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

2. Proof
Define the pulse of width D as

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

6. This integral is called the convolution

7. Another proof for
Sifting properties

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

9. Linear –Time Invariant
Convolution Integral

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

12. Example 2-7
Moving
Fix

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

14. Sep 2 : make the moving function in terms of -l
Sep 2 : add t to to form ( t- l)
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. 2.6 Superposition Integral “convolution” in terms of step response
Impulse response
Now if the input is a step function,
step response
step response

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

44. The system is initially unexcited
and

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

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

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

48. Impulse input
Step input
Ramp input

49. To be Done
From the notes
RC circuit - DFE
X(t) = impulse  h(t)
X(t) is switch u(t)  a(t)
X(t) is ramp r(t)  b(t)
X(t) is
X(t) = r(t)-2r(t-1)+r(t-2)
BIBO condition