1. Time Domain Representation of Linear Time Invariant (LTI).
CHAPTER 2
Time Domain Representation of Linear Time Invariant (LTI).
Mohd Herwan Sulaiman

2. 2.0 Time Domain Representation of Linear Time Invariant (LTI) System.
2.1 Introduction.
2.2 LTI System Properties.
2.3 Convolution Sum.
2.4 Convolution Integral.
2.5 Interconnection of LTI System.
2.6 LTI System Properties and Impulse Response.
2.7 The Step Response.
2.8 Solving Differential & Difference Equation.
2.9 Characteristic of System Natural & Force Response.
2.10 State Variable Representation of LTI System.

3. 2.1 Introduction. Learning Outcome:
Examine several methods for describing the relationship between the input and the output signals of LTI system.
(1) Impulse Response.
(2) Linear Constant-Coefficient Differential.
(3) Block Diagram.
(4) State Variable Description.

4. 2.2 LTI System Properties.
If we know the response of the LTI system to some inputs, we actually know the response to many input.
(1) Commutative Property.

5. Cont’d…
(2) Distributive Property.

6. Cont’d…
(3) Associative Property.

7. 2.3 Convolution Sum.
x[n] is a signal as a weighted sum of basis function; time-shift version of the unit impulse signal. x[k] represents a specific value of the signal x[n] at time k.
The output of the LTI system y[n] is given by a weighted sum of time-shifted impulse response. h[n] is the impulse response of LTI system H.
The convolution of two discrete-time signals y[n ] and h[n] is denoted as

8. Cont’d…
Figure 2.1: Graphical example illustrating the representation of a signal x[n] as a weighted sum of time-shifted impulses.

9. Cont’d… Steps for Convolution Computation.
Step 1: Plot x and h versus k since the convolution sum is on k.
Step 2: Flip h[k] around the vertical axis to obtain h [- k].
Step 3: Shift h [-k] by n to obtain h [n- k].
Step 4: Multiply to obtain x[k] h[n- k].
Step 5: Sum on k to compute
Step 6: Index n and repeat Step 3-6.

10. Example 2.1: Convolution Sum.
For the figure below, compute the convolution, y[n].
Figure 2.2a: Illustration of the convolution sum. (a) LTI system with impulse response h[n] and input x[n], producing y[n] and yet to be determined.

11. Solution: .
Figure 2.2b: The decomposition of the input x[n] into a weighted sum of time-shifted impulses results in an output y[n] given by a weighted sum of time-shifted impulse responses.

12. Example 2.2: Convolution. Solution:
The LTI h[n] having an impulse response of
and the input
Solution:
Details: Explained in class. .

27. 2.3.1 Convolution Sum Evaluation Procedure.
An alternative approach to evaluating the convolution sum.
Recall, the Convolution Sum is expressed as;
define an intermediate signal as;
so,
Time shift n determines the time at which we evaluated the output of the system.
The above formula is the simplified version of Convolution Sum, where we need to determine one signal wn[k] each time.

28. Example 2.4: Convolution Sum Evaluating by using an Intermediate Signal.
Consider a system with impulse response
Use the equation below to determine the output of the system at times n=-5, n =5, and n =10 when the input is x[n]=u[n].
Solution:
In Figure 2.3 below x[k] is superimposed on the reflected and time-shifted impulse response h[n-k].
For n=-5, we have w-5[k]=0. This result in y[-5]=0.
For n=5, we have
The result is in Figure 2.3(c).

29. Cont’d… For n=10, we have The result is in Figure 2.3(d).
Note: for n<0, wk[n]=0, because no overlap occur.

30. (b) The product signal w5[k] used to evaluate y [–5].
Cont’d…
Figure 2.3: (a) The input signal x[k] above the reflected and time-shifted impulse response h[n – k], depicted as a function of k.
(b) The product signal w5[k] used to evaluate y [–5].
(c) The product signal w5[k] used to evaluate y[5]. (d) The product signal w10[k] used to evaluate y[10].

31. In Class Exercise.

32. 2.4 Convolution Integral. Derivation of Convolution Integral.
(a) The operator H denotes the system in which the x(t) is applied.
(b) Use the linearity property.
(c) Define impulse response as unit impulse input.

33. Cont’d…
The time invariance implies that a time-shifted impulse input result in a time shift impulse response output as in Figure 2.4 below.
Figure 2.4: (a) Impulse response of an LTI system H. (b) The output of an LTI system to a time-shifted and amplitude-scaled impulse is a time-shifted and amplitude-scaled impulse response.

34. Cont’d… Step for Convolution Integral Computation.
To compute the superposition integral
Step for Convolution Integral Computation.
Step 1: Plot x and h versus t since the convolution sum is on t.
Step 2: Flip h(t) around the vertical axis to obtain h(-t).
Step 3: Shift h(t) by n to obtain h(t-t).
Step 4: Multiply to obtain x(t) h(t-t).
Step 5: Integrate on t to compute
Step 6: Increase t and repeat Step 3-6.

47. Example 2.5: Convolution Integral.
Given a RC circuit below (RC=1s). Use convolution to determine the voltage across the capacitor y(t). Input voltage x(t)=u(t)-u(t-2).
Solution: t a
y(t)=x(t)*h(t)
- capacitor start charging at t=0 and discharging at t=2. b

48. Cont’d…

49. Cont’d… .

50. 2.5 Interconnection of LTI System.
The objective of this section is to develop the relationship between the impulse response of an interconnection of LTI systems and impulse response of the constituent systems.
Parallel Connection of LTI System.
Cascade Connection of LTI System.

51. 2.5.1 Parallel Connection of LTI Systems.
Two LTI systems with impulse response h1(t) and h2(t) connected in parallel, as in Figure 2.5 below.
Figure 2.5: Interconnection of two LTI systems.
(a) Parallel connection of two systems. (b) Equivalent system.
Derivation;

52. Cont’d… Where h(t) = h1(t)+h2(t).
The impulse response of the overall system represented by the two LTI systems connected in parallel is the sum of their individual impulse response.
Distributive property of convolution (CT and DT),

53. 2.5.2 Cascade Connection of Systems.
The impulse response an equivalent system representing two LTI systems connected in cascade is the convolution of their individual impulse responses.
Figure 2.6: Interconnection of two LTI systems. (a) Cascade connection of two systems. (b) Equivalent system. (c) Equivalent system: Interchange system order.

54. Cont’d…
Derivation;

55. Cont’d…

56. Cont’d…

57. Cont’d… Continuous Time Associative properties
Commutative properties. It is often used to simplify the evaluation or interpretation of the convolution integral.
Discrete Time
Commutative properties.

58. Example 2.6: Equivalent System to Four Interconnected System.
The interconnected of four LTI system is shown in figure below. Given the impulse response of the system,
h1[n]=u[n]
h2[n]=u[n+2]- u[n]
h3[n]=d[n-2] and h4[n]=anu[n]
Find the impulse response of the overall system.
Solution:
h[n]=(h1[n]+h2[n])*h3[n]-h4 [n],
Substitute the specific form of h1[n] and h2[n] to obtain.
h12[n]=u[n]+u[n+2]-u[n]
=u[n+2]

59. Figure 2.7 : Interconnection of systems.
Cont’d…
Convolving h12[n] with h3[n].
h123 [n]= u[n]+u[n+2]*d[n]
= u[n]
Finally, we sum h123[n] and -h4[n] to obtain the overall impulse response:
h[n]= {1-an}u[n].
Figure 2.7 : Interconnection of systems. .

60. 2.6 Relations between LTI System Properties and Impulse Response.
The impulse response characterized the input output behavior of an LTI system. Below are the properties of the system that relates to the impulse response.
2.6.1 Memoryless LTI Systems.
2.6.2 Causal LTI Systems.
2.6.3 Stable LTI Systems.
2.6.2 Invertible LTI Systems.

61. 2.6.1 Memoryless LTI Systems.
The output of a memoryless system depends only on the present input.
Continuous Time;
Discrete Time;

62. 2.6.2 Causal LTI Systems.
The output of a causal system depends only on the past input.
Continuous Time;
Discrete Time;

63. 2.6.3 Stable LTI Systems.
Bounded Input Bounded Output (BIBO) stable system.
Discrete Time: absolute summability of the impulse response.
Continuous Time;

64. 2.6.4 Invertible System and Deconvolution.
Deconvolution is the process of recovering x(t) from h(t)*x(t).
An exact inverse system may be difficult to find or implement. Determination of an approximate solution is often sufficient.
Figure 2.8: Cascade of LTI system with impulse response h(t) and inverse system with impulse response h-1(t).

65. 2.7 Step Response.
Step response is defined as output due to a unit step input signal.
Step response s[t] of the discrete time is the running sum of the impulse response.
Step response s(t) of the continuous time system is the running integral of the impulse response.
Step response can be inverted to express in term of impulse response.

66. Example 2.7: RC Circuit of Step Response.
The impulse response of the RC circuit is,
Find the step response of the circuit.
Solution:
The step represented a switch that turns on a constant voltage source at time t=0. We expect the capacitor voltage to increase toward the value of the source in an exponential manner.

67. Figure 2.9: RC circuit step response for RC = 1 s.
Cont’d…
We simplify the integral to get
Figure 2.9: RC circuit step response for RC = 1 s. .

68. 2.8 Solving Differential and Difference Equation.
The output of Differential and Difference Equation can be described as the sum of two components;
(1) Homogeneous Solution y(h).
(2) Particular Solution y(p).
The complete solution is y = y(h) + y(p)
**Note: ( ) and [ ] are omit when referring to continuous and discrete time.

69. 2.8.1 Homogeneous Solution.
Homogeneous form of difference and differential equation is obtained by setting all the terms involving the input to zero.
Continuous-time System
Solution of homogeneous equation
Equation (1)
Where the r1 are the N roots of the system’s characteristic equation
Substitution of equation (1) into the homogeneous equation results in y(h)(t) is a solution for any set of constant ci

70. Cont’d… Discrete-time System Solution of homogeneous equation
Where the r1 are the N roots of the system’s characteristic equation
Substitution of equation (2) into the homogeneous equation results in y(h)[t] is a solution for any set of constant ci. Note the CT and DT characteristic equations are different.

71. 2.9 Characteristic of System Natural and Force Response.
The output of a system described by a differential or difference equation is the sum of two components;
(1) Natural Response y(n): the one associated with the initial condition.
(2) Forced Response y(f): the one associated with only input signal.
The complete output is y = y(n)+y(f)

72. 2.9.1 Natural Response.
The natural response is the system output for zero input by which the system dissipates any unstored energy or memory of the past represented by non-zero initial condition.
The system output is obtained from the homogeneous solution.

73. Example 2.8: Natural Response.
The system is describe by the differential equation,
Find the natural response of the system, given y(0)=2V, r=1W, and C=1F.
Solution:
The homogeneous equation is
The natural response is .

74. 2.9.2 Force Response.
The force response is the system output due to the input signal assuming zero initial conditions.
Zero initial condition indicates that the system is “at rest”, there is no stored energy or memory in the system.
The force response describes the system behavior that is “forced” by the input when the system is at rest.

75. Example 2.9: Forced Response.
The system is describe by the first order recursive system,
Find the force response of the system if the input x[n]=(1/2)nu[n].
Solution:
The difference between the previous example is the initial condition. Recall the complete solution is of the form
To obtain c1, we translate the at-rest condition y[-1]=0 to time n=0 by noting that
y[0]=1+(1/4)x0, now we know y[0]=1 and use to solve for c1.
1=2(1/2) 0 + c1(1/4)0 ; c1= -1
The force response is, .

76. Example 2.10: Complete Response.

77. 2.10 State Variable Description of LTI System.
The state-variable description for a LTI system consists of a series of coupled first-order differential or difference equations that describe how the state of the system evolves and an equation that relates the output of the system to the current state variables and input. 􀂄
The state of a system may be defined as a minimal set of signals that represent the system’s entire memory of past.
The state-variable description is the only analytic system representation capable of specifying the internal structure of the system. 􀂄
Thus state-variable description is used in any problem in which the internal system structure needs to be considered.

78. Cont’d… Block Diagram Representation.
Block diagram is an interconnection of elementary operations that act on the input signal.
A more detailed representation of the system than the impulse response or differential (difference) equation description since it describes how the system’s internal computations or operations are ordered.
Block diagram representations consists of an interconnection of three elementary operations on signals;
(1) Scalar Multiplication.
(2) Addition.
(3) Integration for continuous-time LTI system.

79. Cont’d… Block Diagram Representation.
Figure 2.10: Symbols for elementary operations in block diagram descriptions of systems. (a) Scalar multiplication. (b) Addition. (c) Integration for continuous-time systems and time shifting for discrete-time systems.

80. Cont’d… Discrete-Time System: Second Order Difference Equation.
Figure 2.11: Block Diagram Representation of Discrete Time LTI system Described by second order Differential Equation.

81. Cont’d… Discrete-Time System: Second Order Difference Equation.
Figure 2.12: Direct form II representation of an LTI system described by a second-order difference equation.

82. Cont’d…
Continuous-Time System.

83. 2.10.1 State Variable Description.
State Variable Representation: General Procedure.
Define the system equation.
Identity the system inputs, outputs and states.
Rewrite the new system equation in standard state variable vector-matrix notation. 􀂄
A = system matrix,
B = input matrix,
C = output matrix
D = direct feed through matrix.

84. Cont’d…
The next value of the state is obtained from the current state and the input via the two equations.
q1[n+1]=-a1q1[n]-a2q2 [n]+x[n]
The system output is expressed as,
y[n]=x[n]-a1q1[n]-a2q2[n]
Figure 2.13: Direct form II representation of a second-order discrete-time LTI system depicting state variables q1[n] and q2[n].

85. Cont’d…

86. Cont’d…
In matrix form,

87. Example 2.11: State Variable Description of a Second Order System.
Find the state variable description corresponding to the system in figure below, by choosing the state variables to be the outputs of the unit delay.
Solution:
q1[n+1]= aq1[n] + d1x [n]
q2[n+1]= gq1[n] + bq2 [n] + d2x [n]
The output is given by
y[n]= h1q1[n] + h2q2 [n]
The equation can be expressed in state variable forms, .

88. Example 2.12: State Variable Description of an Electrical Circuit.
Given an electrical circuit in Figure below. Derive the state-variable description of this system if the input is x(t) and the output is the current y(t) through the resistor.
Solution:

89. Cont’d…
KVL
The state variable .

90. Example 2.13: State Variable Description from Block Diagram.
Determine the state variable description of the block diagram below.
Solution: .

91. 2.10.2 Transformation of the State.
There is no unique state-variable description for a system with a given input-output characteristics. 􀂄
Different state-variable descriptions may be obtained by transforming the state variables.

92. Example 2.14: Transforming the State.
A discrete variable description A’, b’, c’ and D’ corresponding to the new states
q’1[n]=-1/2q1[n]+1/2q2[n], q’2[n]=1/2q1 [n]+1/2q2[n].
Solution: .