo Lam © 2010-2012
Differential Equation: Linearity
15
• Define:
• Can we show that:
• What do we need to prove?
dt
tytydbtxktxkaty
))()(()()()( 21
2211
Leo Lam © 2010-2012
Differential Equation: Time Invariance
16
• System works the same whenever you use it• Shift input/output – Proof• Example:
• Time shifted system:• Time invariance?• Yes: substitute t for t (time shift the input)
dt
tdxty
)()(
dt
ttdxtty
)()( 00
d
dxy
)()(
Leo Lam © 2010-2012
Differential Equation: Time Invariance
17
• Any pure differential equation is a time-invariant 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
LTI System response
18
• A little conceptual thinking• Time: t=0
• Linear system: Zero-input response and Zero-state output do not affect each other
TUnknown past Initial conditionzero-input response (t)
TInput x(t) zero-state output (t)
Total response(t)=Zero-input response (t)+Zero-state output(t)
Leo Lam © 2010-2012
Zero input response
19
• General nth-order differential equation
• Zero-input response: x(t)=0
• Solution of the Homogeneous Equation is the natural/general response/solution or complementary function
Homogeneous Equation
Leo Lam © 2010-2012
Zero input response (example)
20
• Using the first example:
• Zero-input response: x(t)=0
• Need to solve:
• Solve (challenge)n for “natural response”
Leo Lam © 2010-2012
Zero input response (example)
21
• Solve
• Guess solution:
• Substitute:
• One term must be 0:
Characteristic Equation
Leo Lam © 2010-2012
Zero input response (example)
22
• Solve
• Guess solution:
• Substitute:
• We found:
• Solution:
Characteristic roots = natural frequencies/
eigenvalues
Unknown constants:Need initial conditions
Leo Lam © 2010-2012
Summary
• Differential equation as LTI system• Complete example tomorrow
