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

2. Leo Lam © 2010-2012 Today’s menu Homework 2 due now Convolution!

3. y(t) at all t Leo Lam © 2010-2011 3 At all t t<0 The product of these two signals is zero where they don’t overlap ShiftMultiplyIntegrate

4. y(t) at all t Leo Lam © 2010-2011 4 At all t 0≤t<1 ShiftMultiplyIntegrate

5. y(t) at all t Leo Lam © 2010-2011 5 At all t 1≤t<2 y(t)=2-t for 1≤t<2 ShiftMultiplyIntegrate

6. y(t) at all t Leo Lam © 2010-2011 6 At all t t≥2 y(t)=0 for t≥2 (same as t<0, no overlap) ShiftMultiplyIntegrate

7. y(t) at all t Leo Lam © 2010-2011 7 Combine it all –y(t)=0 for t 2 –y(t)=t for 0≤t<1 –y(t)=2-t for 1≤t<2

8. Another example Leo Lam © 2010-2011 8 At all t t<0 The product of these two signals is zero where they don’t overlap ShiftMultiplyIntegrate

9. Another example Leo Lam © 2010-2011 9 At all t 0≤t<0.5 ShiftMultiplyIntegrate h(t) moving right

10. Another example Leo Lam © 2010-2011 10 At all t 0.5≤t<1 h(t) moving right ShiftMultiplyIntegrate

11. Another example Leo Lam © 2010-2011 11 At all t 1≤t<1.5 ShiftMultiplyIntegrate h(t) moving right

12. Another example Leo Lam © 2010-2011 12 At all t 1.5≤t? ShiftMultiplyIntegrate y(t)=0 because there is no more overlapping

13. Another example Leo Lam © 2010-2011 13 At all t Combining Can you plot and formulate it?

14. Another example Leo Lam © 2010-2011 14 At all t

15. Few things to note Leo Lam © 2010-2011 15 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?

16. From yesterday Leo Lam © 2010-2011 16 Stretching the thinking –What if one signal has infinite width? Width = infinite (infinite overlapping) Start time and end time all infinite

17. One more example Leo Lam © 2010-2011 17 For all t: x(t) 2 1 t FlipShift Can you guess the “width” of y(t)?

18. One more example Leo Lam © 2010-2011 18 For all t: x(t) 2 1 t Multiply & integrate