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

3. Leo Lam © 2010-2013 Working with computers

4. Leo Lam © 2010-2013 Today’s menu Vote for Midterm Date Convolution properties LTI properties

5. Convolution Properties Leo Lam © 2010-2013 4

6. Commutative Leo Lam © 2010-2013 5 Commutative: Doesn’t matter which signal to flip, it’s the same Pick the easier one!

7. Associative Leo Lam © 2010-2013 6 Associative: Order doesn’t matter h 1 (t)h 2 (t) x(t)y(t) The overall response of two LTI systems in series is given by

8. Distributive Leo Lam © 2010-2013 7 Distributive: Two types h 1 (t) h 2 (t) x(t) y(t) + The overall response of two systems in parallel is h(t) x 1 (t) y(t) + x 2 (t) “Divide and conquer” for input signals

9. More Convolution Properties Leo Lam © 2010-2013 8 Convolution of any signal with an impulse, gives the same signal Convolution of any signal with a shifted impulse, shifts the signal

10. More Convolution Properties Leo Lam © 2010-2013 9 Convolution of the impulse response of an LTI system with a unit step, gives its step response s(t).

11. Another implication for LTI Leo Lam © 2010-2013 10 Recall: d/dt u(t) h(t) s(t) h(t) d/dt u(t)  (t) “taking the derivative” is an LTI system, and using associative properties: We can find the impulse response of a system from its step response s(t)

12. More Convolution Properties Leo Lam © 2010-2013 11 Convolution with a time-shifted signal, gives a time shifted output: –If –then

13. Summary: Leo Lam © 2010-2013 12 Convolution properties –Commutative –Associative –Distributive –Convolve with impulse –Convolve with shifted impulse –Convolve h(t) with u(t) gives s(t)

14. Echo Properties Leo Lam © 2010-2013 13 Echo properties of impulse * 3 x(t)  (t-3) t t t 3 = What does this system do?

15. Echo Properties Leo Lam © 2010-2013 14 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)

16. Echo Properties Leo Lam © 2010-2013 15 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

17. Echo Properties Leo Lam © 2010-2013 16 More… With multiple time shifts, add them all up.

18. Finding Impulse Response Leo Lam © 2010-2013 17 Example: find h(t) when 1) Plug in  (t) for x(t)

19. System properties testing given h(t) Leo Lam © 2010-2013 18 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

20. Causality for LTI Leo Lam © 2010-2013 19 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:

21. Causality for LTI Leo Lam © 2010-2013 20 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

22. Convolution of two causal signals Leo Lam © 2010-2013 21 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

23. Step response of LTI system Leo Lam © 2010-2013 22 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)

24. Step response example for LTI system Leo Lam © 2010-2013 23 If the impulse response to an LTI system is: First: is it causal? Find s(t)