1. Time Domain Representation of Linear Time Invariant (LTI) Systems.
CHAPTER 2
Time Domain Representation of Linear Time Invariant (LTI) Systems.
EMT Signal Analysis

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 Characteristics of system natural and forced
response.
2.10 State variable description of LTI systems

3. 2.1 Introduction.
This chapter will examine several methods for describing the relationship between the input and the output signals of LTI system.
(1) Impulse Response.(the output of an LTI system due to a unit impulse signal input t = 0 or n = 0.
(2) Linear Constant-Coefficient Differential (Continuous-time Systems) or Difference Equation (Discrete-time systems)
(3) Block Diagram(represent systems as an interconnection of three elementary operations: scalar multiplication, addition and a time shift for discrete-time signal or integration for continuous-time signal)

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 x[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. 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.

10. 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.

11. 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.

12. 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).

13. 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.

14. (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].

15. 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.

16. In Class Exercise.

17. 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.

18. 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.

19. 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 t 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.

20. 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 t

21. Cont’d…

22. Cont’d… .

23. 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.

24. 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;

25. 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),

26. 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.

27. Cont’d…
Derivation;

28. Cont’d…

29. 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.

30. 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]=αnu[n]
Find the impulse response of the overall system, h[n].
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]
h12[n]= =u[n+2]

31. Figure 2.7 : Interconnection of systems.
Cont’d…
Convolving h12[n] with h3[n].
h123 [n]= u[n+2]*d[n-2]
h123 [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. .

32. 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.

33. 2.6.1 Memoryless LTI Systems.
The output of a memoryless system depends only on the present input.
Continuous Time;
Discrete Time;

34. 2.6.2 Causal LTI Systems.
The output of a causal system depends on the present or past input.
Continuous Time;
Discrete Time;

35. 2.6.3 Stable LTI Systems.
Bounded Input Bounded Output (BIBO) stable system.
Discrete Time: absolute summability of the impulse response.
Continuous Time;

36. 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).

37. 2.7 Step Response.
Step response is defined as output due to a unit step input signal.
Step response s[n] 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.

38. 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.

39. 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. .

40. 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.

41. 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 (2.1)
Where the ri are the N roots of the system’s characteristic equation
Equation (2.2)
Substitution of equation (1) into the homogeneous equation results in y(h)(t) is a solution for any set of constant ci

42. Cont’d… Discrete-time System Solution of homogeneous equation
Where the ri are the N roots of the system’s characteristic equation
Equation (2.4)
Substitution of equation (2) into the homogeneous equation results in y(h)[n] is a solution for any set of constant ci. Note the CT and DT characteristic equations are different.

43. Solution: The homogeneous equation is
Example 2.8: Homogeneous Solution
Find the homogeneous solution for the first order recursive system described by the difference equation
Solution: The homogeneous equation is
And its solution is given by Equation (2) for N=1:
The parameter r1 is obtained from the root of the characteristic equation given by Equation (4) with N=1:
r1 - p=0
Hence, r1 = p and the homogeneous solution is

44. 2.8.2 Particular Solution
The particular solution y(p) represent any solution of the differential and difference equation for the given input.
Continuous Time
Discrete Time
Input, x(t)
Particular Solution, y(p) (t)
Input, x[n]
Particular Solution, y(p) [n] 1 c t
c1t + c2 n
c1n + c2
e-at
ce-at αn
cαn
cos(ωt + ø)
c1cos(ωt) +c2sin(ωt)
cos(Ωn + ø)
c1cos(Ωn) +c2sin(Ωn)

45. Example 2.9: Particular Solution Find the particular solution for the first order recursive system described by the difference equation If the input x[n]=(1/2)n Solution: We assume particular solution of the form Substitute y(p)[n] and x[n] to the given difference equation yields Multiply both sides of the equation by(1/2)-n to obtain cp(1-2p)=1 Equation (2.5) Solving this equation for cp gives the particular solution

46. Cont’d…
If p=(1/2), then particular solution has the same form as homogeneous solution found in Example 2.8. Note that in this case no coefficient cp satisfies Equation (2.5), and must assume particular solution of the form of
Substituting this particular solution into the difference solution gives
Using p=(1/2), cp=1

47. 2.8.3 Complete Solution
The complete solution of the differential or difference equation id obtained by summing the particular solution and the homogeneous solution and finding the unspecified coefficients in the homogeneous solution so that the complete solution satisfies the prescribed initial conditions. The procedure (solving a Differential or Difference Equation) is summarizes as follows:
1. Find the form of the homogeneous solution y(h) from the roots of the characteristic equation.
2. Find a particular solution y(p) by assuming that it is of the same form as the input, yet is independent of all terms in the homogeneous solution.
3. Determine the coefficients in the homogeneous solution so that the complete solution y = y(p) + y(h) satisfies the initial conditions

48. Example 2.10: Complete Solution Find the solution for the first order recursive system described by the difference equation Equation (2.6) If the input x[n]=(1/2)nu[n] and initial condition y[-1] =8. Solution: The form of the solution is obtained by summing the homogeneous solution determined in Example 2.8 with the particular solution in Example 2.9 after setting р =1/4 Equation (2.7) The coefficient c1 is obtained from initial condition. First, translate initial condition to time n=0 by rewrite Equation (2.6) in recursive form and substituting n=0 to obtain which implies that y[0]=1+(1/4)x8=3. Then, substitute now y[0]=3 into Equation (2.7) 3=2(1/2) 0 + c1(1/4)0 ; c1= 1 The complete solution is

49. 2.9 Characteristics of systems : natural and forced response.
2.9.1 Natural Response
The natural response is the system output for zero input thus
describes the manner in which the system dissipates any stored
energy or memory of the past represented by non-zero initial
conditions.
The natural response assumes zero input, it is obtained from the
homogenous solution in Equation (1) and (2) by choosing the
coefficients ci so that the initial conditions are satisfied.
The natural response assumes zero input and thus does not
involve a particular solution.

50. 2.9.1 Forced Response
The force response is the system output due to the input signal assuming zero initial conditions.
The forced response is the same form as the complete solution of the differential or difference equation.
A system with zero initial conditions is said to be “at rest”, since there is no stored energy or memory in the system.
The forced response describes the system behavior that is “forced” by the input when the system is at rest.
The forced response depends on particular solution, which is valid only for times t>0 or n>0.

51. Example 2.11: 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 this example and Example 2.10 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
which implies that y[0]=1+(1/4)x0, now we use y[0]=1 to solve for c1 from the equation. 1=2(1/2) 0 + c1(1/4)0 ; c1= -1
The force response of the system is, .

52. 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: y(t) = cx(t) or y[n] = cx[n] ;
c is a scalar
(2) Addition: y(t) = x(t) + w(t) or y[n] =x[n] + w[n]
(3) Integration for continuous-time LTI system:
y(t) = ∫t -∞ x(τ) dτ
and a time shift for discrete-time LTI systems:
y[n] = x[n-1]

53. 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.

54. Example of Difference Equation:
Discrete-Time System: Second Order Difference Equation.
Figure 2.11: Block Diagram Representation of Discrete Time LTI system Described by second order Differential Equation.

55. 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.