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Pseudo Random Binary Sequence on Second Order System

Maimun Huja Husin, Mohamad Faizrizwan Mohd Sabri, Ade Syaheda Wani Marzuki, Kasumawati

Lias Faculty of Engineering

Universiti Malaysia Sarawak Kota Samarahan, Sarawak, Malaysia e-mail: hhmaimun@feng.unimas.my

Mohd Fuaad Rahmat Faculty of Electrical Engineering

Universiti Teknologi Malaysia UTM Skudai, Johor, Malaysia

e-mail: fuaad@fke.utm.my

AbstractPseudo random binary sequence (PRBS) signal was developed using MATLAB software and used as a forcing function in simulated second order system. The autocorrelation function (ACF) of the input signal and cross correlation function (CCF) between input and output signal were performed using MATLAB software and transfer function of the system was estimated from the correlograms. For verification of the simulation work, PRBS generator circuit was build using transistor-transistor logic and analyzed using Dynamic Signal Analyzer (DSA). PRBS is used as forcing function to an unknown system. The ACF of the input signal and CCF between input and output signal were performed using DSA and the transfer function model of the unknown system was estimated.

Keywords-PRBS, ACF, CCF, correlograms

I. INTRODUCTION 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 and furthermore it is sufficient for test purposes [1]. 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.

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 [2]. 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 (MLS) is N = 2n - 1 where n is the order of the PRBS.

MLS can be generated by an n stage shift register with

the first 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 linear feedback shift register and its general structure is shown in Figure 1.

Figure 1. Linear feedback shift register

The output can be taken from any stage and is a serial

sequence of logic states having cyclic period Nt. 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 1.

Table 1: Feedback configuration of linear feedback shift

register [3] 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

II. ACF AND CCF The ACF of a signal x(t) is given the symbol xx() and is

defined as,

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399

T

T)dtx(t)x(t

2T1

Tlimxx

(1)

or

T

T)x(t)dtx(t

2T1

Tlimxx (2)

where )( tx and )( tx are displacement of signal )(tx .

CCF is a measure of dependence of one signal on the

other. CCF is defined as,

)dtx(t)y(t2T1lim

T

TTxy

(3)

or

T

TTxy

)dty(t)x(t2T1lim (4)

where )( ty and )( tx are displacement of signal )(ty and )(tx respectively.

III. SECOND ORDER RC LOW PASS FILTER CIRCUIT The second order system used as unknown system in the

hardware implementation of PRBS signal is shown in Figure 2.

Figure 2. Second order RC circuit

The values for each components in the second order RC

circuit are R1 = R2 = 470 k, R3 = 4.7 k, C1 = C2 = 0.1 F and R4 is a potentiometer of 10 k. Value of R4 is varies according to type of second order system being tested as shown in Table 2.

Table 2. RC low pass filter second order system transfer function

No Type of second order

system

R4 value Transfer function

1 Critically damped

R4 = 0

452.742.6ss452.7

2

2 Underdamped 0

400

Figure 4. CCF of output signal using PRBS signal

From the CCF and ACF graphs of the measurement

curves, model parameter for the hardware analysis can be calculated using the following steps: a) The height of ACF triangle shown in Figure 3 is

V2=0.95V and the bit interval is 0.1281s. The impulse strength is V2 times the bit interval which evaluates to 0.95V 0.1281s = 0.12 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 0.10056s (decay) and 0.02086s (rise). So, = 9.94 and = 47.94 .

d) A is obtained from value of peak height, A = 1.103 . e) Divide by the unit impulse response,

)(19.9)( 94.4794.9 tt eetf

f) 52.47688.57

22.34994.47

19.994.9

19.9)(2

sssssF

B. Under damped response Figure 5 shows the ACF graph for PRBS signal for MLS

of N = 63. Measurement curve represents the hardware analysis result while prediction curve represents the software analysis result. It can be shown from the graph of the measurement curve, height of the ACF triangle is V2 = 0.95V and bit interval is 0.1281s. For the software analysis (shown by the prediction curve), the ACF triangle, V2 = 1V and the bit interval is 0.1s.

Figure 5. ACF of PRBS signal for MLS of N = 63

Figure 6 shows the CCF of the output signal obtained using PRBS signal as the input to the RC second order under damped system for both hardware and software analysis.

Figure 6. Cross correlation function of output signal using

PRBS signal From the CCF and ACF graphs, model parameter for the

hardware analysis can be calculated. For the hardware analysis, the following steps are used to determine the transfer function of the prediction curves: a) The height of ACF triangle shown in Figure 5 is V2 =

0.95V and the bit interval is 0.1281s. The impulse strength is V2 times the bit 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) is obtained from cycle time,

rad/s39.173612.02

d) is obtained from peak decay ratio,

96.47592.03098.0ln

3612.02

e) A is obtained from the first peak height, A = 1.073 f) Divide by the unit impulse response,

tetf t 39.17sin94.8)( 96.4

g) 01.32792.9

47.15539.17)96.4(

)39.17(94.8)( 222

ssssF

VI. DISCUSSION The transfer function obtained in hardware analysis for

both critically damped and under damped responses are not very close to the actual transfer function used in the hardware analysis. This is due to the difficulty in obtaining the correct transfer function using correlation technique for a CCF graph of the second order critically damped response which does not yield a good approximation to an impulse response.

The correlations for both critically damped and under

damped responses are carried for a short period of time. This has affected the result obtained from the correlation analysis. In order to help smoother the curves, longer period of correlation is needed, provided the dynamic characteristic of

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the system being tested remained unchanged over long period of time span involved.

VII. CONCLUSION 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.

ACKNOWLEDGEMENT This is a self-funded work and the author wishes to thank

Universiti Malaysia Sarawak for supporting the dissemination of this research.

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

[3] Mohd Fuaad Rahmat, KH Yeoh, Sahnius Usman and Norhaliza Abdul Wahab, Modelling of PT326 Hot Air Blower Trainer Kit Using PRBS Signal and Cross Correlation Technique, Journal of Technology Part D, UTM Publisher, June 2005, volume 42, pp 9 22

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