1. ( II ) This property is known as “convolution” ( الإلتواء التفاف ) From Chapter 2, we have Proof is shown next Chapter 3

2. Define the pulse of width  as Proof

3. We now can approximate the function In terms of the pulse function Approximation

6. This integral is called the convolution (الإلتواء التفاف ) integral

7. Another proof for Sifting properties

8. Impulse Input Impulse response Shifted Impulse Input Linear –Time Invariant Shifted Impulse Response

9. Linear –Time Invariant Convolution Integral

10. Linear –Time Invariant Operator with respect to t Integration with respect to constant with respect to t

12. Moving Fix Example 2-7

13. Sep 1 : make the functions or signals in terms of the variable

14. Sep 2 : make the moving function in terms of  Sep 2 : add t to to form ( t  ) Moving to the right

15. For t ≤ 4 there is no overlapping between the functions

19. For t ≥ 10 For t ≥ 10 there is no overlapping between the functions

29. TO be down

40. Now if the input is a step function, 2.6 Superposition Integral “convolution” in terms of step response Impulse response step response

41. Now if the input is x(t), The output in terms of the impulse response h(t) Objective is to write y(t) in terms of the step response a(t)

42. Now if the input is x(t), Integrating by parts, step response Over dot denotes differentiation

43. Integrating by parts, Now we can write y(t) in terms of the step response a(t)

44. The system is initially unexcitedand

45. In term of impulse response In term of step response Note

46. Impulse input Step input Ramp input Ramp response Impulse response step response Objective is the ramp response b(t)

47. Now if x(t) is the ramp r(t)

48. Impulse input Step input Ramp input