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

2. Summary: Convolution Leo Lam © 2010-2012 2 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)

3. y(t) at specific time t 0 Leo Lam © 2010-2012 3 Flip Shift Multiply Integrate Here t 0 =3/4 y(t 0 =3/4)= ?3/4

4. y(t) at all t Leo Lam © 2010-2012 4 At all t t<0 The product of these two signals is zero where they don’t overlap ShiftMultiplyIntegrate

5. y(t) at all t Leo Lam © 2010-2012 5 At all t 0≤t<1 ShiftMultiplyIntegrate

6. y(t) at all t Leo Lam © 2010-2012 6 At all t 1≤t<2 y(t)=2-t for 1≤t<2 ShiftMultiplyIntegrate

7. y(t) at all t Leo Lam © 2010-2012 7 At all t t≥2 y(t)=0 for t≥2 (same as t<0, no overlap) ShiftMultiplyIntegrate

8. y(t) at all t Leo Lam © 2010-2012 8 Combine it all –y(t)=0 for t 2 –y(t)=t for 0≤t<1 –y(t)=2-t for 1≤t<2

9. Another example Leo Lam © 2010-2012 9 At all t t<0 The product of these two signals is zero where they don’t overlap ShiftMultiplyIntegrate

10. Another example Leo Lam © 2010-2012 10 At all t 0≤t<0.5 ShiftMultiplyIntegrate h(t) moving right

11. Another example Leo Lam © 2010-2012 11 At all t 0.5≤t<1 h(t) moving right ShiftMultiplyIntegrate

12. Another example Leo Lam © 2010-2012 12 At all t 1≤t<1.5 ShiftMultiplyIntegrate h(t) moving right

13. Another example Leo Lam © 2010-2012 13 At all t 1.5≤t? ShiftMultiplyIntegrate y(t)=0 because there is no more overlapping

14. Another example Leo Lam © 2010-2012 14 At all t Combining Can you plot and formulate it?

15. Another example Leo Lam © 2010-2012 15 At all t

16. Few things to note Leo Lam © 2010-2012 16 Three things: –Width of y(t) = Width of x(t)+Width of h(t) –Start time adds –End time adds –y(t) is smoother than x(t) and h(t) (mostly) Stretching the thinking –What if one signal has infinite width?

17. Leo Lam © 2010-2012 Summary Convolution examples Convolution properties