Electrical systems identification using Golay complementary series

M.K. Oolun

Abstract: Various techniques for interrogating systems are in current use, each having its own advantages and limitations. The author presents the design, implementation and testing of a PC and DSP based system that combines the well established properties of Golay complementary series with a cross-correlation technique, to perform electrical systems identification. Initially, the author develops the theoretical background and presents Golay codes adapted in a way to suit the application. Details of the design for the prototype model implemented in the work are given, and results of the tests carried out to evaluate the system’s performance are reported. It is shown that this new system is capable of working at very high speed and covering a wide frequency spectrum.

1 Introduction

System identification is the process of determining by means of practical testing the dynamic response per- formance of a system [l], and is considered to be a very powerful tool for the design and analysis of systems [2]. Traditional experimental procedures for system interro- gation involve stimulating the system under test with one of the fundamental signals [3] , and then performing an analysis of the output response. The advantage of these test inputs is that they are the basis of most nor- mally occurring system inputs in real life situations; but they are unfortunately very difficult to handle in the presence of a high noise level. In addition, using an impulse as the input means running the risk of forcing the system under test into nonlinear behaviour.

Later works based on a statistical identification approach proposed the use of white Gaussian noise [4] and the pseudo random binary sequence (PRBS) [5] in conjunction with cross-correlation. In the former case, the disadvantage is the need to provide for a long inte- gration and averaging time to obtain an accurate and consistent estimate of the auto-correlation function ~

necessary to ensure that changes in the auto-correlation function of the input do not affect the cross-correlation function. In practice, this may be unacceptable, as the

0 IEE, 1997 ZEE Proceedings online no. 19971531 Paper first received 29th January and in final revised form 30th July 1997 The author is with the Department of Electrical and Elcctronics Engineer- ing, Faculty of Engineering, University of Mauritius, Reduit, Mauritius

system’s characteristics are not guaranteed to remain constant throughout that time. As for the PRBS approach, extreme care [5] is recommended due to the low amplitude PRBS signals that are normally used. This means that a considerably long integration time is required to achieve reasonably good signal-to-noise ratio (SNR) performance, because it has been shown that [6]:

SNR = h , ( t ) , ~ . A f i (1)

where T: integration time p: power spectral density (psd) of the PRBS signal U : amplitude of the stimulating (PRBS) signal A: clocking period. Furthermore, the Fourier domain representation of the auto-correlation function of a PRBS [7] given in eqn. 2 shows the presence of a zero-frequency term which causes the signal to deviate from the desirable Dirac- like characteristics.

2 0 0 a + 1 sin(wAtl2) +(U) = ~ [ ]

n=-m U wAt/2

1 U 2

+ -S(U)

In most cases this DC term is neglected but for accu- rate measurements we need to cancel or compensate for it, for example by using inverse repeat PRBS. A more elegant solution however, is to use signals such as Golay codes [8] or the Huffman sequence [3] , which have the inherent property of performing the cancella- tion with practically no additional effort. The objective of this work was thus to develop a system that uses Golay codes for system identification, and in this paper we describe the design, implementation, and testing stages for the prototype system that was developed.

Fig. 1 Block dugmm repr~~entintion of a system

2 Theory

For the block diagram representation of a general sys- tem shown in Fig. 1, eqn. 3 gives the relationship that exists between the various parameters, in the time and frequency domains, respectively.

where 0 denotes the convolution operation. y ( t ) = h( t ) c3 z ( t ) H Y(w) = H(w).X(LJ) (3)

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The cross-correlation function Rxy(z) of the input and the output over an integration time T, which is given by eqii. 4 can be simplified to eqn. 5 for a finite record length of data taken over an interval -T to +T,

1 J o where z is the correlation delay.

is the discretised version of eqn. 5, is more relevant. In the sampled-data domain however, eqn. 6, which

n=O

where N is the number of sampled data points, k is the correlation delay, which for most applications, is equal to the sampling period At, and n is the discrete-time step.

Now, eqn. 7 can be generated from eqn. 3; and the Inverse Fourier transform of eqn. 7 gives eqn. 8.

Y ( U ) .x* ( U ) = H ( U ) .x ( U ) .x * ( U )

= h(w) / G ( w ) I (7) where lG(.1->1 and ~ ' ( c o ) are, respectively, the power spectral density, and the complex conjugate of the input signal.

Rzy(7) = h(t).Rz,(7 - t ) d t (8) .6* where R,,(z): cross-correlation function of input and output signals. R,,(z): 'tuto-correlation function of the input signal.

The discrete-time version of eqn. 8 is given by eqn. 9.

Rzy[kI = h[nI 8 RZZ[kI (9) By putting IG(co)o)l = 1 in eqii. 7, eqn. 8 is simplified to eqn. 10.

RTY(7) = im h(t).S(.r) d t = h(r) (10)

This means that if we choose an input signal that has a uniform power spectral density, of magnitude unity over a wide band, the cross-correlation of the generated output and the input will produce the impulse response h(t). Noise signals and PRBS have been used in this way in previous work [4, 51. It is shown in the next Sec- tion that a pair of Golay codes, as opposed to the other two signals, makes a better candidate.

3 Golay codes

Golay defines a set of complementary series [SI as a pair of equally long finite sequences of two kinds of element which has the property that the number of pairs of like elements with any given separation in one series, IS equal to the number of pairs of unlike ele- ments with the same separation in the other series. Golay also proposed an algorithm [8] to generate a pair of complementary series A and B, for which the sequence lengths are a power of two; such that A = a l , a2, .. U,] and B = bl, b2, . . b,?, respectively, and the ele- ments U, and 6, are members of the binary number set

By assigning opposite polarity of unity strength to A and B; I e. { a I , 6, E [-l,+l]}, and defining C, and 0, as the auto-correlative series for A and B, respectively;

268

such that %=n- 7

c, = %a%+, a=l

?=n--.l

z=1 We see that

C , + D , = O J # O

c, f Do = 2rl Functions Cj and D, are shown in Figs. 2 and 3, respec- tively. Eqn. 12 shows that the sum of the auto-correla- tion functions from a pair of Golay codes produces a large central peak of magnitude equal to twice the value of the sequence length, with no range sidelobes [9], as shown in Fig. 4. The large central peak, thus obtained, has ideal Dirac-like characteristics.

(12)

Fig. 2 Scales: 14.6w/div. and IbV/div

Autocorrelation function .for computer generated Golay code A

Fig. 3 Scales: 14.6,midiv. and l6Vidiv

Autocorrelation junction for computer generated Golay code B

Fig.4 Scales: 0.912pddiv. and 32Vidlv

Sun? of autocorrelation junctions of codes A and B

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This special property of Golay codes has been widely exploited in applications such as improved ultrasonic pulse echo measurement [9], flow measurement of pul- verised coal [lo], and non-invasive medical measure- ments [ I l l to mention but a few. In this work we conceive a system that combines this same property with the high-speed computational capabilities of mod- ern signal processors, to perform cross-correlation, for electrical systems identification. Such a system presents no risk of saturation to the system under test, as in the Dirac pulse approach. Furthermore the offset error encountered with the conventional PRBS method is completely obviated, at practically no extra effort.

Now, for a given pair of Golay codes A and B that are applied to stimulate a system under test, and the corresponding output responses thus obtained, eqns. 13 and 14, respectively, can be deduced from eqn. 9.

R Z B Y B [at] ZZ h[n] @ RZ,ZB [at] (14) By adding eqns. 13 and 14, we obtain eqn. 15;

= h[n] 8 K.6(t) (15) where K is the height of the central peak. Hence, the impulse response h[n] can be obtained directly by mak- ing K equal to unity; a normalisation exercise which was achieved purely in software for this work.

4 System design specifications

The choice of the clocking period and sequence length for the stimulating Golay codes, the sampling fre- quency for the output response, and the length of the sample record to be processed, must be made with extreme care, since poor selection of any of these adversely affects the accuracy of the results. When we have set these parameters to convenient values we can then specify the frequency range over which the system will perform interrogation.

To be able to calculate these parameters, we need to analyse the Fourier transform for the sharp pulse shown in Fig. 4. This is given analytically by eqn. 16, and is shown graphically in Fig. 5.

(16) sin wAt/2

q w ) = V2At

where V is the amplitude of the Golay codes.

2x At -

2n 2a G 3x w, radls

- 4% At

Fi .5 co% pair

Fourier transform of sum of autocorrelation junctions of Golay

We observe that the lowest frequency component has a value of 2zlNAt radians per second, which also defines the frequency resolution for N sampled data points, and a sampling period of At. In addition, the highest harmonic found within the 3dB range of the power spectrum has a value of 2nl3At rads-'. This 3dB range defines a window over which the frequency response of the system under test should lie to ensure that all its frequency components have reasonable power levels. As such, values of N and At are initially chosen so that they meet a desirable frequency range for interrogation, and the remaining two parameters are then derived. Alternatively we can set the parame- ters to their maximum allowable values as supported by the hardware and software, and then specify the fre- quency range over which the system works reliably.

In this work, we have set N to 1024, since the FFT algorithm, in practice, requires a number of samples which is a power of 2; At is set to 1 p to provide a sam- pling frequency of 1MHz. To be within the Nyquist sampling threshold we have chosen a sequence length of 512 for the Golay codes which are clocked at a fre- quency of 500 kHz. These parameters yield a frequency resolution of 0.98kHz and a 3dB window of 330kHz, which effectively corresponds to the frequency range over which a system of unknown response can be inter- rogated.

5 The Golay codes system

The prototype system developed for this work is mostly software-oriented and is thus based around a PC and a DSP. It is shown in block diagram form in Fig. 6. The PC, used to make the system user-friendly, is a 386 machine working at 33MHz. It also generates the Golay codes of a specified sequence length, and out- puts them, via its printer port, to stimulate the system under test. The DSP, a TMS320C30 system in this case, operates in parallel with the PC to provide the required high sampling rate of 1MHz. It is equally used to perform the required cross-correlation opera- tion after the data capture, at high speed, since it is known that the hardware solution for the same is either intensive or slow [12]. An interface circuit comprising a high speed analogue to digital converter (ADC), and a timing and control unit, is equally part of the system. The ADC provides the required high digitising rate for eventual sampling and storage by the DSP system. The timing and control unit is necessary to co-ordinate all the activities taking place, generate the timing signal for the A to D conversion, and match the difference in signal characteristics from the different parts of the sys- tem. The program for the PC is developed in high-level C while that of the DSP system is developed in TMS320C30 Assembly codes.

269 IEE Proc.-Sci. Meas Technol.. Vol 144, No. 6, November 1997

input

M I-

> nC

+J AD826 AD826 I

I I

w m \ a

3 YI %i .-(

0

-6 +

I 1

Fig. 7 hiwface circuit diagram for the prototype system

m Hi1 +

The system operates as follows: (i) The PC downloads the control program for the DSP in its program memory area and requests the latter to wait until a synchronisation signal arrives. (ii) The PC then feeds the first sequence of Golay codes of pre-defined sequence length to the input of the system, via the printer port. Simultaneously, the syn-

270

chronisation signal is sent to the DSP and the ADC is started. (iii) The digital output of the ADC is read at high speed by the DSP chip at 1 MHz rate and stored in the on-chip memory. (iv) The complementary sequence of the Golay codes pair is then sent together with some control signals.

IEE Pioc -SCI Mear Tethnol, Vol 144, No 6, November 1997

The response to this code sequence is again digitised and stored as described above. (v) On completion of the acquisition process, the DSP performs the cross-correlation of the sampled values with the respective input stimuli. (vi) The cross-correlation functions obtained for the pair of Golay codes are added together and then nor- malised so as to produce the impulse response. The PC is informed that the computation is over and that the results can be read. (vi;) The PC program finally performs an FFT on the acquired data and displays the impulse and frequency responses of the system under test, on the screen. The circuit diagram for the interface circuit is given in Fig. 7, where the left hand side shows the connection points to the DSP system board, DSPLINK connector ~ 3 1 .

6 Tests and results

The prototype system was tested with electrical circuits containing both passive and active elements. We dis- cuss results of the test carried out on a simple low-pass RC filter with a cut-off frequency of SOkHz, and which is known to have a frequency response as shown in Fig. 8; where w, is the cut-off frequency and A is slightly less than unity. The impulse and frequency responses that we obtained from the Golay codes system are shown in Figs. 9 and 10.

lH(j w ) l

t

. I . I . . , , . I . , . , . , .

. , / . . . I . . I ____.._._..___ :. ..... ~ ..._._........ i .

. . . . . .

. , . .

Fi .9 hiipdse response of low-puss RC filter ohtuiiiedfionz new Goluy coz3 system Scales: 25.6ys/div. and 0.563V/div

From the shape of the frequency response in Fig. 10, we can clearly observe the low-pass nature of the filter. Some spikes, mainly due to noise pick-up along the unscreened cable used to connect the interface circuit and the DSPLINK connector of the DSP system, can also be observed. The cut-off frequency can be read as

IEE ProcSci . Meas. Teciiizol., Vol. 144, No. 6, Novenzhei I997

slightly less than SOkHz. We also observe that at fre- quencies lower than cut-off; the amplitude for the vari- ous frequency components remains within a 3dB range. We have a sharp roll-off around the cut-off frequency followed by a drastic decrease in the amplitude, that becomes more and more important as the frequency increases beyond cut-off point. This is a clear evidence that system identification of the filter has been per- formed.

frequency buffer 1

37 46 1 I

0 100 200 300 400

frequency, kHz Fi . 10 GO?UJ code,r .ry.ctet?l

Frequency re,sponsc~ oj 1oi.o-p.s~s RC ,filter. obtuined ,from neu

In real life applications, the system to be interrogated will have a time or frequency response unknown to the user. Just as in this test case, where the expected time and frequency responses of the system under test have been reproduced faithfully, we would expect the Golay codes system to produce the same for any other unknown electrical systems, since in most practical cases the latter can be modelled in terms of active and passive elements connected together in either series or parallel, or a combination of the two, using the Foster synthesis principle [ 141.

7 Conclusion

A system that combines the well established properties of Golay codes and the high-speed computational capabilities of a fast signal processor, for electrical sys- tem identification, has been presented. The prototype system, designed around a 386 PC and a TMS32OC30 DSP system, was set to operate with Golay codes of sequence length 512, and a sampling rate of 1MHz; thereby resulting in a resolution of 0.98kHz. Results of the tests carried out on a low-pass filter have been reported; they show that, although the system does require some minor refinements, it is very suitable for electrical systems identification.

8 Further work

Future improvements that it is thought can be brought to the present system are mainly in the following direc- tions: (i) To upgrade the frequency range over which we can interrogate systems by using a second DSP system to generate the Golay codes. Consideration can also be given to using DSP systems that provide faster transfer

(ii) To design a universal Golay codes system, since the present system is suited only for electrical systems iden- tification. This will require the capability to generate

rate to the outside world.

27 1

Golay codes in other forms than electrical. For instance, if we are to interrogate a pressure sensor, the stimulating Golay codes sequence should be in the Form of a pressure signal, and so on. (iii) To adapt the system to cope with long distances. This is of particular interest when we do real time interrogation, where the system to be interrogated may be placed in a different environment far from the inter- rogating system. Encoding to match the channel char- acteristics should also be considered in this case, and care should be taken so that the complementary nature of the Golay codes is preserved. (iv) To consider the possibility of using other types of codes that exhibit similar complementary properties to Golay codes, or which can be tailored in some way to do so, and applying the same principle or a modified version if need be. The Huffman sequence [3] can be considered for example.

9 Acknowledgment

The facilities provided by DIAS, UMIST in the devel- opment of this system are gratefully acltnowledged.

10 References

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11 O’DONOVAN, T.L., CONTLA, P.A., and DAS- GUPTA, D.K.: ‘Application of Golay codes and piezoelectric ultrasonic transducers for medical non-invasive measurements’, IEEE Trans. Electr. Insul., 1993, 28, ( l ) , pp. 93-100

12 PAYNE, P.A.: ‘Correlation methods applied to instrumentation’ in GARDNER, J.W., and HINGLE, H.T. (Eds.): ‘Development in nanotechnology. Vol 1 : From instrumentation to nanotechnol- ogy’ (Gordon and Breach Science Publishers, 1991), pp. 35-84

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