application of pseudo random binary sequence (prbs

APPLICATION OF PSEUDO RANDOM BINARY SEQUENCE (PRBS) SIGNAL

IN SYSTEM IDENTIFICATION

MAIMUN BINTI HUJA HUSIN

A project report submitted in partial fulfilment of the

requirements for the award of the degree of

Master of Engineering (Electrical – Mechatronics and Automatic Control)

Faculty of Electrical Engineering

Universiti Teknologi Malaysia

MAY 2008

iii

To my family who loves me, especially to my beloved mother and father for

education they give me and also for their supports and

understandings

iv

ACKNOWLEDGEMENT

First of all, thanks to Allah SWT for giving me strength and chances incompleting this project.

Secondly, I wish to express my sincere appreciation to my supervisor,Associate Professor Dr Mohd Fua’ad bin Rahmat, for encouragement and guidance. Igreatly appreciate his dedication in constructively criticizing my work, including mythesis. I have truly enjoyed working with him.

I wish to thank Universiti Malaysia Sarawak (UNIMAS) and Malaysiangovernment, for a study leave and financial support, through SLAB-JPA scholarship.

Finally, I would like to thank my parents and family for their constantsupport, encouragement and understanding during my struggle away from home,friends in Universiti Teknologi Malaysia (UTM for coloring my life in UTM.

v

ABSTRACT

This project emphasized on both software and hardware analysis. Pseudo

random binary sequence (PRBS) signal of 15 different maximum length sequences

were developed using MATLAB software and were used as forcing function in

simulated second order. There are four second order system responses that were

examined; overdamped, underdamped, undamped and critically damped. For each

response, traces of the output response of system forced by PRBS or without PRBS

in the absence or presence of noise were analyzed. The autocorrelation function of

the input signal and cross correlation function between input and output signal were

performed using MATLAB software. From the correlograms of autocorrelation and

cross correlation, the transfer function of the system was estimated. For verification

of the simulation work, PRBS generator circuit was build using Transistor-transistor

logic. The PRBS signal generated was analyzed using Dynamic Signal Analyzer.

An experiment using PRBS as the forcing function to an unknown system was

performed. The autocorrelation function of the input signal and cross correlation

function between input and output signal were performed using Dynamic Signal

Analyzer and the transfer function model of the unknown system was estimated.

Results from this experiment were used to validate the simulation work previously.

vi

ABSTRAK

Projek ini tertumpu kepada penganalisaan aturcara dan juga perkakasan.

Isyarat Perduaan Jujukan Rawak (PRBS) sebanyak 15 panjang jujukan maksima

dihasilkan menggunakan aturcara MATLAB dan ianya digunakan sebagai fungsi

pemaksa di dalam pengujian sistem tertib kedua. Empat jenis sambutan sistem tertib

kedua telah dianalisa; redaman lampau, teredam, sambutan tanpa redaman dan

redaman genting. Untuk setiap jenis sambutan tertib kedua, analisis terhadap

sambutan sistem yang dipaksa oleh PRBS atau yang tidak dipaksa oleh PRBS, dalam

kehadiran gangguan atau tidak telah dilaksanakan. Fungsi sekaitan auto untuk

isyarat masukan dan fungsi sekaitan silang antara isyarat masukan dan keluaran akan

dilaksanakan menggunakan aturcara MATLAB. Dari graf sekaitan auto melawan

masa lengah dan sekaitan silang melawan masa lengah, rangkap pindah untuk model

sistem tersebut dikenalpasti. Untuk pembuktian keputusan analisa menggunakan

aturcara MATLAB, penjana isyarat PRBS dibina menggunakan IC TTL. Isyarat

PRBS yang dihasilkan dianalisis menggunakan Penganalisis Isyarat Dinamik. Satu

ujikaji menggunakan isyarat PRBS sebagai fungsi pemaksa kepada satu sistem yang

tidak diketahui telah dijalankan. Fungsi sekaitan auto bagi isyarat masukan dan

fungsi sekaitan silang di antara isyarat masukan dan isyarat keluaran dilaksanakan

menggunakan Penganalisis Isyarat Dinamik dan seterusnya rangkap pindah untuk

model sistem yang tidak diketahui dikenalpasti. Keputusan ujikaji tersebut

digunakan untuk membuktikan keputusan analisa menggunakan aturcara MATLAB

yang sebelum ini.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENT vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF ABBREVIATIONS xiv

LIST OF APPENDICES xv

1 INTRODUCTION 1

1.1 Introduction 1

1.2 Rational, Significance and Need for the Study 1

1.3 Research Objectives 2

1.4 Scope of project 2

1.5 Project Outline 3

2 LITERATURE REVIEW 4

2.1 Previous research 4

2.2 System Identification 5

2.3 Input signal 7

2.4 Types of PRBS 8

viii

2.4.1 MLS signals 8

2.4.2 QRB signals 9

2.4.3 HAB signals 9

2.4.4 TPB signals 10

2.4.5 QRT signals 10

2.5 Linear feedback shift register (LFSR) 10

2.6 Feedback configuration 11

2.7 Properties of PRBS 12

2.7.1 Modulo-2 13

2.7.2 Correlation 13

2.7.2.1 Autocorrelation Function 14

2.7.2.2 Cross Correlation Function 16

2.7.3 Power Spectral Density 17

2.8 Summary 18

3 METHODOLOGY 19

3.1 Introduction 19

3.2 Software analysis 19

3.2.1 PRBS generator 19

3.2.2 PRBS signal as test signal to second 20

order system

3.3 Hardware analysis 27

3.3.1 PRBS generator 27

3.3.1.1 Clock circuit 27

3.3.1.2 Feedback circuit 29

3.3.1.3 Shift register circuit 29

3.3.2 PRBS signal as test signal to second 31

order system

4 RESULT 33

4.1 Introduction 33

4.2 PRBS signal (Simulation result) 33

ix

4.3 PRBS signal as forcing function in a second 36

order system (Simulation result)

4.31 Critically damped response 37

4.3.2 Underdamped response 40

4.3.3 Overdamped response 44

4.3.4 Undamped response 48

4.4 PRBS signal (Hardware result) 51

4.5 PRBS signal as test input to a second order system 53

(Hardware result)

4.5.1 Critically damped response 53

4.5.2 Underdamped response 56

5 CONCLUSIONS AND FUTURE WORKS 60

5.1 Conclusion 60

5.2 Future Works 61

REFERENCES 62

Appendices A – C 64 - 111

x

LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Feedback configuration of LFSR 12

2.2 “Exclusive or” operation 13

3.1 Second order system being identified 21

3.2 List of components for clock circuit 27

3.3 List of components for shift register circuit 29

3.4 List of components for RC low pass filter circuit 31

3.5 RC low pass filter second order system transfer function 32

4.1 Successive states of shift register 34

4.2 Transfer function for several different PRBS maximum 40

length


length


length

4.5 Transfer function obtained for hardware analysis 59

5.1 Transfer function obtained for each system (simulation) 60

5.2 Transfer function obtained for each system (hardware) 61

xi

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Dynamic system 5

2.2 Schematic flowchart of system identification 7

2.3 LFSR 11

2.4 Autocorrelation function of PRBS signal 16

2.5 Autocorrelation function of periodic white noise 16

2.6 Power spectral density of PRBS signal 18

3.1 SIMULINK block diagram of PRBS generator circuit 20

for MLS of N = 15

3.2 Block diagram of system (critically damped) being 23

identified

3.3 Block diagram of system (overdamped) being identified 24

3.4 Block diagram of system (underdamped) being identified 25

3.5 Block diagram of system (undamped) being identified 26

3.6 Block diagram of PRBS generator circuit 27

3.7 Clock circuitry 28

3.8 Block diagram of PRBS generator for MLS of N = 255 30

3.9 Second order system RC circuit 31

4.1 (a) Clock signal, (b) PRBS signal, 34

(c) Autocorrelation function, and

(d) Power spectral density for MLS of N = 15






xii


4.4 (a) PRBS signal and traces of output response of system 37

(b) forced by PRBS in the absence of noise

(c) without PRBS in the presence of noise

(d) forced by PRBS in the presence of noise

4.5 Autocorrelation functions of input and output signals 38

4.6 Cross correlation functions of output signals 38

4.7 Power spectral density of input and output signals 40






















4.20 Dynamic Signal Analyzer (HP35670A DSA) 51

4.21 PRBS signal for MLS of N = 63 51

4.22 Autocorrelation function of PRBS signal for MLS of 52

N = 63

4.23 Power spectral density of PRBS signal for MLS of N = 63 52

xiii

4.24 Block diagram of PRBS testing 53

4.25 Schematic circuits for critically damped response 54

4.26 Output signal using PRBS signal 54

4.27 Autocorrelation function of output signal using PRBS 55

signal

4.28 Cross correlation function of output signal using PRBS 55

signal

4.29 Schematic circuits for underdamped response 56

4.30 Output signal using PRBS signal 57

4.31 Autocorrelation function of output signal using PRBS 58

signal

4.32 Cross correlation function of output signal using PRBS 58

signal

xiv

LIST OF ABBREVIATIONS

HAB – Hall Binary

LFSR – Linear feedback shift register

MLS – Maximum length sequence

PRBS – Pseudo random binary sequence

QRB – Quadratic residue binary

QRT – Quadratic residue ternary

TPB – Twin Prime Binary

xv

LIST OF APPENDICES

APPENDIX TITLE PAGE

A Computer Programs 65

B Datasheets 68

C Presentation Slide 89

CHAPTER 1

INTRODUCTION

1.1 Introduction

Pseudo random signal has been widely used for system identification (A.H.

Tan and K.R. Godfrey, 2002). Maximum length sequence (MLS) signals are the

known class of pseudo random signals (N. Zierler, 1959); because it can be easily

generated using feedback shift registers (A.H. Tan and K.R. Godfrey, 2002). There

are several other classes of binary and near-binary signal but are less well known

such as quadratic residue binary (QRB), Hall binary (HAB), Twin Prime binary

(TPB) and quadratic residue ternary (QRT).

1.2 Rational, Significance and Need for the Study

In the 1960’s and early 1970’s, there was a fairly large amount of research

into the design and application of pseudo random signals. Pseudo random binary

signals based on maximum length sequences are easy to generate using simple shift

register circuitry with appropriate feedback, and this has resulted in their

incorporation as a routine facility in a number of signal generators and their use in a

wide range of system dynamic testing (K.R. Godfrey, 1991).

It is important to study and generate PRBS because of the difficulty faced in

generating a truly random sequence. A PRBS is not a truly random sequence but

with long sequence lengths, it can show close resemblance to truly random signal

2

and furthermore it is sufficient for the test purposes. PRBS have well known

properties and the most important point is its generation is rather simple. Moreover,

knowing how a PRBS signal is generated make it is possible to predict the sequence.

Outermost it makes error that might occur in the sequence is possible to register and

count.

1.3 Research Objectives

There are four main objectives of this research, as stated below:

(i) To design and generate PRBS generator with different MLS using

MATLAB,

(ii) To design PRBS generator using hardware (Transistor-transistor

logic-TTL),

(iii) To analyze the characteristic of PRBS signal such as auto correlation

function, cross correlation function, and power spectral density using

MATLAB and dynamic signal analyzer,

(iv) To perform an experiment using real system where PRBS is the test

input.

1.4 Scope of project

This project emphasized on both software and hardware analysis. PRBS

generator with 15 different MLS (n=2, 3…, 16) were designed using MATLAB

(SIMULINK) software. The signals obtained were used as forcing function in

second order system. Four second order system responses were examined;

overdamped, critically damped, undamped and critically damped. For each category,

the response curves, autocorrelation function, cross correlation function and power

spectral density are observed for three different conditions; system forced by PRBS

signal in absence of noise, noisy system forced by PRBS signal and noisy system

without PRBS signal as forcing function. The autocorrelation function of the input

3

signal and cross correlation function between input and output signal were used to

estimate the transfer function model of the system.

Hardware analysis is done for the purpose of validation. PRBS generator was

constructed using TTL. PRBS signal generated was tested using dynamic signal

analyzer. An experiment using real second order system using PRBS as the test

input was performed. The autocorrelation function of the input signal and cross

correlation function between input and output signal were performed using Dynamic

Signal Analyzer. The correlograms of these two functions were used to determine the

transfer function model of the real second order system.

1.5 Project Outline

The preceding sections briefly summarized the contributions of the thesis.

This section outlines the structure of the thesis and summarizes each of the chapters.

Chapter 2 describes the relevant literature and previous work regarding PRBS

and its application in system identification. Overview of several classes of binary

and near binary signals such as MLS, QRB, HAB, TPB and QRT will be explore,

and characteristic of PRBS signal such as autocorrelation function, cross correlation

function and power spectral density will be explained.

Chapter 3 introduces method or approach taken in order to achieve the four

objectives set earlier in Chapter 1. This chapter describes the design for PRBS

generator for both approaches, software simulation using MATLAB SIMULINK and

hardware implementation using TTL.

Chapter 4 presents the results obtained from the simulation and experimental

work done. Analyses were done on the results. Experimental results obtained

validated the simulation result. Chapter 5 consists of conclusion and suggestions for

future improvement.

CHAPTER 2

LITERATURE REVIEW

2.1 Previous research

In the 1960’s and early 1970’s, there was a substantial amount of research

into the design and application of pseudo random signals (Godfrey, 1990). Periodic

signals have been widely used in the field of system identification. These signals can

be split into two main categories, computer – optimized signals and pseudo random

signals.

Periodic, multiharmonic test signals are extremely suitable for linear system

identification (Van Den Bos, 1993). There are many research are done on periodic,

multisine, multilevel multi harmonic signals.

Pseudo random binary signals based on MLS are widely used in system

dynamic testing and also incorporating as a routine facility in number of signal

generator because they are easy to generate using simple shift register (Godfrey,

1991). One research is done on generating pseudo random sequence longer than

maximum length sequence by subdividing the 1-stage shift register into two parts

and clocking each part at different speeds (Mouine and Boutin, 1998). There is

research done on other classes of binary and near – binary pseudo random signals

(Tan and Godfrey, 2002). Appropriately chosen pseudo random signals provide

highly acceptable alternatives to multisine signals in applications requiring uniform

power in the frequency spectrum (Godfrey, Barker and Tucker, 1999).

5

2.2 System Identification

System identification is a field of modeling dynamic systems form

experimental data (Sodestrom and Stoica, 1989). A dynamic system can be

described as in Figure 2.1, with u (t) is the input variable, v (t) is the disturbance and

y (t) is the output signal. The output signal is a variable provides useful information

about the system.

Figure 2.1 Dynamic system

There are two ways of constructing mathematical models:

(i) Mathematical modeling

Mathematical modeling is an analytic approach. In order to describe the

dynamic behavior of the process, basic laws from physics are used. For

example, balance equations are used in stirred tank modeling.

(ii) System identification

System identification is an experimental approach. This approach requires

some experiments to be performed on the system. Then, a model is fitted to

the recorded data by assigning suitable numerical values to its parameters.

In many cases where a complex processes involved, mathematical model

cannot be used. In such cases, only identification technique can be applied. System

identification usually applied when a model based on physical insight contains a

number of unknown parameters (even though the structure is derived from some

physical laws). Identification methods can be applied to estimate unknown

parameters.

Disturbancev (t)

Outputy (t)

Inputu (t)

System

6

The models obtained by system identification have the following properties

(Sodestrom and Stoica, 1989):

(i) Limited validity (valid for certain working point, certain type of input, certain

process, etc.)

(ii) Little physical insight

(iii) Easy to construct and use

Without interaction from the user, identification cannot be used. The reasons

for this include:

(i) Appropriate model must be found

(ii) No perfect data in real life

(iii) Process may vary with time, which can cause problems if an attempt is made

to describe it with a time-invariant model

(iv) May be difficult to measure some variables or signal which are important for

the model

An identification experiment is performed by exciting the system using some

input signal (such as step, sinusoid or random signal) and its input and output is

observed over a time interval. These signals are recorded. Then a parametric model

is choosing in order to fit the recorded signals. In order to do this, the first step to be

taken is to determine an appropriate form of the model. Then, the second step is to

estimate the unknown parameters of the model. Finally, the model is tested to check

whether it is an appropriate representation of the system. The summary of

identification experiment is shown in Figure 2.2.

7

Modelaccepted?

Modelvalidation

ChoosemethodEstimate

parameters

Determine/choose model

structure

PerformexperimentCollect data

Design ofexperiment

Start

End

A prioriknowledgePlanned useof the model

New data set

YES

NO

Figure 2.2 Schematic flowchart of system identification

2.3 Input signal

The input signal used in an identification experiment can have a significant

influence on the resulting parameter estimates (Sodestrom and Stoica, 1989).

Traditional experiment procedures involve subjecting the system to input signals

8

such as step, ramp, impulse or sinusoidal input. These types of inputs have simple

analysis of the output response curves.

The advantages of these input signals are:

(i) Ease of signal generation

(ii) Ease of analysis

(iii) The physical understanding of system response which result

The only disadvantage of these input signals is it is not practical because of

limitations imposed by the existence of system noise.

A PRBS signal is a popular input signal for system identification because it is

persistently exciting to the order of the period of the signal. A maximum length

PRBS signal has a correlation function that resembles a white noise correlation

function. This property does not hold for non-maximum length sequences. Thus the

PRBS signal used in identification processes should be a maximum length PRBS

signal. The maximum possible period for a maximum length sequence is N = 2n - 1

where n is the order of the PRBS.

2.4 Types of PRBS

There are several types of PRBS such as MLS, QRB, HAB, TPB and QRT.

In this research, MLS will be used in designing the PRBS generator due to its

simplicity in construction.

2.4.1 MLS signals

MLS signals exist for N = 2n – 1 (Zapernick and Finger, 2005), where n is an

integer > 1, that is N = 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, etc. They can be

generated in hardware using shift registers consisting of n stages (Tan and Godfrey,

2002).

9

MLS is one of the most important classes of pseudo random binary sequence.

It has excellent pseudo randomness properties and fulfills all randomness criteria

[Section 2.7].

2.4.2 QRB signals

QRB signals exist for N = 4k – 1, where k is an integer and N is prime

(Zapernick and Finger, 2005), that is N = 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79,

etc. The sequence rx , Nr ,,2,1 is formed from the rule (Tan and Godfrey,

2002)

otherwise1

modulosquare,aisif1

r

r

x

Nrx

1or1 Nx

2.4.3 HAB signals

HAB signals exists for periods N = 4k2 + 27, where k is an integer and N is

prime (Zapernick and Finger, 2005), that is N = 31, 43, 127, 223, 283, 811, 1051,

1471, 1627, etc. A primitive root u of N is first chosen. These sequence is formed

from the rule that (Tan and Godfrey, 2002)

6)(modulo3or1,0where

modulo,if1

t

Nurx tr

otherwise1rx

10

2.4.4 TPB signals

TPB signals exist for N = k (k + 2), where k and k + 2 are both prime

(Zapernick and Finger, 2005), that is N = 15, 35, 143, 323, 899, 1763, 3599, 5283,

etc. First, QRB sequences are generated for lengths k and k + 2; these sequences are

denoted by ra and rb respectively [1]. Then the TPB sequence rx is defined

by (Tan and Godfrey, 2002)

2)(kmodulo0but

k,modulo0if1

2)k(modulo0if1

2)k(moduloorkmodulo,0for

r

rx

rx

rbax

r

r

rrr

2.4.5 QRT signals

QRT signals exist for N = 4k ± 1 (Zapernick and Finger, 2005), where k is an

integer and N is prime, that is N = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, etc.

This class of pseudorandom signals has a large number of possible values of N.

They are generated using the same formula as for QRB signal except that Nx is set to

0, resulting in a ternary signal with (N – 1) / 2 elements + 1, (N – 1) / 2 elements – 1,

and one element zero (Tan and Godfrey, 2002).

The autocorrelation function of a QRT signal is nearly identical to that of

MLS signal, and for a QRT signal with signal levels – 1, 0, and + 1, the on – peak

value of the autocorrelation is (N – 1) / N and the off – peak value is – 1 / N.

2.5 Linear feedback shift register (LFSR)

Length of MLS is given by 12 nN where n is an integer (i.e. N +15, 31,

63, 127, 255…). MLS can be generated by an n stage shift register with the first

11

stage determined by feedback of the appropriate modulo two sum of the last stage

and one or two earlier stage. This structure is usually called LFSR and its general

structure is shown in Figure 2.3.

+

Flip flops

Modulo 2 addition

second)teverycontentsshift(topulseClock

Figure 2.3 LFSR

2.6 Feedback configuration

The logic contents of the shift register are moved one stage to the right every

∆t seconds by simultaneous triggering by a clock pulse. All possible states of the

shift register are passed through except that of all zeros. The output can be taken

from any stage and is a serial sequence of logic states having cyclic period N ∆t. If

feedback is taken from the modulo 2 sum of the wrong register stages, then the

resulting cyclic sequence has length less than the maximum length, and will not be

suitable.

The correct stages the most commonly used lengths are shown in Table 2.1.

12

Table 2.1 Feedback configuration of LFSR

No. n N = 2n – 1 Feedback

1 2 3 2, 1

2 3 7 3, 1

3 4 15 1, 4 / 3, 4

4 5 31 2, 5 / 3, 5

5 6 63 1, 6 / 5, 6

6 7 127 1, 7 / 4, 7

7 8 255 2, 3, 4, 8

8 9 511 4, 9 / 5, 9

9 10 1023 3, 10

10 11 2047 2, 11

11 12 4095 1, 2, 10, 12

12 13 8191 1, 2, 12, 13

13 14 16383 1, 2, 12, 14

14 15 32767 1, 15

15 16 65535 2, 3, 5, 16

2.7 Properties of PRBS

MLS is one of the most important classes of pseudo random binary sequence.

It has excellent pseudo randomness properties and fulfills all randomness criteria

below (Zapernick and Finger, 2005):

(i) Balance property,

In each period of random sequence the number of logic zeros should not

differ from the number of logic ones by at most one.

(ii) Run property,

Let a run refer to a string of consecutive ones. The 0-runs and 1-runs

alternate with equally many 0-runs and 1-runs of the same length. The

lengths of runs in each period are distributed such that one-half the runs are

13

of length 1, one-quarter the runs are of length 2, one-eight the runs are of

length 3, etc.

(iii) Correlation property

If a period of the random sequence is compared term by term with any cyclic

shift of itself, then the number of agreements and disagreements should not

differ by more than one.

2.7.1 Modulo-2

Modulo 2 addition is the logic function “exclusive or”. In “exclusive or”

operation, if the inputs are the same, the output is logic 0; if the inputs are different,

the output is logic 1. Table 2.2 illustrates the “exclusive or” operation.

Table 2.2 “Exclusive or” operation

Inputs Output

A B Q

0 0 0

0 1 1

1 0 1

1 1 0

2.7.2 Correlation

A non – deterministic signal cannot be defined by means of an explicit

function of time but must instead be described in some probabilistic manner. Term

correlation functions are used to describe the appropriate statistical descriptions for

the signals when undertaking system identification with non – deterministic forcing

functions and carrying out the analysis in the time domain.

14

The correlation of two random variables is the expected value of their

product; showing the dependency of one variable with another. A high correlation

might be expected when the two time instants are very close together, but much less

correlation when the time instants are widely separated.

If the random variables come from the same signal the function is called an

autocorrelation function. If the random variables come from the different signal the

function is called a cross correlation function.

2.7.2.1 Autocorrelation Function

The autocorrelation function of a signal x(t) is given the symbol )(xx and is

defined as,

)(signalofntdisplacemeis)(and)(

where

)()(2

1lim)(

)()(2

1lim)(

xx

xx

txtxtx

dttxtxT

or

dttxtxT

T

TT

T

TT

)(xx is the time average of the product of the value of the function

seconds apart as is allowed to vary from zero to some large value, the averaging

being carried out over a long period 2T.

Some of the properties of autocorrelation function )(xx of a signal x(t) are

outlined below:

(i) The autocorrelation function is an even function of , i.e. )()( xxxx ,

because the same set of product values is averaged regardless of the direction

of translation in time.

(ii) )0(xx is the mean square value, or average power of x(t).

15

(iii) )0(xx is the largest value of autocorrelation function, but if x(t) is periodic,

then )(xx will have the same maximum value when is an integer

multiple of the period.

(iv) If x(t) has a d.c. component or mean value, then )(xx also has a d.c.

component, the square of the mean value.

(v) If x(t) has a periodic component, then )(xx also has a component with the

same period, but with a distorted shape resulting from the lack of

discrimination between differing phase relationship of the constituent

sinusoidal components.

(vi) If x(t) has only random components, 0)(xx as .

(vii) A given autocorrelation function may correspond to many time functions, but

any one time function has only one autocorrelation function.

For PRBS, first value is considered at tk where k is an integer. Let value

of the sequence for successive intervals ∆t to be N)x(3),...x(x(2),x(1), . The

autocorrelation function of PRBS is

N

j

kjxjxN

k1

xx )()(1

)(

)digitsdifferingofnumber-digitsmatchingofnumber()(2

xx N

ak

0if

0if)(

2

2

xx

ka

kN

ak

It can be shown by considering area changes that autocorrelation function is

linear between these points. Hence the form of the autocorrelation function is as

shown in Figure 2.4.

16

N

a2

tNtt

2a

)(xx k

Figure 2.4 Autocorrelation function of PRBS signal

As 0t and N becomes large the autocorrelation function tends closer to that of

true periodic white noise as shown in Figure 2.5.

shifttime

)(xx

Figure 2.5 Autocorrelation function of periodic white noise

2.7.2.2 Cross Correlation Function

Process of comparing one signal with another by multiplication of

corresponding instantaneous values and taking the average is called cross correlation

function. Cross correlation function is a graph of the value of the coefficient against

parametric time shift. Cross correlation function is a measure of the similarity

between two different signals.

17

Frequently, there exist two signals x(t) and y(t) which are not completely

independent. Cross correlation function is a measure of dependence of one signal on

the other. Cross correlation function is defined as,

lyrespective)(and)(signalofntsdisplacemeare)(and)(

where

)()(2

1lim)(

)()(2

1lim)(

xy

xy

txtytxty

dttxtyT

or

dttytxT

T

TT

T

TT

2.7.3 Power Spectral Density

It is convenient to describe the signals in terms of frequency domain

characteristics. The function used is the power density spectrum or Power spectral

density )(xx which is the Fourier transform of the autocorrelation function:

functionationautocorrelis)(

where

)()(xx

xx

jxx de

The power spectrum of a PRBS is shown in Figure 2.6. The difference

between a true random signal and that of maximal length PRBS, is that the spectrum

of the true random signal is continuous, while that of a PRBS is discrete. But by

choosing a PRBS with a long period, close resemblance to a true random signal can

be obtained. This property makes PRBS ideal as test signals.

18

tN

2

t3

2 t

2

t

4

)1

(2

N

Nta

)(xx

3dB

Figure 2.6 Power spectral density of PRBS signal

2.8 Summary

A PRBS is a random bit sequence that repeats itself. The properties of PRBS

hold, together with the simple generation and acquisition scheme makes them ideal

for test purposes. If the sequence length of a PRBS is chosen long enough, the

power spectrum of the sequence will show very close resemblance to that of a truly

random sequence.

CHAPTER 3

METHODOLOGY

3.1 Introduction

This chapter illustrates the approaches taken to fulfill the objectives set for

this project. The approaches are divided into two main parts. The first part is the

design procedure for software analysis, and the second part is the design procedure

for hardware analysis. There is an additional part on the procedures to obtain a

transfer function from correlograms of autocorrelation and cross correlation for both

software and hardware analysis.

3.2 Software analysis

Software used in this project is MATLAB SIMULINK. There are two sub

topics describe in this part; PRBS generator circuit and PRBS signal as test signal to

a second order system.

3.2.1 PRBS generator

PRBS generator circuit consists of few stages of flip-flops depends on the

maximum length sequence chosen, a feedback circuit, and a clocking circuitry. By

using MATLAB SIMULINK block sets, the block diagram of PRBS generator is

shown in Figure 3.1.

20

Figure 3.1 SIMULINK block diagram of PRBS generator circuit for MLS of N = 15

From Figure 3.1, for maximum length sequence of N = 15, the first stage of

shift register is determined by feedback of the appropriate modulo two sum of the

last stage and one earlier stage. Modulo two sum is represents by the logic function

‘exclusive or’. The logic contents of the shift register are moved one stage to the

right every t seconds by simultaneous triggering by a clock pulse. The output can

be taken from any stage and is a serial sequence of logic states having cyclic period

tN .

3.2.2 PRBS signal as test signal to second order system

PRBS signal is used as test signal or forcing function in a second order

system. There are four systems being examined; critically damped, overdamped,

underdamped and undamped. The transfer function and the corresponding damping

ratios for these systems are shown in Table 3.1.

21

Table 3.1 Second order system being identified

No. Type of second order system Damping ratio, ξ Transfer function

1 Critically damped ξ = 1

96

92 ss

2 Underdamped 0 < ξ < 1

92

92 ss

3 Overdamped ξ > 1

99

92 ss

4 Undamped ξ = 0

9

92 s

For overdamped response,

)146.1)(854.7(

9

)99(

9)(

2

sssssssC

This response has a pole at the origin that comes from the unit step input and

two real poles that come from the system. The input pole at the origin generates the

constant forced response; each of the two system poles on the real axis generates an

exponential natural response whose exponential frequency is equal to the pole

location. This response is called overdamped.

For underdamped response,

)81)(81(

9

)92(

9)(

2jsjsssss

sC

This function has a pole at the origin that comes from the unit step input and

two complex poles that come from the system. The real part of the system pole

generates exponentially decaying amplitude while the imaginary part of the system

pole generates sinusoidal waveform. The time constant of the exponential decay is

equal to the reciprocal of the real part of the system pole. The value of the imaginary

part is the actual frequency of the sinusoid. This sinusoidal frequency is given by the

name damped frequency of oscillation, d . Finally, the steady-state response (unit

step) was generated by the input pole located at the origin. This type of response is

22

called an underdamped response, one which approaches a steady-state value via a

transient response that is a damped oscillation.

For undamped response,

)3)(3(

9

)9(

9)(

2 jsjsssssC


two imaginary poles that come from the system. The input pole at the origin

generates the constant forced response, and the two system poles on the imaginary

axis at 3j generate a sinusoidal natural response whose frequency is equal to the

location of the imaginary poles. This type of response is called undamped. The

absence of a real part in the pole pair corresponds to an exponential that does not

decay.

For critically damped response,

)3)(3(

9

)96(

9)(

2

sssssssC


two multiple real poles that come from the system. The input pole at the origin

generates the constant forced response, and the two poles on the real axis at -3

generate a natural response consisting of an exponential and an exponential

multiplied by time, where the exponential frequency is equal to the location of the

real poles. This type of response is called critically damped. Critically damped

responses are the fastest possible without the overshoot that is characteristic of the

undamped response.

The SIMULINK block diagrams for each type of second order systems are

shown in Figure 3.2 to Figure 3.5.

23

Fig

ure

3.2

Blo

ckdia

gra

mof

syst

em(c

riti

call

ydam

ped

)bei

ng

iden

tifi

ed

24

Fig

ure

3.3

Blo

ckdia

gra

mof

syst

em(o

ver

dam

ped

)bei

ng

iden

tifi

ed

25

Fig

ure

3.4

Blo

ckdia

gra

mof

syst

em(u

nder

dam

ped

)bei

ng

iden

tifi

ed

26

Fig

ure

3.5

Blo

ckdia

gra

mof

syst

em(u

ndam

ped

)bei

ng

iden

tifi

ed

27

3.3 Hardware analysis

Hardware analysis is divided into two; PRBS generator circuit using

transistor – transistor logic and PRBS signal as the test input for a second order

system. A simple second order RC low pass filter is designed for test purposes.

3.3.1 PRBS generator

PRBS generator circuit consists of four main circuits; supply voltage, clock

circuit, feedback circuit, and PRBS generator circuit. Supply voltage of 5V is

required to supply the clock circuit, feedback circuit and shift register circuit. The

overall block diagram for PRBS generator is shown in Figure 3.6.

Figure 3.6 Block diagram of PRBS generator circuit

3.3.1.1 Clock circuit

Clock circuit consists of a basic oscillator circuit using LM555 timer chip.

The circuit diagram for clock circuit is as shown in Figure 3.7. List of components

used to construct the clock circuit is as shown in Table 3.2.

Supply voltage

Shift RegisterClock circuit

Feedback circuit

PRBS Signal

28

Table 3.2 List of components for clock circuit

No. Description Quantity

1 IC LM555 Timer 1

2 Resistor 2 k 1

3 Resistor 10 k 1

4 Resistor 47 k 1

5 Resistor 47 k 1

6 Capacitor 10 ηF 1

7 Capacitor 100 ηF 1

8 IC 7805 regulator 1

R12.0kR2

47k

R3470k

C1100nF C2

10nF

R410k

555

NET_8

2

3

4 5

1

7

8

65

2

7

6

9

3

4

Figure 3.7 Clock circuitry

The output for this clock circuitry is clock pulses of frequency 15Hz. To

calculate the frequency:

Hzf

Fkkf

CRRf

15

1.0)470247(693.0

1

)221(693.0

1

29

3.3.1.2 Feedback circuit

Feedback circuit is used to determine different maximum length sequence of

the PRBS signal. Different maximum length sequence has different feedback

configuration. Feedback circuit consist of IC 74LS86 (EX-OR).

3.3.1.3 Shift register circuit

The first stage of shift register is determined by the feedback circuit. The

output can be taken from any stage of the shift register. List of components used to

construct the shift register circuit is as shown in Table 3.3.

Table 3.3 List of components for shift register circuit


1 IC 74LS112 (J – K flip flop) 8

2 IC 74LS04 (Inverter flip – flop) 1

3 Light Emitting Diode (Red) 16

4 IC 7805 regulator 1

Block diagram of PRBS generator circuit is shown in Figure 3.8.

30

U1A

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

U1B

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

U2A

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

U2B

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

U3A

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

U3B

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

U4A

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

U4B

74LS

112D

1Q

5

~1Q

6

~1PR4

1K

2

~1CLR

15

1J

3

1CLK

1

24

6

7

12

13

14

U5A

74LS

04D

15

U6A

74LS

86D

U6B

74LS

86D

U6C

74LS

86D

21

3

5

17

20

18

1

19

11

10

9

VC

C

5V

VCC

V1

10

Hz

2V

22

Fig

ure

3.8

Blo

ckdia

gra

mof

PR

BS

gen

erat

or

for

ML

Sof

N=

255

31

3.3.2 PRBS signal as test signal to second order system

The second order system used as unknown system in the hardware

implementation of PRBS signal as test signal is shown in Figure 3.9. It is actually an

RC low pass filter circuit.

Figure 3.9 Second order RC circuit

List of components used to construct the second order RC low pass filter

circuit is as shown in Table 3.4. The values for each components in the second order

RC circuit are R1 = R2 = 470 k, R3 = 4.7 k, C1 = C2 = 0.1 μF and R4 is a

potentiometer of 10 k. Value of R4 is varies according to type of second order

system being tested.

Table 3.4 List of components for RC low pass filter circuit


1 IC LM741 (Op – amp) 1

2 Resistor 470 k 2

3 Resistor 4.7 k 1

4 Potentiometer 10 k 1

5 Capacitor 0.1 μF 2

R2R1

C1

C2

R3

R4

VOUT

VIN

32

The transfer function for the above second order system is:

21

21

1

311

4

11

22

21

21

1

)(

CRs

RCR

R

CRs

CRsT

Potentiometer values determined type of second order system of the RC

second order system. The transfer function obtained for different set of

potentiometer values are shown in Table 3.5. According to these values, testing on

different types of second order systems was performed.

Table 3.5 RC low pass filter second order system transfer function

No. Type of second order system Potentiometer (R4) value Transfer function

1 Critically damped R4 = 0

7.4526.42

7.4522 ss

2 Underdamped 0 < R4 < 9.4 k

R4 = 5 k 7.4529.19

7.4522 ss

3 Overdamped R4 < 0 -

4 Undamped R4 = 9.4 k

7.452

7.4522 s

It is shown from Table 3.5; the calculated value for R4 to obtain the

overdamped response is less than 0. So, this type of response is rule out since it is

impossible to be implemented using the proposed RC low pass filter circuit.

CHAPTER 4

RESULT

4.1 Introduction

This chapter discuss on the results obtained in both software and hardware

analysis. In the software analysis, the PRBS generator and its application as test

input to a second order system were examined. For the hardware analysis, PRBS

signal obtained and it is used as test input to a second order system were elaborated.

4.2 PRBS signal (Simulation result)

PRBS signal obtained from the simulation analysis is studied. The

autocorrelation and power spectral density of the PRBS signal are observed. These

results were confirmed with the theory.

PRBS signal is successfully generated. The sequence / pattern will be

repeated after a complete cycle of N value. The PRBS signal, autocorrelation

function and power spectral density of three different maximum length sequences are

shown in Figure 4.1 to Figure 4.3.

34

Figure 4.1 (a) Clock signal, (b) PRBS signal, (c) Autocorrelation function, and (d)

Power spectral density for MLS of N = 15

Figure 4.1 shows the PRBS signal, autocorrelation function and power

spectral density for a four stage shift register with feedback from stages 1 and 4. The

successive states of the shift register, starting all ones, are:

Table 4.1 Successive states of shift register

Stage

1 1 0 1 0 1 1 0 0 1 0 0 0 1 1 1 1

2 1 1 0 1 0 1 1 0 0 1 0 0 0 1 1 1

3 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 1

4 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1

and

the

pattern

repeats

Hence, the sequence length is 15, which is 2n – 1 with n = 4. The three

properties of randomness when applied to the full 15 bit sequence are:

(a) Balance property:

Number of ones = 8

Number of zeros = 7

Difference = 1

35

(b) Run property:

Length of run 1 2 3 4

Number of runs 4 2 1 1

Actual ratio 84

82

81

81

Ideal ratio 21

41

81

161

(c) Correlation property:

Compare stages 1 and 4, say

Number of agreements = 7

Number of disagreements = 8

Difference = 1

From the analysis above, the PRBS signal generated satisfy all three

conditions of randomness.



36



From Figure 4.1 to Figure 4.3, it can be shown that the average power or

mean square value of PRBS signal is at t = 0 second. During this time also the

autocorrelation function value is at the largest value, and because it is periodic, the

same maximum value of autocorrelation function will be obtained at τ, where τ is an

integer multiple of the period.

Power spectral density of PRBS signal is a line spectrum and not a

continuous spectrum (shown in Figure 4.1 to Figure 4.3). The lowest frequency

component in the PRBS signal is that corresponding to the period, 2 π / (N Δ t)

radians / second, and all other frequencies present are integer multiples of this value.

4.3 PRBS signal as forcing function in a second order system (Simulation

result)

There are four responses of second order system examined in this project;

they are underdamped, critically damped, undamped and overdamped response. All

the responses are analyzed in terms of the autocorrelation function, cross correlation

function and finally power spectral density.

37

4.3.1 Critically damped response

Figure 4.4a shows the form of PRBS input and Figure 4.4b shows the

resulting system output in the absence of noise. Figure 4.4c shows a typical sample

trace of the output response of the system in the presence of noise. The response of

the system to the PRBS signal in the presence of noise is shown in Figure 4.4d. A

clear difference can be seen between this and the normal noise output shown in

Figure 4.4c and this response curve show close resemblance of output response of

system forced by PRBS in the absence of noise.

Figure 4.4 (a) PRBS signal and traces of output response of system (b) forced by

PRBS in the absence of noise (c) without PRBS in the presence of noise (d) forced

by PRBS in the presence of noise

Auto correlation functions of input and output signals are shown in Figure

4.5.

38

Figure 4.5 Autocorrelation functions of input and output signals

The autocorrelation function of PRBS signal has the form theoretically

expected, whilst that of the system output in the absence of noise shows a reduction

in signal power to somewhat less than a quarter of the input power. The

autocorrelation function of the noise signal forced by PRBS input shows that there is

a significant component of the signal which approximates to white noise, show an

increased in signal power compared to system in the absence of noise. The

autocorrelation function of noisy system in absence of PRBS input shows a reduction

in signal power to almost a quarter of the input power.

Cross correlation functions of output signals are shown in Figure 4.6.

Figure 4.6 Cross correlation functions of output signals

From the cross correlation function and autocorrelation function graphs,

model parameter can be calculated using the following steps:

39

(a) The height of autocorrelation triangle shown in Figure 4.5 is V2 = 1V and

the bit interval is 0.1s. The impulse strength is V2 times the bit interval

which evaluates to 1 × 0.1s = 0.1 Vs.

(b) The response appears to be a combination of rise and decay wave. The

general form is )( tt eeA . This response curve is difficult to analyze

using correlation technique. It is easier by using frequency response

method.

(c) The time constant to be 0.7040s (decay) and 0.1908s (rise). So, 42.1

and 24.5 .

(d) A is obtained from value of peak height, 276.0A .

(e) Divide by the unit impulse response, )(76.2)( 24.542.1 tt eetf .

(f)44.766.6

54.10

24.5

76.2

42.1

76.2)(

2

sssssF

The chosen time interval, st 1.0 used in the simulation gives adequate

approximation to white noise for this system. The period of 6.3s correctly exceeds

the system settling time. This shows that the sequence of N = 31 could have been

used instead. Table 4.2 shows the transfer function obtained using several different

PRBS maximum length.

It is shown from Table 4.2 that the transfer function obtained is not very close

to the actual transfer function used in the simulation. This is due to the difficulty in

obtaining the correct transfer function using correlation technique for a cross

correlation function graph which does not yield a good approximation to an impulse

response (decaying sine wave).

Power spectral density curves of input and output signals are shown in Figure

4.7. It can be shown in this figure that the systems with PRBS input, almost the

entire power of the output signals are contained in the frequency range of 1 to 10Hz.

The power spectral density curve for PRBS input shows that over this frequency

range, the PRBS input has a substantially constant power spectral density values.

This has confirms that st 1.0 used in this simulation gives an excitation signal

which is good approximation to true white noise for the system tested.

40

Table 4.2 Transfer function for several different PRBS maximum length

Length, N Transfer function

63

44.766.6

54.102 ss

255

00.694.5

21.92 ss

1023

17.941.7

10.112 ss

Average transfer function model using 3 different length of PRBS

54.767.6

18.102 ss

Figure 4.7 Power spectral density of input and output signals

4.3.2 Underdamped response








41




Auto correlation functions of input and output signals are shown in Figure

4.9.


The autocorrelation function of PRBS signal has the form theoretically

expected, whilst that of the system output in the absence of noise shows a reduction

in signal power to somewhat equal to a quarter of the input power. The

autocorrelation function of the noise signal forced by PRBS input shows an increase

in signal power to somewhat half of the input power. The autocorrelation function of

42

noisy system in absence of PRBS input shows a reduction in signal power to less

than a quarter of input power.





(a) The height of autocorrelation triangle shown in Figure 4.9 is V2 = 1V and

the bit interval is 0.1s. The impulse strength is V2 times the bit interval

which evaluates to 1 × 0.1s = 0.1 Vs.

(b) The response appears to be a decaying sine wave. The general form is

tAe t sin . This response yields a good approximation to impulse

response.

(c) ω is obtained from cycle time, rad/s73.23.2

2

(d) α is obtained from peak decay ratio, 9787.02009.0

06518.0ln

3.2

2

.

(e) A is obtained from the first peak height, 312.0A

(f) Divide by the unit impulse response, tetf t 73.2sin12.3)( 9787.0 .

(g)41.896.1

52.8

73.2)9787.0(

)73.2(12.3)(

222

ssssF

43





PRBS maximum length. It is shown from this table that the transfer function

obtained is closed to the actual transfer function used in the simulation. This is due

to the cross correlation function graph yield a good approximation to impulse

response and thus easier to analyze using correlation technique.










63

41.896.1

52.82 ss

255

68.887.1

04.92 ss

1023

39.894.1

60.82 ss


49.892.1

72.82 ss

44


4.3.3 Overdamped response








The autocorrelation function of PRBS signal shown in Figure 4.13 has the

form theoretically expected, whilst that all of the system outputs show a reduction in

signal power to somewhat less than a quarter of the input power.

45






46




(a) The height of autocorrelation triangle shown in Figure 4.13 is V2 = 1V

and the bit interval is 0.1s. The impulse strength is V2 times the bit

interval which evaluates to 1 × 0.1s = 0.1 Vs.




method.

(c) The time constant to be 0.8200s (decay) and 0.1640s (rise). So,

2195.1 and 0976.6 .


(e) Divide by the unit impulse response, )(68.1)( 0976.62195.1 tt eetf .

(f)44.732.7

20.8

0976.6

68.1

2195.1

68.1)(

2

sssssF





PRBS maximum length.

47



63

44.732.7

20.82 ss

255

79.582.6

13.72 ss

1023

25.615.7

16.72 ss


49.610.7

50.72 ss

It is shown from Table 4.4 that the transfer function obtained is not very close

to the actual transfer function used in the simulation. This is due to the difficulty in

obtaining the correct transfer function using correlation technique for a cross

correlation function graph which does not yield a good approximation to an impulse

response (decaying sine wave).








48


4.3.4 Undamped response








Figure 4.17 shows there is a fluctuation of large signal power in the system

outputs forced by PRBS input in the presence and absence of noise. The

autocorrelation function of PRBS input and noisy system in absence of PRBS input

shows a very small signal power compared to the system output forced by PRBS

input in the presence and absence of noise.

49





Cross correlation functions of output signals are shown in Figure 4.18. The

analysis of the cross correlation function graph is difficult to perform since the

response does not yield a good approximation to an impulse response (decaying sine

wave).

50



4.19.


It can be observed that that the systems with PRBS input, almost the entire

power of the output signals are contained in the frequency range of 2 to 4Hz. The

power spectral density curve for PRBS input shows that over this frequency range,

the PRBS input has a low power spectral density values. This is not good since most

of the entire power of the output signal does not contain within power spectral

density curve for PRBS input.

51

4.4 PRBS signal (Hardware result)

PRBS signal, autocorrelation function and power spectral density is analyze

using Dynamic Signal Analyzer (HP35670A DSA). HP35670A DSA is shown in

Figure 4.20. About 512 data of the PRBS signal, autocorrelation function and power

spectral density are captured using Dynamic Signal Analyzer for every maximum

length sequence of PRBS signal. MATLAB software is used to plot the PRBS

signal, autocorrelation function and power spectral density.

Figure 4.20 Dynamic Signal Analyzer (HP35670A DSA)

Figure 4.21 shows the PRBS signal for maximum length sequence of N = 63.

It is shown that the measurement values are closed to the prediction values.

Figure 4.21 PRBS signal for MLS of N = 63

52

Figure 4.22 shows the autocorrelation function graph for PRBS signal for

maximum length sequence of N = 63. It can be shown from the graph that the height

of the autocorrelation function triangle, V2 = 0.95V and the bit interval is 0.1281s.

Figure 4.22 Autocorrelation function of PRBS signal for MLS of N = 63

Figure 4.23 shows power spectral density curve for PRBS signal of maximum

length sequence equal to 63. It can be observed from the graph that the lowest

frequency component is 70Hz, which is a bit higher than the calculated value, 57Hz.

Figure 4.23 Power spectral density of PRBS signal for MLS of N = 63

53

4.5 PRBS signal as test input to a second order system (Hardware result)

A PRBS signal is used as an input to determine the model of second order

system. The autocorrelation of the input signal (PRBS signal) and cross correlation

between the input and output signal is performed using the Dynamic Signal Analyzer

(HP35670A DSA). There are two responses observed in this part; critically damped

and underdamped responses. The underdamped response is precluded in this

analysis because the analyzing process for this response is difficult. For the

overdamped response, it does not include in the analysis since the implementation

wise of this response is impossible using the proposed RC second order circuit.

Second ordersystem

g(t)

PRBS signal

x(t)

Output response

y(t)

Figure 4.24 Block diagram of PRBS testing

4.5.1 Critically damped response

Figure 4.25 shows the schematic circuit for RC low pass filter second order

system critically damped. Transfer function of the second order critically damped

response is obtained using this equation:

7.4526.42

7.452)(

)1.0()470(1

)1.0)(470(22

)1.0()470(1

)(

2

22

22

sssT

Fks

Fks

FksT

54

Figure 4.25 Schematic circuits for critically damped response

Figure 4.26 shows the output signal obtained using PRBS signal as the input

to the RC second order system (critically damped response). It is clearly shown that

the measurement result is close to the prediction. Figure 4.27 shows the

autocorrelation function of the output signal obtained using PRBS signal as the input

to the RC second order system while Figure 4.28 shows the cross correlation

function of the output signal obtained using PRBS signal as the input to the RC

second order system. The measurement result of autocorrelation function of the

output signal has the value close to the prediction value.

Figure 4.26 Output signal using PRBS signal

470k470k

0.1u

0.1u

4.7k

VOUT

VIN

55

Figure 4.27 Autocorrelation function of output signal using PRBS signal

Figure 4.28 Cross correlation function of output signal using PRBS signal



(a) The height of autocorrelation triangle shown in Figure 4.22 is V2 = 0.95V


interval which evaluates to 0.95V × 0.1281s = 0.12 Vs.




method.

(c) The time constant to be 0.10056s (decay) and 0.02086s (rise). So,

94.9 and 94.47 .


56

(e) Divide by the unit impulse response, )(19.9)( 94.4794.9 tt eetf

(f)52.47688.57

22.349

94.47

19.9

94.9

19.9)(

2

sssssF

The transfer function obtained is not very close to the actual transfer function

used in the hardware analysis. This is due to two reasons; first reason is the

difficulty in obtaining the correct transfer function using correlation technique for a

cross correlation function graph which does not yield a good approximation to an

impulse response and the second reason is the correlation is carried out for short

time. Longer the period of correlation could help smoother the curves, provided

dynamic characteristic of the system being tested remained unchanged over long

period of time span involved.

4.5.2 Underdamped response

Figure 4.29 shows the schematic circuit for RC low pass filter second order system

underdamped.

Figure 4.29 Schematic circuits for underdamped response

Transfer function of the second order underdamped response is obtained

using this equation:

470k470k

0.1u

0.1u

4.7k

5k

VOUT

VIN

57

7.4529.19

7.452)(

)1.0()470(1

)7.4)(1.0)(470(5

)1.0)(470(22

)1.0()470(1

)(

2

22

22

sssT

Fks

kFkk

Fks

FksT

Figure 4.30 shows the output signal obtained using PRBS signal as the input

to the RC second order system (underdamped response). It is clearly shown that the

measurement result is close to the prediction.

Figure 4.30 Output signal using PRBS signal for MLS

Figure 4.31 shows the autocorrelation function of the output signal obtained

using PRBS signal as the input to the RC second order system while Figure 4.32

shows the cross correlation function of the output signal obtained using PRBS signal

as the input to the RC second order system. The measurement result of

autocorrelation function of the output signal has the value close to the prediction

value.

58

Figure 4.31 Autocorrelation function of output signal using PRBS signal

Figure 4.32 Cross correlation function of output signal using PRBS signal



(a) The height of autocorrelation triangle shown in Figure 4.22 is V2 = 0.95V


interval which evaluates to 0.95V × 0.1281s = 0.12 Vs.

(b) The response appears to be a decaying sine wave. The general form is

tAe t sin . This response yields a good approximation to impulse

response.

(c) ω is obtained from cycle time, rad/s39.173612.0

2

(d) α is obtained from peak decay ratio, 96.47592.0

3098.0ln

3612.0

2

.

59

(e) A is obtained from the first peak height, 073.1A

(f) Divide by the unit impulse response, tetf t 39.17sin94.8)( 96.4 .

(g)01.32792.9

47.155

39.17)96.4(

)39.17(94.8)(

222

ssssF

The transfer function obtained is not very close to the actual transfer function

used in the hardware analysis. This is due to the shorter time duration for

correlation. Longer the period of correlation could help smoother the curves,

provided dynamic characteristic of the system being tested remained unchanged over

long period of time span involved.

It can be summarized in Table 4.5 the transfer function obtained using

correlation technique for both responses; critically damped and underdamped.

Table 4.5 Transfer function obtained for hardware analysis

Type of second order

system

Transfer function used in

hardware implementation

Transfer function obtained

using correlation technique

Critically damped

7.4526.42

7.4522 ss 52.47688.57

22.3492 ss

Underdamped

7.4529.19

7.4522 ss 01.32792.9

47.1552 ss

CHAPTER 5

CONCLUSION AND FUTURE WORKS

5.1 Conclusion

Pseudo random binary sequence (PRBS) signal of 15 different maximum

length sequences has successfully developed using MATLAB software. The

generated signal was used as forcing function in simulated overdamped,

underdamped, undamped and critically damped second order. The transfer functions

of the each system obtained from the correlograms of autocorrelation and cross

correlation are shown in Table 5.1.

Table 5.1 Transfer function obtained for each system (simulation)

No. Type of second

order system


simulation


from correlograms

1 Critically damped

96

92 ss 54.767.6

18.102 ss

2 Underdamped

92

92 ss 49.892.1

72.82 ss

3 Overdamped

99

92 ss 49.610.7

50.72 ss

4 Undamped

9

92 s

Difficult to obtained using

correlation technique

61

PRBS generator circuit has successfully built using TTL. The PRBS signal,

autocorrelation function and power spectral density observed using Dynamic Signal

Analyzer are as theoretically expected. The experiment using PRBS as the forcing

function to an unknown system has successfully performed. The transfer function of

the unknown system has successfully estimated using correlograms of

autocorrelation and cross correlation. The transfer functions obtained are shown in

Table 5.2. The results from this experiment have validated the simulation work

previously.

Table 5.2 Transfer function obtained for each system (hardware)

No. Type of second

order system


hardware implementation


from correlograms

1 Critically damped

6.4526.42

7.4522 ss 52.47688.57

22.3492 ss

2 Underdamped

7.4529.19

7.4522 ss 01.32792.9

47.1552 ss

For overdamped system, the hardware implementation is difficult since the

calculated value for the potentiometer is negative. For undamped system, the

analyzing process for this response is difficult.

5.2 Future Works

As for future works, an improvement on the hardware part of PRBS signal as

test input to undamped and overdamped second order system can be done. Graphic

User Interface (GUI) for PRBS signal and its application can be designed for more

organize and convenience while testing the PRBS signal. Lastly, another type of

PRBS signal such as QRB, HAB, TPB and QRT can be used instead to generate the

PRBS signal.

62

REFERENCES

1. Tan, A.H. and Godfrey, K.R. (2002). The generation of binary and near-

binary pseudorandom signals: an overview. IEEE Trans. Instrum. Meas. 51

(4), 583-588.

2. Van Den Bos, A. (1993). Periodic test signals – Properties and use.

Godfrey, K. Perturbation Signals for System Identification. (ch.4). Ed.

London, U.K.: Prentice Hall.

3. Darnell, M. (1993). Periodic and nonperiodic, binary and multi-level

pseudorandom signals. Godfrey, K. Perturbation Signals for System

Identification. (ch.5). Ed. London, U.K.: Prentice-Hall.

4. Godfrey, K. (1993). Introduction to perturbation signals for frequency-

domain system identification. Godfrey, K. Perturbation Signals for System

Identification. (ch.2). Ed. London, U.K.: Prentice-Hall.

5. Godfrey, K. R., Barker, H. A. and Tucker, A. J. (1999). Comparison of

perturbation signals for linear system identification in the frequency domain.

Proc. Inst. Elect. Eng. – Control Theory Applicat. 146(6), 535–548.

6. Kollár, I. (1994). Frequency Domain System Identification Toolbox for use

With MATLAB. Natick, MA: The MathWorks Inc.

7. McCormack, A. S., Godfrey, K. R. and Flower, J. O. (1995). Design of

multilevel multiharmonic signals for system identification. Proc. Inst. Elect.

Eng. – Control Theory Applicat. 142(3), 247–252.

8. Zierler, N. (1959). Linear recurring sequences. J. Soc. Ind. Appl. Math. 7,

31–48.

9. Godfrey, K. (1993). Introduction to perturbation signals for time-domain

system identification. Godfrey, K. Perturbation Signals for System

Identification. (ch.1). Ed. Englewood Cliffs, NJ: Prentice Hall.

63

10. Godfrey, K.R. (1991). Introduction to binary signals used in system

identification. Control 1991. Control '91, International Conference on, vol.,

no., pp.161-166 vol.1, 25-28.

11. Zapernick, H.-J. and Finger, A. (2005). Pseudo Random Signal Processing –

Theory and Application. Chichester: John Wiley & Sons, Ltd.

12. Sodestrom, T. and Stoica, P. (1989). System Identification. Hertfordshire:

Prentice Hall International (UK) Ltd.

13. Godfrey, K. R. and Briggs, P. A. N. (1972). Identification of processes with

direction-dependent dynamics responses. Proc. Inst. Elect. Eng. – Control

Sci. 119(12), 1733–1739.

14. Godfrey, K. R. and D. J. Moore (1974). Identification of processes having

direction-dependent responses, with gas – turbine engine applications.

Automatica, 10(5), 469–481.

15. Tan, A. H. and Godfrey, K. R. (2001). Identification of processes with

direction-dependent dynamics. Proc. Inst. Elect. Eng. – Control Theory

Applicat. 148(5), 362–369.

16. Barker, H. A., Godfrey, K. R. and Tan, A. H. (2000). Identification of

systems with direction-dependent dynamics. Proc. 39th IEEE Conf. Decision

Control (CDC 2000), 2843–2848.

17. Mouine, J. and Boutin, N (1998). A novel way to generate pseudo – random

sequences longer than maximal length sequences. Proc. Inst. Elect. & Comp.

Eng. 2, 529-532.

18. Rahmat, M. F. (2007). Pseudo random binary sequence. System

Identification & Parameter Estimation Lecture Note, UTM Skudai.

APPENDIX

APPENDIX A

COMPUTER PROGRAMS

66

%Plot autocorrelation function (ACF)vector = (ifft(abs(fft(prbs)).^2))/length(prbs);Rxx = real(vector); %real=Real part of complex number

vector1 =(ifft(abs(fft(forced_by_prbs_absence_noise)).^2))/length(forced_by_prbs_absence_noise);Rxx1 = real(vector1); %real=Real part of complex number

vector2 =(ifft(abs(fft(without_prbs_presence_noise)).^2))/length(without_prbs_presence_noise);Rxx2 = real(vector2); %real=Real part of complex number

vector3 =(ifft(abs(fft(forced_by_prbs_presence_noise)).^2))/length(forced_by_prbs_presence_noise);Rxx3 = real(vector3); %real=Real part of complex number

figure (1)plot(tout, Rxx, 'magenta'); grid;hold onplot(tout, Rxx1, 'k'); grid;hold onplot(tout, Rxx2, 'b'); grid;hold onplot(tout, Rxx3, 'r'); grid;

%Plot crosscorrelation function (CCF)

Rxy1 = xcorr(prbs, forced_by_prbs_absence_noise);

Rxy2 = xcorr(prbs, without_prbs_presence_noise);

Rxy3 = xcorr(prbs, forced_by_prbs_presence_noise);t=-length(prbs)+1:1:length(prbs)-1;figure (2)plot(t, Rxy1, 'k'); grid;hold onplot(t, Rxy2, 'b'); grid;hold onplot(t, Rxy3, 'r'); grid;hold on

%Power Spectral Density function (PSD)harmonic = [1:3*length(prbs)];harmonic1 = [1:3*length(forced_by_prbs_absence_noise)];harmonic2 = [1:3*length(without_prbs_presence_noise)];harmonic3 = [1:3*length(forced_by_prbs_presence_noise)];

DFT = abs(fft(prbs));three_periods = [DFT; DFT; DFT];%calculate power prbsamp(1) = DFT(1)/length(prbs);power(1) = amp(1)^2;for k = 2: length(three_periods)

angle(k) = pi*(k-1)/length(prbs);amp(k) = sqrt(2)/length(prbs)*abs(sin(angle(k))*three_periods(k)/angle(k));power(k) = amp(k)^2;

end

DFT1 = abs(fft(forced_by_prbs_absence_noise));three_periods1 = [DFT1; DFT1; DFT1];

%calculate power forced_by_prbs_absence_noiseamp1(1) = DFT1(1)/length(forced_by_prbs_absence_noise);power1(1) = amp1(1)^2;for k = 2: length(three_periods1)

angle1(k) = pi*(k-1)/length(forced_by_prbs_absence_noise);amp1(k) =

sqrt(2)/length(forced_by_prbs_absence_noise)*abs(sin(angle1(k))*three_periods1(k)/angle1(k));

power1(k) = amp1(k)^2;end

DFT2 = abs(fft(without_prbs_presence_noise));three_periods2 = [DFT2; DFT2; DFT2];

67

%calculate power without_prbs_presence_noiseamp2(1) = DFT2(1)/length(without_prbs_presence_noise);power2(1) = amp2(1)^2;for k = 2: length(three_periods2)

angle2(k) = pi*(k-1)/length(without_prbs_presence_noise);amp2(k) =

sqrt(2)/length(without_prbs_presence_noise)*abs(sin(angle2(k))*three_periods2(k)/angle2(k));


DFT3 = abs(fft(forced_by_prbs_presence_noise));three_periods3 = [DFT3; DFT3; DFT3];

%calculate power forced_by_prbs_presence_noiseamp3(1) = DFT3(1)/length(forced_by_prbs_presence_noise);power3(1) = amp3(1)^2;for k = 2: length(three_periods3)

angle3(k) = pi*(k-1)/length(forced_by_prbs_presence_noise);amp3(k) =

sqrt(2)/length(forced_by_prbs_presence_noise)*abs(sin(angle3(k))*three_periods3(k)/angle3(k));


%plot power againts harmonic number

figure (1)plot(harmonic -1, power, 'magenta')hold onplot(harmonic1 -1, power1, 'k')hold onplot(harmonic2 -1, power2, 'b')hold onplot(harmonic3 -1, power3, 'r')hold on

APPENDIX B

DATASHEETS

© 2000 Fairchild Semiconductor Corporation DS006382 www.fairchildsemi.com

August 1986

Revised March 2000

DM

74LS

112A D

ual N

egative-E

dg

e-Trigg

ered M

aster-Slave J-K

Flip

-Flo

p w

ith P

reset, Clear, an

d C

om

plem

entary

Ou

tpu

ts

DM74LS112ADual Negative-Edge-Triggered Master-Slave J-K Flip-Flopwith Preset, Clear, and Complementary Outputs

General DescriptionThis device contains two independent negative-edge-trig-gered J-K flip-flops with complementary outputs. The J andK data is processed by the flip-flop on the falling edge ofthe clock pulse. The clock triggering occurs at a voltagelevel and is not directly related to the transition time of thefalling edge of the clock pulse. Data on the J and K inputsmay be changed while the clock is HIGH or LOW withoutaffecting the outputs as long as the setup and hold timesare not violated. A low logic level on the preset or clearinputs will set or reset the outputs regardless of the logiclevels of the other inputs.

Ordering Code:

Devices also available in Tape and Reel. Specify by appending the suffix letter “X” to the ordering code.

Connection Diagram Function Table

H = HIGH Logic LevelL = LOW Logic LevelX = Either LOW or HIGH Logic Level↓ = Negative Going Edge of PulseQ0 = The output logic level before the indicated input conditions were

established.Toggle = Each output changes to the complement of its previous level on

each falling edge of the clock pulse.

Note 1: This configuration is nonstable; that is, it will not persist whenpreset and/or clear inputs return to their inactive (HIGH) level.

Order Number Package Number Package Description

DM74KS112AM M16A 16-Lead Small Outline Integrated Circuit (SOIC), JEDEC MS-012, 0.150 Narrow

DM74LS112AN N16E 16-Lead Plastic Dual-In-Line Package (PDIP), JEDEC MS-001, 0.300 Wide

Inputs Outputs

PR CLR CLK J K Q Q

L H X X X H L

H L X X X L H

L L X X X H (Note 1) H (Note 1)

H H ↓ L L Q0 Q0

H H ↓ H L H L

H H ↓ L H L H

H H ↓ H H Toggle

H H H X X Q0 Q0

www.fairchildsemi.com 2

DM

74L

S11

2A Absolute Maximum Ratings(Note 2)Note 2: The “Absolute Maximum Ratings” are those values beyond whichthe safety of the device cannot be guaranteed. The device should not beoperated at these limits. The parametric values defined in the ElectricalCharacteristics tables are not guaranteed at the absolute maximum ratings.The “Recommended Operating Conditions” table will define the conditionsfor actual device operation.

Recommended Operating Conditions

Note 3: CL = 15 pF, RL = 2 kΩ, TA = 25°C and VCC = 5V.

Note 4: The symbol (↓) indicates the falling edge of the clock pulse is used for reference.

Note 5: CL = 50 pF, RL = 2 kΩ, TA = 25°C and VCC = 5V.

Supply Voltage 7V

Input Voltage 7V

Operating Free Air Temperature Range 0°C to +70°C

Storage Temperature Range −65°C to +150°C

Symbol Parameter Min Nom Max Units

VCC Supply Voltage 4.75 5 5.25 V

VIH HIGH Level Input Voltage 2 V

VIL LOW Level Input Voltage 0.8 V

IOH HIGH Level Output Current −0.4 mA

IOL LOW Level Output Current 8 mA

fCLK Clock Frequency (Note 3) 0 30 MHz

fCLK Clock Frequency (Note 5) 0 25 MHz

tW Pulse Width Clock HIGH 20

(Note 3) Preset LOW 25 ns

Clear LOW 25

tW Pulse Width Clock HIGH 25

(Note 5) Preset LOW 30 ns

Clear LOW 30

tSU Setup Time (Note 3)(Note 4) 20↓ ns

tSU Setup Time (Note 4)(Note 5) 25↓ ns

tH Hold Time (Note 3)(Note 4) 0↓ ns

tH Hold Time (Note 4)(Note 5) 5↓ ns

TA Free Air Operating Temperature 0 70 °C

3 www.fairchildsemi.com

DM

74LS

112AElectrical Characteristics over recommended operating free air temperature range (unless otherwise noted)

Note 6: All typicals are at VCC = 5V, TA = 25°C.

Note 7: Not more than one output should be shorted at a time, and the duration should not exceed one second. For devices, with feedback from the outputs,where shorting the outputs to ground may cause the outputs to change logic state an equivalent test may be performed where VO = 2.125V with the minimum

and maximum limits reduced by one half from their stated values. This is very useful when using automatic test equipment.

Note 8: With all outputs OPEN, ICC is measured with the Q and Q outputs HIGH in turn. At the time of measurement the clock is grounded.

Switching Characteristics at VCC = 5V and TA = 25°C

Symbol Parameter Conditions MinTyp

Max Units(Note 6)

VI Input Clamp Voltage VCC = Min, II = −18 mA −1.5 V

VOH HIGH Level VCC = Min, IOH = Max2.7 3.4 V

Output Voltage VIL = Max, VIH = Min

VOL LOW Level VCC = Min, IOL = Max0.35 0.5

Output Voltage VIL = Max, VIH = Min V

IOL = 4 mA, VCC = Min 0.25 0.4

II Input Current @ Max VCC = Max, VI = 7V J, K 0.1

Input Voltage Clear 0.3mA

Preset 0.3

Clock 0.4

IIH HIGH Level Input Current VCC = Max, VI = 2.7V J, K 20

Clear 60µA

Preset 60

Clock 80

IIL LOW Level Input Current VCC = Max, VI = 0.4V J, K −0.4

Clear −0.8mA

Preset −0.8

Clock −0.8

IOS Short Circuit Output Current VCC = Max (Note 7) −20 −100 mA

ICC Supply Current VCC = Max (Note 8) 4 6 mA

From (Input) RL = 2 kΩ

Symbol Parameter To (Output) CL = 15 pF CL = 50 pF Units

Min Max Min Max

fMAX Maximum Clock Frequency 30 25 MHz

tPLH Propagation Delay TimePreset to Q 20 24 ns

LOW-to-HIGH Level Output

tPHL Propagation Delay TimePreset to Q 20 28 ns

HIGH-to-LOW Level Output

tPLH Propagation Delay TimeClear to Q 20 24 ns


tPHL Propagation Delay TimeClear to Q 20 28 ns


tPLH Propagation Delay TimeClock to Q or Q 20 24 ns


tPHL Propagation Delay TimeClock to Q or Q 20 28 ns



DM

74L

S11

2A Physical Dimensions inches (millimeters) unless otherwise noted

16-Lead Small Outline Integrated Circuit (SOIC), JEDEC MS-012, 0.150 NarrowPackage Number M16A


DM

74LS

112A D

ual N

egative-E

dg

e-Trigg

ered M

aster-Slave J-K

Flip

-Flo

p w

ith P

reset, Clear, an

d C

om

plem

entary

Ou

tpu

tsPhysical Dimensions inches (millimeters) unless otherwise noted (Continued)

16-Lead Plastic Dual-In-Line Package (PDIP), JEDEC MS-001, 0.300 WidePackage Number N16E

Fairchild does not assume any responsibility for use of any circuitry described, no circuit patent licenses are implied andFairchild reserves the right at any time without notice to change said circuitry and specifications.

LIFE SUPPORT POLICY

FAIRCHILD’S PRODUCTS ARE NOT AUTHORIZED FOR USE AS CRITICAL COMPONENTS IN LIFE SUPPORTDEVICES OR SYSTEMS WITHOUT THE EXPRESS WRITTEN APPROVAL OF THE PRESIDENT OF FAIRCHILDSEMICONDUCTOR CORPORATION. As used herein:

1. Life support devices or systems are devices or systemswhich, (a) are intended for surgical implant into thebody, or (b) support or sustain life, and (c) whose failureto perform when properly used in accordance withinstructions for use provided in the labeling, can be rea-sonably expected to result in a significant injury to theuser.

2. A critical component in any component of a life supportdevice or system whose failure to perform can be rea-sonably expected to cause the failure of the life supportdevice or system, or to affect its safety or effectiveness.

www.fairchildsemi.com

5-1

FAST AND LS TTL DATA

QUAD 2-INPUTEXCLUSIVE OR GATE

14 13 12 11 10 9

1 2 3 4 5 6

VCC

8

7

GND

TRUTH TABLE

IN OUT

A B Z

L L LL H HH L HH H L

GUARANTEED OPERATING RANGES

Symbol Parameter Min Typ Max Unit

VCC Supply Voltage 5474

4.54.75

5.05.0

5.55.25

V

TA Operating Ambient Temperature Range 5474

–550

2525

12570

°C

IOH Output Current — High 54, 74 –0.4 mA

IOL Output Current — Low 5474

4.08.0

mA

SN54/74LS86

QUAD 2-INPUTEXCLUSIVE OR GATE

LOW POWER SCHOTTKY

J SUFFIXCERAMIC

CASE 632-08

N SUFFIXPLASTIC

CASE 646-06

141

14

1

ORDERING INFORMATION

SN54LSXXJ CeramicSN74LSXXN PlasticSN74LSXXD SOIC

141

D SUFFIXSOIC

CASE 751A-02

5-2

FAST AND LS TTL DATA

SN54/74LS86

DC CHARACTERISTICS OVER OPERATING TEMPERATURE RANGE (unless otherwise specified)

S b l P

Limits

U i T C di iSymbol Parameter Min Typ Max Unit Test Conditions

VIH Input HIGH Voltage 2.0 VGuaranteed Input HIGH Voltage forAll Inputs

VIL Input LOW Voltage54 0.7

VGuaranteed Input LOW Voltage for

VIL Input LOW Voltage74 0.8

Vp g

All Inputs

VIK Input Clamp Diode Voltage –0.65 –1.5 V VCC = MIN, IIN = –18 mA

VOH Output HIGH Voltage54 2.5 3.5 V VCC = MIN, IOH = MAX, VIN = VIHVOH Output HIGH Voltage74 2.7 3.5 V

CC , OH , IN IHor VIL per Truth Table

VOL Output LOW Voltage54, 74 0.25 0.4 V IOL = 4.0 mA VCC = VCC MIN,

VIN = VIL or VIHVOL Output LOW Voltage74 0.35 0.5 V IOL = 8.0 mA

VIN = VIL or VIHper Truth Table

IIH Input HIGH Current40 µA VCC = MAX, VIN = 2.7 V

IIH Input HIGH Current0.2 mA VCC = MAX, VIN = 7.0 V

IIL Input LOW Current –0.8 mA VCC = MAX, VIN = 0.4 V

IOS Short Circuit Current (Note 1) –20 –100 mA VCC = MAX

ICC Power Supply Current 10 mA VCC = MAX

Note 1: Not more than one output should be shorted at a time, nor for more than 1 second.

AC CHARACTERISTICS (TA = 25°C)

S b l P

Limits

U i T C di iSymbol Parameter Min Typ Max Unit Test Conditions

tPLHtPHL

Propagation Delay,Other Input LOW

1210

2317 ns

VCC = 5.0 V

tPLHtPHL

Propagation Delay,Other Input HIGH

2013

3022 ns

CCCL = 15 pF

© 2000 Fairchild Semiconductor Corporation DS006345 www.fairchildsemi.com

August 1986

Revised March 2000

DM

74LS

04 Hex In

verting

Gates

DM74LS04Hex Inverting Gates

General DescriptionThis device contains six independent gates each of whichperforms the logic INVERT function.

Ordering Code:

Devices also available in Tape and Reel. Specify by appending the suffix letter “X” to the ordering code.

Connection Diagram Function TableY = A

H = HIGH Logic LevelL = LOW Logic Level

Order Number Package Number Package Description

DM74LS04M M14A 14-Lead Small Outline Integrated Circuit (SOIC), JEDEC MS-120, 0.150 Narrow

DM74LS04SJ M14D 14-Lead Small Outline Package (SOP), EIAJ TYPE II, 5.3mm Wide

DM74LS04N N14A 14-Lead Plastic Dual-In-Line Package (PDIP), JEDEC MS-001, 0.300 Wide

Input Output

A Y

L H

H L


DM

74L

S04 Absolute Maximum Ratings(Note 1)

Note 1: The “Absolute Maximum Ratings” are those values beyond whichthe safety of the device cannot be guaranteed. The device should not beoperated at these limits. The parametric values defined in the ElectricalCharacteristics tables are not guaranteed at the absolute maximum ratings.The “Recommended Operating Conditions” table will define the conditionsfor actual device operation.

Recommended Operating Conditions

Electrical Characteristics over recommended operating free air temperature range (unless otherwise noted)

Note 2: All typicals are at VCC = 5V, TA = 25°C.

Note 3: Not more than one output should be shorted at a time, and the duration should not exceed one second.

Switching Characteristics at VCC = 5V and TA = 25°C

Supply Voltage 7V

Input Voltage 7V

Operating Free Air Temperature Range 0°C to +70°C

Storage Temperature Range −65°C to +150°C

Symbol Parameter Min Nom Max Units

VCC Supply Voltage 4.75 5 5.25 V

VIH HIGH Level Input Voltage 2 V

VIL LOW Level Input Voltage 0.8 V

IOH HIGH Level Output Current −0.4 mA

IOL LOW Level Output Current 8 mA

TA Free Air Operating Temperature 0 70 °C

Symbol Parameter Conditions MinTyp

Max Units(Note 2)

VI Input Clamp Voltage VCC = Min, II = −18 mA −1.5 V

VOH HIGH Level VCC = Min, IOH = Max,2.7 3.4 V

Output Voltage VIL = Max

VOL LOW Level VCC = Min, IOL = Max,0.35 0.5

Output Voltage VIH = Min V

IOL = 4 mA, VCC = Min 0.25 0.4

II Input Current @ Max VCC = Max, VI = 7V 0.1 mA

Input Voltage

IIH HIGH Level Input Current VCC = Max, VI = 2.7V 20 µA

IIL LOW Level Input Current VCC = Max, VI = 0.4V −0.36 mA

IOS Short Circuit Output Current VCC = Max (Note 3) −20 −100 mA

ICCH Supply Current with Outputs HIGH VCC = Max 1.2 2.4 mA

ICCL Supply Current with Outputs LOW VCC = Max 3.6 6.6 mA

RL = 2 kΩ

Symbol Parameter CL = 15 pF CL = 50 pF Units

Min Max Min Max

tPLH Propagation Delay Time3 10 4 15 ns


tPHL Propagation Delay Time3 10 4 15 ns



DM

74LS

04Physical Dimensions inches (millimeters) unless otherwise noted

14-Lead Small Outline Integrated Circuit (SOIC), JEDEC MS-120, 0.150 NarrowPackage Number M14A


DM

74L

S04 Physical Dimensions inches (millimeters) unless otherwise noted (Continued)

14-Lead Small Outline Package (SOP), EIAJ TYPE II, 5.3mm WidePackage Number M14D


DM

74LS

04 Hex In

verting

Gates

Physical Dimensions inches (millimeters) unless otherwise noted (Continued)

14-Lead Plastic Dual-In-Line Package (PDIP), JEDEC MS-001, 0.300 WidePackage Number N14A

Fairchild does not assume any responsibility for use of any circuitry described, no circuit patent licenses are implied andFairchild reserves the right at any time without notice to change said circuitry and specifications.

LIFE SUPPORT POLICY

FAIRCHILD’S PRODUCTS ARE NOT AUTHORIZED FOR USE AS CRITICAL COMPONENTS IN LIFE SUPPORTDEVICES OR SYSTEMS WITHOUT THE EXPRESS WRITTEN APPROVAL OF THE PRESIDENT OF FAIRCHILDSEMICONDUCTOR CORPORATION. As used herein:

1. Life support devices or systems are devices or systemswhich, (a) are intended for surgical implant into thebody, or (b) support or sustain life, and (c) whose failureto perform when properly used in accordance withinstructions for use provided in the labeling, can be rea-sonably expected to result in a significant injury to theuser.

2. A critical component in any component of a life supportdevice or system whose failure to perform can be rea-sonably expected to cause the failure of the life supportdevice or system, or to affect its safety or effectiveness.

www.fairchildsemi.com

TL/H/9341

LM

741

Opera

tionalA

mplifie

r

November 1994

LM741 Operational Amplifier

General DescriptionThe LM741 series are general purpose operational amplifi-

ers which feature improved performance over industry stan-

dards like the LM709. They are direct, plug-in replacements

for the 709C, LM201, MC1439 and 748 in most applications.

The amplifiers offer many features which make their appli-

cation nearly foolproof: overload protection on the input and

output, no latch-up when the common mode range is ex-

ceeded, as well as freedom from oscillations.

The LM741C/LM741E are identical to the LM741/LM741A

except that the LM741C/LM741E have their performance

guaranteed over a 0§C to a70§C temperature range, in-

stead of b55§C to a125§C.

Schematic Diagram

TL/H/9341–1

Offset Nulling Circuit

TL/H/9341–7

C1995 National Semiconductor Corporation RRD-B30M115/Printed in U. S. A.

Absolute Maximum RatingsIf Military/Aerospace specified devices are required, please contact the National Semiconductor Sales Office/

Distributors for availability and specifications.

(Note 5)

LM741A LM741E LM741 LM741C

Supply Voltage g22V g22V g22V g18V

Power Dissipation (Note 1) 500 mW 500 mW 500 mW 500 mW

Differential Input Voltage g30V g30V g30V g30V

Input Voltage (Note 2) g15V g15V g15V g15V

Output Short Circuit Duration Continuous Continuous Continuous Continuous

Operating Temperature Range b55§C to a125§C 0§C to a70§C b55§C to a125§C 0§C to a70§CStorage Temperature Range b65§C to a150§C b65§C to a150§C b65§C to a150§C b65§C to a150§CJunction Temperature 150§C 100§C 150§C 100§CSoldering Information

N-Package (10 seconds) 260§C 260§C 260§C 260§CJ- or H-Package (10 seconds) 300§C 300§C 300§C 300§CM-Package

Vapor Phase (60 seconds) 215§C 215§C 215§C 215§CInfrared (15 seconds) 215§C 215§C 215§C 215§C

See AN-450 ‘‘Surface Mounting Methods and Their Effect on Product Reliability’’ for other methods of soldering

surface mount devices.

ESD Tolerance (Note 6) 400V 400V 400V 400V

Electrical Characteristics (Note 3)

Parameter ConditionsLM741A/LM741E LM741 LM741C

UnitsMin Typ Max Min Typ Max Min Typ Max

Input Offset Voltage TA e 25§CRS s 10 kX 1.0 5.0 2.0 6.0 mV

RS s 50X 0.8 3.0 mV

TAMIN s TA s TAMAX

RS s 50X 4.0 mV

RS s 10 kX 6.0 7.5 mV

Average Input Offset15 mV/§C

Voltage Drift

Input Offset Voltage TA e 25§C, VS e g20Vg10 g15 g15 mV

Adjustment Range

Input Offset Current TA e 25§C 3.0 30 20 200 20 200 nA

TAMIN s TA s TAMAX 70 85 500 300 nA

Average Input Offset0.5 nA/§C

Current Drift

Input Bias Current TA e 25§C 30 80 80 500 80 500 nA

TAMIN s TA s TAMAX 0.210 1.5 0.8 mA

Input Resistance TA e 25§C, VS e g20V 1.0 6.0 0.3 2.0 0.3 2.0 MX

TAMIN s TA s TAMAX,0.5 MX

VS e g20V

Input Voltage Range TA e 25§C g12 g13 V

TAMIN s TA s TAMAX g12 g13 V

Large Signal Voltage Gain TA e 25§C, RL t 2 kX

VS e g20V, VO e g15V 50 V/mV

VS e g15V, VO e g10V 50 200 20 200 V/mV

TAMIN s TA s TAMAX,

RL t 2 kX,


VS e g15V, VO e g10V 25 15 V/mV


2

Electrical Characteristics (Note 3) (Continued)

Parameter ConditionsLM741A/LM741E LM741 LM741C

UnitsMin Typ Max Min Typ Max Min Typ Max

Output Voltage Swing VS e g20V

RL t 10 kX g16 V

RL t 2 kX g15 V

VS e g15V

RL t 10 kX g12 g14 g12 g14 V

RL t 2 kX g10 g13 g10 g13 V

Output Short Circuit TA e 25§C 10 25 35 25 25 mA

Current TAMIN s TA s TAMAX 10 40 mA

Common-Mode TAMIN s TA s TAMAX

Rejection Ratio RS s 10 kX, VCM e g12V 70 90 70 90 dB

RS s 50X, VCM e g12V 80 95 dB

Supply Voltage Rejection TAMIN s TA s TAMAX,

Ratio VS e g20V to VS e g5V

RS s 50X 86 96 dB

RS s 10 kX 77 96 77 96 dB

Transient Response TA e 25§C, Unity Gain

Rise Time 0.25 0.8 0.3 0.3 ms

Overshoot 6.0 20 5 5 %

Bandwidth (Note 4) TA e 25§C 0.437 1.5 MHz

Slew Rate TA e 25§C, Unity Gain 0.3 0.7 0.5 0.5 V/ms

Supply Current TA e 25§C 1.7 2.8 1.7 2.8 mA

Power Consumption TA e 25§CVS e g20V 80 150 mW

VS e g15V 50 85 50 85 mW

LM741A VS e g20V

TA e TAMIN 165 mW

TA e TAMAX 135 mW

LM741E VS e g20V

TA e TAMIN 150 mW

TA e TAMAX 150 mW

LM741 VS e g15V

TA e TAMIN 60 100 mW

TA e TAMAX 45 75 mW

Note 1: For operation at elevated temperatures, these devices must be derated based on thermal resistance, and Tj max. (listed under ‘‘Absolute Maximum

Ratings’’). Tj e TA a (ijA PD).

Thermal Resistance Cerdip (J) DIP (N) HO8 (H) SO-8 (M)

ijA (Junction to Ambient) 100§C/W 100§C/W 170§C/W 195§C/W

ijC (Junction to Case) N/A N/A 25§C/W N/A

Note 2: For supply voltages less than g15V, the absolute maximum input voltage is equal to the supply voltage.

Note 3: Unless otherwise specified, these specifications apply for VS e g15V, b55§C s TA s a125§C (LM741/LM741A). For the LM741C/LM741E, these

specifications are limited to 0§C s TA s a70§C.

Note 4: Calculated value from: BW (MHz) e 0.35/Rise Time(ms).

Note 5: For military specifications see RETS741X for LM741 and RETS741AX for LM741A.

Note 6: Human body model, 1.5 kX in series with 100 pF.

3

Connection Diagrams

Metal Can Package

TL/H/9341–2

Order Number LM741H, LM741H/883*,

LM741AH/883 or LM741CH

See NS Package Number H08C

Dual-In-Line or S.O. Package

TL/H/9341–3

Order Number LM741J, LM741J/883,

LM741CM, LM741CN or LM741EN

See NS Package Number J08A, M08A or N08E

Ceramic Dual-In-Line Package

TL/H/9341–5

Order Number LM741J-14/883*, LM741AJ-14/883**See NS Package Number J14A

*also available per JM38510/10101

**also available per JM38510/10102

Ceramic Flatpak

TL/H/9341–6

Order Number LM741W/883

See NS Package Number W10A

*LM741H is available per JM38510/10101

4

Physical Dimensions inches (millimeters)

Metal Can Package (H)

Order Number LM741H, LM741H/883, LM741AH/883, LM741CH or LM741EH

NS Package Number H08C

5

Physical Dimensions inches (millimeters) (Continued)

Ceramic Dual-In-Line Package (J)

Order Number LM741CJ or LM741J/883

NS Package Number J08A

Ceramic Dual-In-Line Package (J)

Order Number LM741J-14/883 or LM741AJ-14/883

NS Package Number J14A

6


Small Outline Package (M)

Order Number LM741CM

NS Package Number M08A

Dual-In-Line Package (N)

Order Number LM741CN or LM741EN

NS Package Number N08E

7

LM

741

Opera

tionalA

mplifier


10-Lead Ceramic Flatpak (W)

Order Number LM741W/883

NS Package Number W10A

LIFE SUPPORT POLICY

NATIONAL’S PRODUCTS ARE NOT AUTHORIZED FOR USE AS CRITICAL COMPONENTS IN LIFE SUPPORT

DEVICES OR SYSTEMS WITHOUT THE EXPRESS WRITTEN APPROVAL OF THE PRESIDENT OF NATIONAL

SEMICONDUCTOR CORPORATION. As used herein:

1. Life support devices or systems are devices or 2. A critical component is any component of a life

systems which, (a) are intended for surgical implant support device or system whose failure to perform can

into the body, or (b) support or sustain life, and whose be reasonably expected to cause the failure of the life

failure to perform, when properly used in accordance support device or system, or to affect its safety or

with instructions for use provided in the labeling, can effectiveness.

be reasonably expected to result in a significant injury

to the user.

National Semiconductor National Semiconductor National Semiconductor National SemiconductorCorporation Europe Hong Kong Ltd. Japan Ltd.1111 West Bardin Road Fax: (a49) 0-180-530 85 86 13th Floor, Straight Block, Tel: 81-043-299-2309Arlington, TX 76017 Email: cnjwge@ tevm2.nsc.com Ocean Centre, 5 Canton Rd. Fax: 81-043-299-2408Tel: 1(800) 272-9959 Deutsch Tel: (a49) 0-180-530 85 85 Tsimshatsui, KowloonFax: 1(800) 737-7018 English Tel: (a49) 0-180-532 78 32 Hong Kong

Fran3ais Tel: (a49) 0-180-532 93 58 Tel: (852) 2737-1600Italiano Tel: (a49) 0-180-534 16 80 Fax: (852) 2736-9960

National does not assume any responsibility for use of any circuitry described, no circuit patent licenses are implied and National reserves the right at any time without notice to change said circuitry and specifications.

APPENDIX C

PRESENTATION SLIDE

1

Application of Pseudo Random BinarySequence (PRBS) signal in systemidentification

Prepared by:Maimun binti Huja Husin

ME061188Masters of Electrical Engineering (Mechatronics)

Universiti Teknologi Malaysia

Supervised by:PM. Dr. Mohd Fua’ad Bin Hj. Rahmat

2

Contents

Objectives & Scope of Project Project background, Methodology & Theory Result, analysis & Discussion

PRBS signal as test signal to second order system(simulation)

PRBS signal generator (hardware) PRBS signal as test signal to second order system

(hardware)

Conclusion & Future works References

3

Objectives

To design and generate PRBS generator withdifferent maximum length sequence (MLS) usingsoftware (MATLAB)

To design PRBS generator using hardware(Transistor-transistor logic-TTL)

To analyze the characteristic of PRBS signal suchas ACF, CCF, and PSD using MATLAB anddynamic signal analyzer.

To perform an experiment using real systemwhere PRBS is the test input.

4

Scope of project

Designing PRBS generator with 15 differentmaximum length sequence using MATLAB(SIMULINK) and hardware implementation usingtransistor transistor logic

The response of simulated second order systemsusing PRBS signal as test input will beinvestigated using MATLAB (SIMULINK) and willbe validated using hardware implementation

5

Project background

Most existing test input (e.g. step, ramp,impulse or sinusoidal input)

Characteristics: Ease of signal generation, Ease ofanalysis & The physical understanding of systemresponse which result

Problem: Not practical because of limitationsimposed by the existence of system noise

PRBS Characteristics: Popular input signal for system

identification, Resembles a white noise correlationfunction & Easy to generate using an n stage shiftregister

6

Methodology

Designing PRBS generator usingMATLAB (SIMULINK)

LiteratureReview

Tests the PRBS signal onsimulated second order systems

using MATLAB (SIMULINK)

Build PRBS generator using TTL

Test the PRBS signal on realsecond order system

Verify? EndYesNo

7

Theory

PRBS signals Can take on only two possible states, say +a and –a State can change only at discrete intervals of time Δt Sequence is periodic with period T=NΔt where N is an

integer

The most commonly used type - maximum lengthsequence (length N=2n-1, where n is an integer) Generated by an n shift register

8

Theory

The first stage of the shift register is determined byfeedback of the appropriate modulo two sum (the logicfunction ‘exclusive or’).

The logic contents of the shift register are moved onestage to the right every Δt seconds by simultaneoustriggering by a clock pulse

9

Theory

ACF A measure of the predictability of the signal at some future time

based on knowledge of the present value of signal

CCF A process of comparing one signal with another by multiplication of

corresponding instantaneous values and taking the average

A measure of the similarity between two different signals.

T

TT

T

TT

dttxtxT

or

dttxtxT

)()(2

1lim)(

)()(2

1lim)(

xx

xx

T

TT

T

TT

dttxtyT

or

dttytxT

)()(2

1lim)(

)()(2

1lim)(

xy

xy

10

Theory

Determine transfer function general form

Calculate model parameter

Plug in all the parameters into transfer function generalform

Calculate impulse strength of the input signalImpulse strength = height of ACF triangle x bit interval

Start

End

Steps to determine the transfer functions model of system

11

Result, Analysis & Discussion

On PRBS signal as test signal(simulation)

12

PRBS signal as test signal(simulation)

Four condition of second order system will be examined:overdamped, underdamped, undamped and criticallydamped

Settings: Noise power for band-limited white noise is set to 0.01(1% of the

input magnitude); A step of magnitude unity (1) & N = 63

4

3

2

1

No

9 / (s2+9s+9)1.50Overdamped

9 / (s2+6s+9)1.00Critically damped

0.00

0.33

Damping ratio, ξ Transfer functionType of second ordersystem

9 / (s2+9)Undamped

9 / (s2+2s+9)Underdamped

13

PRBS signal as test signal(simulation) – critically damped

Block diagram of second order system criticallydamped (ξ = 1)

14


Output responses

15

PRBS signal as test signal(simulation) – critically damped ACF of PRBS signal – theoretically expected ACF of system forced by PRBS input in absence of noise – reduction in signal

power ACF of noisy system forced by PRBS input – shows that is a significant

component of signal which approximates to white noise some increase ofsignal power)

Autocorrelation function of input & output signals

16

PRBS signal as test signal(simulation) – critically damped Response of systems forced by PRBS input – rise + decay wave General form : A (e-αt - e-βt)

Chosen Δt = 0.1s – gives adequate approximation to white noise for thissystem

Period of 6.3s correctly exceeds the system settling time sequence ofN = 31 could have been used instead

Cross correlation function of output signals

17

PRBS signal as test signal(simulation) – critically damped With PRBS input, almost entire power of output signals in contained in

frequency range of 1 to 3Hz.

Curve for PSD for PRBS input – shows that over this frequency rangePRBS input has substantially constant PSD Confirms that Δt used gives an excitation signal which is a good

approximation to true white noise for system tested

Power spectral density curves for input & output signals

18


ACF of input signal and CCF of output signals are used todetermine the transfer functions model of system Difficult to obtain correct transfer function – CCF of system output

signal does not yield a good approximation to impulse response(decaying sine wave)

Transfer function obtained using 3 different PRBSmaximum length

9 / (s2+6s+9)

Transfer function used in the simulation:

11.10/(s2+7.41s+9.17)1023

9.21/(s2+5.94s+6.00)255

10.54/(s2+6.66s+7.44)63

Transfer functionLength, N

19

PRBS signal as test signal(simulation) – under damped

Block diagram of second order system underdamped (0 < ξ < 1)

20


Output responses

21

PRBS signal as test signal(simulation) – under damped ACF of PRBS signal – theoretically expected ACF of system forced by PRBS input in absence of noise – reduction in signal

power ACF of noisy system forced by PRBS input – shows that is a significant

component of signal which approximates to white noise some increase ofsignal power)

Autocorrelation function of input & output signals

22

PRBS signal as test signal(simulation) – under damped Response of systems forced by PRBS input – decaying sine wave General form : A e-αt sin ωt

Chosen Δt = 0.1s – gives adequate approximation to white noise for thissystem

Period of 6.3s correctly exceeds the system settling time sequence ofN = 31 could have been used instead

Cross correlation function of output signals

23

PRBS signal as test signal(simulation) – under damped With PRBS input, almost entire power of output signals in contained in

frequency range of 1 to 5Hz. Curve for PSD for PRBS input – shows that over this frequency range

PRBS input has substantially constant PSD Confirms that Δt used gives an excitation signal which is a good

approximation to true white noise for system tested

Power spectral density curves for input & output signals

24


ACF of input signal and CCF of output signals are used todetermine the transfer functions model of system CCF of system output signal yield a good approximation to

impulse response

Transfer function obtained using 3 different PRBS maximumlength

9 / (s2+2s+9)

Transfer function used in the simulation

8.60/(s2+1.94s+8.39)1023

9.04/(s2+1.87s+8.68)255

8.52/(s2+1.96s+8.41)63

Transfer functionLength, N

25


On PRBS signal generator (hardware)

26

PRBS signal generator (hardware)

PRBS generator circuit

Supply voltage

PRBS GeneratorClock circuit

Feedback circuit

PRBS Signal

27

PRBS signal generator(hardware)

PRBS generator circuit for MLS (hardware implementation)

ACF and PSD of PRBS signal is performed using theDynamic Signal Analyzer (HP35670A DSA)

28


512 data of the PRBS signal is captured using Dynamic SignalAnalyzer for every MLS of PRBS signal

MATLAB is used to plot the PRBS signal, autocorrelation and powerspectral density

PRBS signal for MLS of N = 63

29


The height of the ACF triangle, V2 = 0.95V and the bitinterval is 0.1281s

Autocorrelation function for MLS of N = 63

30

PRBS signal generator(hardware)

The lowest frequency component is 70Hz – which is a bithigher than the calculated values 2π/Δt = 57Hz


31


On PRBS signal as test signal(hardware)

32

PRBS signal as test signal(hardware)

A PRBS signal is used as an input signal to determine themodel of second order system

The autocorrelation of the input signal and crosscorrelation between the input and output signal isperformed using the Dynamic Signal Analyzer (HP35670ADSA)

Second ordersystem

g(t)

PRBS signal

x(t)

Output response

y(t)

33

PRBS signal as test signal(hardware)

R2R1

C1

C2

R3

R4

VOUT

VIN

)dampedcritically(7.4526.42

7.452)(0

)dampedunder(7.4529.19

7.452)(5

1.0eter);(potentiom10

;7.4;470

where

21

21

1

311

4

11

22

21

21

1

)(

24

24

214

321

sssAR

sssAkR

FCCkR

kRkRR

CRs

RCR

R

CRs

CRsA

Second order RC circuit

34

PRBS signal as test signal(hardware) – Critically damped

The measurement result has the same shape as prediction output.

Output signal using PRBS of MLS N=63

35


ACF of the measurement result has the value close to the predictionvalue

ACF of the output signal

36


Transfer function obtained: T (s) = 349.22 / (s2 + 57.88s + 476.52)

Transfer function used in hardware implementation: T(s)= 452.7 / (s2 + 42.6s + 452.7)

CCF of the input and output signal

37

PRBS signal as test signal(hardware) – Underdamped

The measurement values obtained follow the prediction values.

Output signal using PRBS of MLS N=63

38


Measurement result has the value close to the prediction value

ACF of the output signal

39


Transfer function obtained: T (s) = 155.47 / (s2 + 9.92s + 327.01)

Transfer function used in hardware implementation: T(s)= 452.7 / (s2 + 19.9s + 452.7)

CCF of the input and output signal

40

Conclusion

PRBS is a good input signal for systemidentification - easy to generate and introduceinto a system

Length of the MLS can be set according to thesystem under test – some system require higherMLS values

PRBS signal as test input has successfully designexcept for the undamped and overdampedsystem

41

Future works

Software: More convenient if GUI can bedesigned for the PRBS generator & its application

Hardware:

Test PRBS signal as test input to undamped andoverdamped second order system

Perform experiment on real system (e.g. suspensionsystem) where PRBS is the test input

42

References Schwarzenbach, J. and Gill, K.F. (1984). System modelling and Control, 2nd

Edition, Edward Arnold (Publishers) Ltd. Tan, A.H. and Godfrey, K.R. (2002). The generation of binary and near-

binary pseudorandom signals: an overview. IEEE Trans. Instrum. Meas.51 (4), 583-588.

Van Den Bos, A. (1993). Periodic test signals – Properties and use.Godfrey, K. Perturbation Signals for System Identification. (ch.4). Ed.London, U.K.: Prentice Hall.

Darnell, M. (1993). Periodic and nonperiodic, binary and multi-levelpseudorandom signals. Godfrey, K. Perturbation Signals for SystemIdentification. (ch.5). Ed. London, U.K.: Prentice-Hall.

Godfrey, K. (1993). Introduction to perturbation signals for frequency-domain system identification. Godfrey, K. Perturbation Signals for SystemIdentification. (ch.2). Ed. London, U.K.: Prentice-Hall.

Godfrey, K. R., Barker, H. A. and Tucker, A. J. (1999). Comparison ofperturbation signals for linear system identification in the frequencydomain. Proc. Inst. Elect. Eng. – Control Theory Applicat. 146(6), 535–548.

43

References Kollár, I. (1994). Frequency Domain System Identification Toolbox for

use With MATLAB. Natick, MA: The MathWorks Inc. McCormack, A. S., Godfrey, K. R. and Flower, J. O. (1995). Design of

multilevel multiharmonic signals for system identification. Proc. Inst.Elect. Eng. – Control Theory Applicat. 142(3), 247–252.

Zierler, N. (1959). Linear recurring sequences. J. Soc. Ind. Appl. Math.7, 31–48.

Godfrey, K. (1993). Introduction to perturbation signals for time-domain system identification. Godfrey, K. Perturbation Signals forSystem Identification. (ch.1). Ed. Englewood Cliffs, NJ: Prentice Hall.

Godfrey, K.R. (1991). Introduction to binary signals used in systemidentification. Control 1991. Control '91, International Conference on,vol., no., pp.161-166 vol.1, 25-28.

Zapernick, H.-J. and Finger, A. (2005). Pseudo Random SignalProcessing – Theory and Application. Chichester: John Wiley & Sons,Ltd.

Sodestrom, T. and Stoica, P. (1989). System Identification.Hertfordshire: Prentice Hall International (UK) Ltd.

44

References Godfrey, K. R. and Briggs, P. A. N. (1972). Identification of processes with

direction-dependent dynamics responses. Proc. Inst. Elect. Eng. – Control Sci.119(12), 1733–1739.

Godfrey, K. R. and D. J. Moore (1974). Identification of processes having direction-dependent responses, with gas – turbine engine applications. Automatica, 10(5),469–481.

Tan, A. H. and Godfrey, K. R. (2001). Identification of processes with direction-dependent dynamics. Proc. Inst. Elect. Eng. – Control Theory Applicat. 148(5),362–369.

Barker, H. A., Godfrey, K. R. and Tan, A. H. (2000). Identification of systems withdirection-dependent dynamics. Proc. 39th IEEE Conf. Decision Control (CDC 2000),2843–2848.

Mouine, J. and Boutin, N (1998). A novel way to generate pseudo – randomsequences longer than maximal length sequences. Proc. Inst. Elect. & Comp. Eng.2, 529-532.

Rahmat, M. F. (2007). Pseudo random binary sequence. System Identification &Parameter Estimation Lecture Note, UTM Skudai.

application of pseudo random binary sequence (prbs

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