1. Time Domain Representations of Linear Time-Invariant Systems
Chapter 2

2. Introduction
Several methods can be used to describe the relationship between the input and output signals of LTI system.
Focus on system description that relate the output signal to the input signal when both are represented as function of time (time domain).
Impulse response defined as output of LTI system due to a unit impulse signal input applied at time t=0 or n=0.

3. Analysis of Continuous-Time Systems

4. Impulse Response Let a system be described by x(t) y(t) H
and let the excitation be a unit impulse at time, t = 0. Then the
response y is the impulse response h.
(t)
h(t) H d
Since the impulse occurs at time t = 0 and nothing has excited
the system before that time, the impulse response before time
t = 0 is zero (because this is a causal system). After time t = 0
the impulse has occurred and gone away. Therefore there is no
excitation and the impulse response is the homogeneous solution
of the differential equation.

5. Impulse Response What happens at time, t = 0?
The left side of the equation must be a unit impulse.
The left side is zero before time t = 0 because the system has never
been excited.
The left side is zero after time t = 0 because it is the solution of the
homogeneous equation whose right side is zero. These two facts
are both consistent with an impulse. The impulse response might
have in it an impulse or derivatives of an impulse since all of
these occur only at time, t = 0. What the impulse response does
have in it depends on the form of the differential equation.

6. Impulse Response
Continuous-time LTI systems are described by differential
equations of the general form,
For all times, t < 0:
If the excitation x(t) is an impulse, then for all time t < 0
it is zero. The response y(t) is zero before time t = 0
because there has never been an excitation before that time.

7. Impulse Response For all time t > 0:
The excitation is zero. The response is the homogeneous
solution of the differential equation.
At time t = 0:
The excitation is an impulse. In general it would be possible
for the response to contain an impulse plus derivatives of an
impulse because these all occur at time t = 0 and are zero
before and after that time. Whether or not the response contains
an impulse or derivatives of an impulse at time t = 0 depends
on the form of the differential equation

8. Convolution A systematic way to find systems respond.
Convolution integral for CT systems.

9. The Convolution Integral
Exact
Excitation
Approximate
Excitation

10. The Convolution Integral
All the pulses are rectangular and the same width, the only differences between pulses are when they occur and how tall they are.
So the pulse responses all have the same form except the delayed by some amount, to account for time of occurrence, and multiplied by a weighting constant, to account for the height.

11. The Convolution Integral
Approximating the excitation as a pulse train can be expressed
mathematically by or

12. The Convolution Integral
Then, invoking linearity, the response to the overall excitation is
(approximately) a sum of shifted and scaled unit pulse responses
of the form

13. The Convolution Integral

14. A Graphical Illustration of the Convolution Integral
The convolution integral is defined by
For illustration purposes let the excitation x(t) and the
impulse response h(t) be the two functions below.

15. A Graphical Illustration of the Convolution Integral
We can begin to visualize this quantity in the graphs below.

16. A Graphical Illustration of the Convolution Integral
The functional transformation in going from h(t) to h(t - t) is

17. A Graphical Illustration of the Convolution Integral
The convolution value is the area under the product of x(t)
and h(t - t). This area depends on what t is. First, as an
example, let t = 5.
For this choice of t the area under the product is zero.
Therefore if
then y(5) = 0.

18. A Graphical Illustration of the Convolution Integral
Now let t = 0.
Therefore y(0) = 2, the area under the product.

19. A Graphical Illustration of the Convolution Integral
The process of convolving to find y(t) is illustrated below.

20. Convolution Integral Properties
Commutativity
Associativity
Distributivity

21. System Interconnections
If the output signal from a system is the input signal to a second
system the systems are said to be cascade connected.
It follows from the associative property of convolution that the
impulse response of a cascade connection of LTI systems is the
convolution of the individual impulse responses of those systems.
Cascade
Connection

22. System Interconnections
If two systems are excited by the same signal and their responses
are added they are said to be parallel connected.
It follows from the distributive property of convolution that the
impulse response of a parallel connection of LTI systems is the
sum of the individual impulse responses.
Parallel
Connection

23. Stability and Impulse Response
A system is BIBO stable if its impulse response is
absolutely integrable. That is if

24. Unit Impulse Response and Unit Step Response
In any LTI system let an excitation x(t) produce the response
y(t). Then the excitation
will produce the response
It follows then that the unit impulse response is the first
derivative of the unit step response and, conversely that the
unit step response is the integral of the unit impulse response.

25. Relations between integrals and derivatives of excitation and responses for an LTI systems
Figure 6.26
Unit impulse
Unit step
Unit ramp

26. Complex Exponential Response
Let an LTI system be excited by a complex exponential of
the form
The response is the convolution of the excitation with the
impulse response or

27. Transfer Functions

28. Transfer Functions

29. Block Diagrams If a system is described by the differential equation
It can also be described by the block diagram below. But this
form of block diagram is not the preferred form.

30. Block Diagrams
Although the previous block diagram is correct, it is not the
preferred way of representing systems in block-diagram
form. A preferred form is illustrated below. This form is
preferred because, as a practical matter, integrators are more
desirable elements for an actual hardware simulation than
differentiators.

31. Block Diagrams

32. Analysis of Discrete-Time Systems

33. Impulse Response
Discrete-time LTI systems are described mathematically
by difference equations of the form
For any excitation x[n] the response y[n] can be found by
finding the response to x[n] as the only forcing function on the
right-hand side and then adding scaled and time-shifted
versions of that response to form y[n].
If x[n] is a unit impulse, the response to it as the only forcing
function is simply the homogeneous solution of the difference
equation with initial conditions applied. The impulse response
is conventionally designated by the symbol h[n].

34. Impulse Response
Since the impulse is applied to the system at time n = 0,
that is the only excitation of the system and the system is
causal. The impulse response is zero before time n = 0.
After time n = 0, the impulse has come and gone and the
excitation is again zero. Therefore for n > 0, the solution of
the difference equation describing the system is the
homogeneous solution.

35. Impulse Response Example
Let a system be described by
Substituting into the homogeneous difference equation,

36. Impulse Response Example
The homogeneous solution is then of the form
The constants can be found be applying initial conditions.
For the case of unit impulse excitation at time n = 0,

37. Impulse Response Example
The impulse response is then
which can also be written in the forms,

38. Impulse Response Example

39. System Response
Once the response to a unit impulse is known, the response of any LTI system to any arbitrary excitation can be found
Any arbitrary excitation is simply a sequence of amplitude-scaled and time-shifted impulses
Therefore the response is simply a sequence of amplitude-scaled and time-shifted impulse responses

40. Simple System Response Example

41. More Complicated System Response Example
Excitation
System
Impulse
Response
System
Response

42. Convolution A systematic way to find systems respond.
Convolution sum for DT systems.
Based on a simple idea, no matter how complicated an excitation signal is, it is simply a sequence of DT impulses.
Therefore, with the assumption that the impulse response to a unit impulse excitation occuring at time n=0 has already found, we will use the convolution technique to find system response.

43. The Convolution Sum
The response y[n] to an arbitrary excitation x[n] is of the
form
where h[n] is the impulse response. This can be written in
a more compact form
called the convolution sum.

44. A Convolution Sum Example

45. A Convolution Sum Example

46. A Convolution Sum Example

47. A Convolution Sum Example

48. Convolution Sum Properties
Convolution is defined mathematically by
The following properties can be proven from the definition.
Let
then
and the sum of the impulse strengths in y is the product of
the sum of the impulse strengths in x and the sum of the
impulse strengths in h.

49. Convolution Sum Properties (continued)
Commutativity
Associativity
Distributivity

50. System Interconnections
The cascade connection of two systems can be viewed as
a single system whose impulse response is the convolution
of the two individual system impulse responses. This is a
direct consequence of the associativity property of
convolution.

51. System Interconnections
The parallel connection of two systems can be viewed as
a single system whose impulse response is the sum
of the two individual system impulse responses. This is a
direct consequence of the distributivity property of
convolution.

52. Stability and Impulse Response
It can be shown that a BIBO-stable system has an
impulse response that is absolutely summable. That is,

53. Unit Impulse Response and Unit Sequence Response
In any LTI system let an excitation x[n] produce the response
y[n]. Then the excitation x[n] - x[n - 1] will produce the
response y[n] - y[n - 1] .

54. Complex Exponential Response
Let an LTI system be excited by a complex exponential of
the form
The response is the convolution of the excitation with the
impulse response or
which can be written as

55. Complex Exponential Response
The response of a discrete-time LTI system to a complex
exponential excitation is another complex exponential of
the same functional form but multiplied by a complex
constant. That complex constant is
Later this will be called the z transform of the impulse
response and will be one of the important transform
methods.

56. Transfer Functions

57. Transfer Functions

58. Block Diagrams
A very useful method for describing and analyzing systems is
the block diagram. A block diagram can be drawn directly
from the difference equation which describes the system. For example, if the system is described by
it can also be described
by a block diagram.