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

2. Leo Lam © 2010-2012 Convergence Two mathematicians are studying a convergent series. The first one says: "Do you realize that the series converges even when all the terms are made positive?" The second one asks: "Are you sure?" "Absolutely!"

3. Leo Lam © 2010-2012 Today’s menu Lab 3 this week Convolution!

4. Convolution Integral Leo Lam © 2010-2012 4 Standard Notation The output of a system is its input convolved with its impulse response

5. Convolution (mathematically) Leo Lam © 2010-2012 5 Use sampling property of delta: Evaluate integral to arrive at output signal: Does this make sense physically?

6. Convolution (graphically) Leo Lam © 2010-2012 6 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

7. Convolution (graphically) Leo Lam © 2010-2012 7 -5 τ t 2 -2 Overlapped at τ =0 y(t=-1)

8. Convolution (graphically) Leo Lam © 2010-2012 8 -5 2 -1 1 Both overlapped y(t=1)

9. Convolution (graphically) Leo Lam © 2010-2012 9 -1 1 3 2 4 Overlapped at τ =2 y(t=3) Does it make sense?

10. Convolution (mathematically) Leo Lam © 2010-2012 10 Using Linearity Let’s focus on this part

11. Convolution (mathematically) Leo Lam © 2010-2012 11 Consider this part: Recall that: And the integral becomes:

12. Convolution (mathematically) Leo Lam © 2010-2012 12 Same answer as the graphically method Apply delta rules:

13. Summary: Convolution Leo Lam © 2010-2012 13 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)

14. Summary: Convolution Leo Lam © 2010-2012 14 Flip Shift Multiply Integrate

15. Leo Lam © 2010-2012 Summary Convolution!

16. y(t) at specific time t 0 Leo Lam © 2010-2011 16 Flip Shift Multiply Integrate Here t 0 =3/4 y(t 0 =3/4)= ?3/4

17. y(t) at all t Leo Lam © 2010-2011 17 At all t t<0 The product of these two signals is zero where they don’t overlap ShiftMultiplyIntegrate

18. y(t) at all t Leo Lam © 2010-2011 18 At all t 0≤t<1 ShiftMultiplyIntegrate

19. y(t) at all t Leo Lam © 2010-2011 19 At all t 1≤t<2 y(t)=2-t for 1≤t<2 ShiftMultiplyIntegrate

20. y(t) at all t Leo Lam © 2010-2011 20 At all t t≥2 y(t)=0 for t≥2 (same as t<0, no overlap) ShiftMultiplyIntegrate

21. y(t) at all t Leo Lam © 2010-2011 21 Combine it all –y(t)=0 for t 2 –y(t)=t for 0≤t<1 –y(t)=2-t for 1≤t<2

22. Leo Lam © 2010-2011 Summary Convolution and first few examples