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

Define the pulse of width  as Proof

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

This integral is called the convolution (الإلتواء التفاف ) integral

Another proof for Sifting properties

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

Linear –Time Invariant Convolution Integral

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

Moving Fix Example 2-7

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

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

For t ≤ 4 there is no overlapping between the functions

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

TO be down

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

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)

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

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

The system is initially unexcitedand

In term of impulse response In term of step response Note

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

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

Impulse input Step input Ramp input