Leo Lam © 2010-2013

Signals and Systems

EE235

Leo Lam © 2010-2013

Happy Tuesday!

• Q: What is Quayle-o-phobia? • A: The fear of the exponential (e).

Leo Lam © 2010-2013

Today’s scary menu

• Wrap up LTI system properties• Onto Fourier Series!

Leo Lam © 2010-2013

Stability of LTI System

4

• 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 © 2010-2013

Stability of LTI System

5

• 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 © 2010-2013

Stability of LTI System

6

• Is h(t)=u(t) stable?• Need to prove that 3( )h d B

Leo Lam © 2010-2013

Invertibility of LTI System

7

• 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 © 2010-2013

LTI System Properties

8

• 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 © 2010-2013

LTI System Properties

9

• 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 © 2010-2013

LTI System Properties

10

• 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 © 2010-2013

LTI System Properties Summary

11

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 © 2010-2013

Exponential response of LTI system

12

• Why do we care?• Convolution = complicated• Leading to frequency etc.

Leo Lam © 2010-2013

Review: Faces of exponentials

13

• 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 © 2010-2013

Exponential response of LTI system

14

• 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 © 2010-2013

Exponential response of LTI system

15

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 © 2010-2013

Eigenfunctions

16

• Definition: An eigenfunction of a system S is any non-zero x(t) such that

• Where is called an eigenvalue.• Example:

• What is the y(t) for x(t)=eat for

• eat is an eigenfunction; a is the eigenvalue

)()( txtxS

( ) ( )d

y t x tdt

Ra)()( taxaety at

S{x(t)}

Leo Lam © 2010-2013

Eigenfunctions

17

• Definition: An eigenfunction of a system S is any non-zero x(t) such that

• Where is called an eigenvalue.• Example:

• What is the y(t) for x(t)=eat for

• eat is an eigenfunction; 0 is the eigenvalue

)()( txtxS

( ) ( )d

y t x tdt

0a)(00)( txty

S{x(t)}

Leo Lam © 2010-2013

Eigenfunctions

18

• Definition: An eigenfunction of a system S is any non-zero x(t) such that

• Where is called an eigenvalue.• Example:

• What is the y(t) for x(t)=u(t)

• u(t) is not an eigenfunction for S

)()( txtxS

( ) ( )d

y t x tdt

)()()( tautty

Leo Lam © 2010-2013

Recall Linear Algebra

19

• Given nxn matrix A, vector x, scalar l• x is an eigenvector of A, corresponding to

eigenvalue l ifAx=lx

• Physically: Scale, but no direction change• Up to n eigenvalue-eigenvector pairs (xi,li)

Leo Lam © 2010-2013

Exponential response of LTI system

20

• 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!

S

Leo Lam © 2010-2013

LTI system transfer function

21

LTIest H(s)est

( ) ( ) sH s h e d

• s is complex• H(s): two-sided Laplace Transform of h(t)

Leo Lam © 2010-2013

LTI system transfer function

22

• 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 © 2010-2013

LTI system transfer function

23

• 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 © 2010-2013

Importance of exponentials

24

• 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 © 2010-2013

Quick note

25

LTIest H(s)est

( )st ste e u t

LTIestu(t) H(s)estu(t)

Leo Lam © 2010-2013

Which systems are not LTI?

26

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 © 2010-2013

Summary

• Eigenfunctions/values of LTI System