1. Examples of Convolution 3.2

2. Graphical Examples Impulse Response of an Integrator circuit due to the unit impulse function Response of an integrator due to a unit step function Response of an Integrator circuit due to the unit ramp input Response of an integrator due to a rectangular pulse

3. Impulse Response of an Integrator =δ(t) 0 (τ)(τ) 0 (τ)(τ) interval of integration

4. Graphical Examples Impulse Response of an Integrator circuit due to the unit impulse function Response of an integrator due to a unit step function Response of an Integrator circuit due to the unit ramp input Response of an integrator due to a rectangular pulse

5. Response of an Integrator Due to a Unit Ramp Function =tu(t) Why is h(t)=u(t) ?

6. Response of an Integrator Due to a Unit Step Function =tu(t) Constant!

7. Graphical Illustration (t=-1) No area of intersection

8. Graphical Illustration (t=1)

9. Response of an Integrator Due to a Unit Step Function =tu(t) Unit step function is only 1 when the argument is greater than 0. It does not make sense to integrate all the way to infinity.

10. A system with rectangular impulse response δ(t) u(t) u(t-2)

11. Example (why not take advatage of linearity ? )

12. Focus on x 1 (t) δ(t)→h(t) δ(t+3)→h(t+3)

13. Understanding h(t-τ)

14. h(t-τ) when t=-1 h(τ) h(-1+τ) h(-1-τ)

15. Integration when t<0 h(t-τ) for t<0

16. h(t-τ) when t=0.5 h(τ) h(0.5+τ) h(0.5-τ)

17. Integration when 0<t<2 h(t-τ) for 0<t<2 (move to the right as t increases )

18. h(t-τ) when t=3.5 h(τ) h(3.5+τ)h(3.5-τ) 3.5 3.5-2

19. Integration when t>2 t t-2