1. Chapter 3 Convolution Representation

2. CT Unit-Impulse Response
Consider the CT SISO system:
If the input signal is and the system has no energy at , the output is called the impulse response of the system
System
System

3. Exploiting Time-Invariance
Let x(t) be an arbitrary input signal with for
Using the sifting property of , we may write
Exploiting time-invariance, it is
System

4. Exploiting Time-Invariance

5. Exploiting Linearity
Exploiting linearity, it is
If the integrand does not contain an impulse located at , the lower limit of the integral can be taken to be 0,i.e.,

6. The Convolution Integral
This particular integration is called the convolution integral
Equation is called the convolution representation of the system
Remark: a CT LTI system is completely described by its impulse response h(t)

7. Block Diagram Representation of CT LTI Systems
Since the impulse response h(t) provides the complete description of a CT LTI system, we write

8. Example: Analytical Computation of the Convolution Integral
Suppose that where p(t) is the rectangular pulse depicted in figure

9. Example – Cont’d
In order to compute the convolution integral
we have to consider four cases:

10. Example – Cont’d
Case 1:

11. Example – Cont’d
Case 2:

12. Example – Cont’d
Case 3:

13. Example – Cont’d
Case 4:

14. Example – Cont’d

15. Properties of the Convolution Integral
Associativity
Commutativity
Distributivity w.r.t. addition

16. Properties of the Convolution Integral - Cont’d
Shift property: define
Convolution with the unit impulse
Convolution with the shifted unit impulse
then

17. Properties of the Convolution Integral - Cont’d
Derivative property: if the signal x(t) is differentiable, then it is
If both x(t) and v(t) are differentiable, then it is also

18. Properties of the Convolution Integral - Cont’d
Integration property: define
then

19. Representation of a CT LTI System in Terms of the Unit-Step Response
Let g(t) be the response of a system with impulse response h(t) when with no initial energy at time , i.e.,
Therefore, it is

20. Differentiating both sides
Representation of a CT LTI System in Terms of the Unit-Step Response – Cont’d
Differentiating both sides
Recalling that
it is
and or