1. Leo Lam © 2010-2011 Signals and Systems EE235 Leo Lam

2. Leo Lam © 2010-2011 Today’s menu Convolution properties Homework 2 Solution posted (by the end of lecture) Homework 3 will be posted tonight

3. Summary: Leo Lam © 2010-2011 3 Convolution properties –Commutative –Associative –Distributive –Convolve with impulse –Convolve with shifted impulse –Convolve h(t) with u(t) gives s(t)

4. Echo Properties Leo Lam © 2010-2011 4 Echo properties of impulse * 3 x(t)  (t-3) t t t 3 = What does this system do?

5. Echo Properties Leo Lam © 2010-2011 5 Multiple echoes (your turn) * 3 x(t) (t) +0.5(t-3)+0.25(t-6) t t 6 3 t 6 = (1) (0.5) (0.25)

6. Echo Properties Leo Lam © 2010-2011 6 Another example * 2 x(t) h(t)=  (t) +0.5  (t-2) t t (1) (0.5) 1 2 t (0.5) (1.5) (1) 13 Solve and plot? Hint: Distribute

7. Echo Properties Leo Lam © 2010-2011 7 More… With multiple time shifts, add them all up.

8. Finding Impulse Response Leo Lam © 2010-2011 8 Example: find h(t) when 1) Plug in  (t) for x(t)

9. System properties testing given h(t) Leo Lam © 2010-2011 9 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

10. Causality for LTI Leo Lam © 2010-2011 10 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:

11. Causality for LTI Leo Lam © 2010-2011 11 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

12. Convolution of two causal signals Leo Lam © 2010-2011 12 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

13. Step response of LTI system Leo Lam © 2010-2011 13 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 for 2 causal signals

14. Step response example for LTI system Leo Lam © 2010-2011 14 If the impulse response to an LTI system is: First: is it causal? Find s(t)

15. Stability of LTI System Leo Lam © 2010-2011 15 An LTI system – BIBO stable Impulse response must be finite Bounded input system Bounded output B 1, B 2, B 3 are constants

16. Stability of LTI System Leo Lam © 2010-2011 16 Is this condition sufficient for stability? Prove it: abs(sum)≤sum(abs) abs(prod)=prod(abs) bounded input if Q.E.D.

17. Stability of LTI System Leo Lam © 2010-2011 17 Is h(t)=u(t) stable? Need to prove that

18. Invertibility of LTI System Leo Lam © 2010-2011 18 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

19. LTI System Properties Leo Lam © 2010-2011 19 Example –Causal? –Stable? –Invertible? YES

20. LTI System Properties Leo Lam © 2010-2011 20 Example –Causal? –Stable? YES

21. LTI System Properties Leo Lam © 2010-2011 21 How about these? Causal/Stable? Stable, not causal Causal, not stable Stable and causal

22. LTI System Properties Summary Leo Lam © 2010-2011 22 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:

23. Leo Lam © 2010-2011 Summary LTI system properties examples