1. Leo Lam © 2010-2012 Signals and Systems EE235

2. Leo Lam © 2010-2012 Surgery Five surgeons were taking a coffee break and were discussing their work. The first said, "I think accountants are the easiest to operate on. You open them up and everything inside is numbered.“ The second said, "I think librarians are the easiest to operate on. You open them up and everything inside is in alphabetical order.“ The third said, "I like to operate on electricians. You open them up and everything inside is color-coded.“ The fourth one said, "I like to operate on lawyers. They're heartless, spineless, gutless, and their heads and their butts are interchangeable." Fifth surgeon said, "I like Engineers...they always understand when you have a few parts left over at the end..."

3. Leo Lam © 2010-2012 Today’s menu Lab 2 this week (posted last week) Convolution

4. Finding Impulse Response Leo Lam © 2010-2012 Knowing T, and let x(t)=(t) What is h(t)? 4 This system is not linear –impulse response not useful.

5. Summary: Impulse response for LTI Systems Leo Lam © 2010-2012 5 T  (t-  )h(t-  ) Time Invariant T Linear Weighted “sum” of impulses in Weighted “sum” of impulse responses out First we had Superposition

6. Summary: another vantage point Leo Lam © 2010-2012 6 LINEARITY TIME INVARIANCE Output! An LTI system can be completely described by its impulse response! And with this, you have learned Convolution!

7. Convolution Integral Leo Lam © 2010-2012 7 Standard Notation The output of a system is its input convolved with its impulse response

8. Convolution Integral Leo Lam © 2010-2012 8 Standard Notation The output of a system is its input convolved with its impulse response

9. Quick recap Leo Lam © 2010-2012 9 x(t) is the sum of the weighted shifted impulses

10. Convolution integral Leo Lam © 2010-2012 10 Function of  =h(- (-t)) Function of  shifted by t , not t

11. Convolution integral Leo Lam © 2010-2012 11 Two ways to evaluate: –Mathematically –Graphically For graphical convolution, see demo in Riskin interactive notes (lesson 6, lesson 7)

12. Convolution (mathematically) Leo Lam © 2010-2012 12 Use sampling property of delta: Evaluate integral to arrive at output signal: Does this make sense physically?

13. Convolution (graphically) Leo Lam © 2010-2012 13 2 -6 τ y(t=-5) -5 t Does not move wrt t -2 Goal: Find y(t) x( τ ) and h(t- τ ) no overlap, y(t)=0

14. Convolution (graphically) Leo Lam © 2010-2012 14 -5 τ t 2 -2 Overlapped at τ =0 y(t=-1)

15. Convolution (graphically) Leo Lam © 2010-2012 15 -5 2 -1 1 Both overlapped y(t=1)

16. Convolution (graphically) Leo Lam © 2010-2012 16 -1 1 3 2 4 Overlapped at τ =2 y(t=3) Does it make sense?

17. Convolution (mathematically) Leo Lam © 2010-2012 17 Using Linearity Let’s focus on this part

18. Convolution (mathematically) Leo Lam © 2010-2012 18 Consider this part: Recall that: And the integral becomes:

19. Convolution (mathematically) Leo Lam © 2010-2012 19 Same answer as the graphically method Apply delta rules:

20. Summary: Convolution Leo Lam © 2010-2012 20 1.Draw x() 2.Draw h() 3.Flip h() to get h(-) 4.Shift forward in time by t to get h(t-) 5.Multiply x() and h(t-) for all values of  6.Integrate (add up) the product x()h(t-) over all  to get y(t) for this particular t value (you have to do this for every t that you are interested in)

21. Summary: Convolution Leo Lam © 2010-2012 21 Flip Shift Multiply Integrate

22. Leo Lam © 2010-2012 Summary Convolution!