NDT.net July 2003, Vol. 8 No.07 |
During automation of industrial processes, imaging of an environment and detection of targets is becoming more and more important. For this purpose ultrasonic systems based on distance measurements may be efficiently exploited. Generic block diagram of such a system is presented in Fig. 1 [1].
Fig 1: Generic block diagram of ultrasonic multi-channel distance measurement system: T_{1}-T_{n} are transmitters, R is a receiver of ultrasonic waves. |
The main idea of this system is that measuring signals simultaneously are sent to and received from different directions. This allows us to reduce measurement time proportionally to the number of measurement channels [1,2]. In this system, during the process of multi-channel distance measurements, the main problem, which is met, is the following: after sending simultaneously several signals in different directions to the receiver arrive several signals from the different directions with different amplitudes. In order to resolve these signals, e.g., to extract only one necessary type of the signal from the mixture of received signals it is necessary to use a set of quasi - orthogonal signals. The number of the required different signals is increasing with the number of distance measurement channels and with the number of simultaneously operating systems.
To solve the above-mentioned problems it was decided to use pseudo-noisy (PN) signals [1, 2].
Single pseudo-noisy signals, such as Gold, Kasami, M-sequences and others are described in [4, 5] and are widely used in telecommunications. But before analysis of the mentioned signals requirements to measuring signals must be formulated.
There are two different measurement tasks met in practice:
From the point of view of measurement it possible to distinguish two different types of an environment [2]:
In order to perform accurate and reliable simultaneous measurements with a good spatial resolution in an environment with many reflectors, the measurement signals must fulfill the following requirements:
In order to fulfill the above formulated requirements investigation of well known PN sequences was carried out. For analysis four PN families of coded sequences were chosen:
These sequences are characterized by following parameters [2, 4]:
Autocorrelation function is given by
(1) |
where: N_{c}T_{c} = T_{s}, N_{c} is the number of elements of PN signal, T_{c} is the length of one element of the signal, T_{s} is the duration of the whole PN signal.
Cross correlation function:
(2) |
After analysis of the Gold, Short Kasami, Large Kasami and M - sequences the following conclusions were made:
After evaluation of the above listed conclusions it was made decision to make optimization of code sequences to find sequences that will match the all requirements for measurement signals. For that reason it five optimization criteria were chosen:
These criteria are given elsewhere [1, 3] and therefore they will not be discussed here. In Fig. 2 - 5 non optimized and optimized, using CO/MSQCC optimization criteria auto and cross correlation functions of Short Kasami PN code sequences are presented. As it can be seen from Fig. 2 5, during optimization of the code sequences reduction of the level of side lobes of an auto correlation function and reduced level of a cross correlation function between two code sequences were achieved.
Fig 2: Normalized auto correlation function of the non optimized Short Kasami sequence, the sequence length 63 elements. | |
Fig 3: Normalized auto correlation function of the optimized Short Kasami sequence, the sequence length 63 elements. | |
Fig 4: Normalized cross correlation function between the two non optimized Short Kasami sequences, the length of sequences is 63 elements. | |
Fig 5: Normalized cross correlation function between the two optimized Short Kasami sequences, the length of sequences is 63 elements. |
In Fig. 6 and 7 cross correlation function values in a set of the analyzed PN sequences are presented.
Fig 6: Normalized maximal cross correlation values between the optimized Gold sequences, all sequences are from the same family set: N is the sequence number, the length of sequences is 63 elements. | Fig 7: Normalized maximal cross correlation values between the optimized Short Kasami sequences, all sequences are from the same family set: N is the sequence number, the length of sequences is 63 elements. |
After investigation of the optimized PN code sequences the following conclusions were made:
During optimization of the PN code sequences quite good results were achieved, e.g. a level of side lobes of an auto correlation function and a level of a cross correlation function of PN code sequences were reduced.. After evaluation of these results it was made decision that in real a situation, during multi-channel measurements each measurement channel will receive measurement signals not only transmitted by this same channel, but also transmitted by other channels, what will create a correlated noise. After evaluation of influence of correlated noise to detection of PN code sequences in a correlated noise it was shown that depending on the length of a sequence it can be simultaneously used from 6 to 12 different PN code sequences. In order to perform high accuracy robust in such a situation it was decided to develop the digital signal processing algorithm.
In order to evaluate the possibility to extract the reference signal x_{atr}(t) from the sum S_{sum}(t) of received signals the calculation of the normalized cross correlation function, between the reference signal and the received signal and a correlated noise is performed:
(3) |
where: E_{xatr}S_{sum} is the energy of the reference signal and the sum of the received signals.
If during this calculation difference between central peak of cross correlation function is mach more bigger than the level of side lobes
then this reference signal can be extracted from a mixture of the signal an a correlated noise and it is not necessarily to make digital signal processing. Otherwise , the reference signal is impossible to extract from the signal and a correlated noise and it is necessarily to apply digital signal processing. This performed in the following way: calculation of the normalized cross correlation function between the signal of correlated noise and all expected PN sequences is performed:
(4) |
and the sequence with the highest max(R_{xztr}S_{sum}(i)) correlation value with the mixture of signal is found. If it is not the reference signal then ten new PN code sequences are calculated x_{j}(t)=x_{i}(t)(0.1n), where n= 1:10. After that calculation the normalized cross correlation function between signal of correlated noise and calculated PN sequences with different amplitudes is calculated:
(5) |
and the PN sequence with the highest correlation value with signal of correlated noise is found max(R_{xztr}S_{sum}(j)). After this calculation it is possible to eliminate this PN sequence from the signal of correlated noise:
(6) |
When elimination of the PN sequence from the signal of correlated noise is done the calculation of cross correlation between a new signal and the reference signal is performed:
(7) |
If during this calculation the difference between central peak of cross correlation function is much bigger than a level of side lobes
The proposed algorithm was checked using both simulated and real signals:
The simulation of the signals was performed in following way. After choosing parameters of PN sequences (family type and length of sequences N) for the digital signal processing, frequency of measurement signal f_{ns}, which can vary form 10kHz to 5MHz, and calculation of period of measurement frequency , calculation of the length of the sequences are made T_{ks} = T_{se}N, where T_{se} is the length of one element of the sequence, which can vary from 1T_{ns} to 20T_{ns}. After that the number of discrete samples is n = T_{ks}/T_{dis}, where T_{dis} = 1/f_{dis} and f_{dis} is sampling frequency, which can vary from 5f_{ns} to 100f_{ns}. Now discrete time scale is introduced t = (n-1)T_{dis} , where n_{dis}=1:n.
After the discrete time and all parameters of the PN sequence are calculated it is possible to generate a sequence:
(8) |
where: a is the amplitude of the signal, KSS_{i} - code sequence, i is the sequence number.
When the coded sequences are calculated, one sequence from the set of PN sequences is marked as a reference PN sequence x_{atr}(t) = x_{i}(t). After that the mixture of the received signals with various amplitudes correlated noise is calculated:
(9) |
where: a_{i} is the amplitude of the PN sequence, which can vary from 0.1 to 1, t is delay time, which can vary from 0 to 5T_{ks}.
Fig 8: Normalized cross correlation function between the mixture of PN signals and the reference signal before digital signal processing. Signal is calculated summing up 10 PN code sequences. | Fig 9: Normalized cross correlation function between mixture of PN signals and the reference signal after digital signal processing. Mixture of signals is calculated summing up 10 PN code sequences. |
Fig 10: Normalized auto correlation function of the reference signal. | Fig 11: The reference signal. |
The results for the first type of signals are presented in Fig. 813. In Fig. 8 the normalized cross correlation function between the mixture at signals, which was calculated summing 10 PN code sequences, and the reference signal, shown in Fig.11 is presented. As we can see from Fig.8 it is impossible to extract the reference signal from the sum of signal, because the central peak is much lover than the level of side lobes . In Fig.9 is shown the normalized cross correlation function between the received mixture of signal and the reference signal after digital signal processing is shown. As we can see now the reference signal can be extracted from the sum of different PN sequences. In this situation the algorithm was run 9 times to eliminate all 9 PN sequences from the received signal. It was done for evaluation of effectiveness of the presented algorithm. For this purpose in Fig.10 the normalized auto correlation function of the reference signal is presented. As we can see Fig.10 and Fig.9 are identical it means, that the presented algorithm is working properly. For more precise evaluation of the algorithm the signal which consists of 60 PN coded sequences was applied. The normalized cross correlation function between this signal and the reference signal before and after digital signal processing is presented correspondingly in Fig.12 and Fig.13.
Fig 12: Normalized cross correlation function between the mixture of 60 PN- signals and the reference signal before digital signal processing. | Fig 13: Normalized cross correlation function between the mixture of 60 PN- signals and the reference signal after digital signal processing. |
Fig 14: Normalized cross correlation function between the mixture of 10 PN- signals and the reference signal before digital signal processing. | Fig 15: Normalized cross correlation function between the mixture of 10 PN- signals and the reference signal after digital signal processing. |
For evaluation of performance of the algorithm in a real situation it the signal consisting of 10 PN code sequences, randomly shifted in the time domain and with random amplitudes was processed. The results of this analysis are presented in Fig.14 and Fig.15. As we can see from the results presented before digital signal analysis it is impossible to extract the reference signal from the mixture of signals because . After digital signal processing we have got a situation when . The achieved results are a little bit worse than in the first case. The main reason of such results is that the PN code sequences are not orthogonal, but quasi orthogonal, so during digital signal processing it is impossible to calculate exact amplitudes of PN sequences in the received signal and for this reason it is impossible to eliminate completely PN sequences from the mixture of the signal.
The last analysis of the algorithm was performed using signals received by the real multi-channel distance measurement system. The reference signal for this case is presented in Fig.16 and the normalized auto correlation function of the reference signal is presented in Fig.17. As we can see the reference signal is much more complicated than the ideal reference signal presented in Fig.11. That happens after bandwidth limitation and distortion of waveforms in ultrasonic transducers. On the other hand the normalized auto correlation function of the reference signal, is almost the same as in an ideal situation (Fig.17).
Fig 16: The reference signal. | Fig 17: Normalized auto correlation function of the reference signal. |
During this analysis the mixture of PN signals was calculated summing up four PN code sequences received by the multi-channel distance measurement system. The results before digital signal processing and after digital signal processing are presented in Fig. 18 and Fig. 19. As we can see from the presented figures after digital signal processing the level of central peak was increased and the level of side lobes was decreased, what indicates a good performance of the algorithm.
Fig 18: Normalized cross correlation function between the mixture of PN- signals and the reference signal before digital signal processing. | Fig 19: Normalized cross correlation function between the mixture of PN- signals and the reference signal after digital signal processing. |
Kvaziortogonaliuju signalu panaudojimas daugiakanaliams atstumo matavimams
Reziume
Kodines sekos daugiakanalese atstumo matavimo sistemose del ju pakankamai geru atskiriamumo savybiu leidia atlikti atstumo matavimus daugeliu krypciu vienu metu. Tai labai supaprastina daugiakanale atstumo matavimo sistema ir pagreitina matavimus. iose sistemose naudojamos kodines sekos, taikomi tokie kokybiniai kriterijai kaip siaura autokoreliacine funkcija, emas autokoreliacines funkcijos oniniu lapeliu lygis, tarpusavio koreliacijos funkcija artima nuliui, kiek imanoma trumpesnes sekos, ilaikant prie tai ivardytus reikalavimus ir kodiniu seku rinkinys ar eima susideda i daugelio seku, tenkinanciu ivardytus reikalavimus. Kodiniu seku analizei sudaryta kompiuterine programa, kuri leidia parinkti kodines sekas, atitinkancias ivardytus kokybinius kriterijus, ir analizuoti gautus seku kokybinius parametrus. Taciau, atlikus kodiniu seku optimizavima bei ianalizavus galimybe panaudoti ias kodines sekas realioje daugiakanaleje atstumo matavimo sistemoje, susidurta su problema, kai vienu metu galima panaudoti tiktai labai nedideli skaiciu iu kodiniu seku. Siekiant ispresti ia problema, buvo sudarytas skaitmeniniu signalu apdorojimo algoritmas, leidiantis kartais padidinti vienu metu naudojamu kodiniu seku skaiciu, kartu ir daugiakanales atstumo matavimo sistemos matavimo kanalu skaiciu.
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