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Leo Lam © 2010-2011
Signals and Systems
EE235
Leo Lam © 2010-2011
Merry Christmas!
• Q: What is Quayle-o-phobia? • A: The fear of the exponential (e).
Leo Lam © 2010-2011
Today’s scary menu
• Wrap up LTI system properties• Onto Fourier Series!
Leo Lam © 2010-2011
System properties testing given h(t)
4
• 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 © 2010-2011
Causality for LTI
5
• 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:
• If LTI system is causal:
( ) ( ) ( )y t h x t d
0
( ) ( ) ( )y t h x t d
Leo Lam © 2010-2011
Causality for LTI
6
• 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 <
( ) ( ) ( )t
y t x h t d
( ) ( ) ( )y t x h t d
Only Integrate to t for causal
systems
Leo Lam © 2010-2011
Convolution of two causal signals
7
• A signal x(t) is a causal signal if x(t)=0 for all t<0
• Consider:
• 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
1 2( ) ( ) ( )y t x x t d
1 2
0
( ) ( ) ( )t
y t x x t d Only Integrate from 0 to t for
2 causal signals
Leo Lam © 2010-2011
Step response of LTI system
8
• Impulse response h(t)
• Step response s(t)
• For a causal system:
T u(t)*h(t)u(t)
T h(t)(t)
0
( ) ( )* ( ) ( )t
s t u t h t h d
( )* ( ) ( ) ( ) ( )h t u t h u t d s t
Only Integrate from 0 to t = Causal! (Proof for causality)
Leo Lam © 2010-2011
Step response example for LTI system
9
• If the impulse response to an LTI system is:
• First: is it causal?• Find s(t)
3( ) 5 ( )th t e u t
3( ) 5 ( ) ( )s t e u u t d
( )* ( ) ( ) ( ) ( )h t u t h u t d s t
0
( ) ( )* ( ) ( )t
s t u t h t h d
3
0
5t
e d 3
0
5
3
t
e
35 5( )
3 3e u t
351 ( )
3te u t
Leo Lam © 2010-2011
Stability of LTI System
10
• An LTI system – BIBO stable
• Impulse response must be finite
3( )h d B
Bounded input
system
Bounded output
B1 , B2, B3 are constants
Leo Lam © 2010-2011
Stability of LTI System
11
• Is this condition sufficient for stability?
• Prove it:
3( )h d B
abs(sum)≤sum(abs)
abs(prod)=prod(abs)
bounded input
if
Q.E.D.
Leo Lam © 2010-2011
Stability of LTI System
12
• Is h(t)=u(t) stable?• Need to prove that 3( )h d B
Leo Lam © 2010-2011
Invertibility of LTI System
13
• A system is invertible if you can find the input, given the output (undo-ing possible)
• 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
( ) (
( )
( ) (( )* ( ))* ( )*( ( )* )
( ) ( )
( (
(
)
)
)
)
i
i i
h t
h t h ty t x t h t x t h t
x t t x t
h t t
Leo Lam © 2010-2011
LTI System Properties
14
• Example
– Causal?– Stable?– Invertible?
( ) 5 ( 1)h t t
1( ) ( 1)
5ih t t YES
5 ( 1) 5t dt
YES
( ) 0 for 0h t t YES
Leo Lam © 2010-2011
LTI System Properties
15
• Example
– Causal?– Stable? YES
YES
2( ) 3 ( )th t e u t( ) 0 for 0h t t
2 2
0
3| 3 ( ) | 3
2t te u t dt e dt
Leo Lam © 2010-2011
LTI System Properties
16
• How about these? Causal/Stable?
| |( ) th t e
( ) ( 1)h t u t
0.5( ) 3 cos(200 ) ( )th t e t u t
Stable, not causal
Causal, not stable
Stable and causal
Leo Lam © 2010-2011
LTI System Properties Summary
17
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
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
Leo Lam © 2010-2011
Exponential response of LTI system
18
• Why do we care?• Convolution = complicated• Leading to frequency etc.
Leo Lam © 2010-2011
Review: Faces of exponentials
19
• Constants for with s=0+j0
• Real exponentials forwith s=a+j0
• Sine/Cosine for
with s=0+jw and a=1/2• Complex exponentials for
s=a+jw
atx )( Rastaetx )(
atetx )( Rastetx )(
)cos()( ttx R
)()( stst eeatx stetx )( Cs
Leo Lam © 2010-2011
Exponential response of LTI system
20
• What is y(t) if ? )(*)( thety st
Given a specific s, H(s) is a constant
S
Output is just a constant times the input
Leo Lam © 2010-2011
Exponential response of LTI system
21
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
