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