1 time domain representation of linear time invariant (lti). chapter 2 school of computer and...

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Time Domain Time Domain Representation Representation of Linear Time of Linear Time Invariant (LTI).Invariant (LTI).

CHAPTER CHAPTER 22

School of Computer and Communication School of Computer and Communication Engineering, UniMAPEngineering, UniMAP

Nordiana Binti Mohamad SaaidNordiana Binti Mohamad [email protected]@unimap.edu.my

EKT 230 EKT 230

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2.1 Introduction.2.1 Introduction.2.2 LTI System Properties.2.2 LTI System Properties.2.3 Convolution Sum. 2.3 Convolution Sum. 2.4 Convolution Integral.2.4 Convolution Integral.2.5 Interconnection of LTI System.2.5 Interconnection of LTI System.2.6 LTI System Properties and Impulse 2.6 LTI System Properties and Impulse

Response.Response.2.7 The Step Response.2.7 The Step Response.2.8 Solving Differential & Difference 2.8 Solving Differential & Difference

Equation.Equation.

2.0 Time Domain 2.0 Time Domain Representation of Linear Representation of Linear Time Invariant (LTI) Time Invariant (LTI) System.System.

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2.1 Introduction.2.1 Introduction.Learning Outcome: Learning Outcome: Examine several methods for describing the relationship between the input and the output signals of LTI system.

(1) Impulse Response. (1) Impulse Response.

(2) Linear Constant-Coefficient Differential.(2) Linear Constant-Coefficient Differential.

(3) Block Diagram.(3) Block Diagram.

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2.2 LTI System 2.2 LTI System Properties.Properties. If we know the response of the LTI system to some

inputs, we actually know the response to many inputs.

(1) (1) Commutative Property.Commutative Property.

txththtxty **

5

(2) (2) Distributive Property.Distributive Property.

thtxthtxty

ththtxty

21

21

**

*

Cont’d…Cont’d…

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(3) (3) Associative Property.Associative Property.

ththtxty

ththtxty

21

21

**

**


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x[n] is a signal as a weighted sum of basic 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 convolutionconvolution of two discrete-time signals y[n ] and h[n] is denoted as

2.3 Convolution Sum.2.3 Convolution Sum.

knhkxnhnxk

*

knkxnxk

knhkxnyk

8

Figure 2.1: Graphical example illustrating the representation of a signal x[n] as a weighted sum of

time-shifted impulses.


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Steps for Convolution Computation.Steps for Convolution Computation.Step 1Step 1: Plot x and h versus k since the convolution

sum is on k.

Step 2Step 2: Flip h[k] around the vertical axis to obtain h [- k].

Step 3Step 3: Shift h [-k] by n to obtain h [n- k].

Step 4Step 4: Multiply to obtain x[k] h[n- k].

Step 5Step 5: Sum on k to compute

Step 6Step 6: Index n and repeat Step 3-6.

knhkxm

knhkxnhnxnyk

*


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Example 2.1:Example 2.1: Convolution Sum.Convolution Sum.For the figure below, compute the convolution, For the figure below, compute the convolution, yy[[nn]]. .

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.

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Solution: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.

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Example 2.2:Example 2.2: Convolution.Convolution.The LTI The LTI hh[[nn] having an impulse response of ] having an impulse response of

and Solution:Solution:

Details: Explained in class.

Otherwise

n

n

nh

,0

0,2

1,1

Otherwise

n

n

n

nx

,0

2,2

1,3

0,2

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Example 2.3:Example 2.3: Convolution.Convolution.The LTI The LTI hh[[nn] having an impulse response of ] having an impulse response of

and the input

Find the convolution of, y[n]=x[n]*h[n].Solution:Solution:

.

nunhn

4

3 6 nununx

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In Class Exercise.In Class Exercise.

6 nununx

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An alternativealternative 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.

2.3.1 Convolution Sum 2.3.1 Convolution Sum Evaluation Procedure.Evaluation Procedure.

knhkxkwn

knhkxnyk

kwnyk

n

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Example 2.4:Example 2.4: Convolution Sum Evaluating Convolution Sum Evaluating by using an Intermediate Signal.by using an Intermediate Signal.Consider a system with impulse responseConsider a system with impulse response

Use the equation below to determine the output of the Use the equation below to determine the output of the system at times system at times nn=-5, =-5, nn =5, and =5, and nn =10 when the input =10 when the input is is xx[[nn]]==uu[[nn]]..

Solution: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).

otherwise

nkknh

kn

,

,

04

3

nunhn

4

3

otherwise

kkw

k

,

50,

04

35

5

17

For n =10, we have

The result is in Figure 2.3(d).

Note: for n<0, wk[n]=0, because no overlap occur.

288.3

34

1

34

1

4

3

3

4

4

35

6

5

5

0

5

k

k

y

otherwise

kkw

k

,

100,

04

310

10

831.3

34

1

34

1

4

3

3

4

4

310

11

10

10

0

10

k

k

y


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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].


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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.

2.4 Convolution Integral.2.4 Convolution Integral.

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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.

tHthwhere

dthxthtx

dthxty

.*


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To compute the superposition integral

Step for Convolution Integral Computation.Step for Convolution Integral Computation.Step 1Step 1: Plot x and h versus since the convolution sum

is on .

Step 2Step 2: Flip h( around the vertical axis to obtain h(-.

Step 3Step 3: Shift h() by t to obtain h(t-).

Step 4Step 4: Multiply to obtain x() h(t-).

Step 5Step 5: Integrate on to compute

Step 6Step 6: Increase and repeat Step 3-6.

thx

thxnhnxny *


22

t

Example 2.5:Example 2.5: Convolution Integral. Convolution Integral.Given a RC circuit below (RC=1s). Use convolution Given a RC circuit below (RC=1s). Use convolution to determine the voltage across the capacitor to determine the voltage across the capacitor yy((tt). ). Input voltage Input voltage xx((tt))=u=u((tt))-u-u((t-2t-2))..

Solution:Solution:y(t)=x(t)*h(t)- capacitor start

charging at t=0 and discharging at t=2.

a

b

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24

.


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2.5 Interconnection of LTI 2.5 Interconnection of LTI System.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.

2.5.1 Parallel Connection of LTI System.2.5.1 Parallel Connection of LTI System.

2.5.2 Cascade Connection of LTI System.2.5.2 Cascade Connection of LTI System.

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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;

2.5.1 Parallel Connection 2.5.1 Parallel Connection of LTI Systems.of LTI Systems.

thtxthtx

tytyty

21

21

**

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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 sum of their individual impulse responseresponse.

Distributive propertyDistributive property of convolution (CT and DT), ththtxthtxthtx 2121 ***

thtx

thx

ththxty

dthxdthxty

*

21

21

ththtxthtxthtx 2121 ***


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The impulse response an equivalent system representing two LTI systems connected in cascade is the convolution of their individual impulse convolution of their individual impulse responsesresponses.

Figure 2.6: Interconnection of two LTI systems. (a) Cascade connection of two systems. (b) Equivalent system. (c)

Equivalent system: Interchange system order.

2.5.2 Cascade Connection 2.5.2 Cascade Connection of Systems.of Systems.

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Derivation;


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Continuous TimeContinuous Time Associative properties

Commutative properties. It is often used to simplify the evaluation or interpretation of the convolution integral.

Discrete TimeDiscrete Time Associative properties

Commutative properties.

ththtxhthtx 2121 ****

thththth 1221 **

thththth 1221 **

ththtxhthtx 2121 ****


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Example 2.6: Example 2.6: Equivalent System to Four Equivalent System to Four Interconnected System.Interconnected System.The interconnected of four LTI system is shown in The interconnected of four LTI system is shown in figure below. Given the impulse response of the figure below. Given the impulse response of the system,system,

hh11[[nn]=]=uu[[nn]]

hh22[[nn]=]=uu[[n+2n+2]- ]- uu[[nn]]

hh33[[nn]=]=[[n-2n-2] and ] and hh44[[nn]=a]=annuu[[nn]]

hh11[[nn]=]=uu[[nn]]

Find the impulse response of the overall system.Find the impulse response of the overall system.

Solution: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]

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Convolving h12[n] with h3[n].

h123 [n]= u[n]+u[n+2]*[n]

= u[n]Finally, we sum h123[n] and -h4[n] to obtain the

overall impulse response:h[n]= {1-n}u[n].

Figure 2.7 : Interconnection of systems.

.


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The impulse response characterized the input output behavior of an LTI system. Below are the propertiesproperties of the system that relates to the impulse response.

2.6.1 Memoryless LTI Systems.2.6.1 Memoryless LTI Systems.

2.6.2 Causal LTI Systems.2.6.2 Causal LTI Systems.

2.6.3 Stable LTI Systems.2.6.3 Stable LTI Systems.

2.6.2 Invertible LTI Systems.2.6.2 Invertible LTI Systems.

2.6 Relations between LTI 2.6 Relations between LTI System Properties and System Properties and Impulse Response.Impulse Response.

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2.6.1 Memoryless LTI 2.6.1 Memoryless LTI Systems.Systems. The output of a memoryless system depends only

on the present input.

Continuous Time;Continuous Time;

Discrete Time;Discrete Time;

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2.6.2 Causal LTI Systems.2.6.2 Causal LTI Systems. The output of a causal system depends only on the

past input.


Discrete Time;Discrete Time;

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2.6.3 Stable LTI Systems.2.6.3 Stable LTI Systems. Bounded Input Bounded Output (BIBO) stable

system.

Discrete Time:Discrete Time: absolute summability of the impulse response.


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2.6.4 Invertible System 2.6.4 Invertible System and Deconvolution.and Deconvolution. Deconvolution is the process of recovering recovering x(t)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).

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2.7 Step Response.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.

n

k

khns

dhtst

tsdt

dth

nsnsnh

1

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Example 2.7:Example 2.7: RC Circuit of Step RC Circuit of Step Response.Response. The impulse response of the RC circuit is, The impulse response of the RC circuit is,

Find the step response of the circuit.Find the step response of the circuit.

Solution: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.

tueRC

th RC

t

1

dueRC

ts RC

t

1

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We simplify the integral to get

Figure 2.9: RC circuit step response for RC = 1 s.

.

0,1

0,0

,1

0,0

te

t

dueRC

t

ts

RC

t

RC


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2.8 Solving Differential and 2.8 Solving Differential and Difference Equation.Difference Equation. The output of Differential and Difference Equation

can be described as the sum of two components;

(1) Homogeneous Solution (1) Homogeneous Solution yy((hh))..

(2) Particular Solution (2) Particular Solution yy((pp)).. The complete solution is y = y(h) + y(p)

**Note: ( ) and [ ] are omit when referring to continuous and discrete time.

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2.8.1 Homogeneous 2.8.1 Homogeneous Solution.Solution. Homogeneous form of difference and differential

equation is obtained by setting all the terms involving the input to zero.

Continuous-time SystemContinuous-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

N

k

kkra

0

0

N

i

tri

h iecty1

00

)(

N

k

hk

k

k tydt

da

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Discrete-time SystemDiscrete-time SystemSolution of homogeneous equation

Equation (2)

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.

N

k

kNkra

0

0

N

i

nii

h rcny1

00

)(

N

k

hk

k

k knydt

da


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Example 2.9:Example 2.9: Forced Response.Forced Response. The system is describe by the first order recursive The system is describe by the first order recursive system, system,

Find the Find the force responseforce response of the system if the input of the system if the input xx[[nn]=(1/2)]=(1/2)nnuu[[nn].].

Solution: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= -1The force response is,.

nxnyny 14

1

0,4

1

2

12 1

ncny

nn

14

100 yxy

0,4

1

2

12)(

nny

nnf

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Example 2.10:Example 2.10: Complete Response.Complete Response.

..

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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.

Block Diagram Representation.Block Diagram Representation.

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Block Diagram Representation.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.


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Discrete-Time System: Second Order Discrete-Time System: Second Order Difference Equation.Difference Equation.

Figure 2.11: Block Diagram Representation of Discrete Time LTI system Described by second order Differential

Equation.

Example of Difference Equation:Example of Difference Equation:

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Discrete-Time System: Second Order Discrete-Time System: Second Order Difference Equation.Difference Equation.

Figure 2.12: Direct form II representation of an LTI system described by a second-order difference equation.


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