leo lam © 2010-2011 signals and systems ee235 lecture 20
Post on 13-Dec-2015
222 Views
Preview:
TRANSCRIPT
Leo Lam © 2010-2011
Signals and SystemsEE235
Lecture 20
Leo Lam © 2010-2011
Today’s menu
• Exponential response of LTI system
Leo Lam © 2010-2011
Exponential response of LTI system
3
• 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
4
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 © 2010-2011
Eigenfunctions
5
• 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 © 2010-2011
Eigenfunctions
6
• 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 © 2010-2011
Eigenfunctions
7
• 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 © 2010-2011
Recall Linear Algebra
8
• 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 © 2010-2011
Exponential response of LTI system
9
• 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 © 2010-2011
LTI system transfer function
10
LTIest H(s)est
( ) ( ) sH s h e d
• s is complex• H(s): two-sided Laplace Transform of h(t)
Leo Lam © 2010-2011
LTI system transfer function
11
• 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 © 2010-2011
LTI system transfer function
12
• 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 © 2010-2011
Importance of exponentials
13
• 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
Quick note
14
LTIest H(s)est
( )st ste e u t
LTIestu(t) H(s)estu(t)
Leo Lam © 2010-2011
Which systems are not LTI?
15
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 © 2010-2011
Summary
• Eigenfunctions/values of LTI System
top related