Transcript h(t
Signals and Systems
EE235
Lecture 16
Leo Lam © 2010-2012
Merry Christmas!
• Q: What is Quayle-o-phobia?
• A: The fear of the exponential (e).
Leo Lam © 2010-2012
Today’s scary menu
• Wrap up LTI system properties (Midterm)
• Midterm Wednesday!
• Onto Fourier Series!
Leo Lam © 2010-2012
System properties testing given h(t)
• Impulse response h(t) fully specifies an LTI
system
• Gives additional tools to test system
properties for LTI systems
• Additional ways to manipulate/simplify
problems, too
Leo Lam © 2010-2012
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Causality for LTI
• A system is causal if the output does not
depend on future times of the input
• An LTI system is causal if h(t)=0 for t<0
• Generally:
y (t )
h( ) x(t )d
• If LTI system is causal:
y (t ) h( ) x(t )d
0
Leo Lam © 2010-2012
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Causality for LTI
• An LTI system is causal if h(t)=0 for t<0
• If h(t) is causal, h(t-)=0 for all (t- )<0 or
all t <
y(t )
x( )h(t )d
t
y(t )
Only
Integrate to t
for causal
systems
x( )h(t )d
Leo Lam © 2010-2012
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Convolution of two causal signals
• A signal x(t) is a causal signal if x(t)=0 for all
t<0
• Consider: y (t ) x1 ( ) x2 (t )d
• If x2(t) is causal then x2(t-)=0 for all (t- )<0
• i.e. x1( )x2(t-)=0 for all t<
• If x1(t) is causal then x1()=0 for all <0
• i.e. x1( )x2(t-)=0 for all <0
t
y (t ) x1 ( ) x2 (t )d
0
Leo Lam © 2010-2012
Only Integrate
from 0 to t for
2 causal
signals
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Step response of LTI system
• Impulse response h(t) (t)
T
h(t)
• Step response s(t)
T
u(t)*h(t)
u(t)
h(t )* u (t )
h( )u(t )d s(t )
• For a causal system:
t
s (t ) u (t ) * h(t ) h( )d
0
Leo Lam © 2010-2012
Only Integrate
from 0 to t =
Causal! (Proof
for causality)
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Step response example for LTI system
• If the impulse response to an LTI system is:
h(t ) 5e3t u (t )
t
• First: is it causal?
• Find s(t)
s (t ) u (t ) * h(t ) h( )d
0
hs((tt)*
()u ()tu(t)d)ds(t )
) u (t ) 5e h3(u
t
5e
0
5 3
e
3
3
t
0
Leo Lam © 2010-2012
d
5
5 3 5
e u (t ) 1 e 3t u (t )
3
3
3
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Stability of LTI System
• An LTI system – BIBO stable
h( ) d B3
• Impulse response must be finite
Bounded
input
system
Bounded
output
B1 , B2, B3 are
constants
Leo Lam © 2010-2012
10
Stability of LTI System
• Is this condition sufficient for stability?
h( ) d B3
• Prove it:
abs(sum)≤sum(abs)
abs(prod)=prod(abs)
bounded input
if
Q.E.D.
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11
Stability of LTI System
• Is h(t)=u(t) stable?
• Need to prove that
h( ) d B3
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12
Invertibility of LTI System
• A system is invertible if you can find the input,
given the output (undo-ing possible)
h(t ) hi (t ) (t )
y (t ) ( x(t )* h(t ))* hi (t ) x(t )*(h(t )* hi (t ))
x(t ) (t ) x(t )
• You can prove invertibility of the system with
impulse response h(t) by finding the impulse
response of the inverse system hi(t)
• Often hard to do…don’t worry for now unless
it’s obvious
Leo Lam © 2010-2012
13
LTI System Properties
• Example
h(t ) 5 (t 1)
– Causal?
– Stable?
– Invertible?
Leo Lam © 2010-2012
YES
YES
YES
h(t ) 0 for t 0
5 (t 1)dt 5
1
hi (t ) (t 1)
5
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LTI System Properties
• Example
2t
h(t ) 3e u (t )
– Causal?
– Stable?
Leo Lam © 2010-2012
YES
YES
h(t ) 0 for t 0
3
| 3e u(t ) | dt 0 3e dt 2
2 t
2 t
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LTI System Properties
• How about these? Causal/Stable?
h(t ) e
|t |
Stable, not causal
h(t ) u (t 1)
h(t ) 3e
0.5t
Leo Lam © 2010-2012
cos(200 t )u (t )
Causal, not stable
Stable and causal
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LTI System Properties Summary
For ALL systems
• y(t)=T{x(t)}
• x-y equation
describes system
• Property tests in
terms of basic
definitions
– Causal: Find time
region of x() used in
y(t)
– Stable: BIBO test or
counter-example
Leo Lam © 2010-2012
For LTI systems
ONLY
• y(t)=x(t)*h(t)
• h(t) =impulse
response
• Property tests on
h(t)
– Causal: h(t)=0 t<0
– Stable:
| h(t ) | dt
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Summary
• LTI system properties
Leo Lam © 2010-2012
Review: Faces of exponentials
• Constants x(t ) a for a R
x(t ) ae st with s=0+j0
• Real exponentials x(t ) e at for a R
x(t ) e st with s=a+j0
• Sine/Cosine x(t ) cos(wt ) for w R
x(t ) a(e st e st ) with s=0+jw and a=1/2
• Complex exponentials x(t ) e st for s C
s=a+jw
Leo Lam © 2010-2012
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Exponential response of LTI system
st
y
(
t
)
e
* h(t ) ?
• What is y(t) if
Given a specific s,
H(s) is a constant
S
Output is just a constant times the input
Leo Lam © 2010-2012
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Exponential response of LTI system
LTI
• Varying s, then H(s) is a function of s
• H(s) becomes a Transfer Function of the
input
• If s is “frequency”…
• Working toward the frequency domain
Leo Lam © 2010-2012
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Eigenfunctions
• Definition: An eigenfunction of a system S is
any non-zero x(t) such that
Sx(t ) x(t )
• Where is called an eigenvalue.
d
• Example:
y (t )
dt
x(t )
S{x(t)}
• What is the y(t) for x(t)=eat for a R
y(t ) ae at ax(t )
• eat is an eigenfunction; a is the eigenvalue
Leo Lam © 2010-2011
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Eigenfunctions
• Definition: An eigenfunction of a system S is
any non-zero x(t) such that
Sx(t ) x(t )
• Where is called an eigenvalue.
d
• Example:
y (t )
dt
x(t )
S{x(t)}
• What is the y(t) for x(t)=eat for a 0
y (t ) 0 0 x(t )
• eat is an eigenfunction; 0 is the eigenvalue
Leo Lam © 2010-2011
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Eigenfunctions
• Definition: An eigenfunction of a system S is
any non-zero x(t) such that
Sx(t ) x(t )
• Where is called an eigenvalue.
d
• Example:
y (t )
dt
x(t )
• What is the y(t) for x(t)=u(t)
y (t ) (t ) au (t )
• u(t) is not an eigenfunction for S
Leo Lam © 2010-2011
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Recall Linear Algebra
• Given nxn matrix A, vector x, scalar
• x is an eigenvector of A, corresponding to
eigenvalue if
Ax=x
• Physically: Scale, but no direction change
• Up to n eigenvalue-eigenvector pairs (xi,i)
Leo Lam © 2010-2011
25
Exponential response of LTI system
S
• Complex exponentials are eigenfunctions of
LTI systems
• For any fixed s (complex valued), the output is
just a constant H(s), times the input
• Preview: if we know H(s) and input is est, no
convolution needed!
Leo Lam © 2010-2011
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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 © 2010-2011
27
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 © 2010-2011
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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 © 2010-2011
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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 © 2010-2011
30
Quick note
e e u(t )
st
est
estu(t)
Leo Lam © 2010-2011
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 © 2010-2011
2 t
cos(3t )
NOT LTI
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Summary
• Eigenfunctions/values of LTI System
Leo Lam © 2010-2011