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
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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
4.3 Transfer function for several different PRBS maximum 43
length
4.4 Transfer function for several different PRBS maximum 47
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
4.2 (a) Clock signal, (b) PRBS signal, 35
(c) Autocorrelation function, and
(d) Power spectral density for MLS of N = 63
4.3 (a) Clock signal, (b) PRBS signal, 36
(c) Autocorrelation function, and
xii
(d) Power spectral density for MLS of N = 255
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
4.8 (a) PRBS signal and traces of output response of system 41
(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
4.9 Autocorrelation functions of input and output signals 41
4.10 Cross correlation functions of output signals 42
4.11 Power spectral density of input and output signals 44
4.12 (a) PRBS signal and traces of output response of system 45
(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
4.13 Autocorrelation functions of input and output signals 45
4.14 Cross correlation functions of output signals 46
4.15 Power spectral density of input and output signals 48
4.16 (a) PRBS signal and traces of output response of system 49
(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
4.17 Autocorrelation functions of input and output signals 49
4.18 Cross correlation functions of output signals 50
4.19 Power spectral density of input and output signals 50
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
%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
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.
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
LOW-to-HIGH Level Output
tPHL Propagation Delay TimeClear to Q 20 28 ns
HIGH-to-LOW Level Output
tPLH Propagation Delay TimeClock to Q or Q 20 24 ns
LOW-to-HIGH Level Output
tPHL Propagation Delay TimeClock to Q or Q 20 28 ns
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
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
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/
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
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 g20V, VO e g15V 32 V/mV
VS e g15V, VO e g10V 25 15 V/mV
VS e g5V, VO e g2V 10 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
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
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
PRBS signal as test signal(simulation) – critically damped
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
PRBS signal as test signal(simulation) – critically damped
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
PRBS signal as test signal(simulation) – under damped
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
PRBS signal as test signal(simulation) – under damped
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
Result, Analysis & Discussion
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
PRBS signal generator (hardware)
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
PRBS signal generator (hardware)
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
Power spectral density for MLS of N = 63
31
Result, Analysis & Discussion
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
PRBS signal as test signal(hardware) – Critically damped
ACF of the measurement result has the value close to the predictionvalue
ACF of the output signal
36
PRBS signal as test signal(hardware) – Critically damped
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
PRBS signal as test signal(hardware) – Underdamped
Measurement result has the value close to the prediction value
ACF of the output signal
39
PRBS signal as test signal(hardware) – Underdamped
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
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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
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