1. Leo Lam © 2010-2013 Signals and Systems EE235

2. Leo Lam © 2010-2013 Happy Tuesday! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).

3. Leo Lam © 2010-2013 Todays scary menu Wrap up LTI system properties Onto Fourier Series!

4. Stability of LTI System Leo Lam © 2010-2013 4 An LTI system – BIBO stable Impulse response must be finite Bounded input system Bounded output B 1, B 2, B 3 are constants

5. Stability of LTI System Leo Lam © 2010-2013 5 Is this condition sufficient for stability? Prove it: abs(sum)sum(abs) abs(prod)=prod(abs) bounded input if Q.E.D.

6. Stability of LTI System Leo Lam © 2010-2013 6 Is h(t)=u(t) stable? Need to prove that

7. Invertibility of LTI System Leo Lam © 2010-2013 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 h i (t) Often hard to do…dont worry for now unless its obvious

8. LTI System Properties Leo Lam © 2010-2013 8 Example –Causal? –Stable? –Invertible? YES

9. LTI System Properties Leo Lam © 2010-2013 9 Example –Causal? –Stable? YES

10. LTI System Properties Leo Lam © 2010-2013 10 How about these? Causal/Stable? Stable, not causal Causal, not stable Stable and causal

11. LTI System Properties Summary Leo Lam © 2010-2013 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:

12. Exponential response of LTI system Leo Lam © 2010-2013 12 Why do we care? Convolution = complicated Leading to frequency etc.

13. Review: Faces of exponentials Leo Lam © 2010-2013 13 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

14. Exponential response of LTI system Leo Lam © 2010-2013 14 What is y(t) if ? Given a specific s, H(s) is a constant S Output is just a constant times the input

15. Exponential response of LTI system Leo Lam © 2010-2013 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

16. Eigenfunctions Leo Lam © 2010-2013 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)=e at for e at is an eigenfunction; a is the eigenvalue S{x(t)}

17. Eigenfunctions Leo Lam © 2010-2013 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)=e at for e at is an eigenfunction; 0 is the eigenvalue S{x(t)}

18. Eigenfunctions Leo Lam © 2010-2013 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

19. Recall Linear Algebra Leo Lam © 2010-2013 19 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 )

20. Exponential response of LTI system Leo Lam © 2010-2013 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 e st, no convolution needed! S

21. LTI system transfer function Leo Lam © 2010-2013 21 LTI e st H(s)e st s is complex H(s): two-sided Laplace Transform of h(t)

22. LTI system transfer function Leo Lam © 2010-2013 22 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

23. LTI system transfer function Leo Lam © 2010-2013 23 Example: For real systems (h(t) is real): where and LTI systems preserve frequency LTI

24. Importance of exponentials Leo Lam © 2010-2013 24 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

25. Quick note Leo Lam © 2010-2013 25 LTI e st H(s)e st LTI e st u(t) H(s)e st u(t)

26. Which systems are not LTI? Leo Lam © 2010-2013 26 NOT LTI

27. Leo Lam © 2010-2013 Summary Eigenfunctions/values of LTI System