Download presentation

Presentation is loading. Please wait.

Published byAlonso Grimes Modified over 2 years ago

1
Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 16

2
Leo Lam © 2010-2012 Merry Christmas! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).

3
Leo Lam © 2010-2012 Today’s scary menu Wrap up LTI system properties (Midterm) Midterm Wednesday! Onto Fourier Series!

4
System properties testing given h(t) Leo Lam © 2010-2012 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

5
Causality for LTI Leo Lam © 2010-2012 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:

6
Causality for LTI Leo Lam © 2010-2012 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 < Only Integrate to t for causal systems

7
Convolution of two causal signals Leo Lam © 2010-2012 7 A signal x(t) is a causal signal if x(t)=0 for all t<0 Consider: If x 2 (t) is causal then x 2 ( t- )=0 for all ( t- )<0 i.e. x 1 ( )x 2 ( t- )=0 for all t< If x 1 (t) is causal then x 1 ( )=0 for all <0 i.e. x 1 ( )x 2 ( t- )=0 for all <0 Only Integrate from 0 to t for 2 causal signals

8
Step response of LTI system Leo Lam © 2010-2012 8 Impulse response h(t) Step response s(t) For a causal system: T u(t)*h(t) u(t) T h(t) (t) Only Integrate from 0 to t = Causal! (Proof for causality)

9
Step response example for LTI system Leo Lam © 2010-2012 9 If the impulse response to an LTI system is: First: is it causal? Find s(t)

10
Stability of LTI System Leo Lam © 2010-2012 10 An LTI system – BIBO stable Impulse response must be finite Bounded input system Bounded output B 1, B 2, B 3 are constants

11
Stability of LTI System Leo Lam © 2010-2012 11 Is this condition sufficient for stability? Prove it: abs(sum)≤sum(abs) abs(prod)=prod(abs) bounded input if Q.E.D.

12
Stability of LTI System Leo Lam © 2010-2012 12 Is h(t)=u(t) stable? Need to prove that

13
Invertibility of LTI System Leo Lam © 2010-2012 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 h i (t) Often hard to do…don’t worry for now unless it’s obvious

14
LTI System Properties Leo Lam © 2010-2012 14 Example –Causal? –Stable? –Invertible? YES

15
LTI System Properties Leo Lam © 2010-2012 15 Example –Causal? –Stable? YES

16
LTI System Properties Leo Lam © 2010-2012 16 How about these? Causal/Stable? Stable, not causal Causal, not stable Stable and causal

17
LTI System Properties Summary Leo Lam © 2010-2012 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:

18
Leo Lam © 2010-2012 Summary LTI system properties

19
Review: Faces of exponentials Leo Lam © 2010-2012 19 Constants for with s=0+j0 Real exponentials for with s=a+j0 Sine/Cosine for with s=0+j and a=1/2 Complex exponentials for s=a+j

20
Exponential response of LTI system Leo Lam © 2010-2012 20 What is y(t) if ? Given a specific s, H(s) is a constant S Output is just a constant times the input

21
Exponential response of LTI system Leo Lam © 2010-2012 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

22
Eigenfunctions Leo Lam © 2010-2011 22 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)=e at for e at is an eigenfunction; a is the eigenvalue S{x(t)}

23
Eigenfunctions Leo Lam © 2010-2011 23 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)=e at for e at is an eigenfunction; 0 is the eigenvalue S{x(t)}

24
Eigenfunctions Leo Lam © 2010-2011 24 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

25
Recall Linear Algebra Leo Lam © 2010-2011 25 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 (x i, i )

26
Exponential response of LTI system Leo Lam © 2010-2011 26 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 e st, no convolution needed! S

27
LTI system transfer function Leo Lam © 2010-2011 27 LTI e st H(s)e st s is complex H(s): two-sided Laplace Transform of h(t)

28
LTI system transfer function Leo Lam © 2010-2011 28 Let s=j LTI systems preserve frequency Complex exponential output has same frequency as the complex exponential input LTI e st H(s)e st LTI

29
LTI system transfer function Leo Lam © 2010-2011 29 Example: For real systems (h(t) is real): where and LTI systems preserve frequency LTI

30
Importance of exponentials Leo Lam © 2010-2011 30 Makes life easier Convolving with e st is the same as multiplication Because e st are eigenfunctions of LTI systems cos(t) and sin(t) are real Linked to e st

31
Quick note Leo Lam © 2010-2011 31 LTI e st H(s)e st LTI e st u(t) H(s)e st u(t)

32
Which systems are not LTI? Leo Lam © 2010-2011 32 NOT LTI

33
Leo Lam © 2010-2011 Summary Eigenfunctions/values of LTI System

Similar presentations

Presentation is loading. Please wait....

OK

CHAPTER 4 Laplace Transform.

CHAPTER 4 Laplace Transform.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on indian english literature Ppt on blood donation in india Ppt on polynomials Ppt on waxes are lipids Ppt on islam and science Ppt on index numbers in excel Ppt on jpeg image compression Ppt on safe construction practices for class 10 Ppt on event management company Download ppt on biodegradable and nonbiodegradable waste