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

Proof Define the pulse of width D as

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

This integral is called the convolution

Another proof for Sifting properties

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

Linear –Time Invariant Convolution Integral

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

Example 2-7 Moving Fix

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

Sep 2 : make the moving function in terms of -l Sep 2 : add t to to form ( t- l) 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

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

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)

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

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)

The system is initially unexcited and

In term of impulse response In term of step response Note

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)

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

Impulse input Step input Ramp input

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