ESTIMATION OF THE NUMBER OF CORRELATED SOURCES WITH COMMON FREQUENCIES BASED ON POWER SPECTRAL DENSITY

1、estimation of the number of correlated sources with common frequencies based on power spectral density*abstract: blind source separation and estimation of the number of sources usually demand that the number of sensors should be greater than or equal to that of the sources, which, however, is very d

2、ifficult to satisfy for the complex systems. a new estimating method based on power spectral density (psd) is presented. when the relation between the number of sensors and that of sources is unknown, the psd matrix is first obtained by the ratio of psd of the observation signals, and then the bound

3、 of the number of correlated sources with common frequencies can be estimated by comparing every column vector of psd matrix. the effectiveness of the proposed method is verified by theoretical analysis and experiments, and the influence of noise on the estimation of number of source is simulated. k

4、ey words: blind signal number of sources power spectral density0introductionblind source separation (bss), a new signal processing method, consists of recovering unobserved sources from several observed mixture signals. though bss can be solved using various algorithms, it must be compensated by con

5、sidering some special assumptions on source signals or mixing system due to the lack of priori knowledge on the sources. one of the main assumptions is that the number of sensors must be greater than or equal to that of the sources, which is very difficult to satisfy f:ir the unknown blind sources.

6、for this reason, estimating the /itunber of sources becomes an important reseach topic.at present the estimating methods foi (he number of sources are mainly based on principal component analysis (pca) and singular value decomposition (svd)m1. in these methods, the number of sources is expected to b

7、e equal to the number of non-zero eigenvalues or that of non-zero singular values. that is to say, the number of sensors must be greater than or equal to that of the sources, which is as difficult as bss to satisfy.in this paper, a new estimating method based on power spectral density (psd) is propo

8、sed without imposing any special requirements on the signals or mixing matrix. when the relation between the number of sensors and that of sources is unknown (either greater than or smaller than or equal to), the bound of the number of correlated sources with common frequency can be estimated by com

9、paring the column vector of the psd matrix consisting of the ratio of psd of the observation signals. the simulation and an example of a pump assembly at normal and fault conditions are used to illustrate the method.1problem formulationin bss, when the noise is not considered, the classical model is

10、 the linear instantaneous combinations of sources, that iswhere x(t) is an m-dimensional vector, s(t) is an n-dimensional unknown original source vector, a is an mx dimensional unknown mixing matrix. when m is greater than or equal to , a is with full-column rank, and when m is smaller than n, a is

11、with full-row rank and its column vectors are not proportional. the element s/,(t) of the source signal s(t) can be given bywhere am denotes a non-commor. jreqviency in sh(t), and non-common frequencies diffrr in vaiue from each other; aydenotes a common frequency of ah sources, and common frequenci

12、es also differ in value frc-. each other; 6*, ckv respectively denote coefficient: of non-common and common frequency in sjt); a denotes the number of non-common frequencies in .*(/); kdopjtts the number of all common frequencies. so, xfc) can be expressed by where s,(t) is an n-dimensional (n = nt

13、+ n2 +- + nn) sine function vector corresponding to non-common frequencies; m is an nxn dimensional coefficient matrix; s2(t) is a -dimensional sine function vector corresponding to common frequencies; m2 is an nx.v dimensional coefficient matrix; d=av-mi is an m%v dimensional coefficient matrix. th

14、e column vectors in d are not proportional to each other and they are not proportional to the column vectors of a either, for the convenient estimation of the number of sources.formerly, when the number of sensors is smaller than that of sources in real signals, it is extremely difficult to estimate

15、 the bound of the number of sources, and this is the problem to be solved in the paper.2 estimating the number of correlated sources with common frequencies2.1 power spectral densityconsidering eq. (3), the cross correlation function /?*() of x,t) and xjt) is given bynote that the non-common frequen

16、cies are not equal to the common, and non-common frequencies are not equal to each other, which is also true for the common frequencies. so, in eq. (6), only when a=r, k=s, and v=p, the first and fourth elements are not equal to zero while the others are zero, and therefore the simplified eq. (6) is

17、2.2 psd matrix at the non-common frequencynow suppose that av is a non-common frequency of another source s(0- according to eqs. (9)(12), the ratio between pu(,) (i=lm)and pa) atevisthe ratio of psd in eq. (12) is not equal to that in eq. (13).now, for all m observation signals, if there are n non-c

18、ommon frequencies, using eq. (13), the psd matrix pi at the n non-common frequencies can be obtainedfrom eq. (16), it can be seen as follows.(1) in matrix pu every element has only relation to mixing matrix a, not to any others.(2) in matrix p, if the denominator of every element is not zero, the co

19、lumn vectors which the non-common frequencies coming from the same source correspond to are equal.(3) because the column vectors of mixing matrix a are not proportional to each other, and suppose the denominator of every element in eq. (16) is not zero, the column vectors which the non-common freque

20、ncies coming from different sources correspond to are not equal.similarly, the other psd matrix p2, p3,-, pm at nnon-common frequencies which are similar to p can also be obtained, here are not described.2.3 psd matrix at the common frequencysuppose ) is as in section 2.2, the ratio between /() k, i

21、-lm) and eq. (18) shows that the ratio between/4*(5(i)andp,(q is djdt, which is related only to the matrix d in eq. (5), not to it or any others.now, if there are v common frequency components in m observation signals, similar to the psd matrix at non-common frequency, the psd matrix ft at common fr

22、equency can be expressed as followsfrom eq.(19), it can be found as follows.(1) in matrix ft, every element is related only to the matrix d, not to any others.(2) because the column vectors of matrix d are not proportional to each other, suppose the denominator of every element in eq. (19) is not ze

23、ro, the column vectors which different common frequencies correspond to are then not equal.similarly, the other psd matrix ft, ft, , ft, at vcommon frequencies which are similar to ft can also be obtained, and is not fully described here.2.4 estimating methodsuppose that there are k frequency pulses

24、 in all m observation signals, k=n+v, and for avoiding estimation error, the very small values are not considered. using eq. (16) and eq. (19), the psd matrix hi at k frequency pulses which is composed of p and ft can be obtainedfinally, from the basic principle for estimating the number cf correlat

25、e sources with the common frequency can be summarized as follows.(1) if the denominator of every element in j?i is not zero, the column vectors which the non-common frequencies coming from the same source correspond to are equal and the column vectors which the non-common frequencies coming from dif

26、ferent sources correspond to are not equal.(2) if the denominator of every element in hi is not zero, the column vectors which the different common frequencies correspond to are not equal. and the column vectors which the non-common and common frequencies coming from the same source correspond to ar

27、e not equal.(3) in rit if two or more column vectors are equal, the frequencies they correspond to must be non-common and come from the same source. accordingly, the number of the determinate sources can be obtained and taken as lower-bound. for the column vectors where there are not adjacent column

28、 vectors, the frequencies they correspond to may be common or non-common with only one frequency. here, these frequencies are all taken as non-common frequencies with only one frequency. added by the lower-bound, the upper-bound can be obtained.(4) when there are the zero-denominators in r, the numb

29、er of sources cannot be estimated by r. through calculating other psd matrix such as r2, rj, , r (similar to rt), the number of sources can be estimated by the method given above.3 experimental resultsin this section, the performance of the proposed method is first investigated via computer simulati

30、on and the influence of noise on the estimation of number of sources is also simulated. then, the method is further investigated via the data of a pump assembly (including motor, gear box and pump) at normal and fault conditions61. it must be pointed out that because of the calculation error, sampli

31、ng error, the influence of noise, and so on, if the relative error of two or more column vectors in simulation experiment is less than 10%, they are taken to be equal, and if the relative error of two or more column vectors in the real observation signals is less than 20%, they are also taken to be

32、the equal.experiment 1: this is a simulation experiment free of noise, and all sources have only one common frequency. the details are as followsfig. 1 shows that 60 hz and 90 hz are from the same source; 40 hz, 80 hz and 120 hz are from the same source; 10 hz, 30 hz and 70 hz are from the same sour

33、ce; 70 hz, 100 hz and 130 hz are from the same source. therefore, 70 hz is a common frequency while the others are non-common frequencies. in fig. 2, there are 10 frequency pulses. according to the method proposed in the paper, the ratios of psd at every frequency pulse are given in table 1.in table

34、 1, the value greater than 20 is written as a, and the value smaller than 0.01 is written as 0. comparing every column vector, it is obvious that the vectors 10 hz and 30 hz correspond to are equal; the vectors 40 hz, 80 hz and 120 hz, 60 hz and 90 hz, 100 hz and 130 hz correspond to are equal, too.

35、 so, the lower-bound of the number of sources is 4. in table 1, the vector not close to any others is the one 70 hz corresponds to. it may be common frequency or non-common with only one frequency. here, it is regarded as the non-common with only one frequency. now, it is certain that the upper-boun

36、d of the number of sources isexperiment 2: adding the noises to the experiment 1, the mixing signals will be obtained where n(t) is the white noise whose mean is zero and whose variance is 1; a is the weighting coefficient of noise. the snr is defined as where n, r, are the numbers of sources and th

37、e mixing signals respectively. the tsiimating result at different snr is presented in table 2.table 2 shows the degree of noises influence on the estimation. it is obvious that the scope of the bound is getting bigger and bigger with the decrease of snr. however, the true value is inside this bound.

38、experiment 3: this is a laboratory experiment of the pump assembly (including motor, gear box and pump). the gear box is measured synchronously with the 4 vibration measurements at the same direction. the sampling frequency is 12 800 hz, and the data length of every channel is 77 824. for demonstrat

39、ing the validity of the proposed method, the number of sources is estimated under two different conditions of gear box. the 4 power spectrums for the gear box at normal condition are shown in fig. 3.in fig. 3, the 11 frequency pulses can be easily picked out. as is known, in theory, for exactly esti

40、mating the number of sources, all the ratios of psd should be calculated. yet because a(,j(g) is the reciprocal value of a/,) which calls for but repetitive calculation for the estimating result, in table 3 only either of af,/o) and i(w) is considered.in table 3, comparing every column vector, it ca

41、n be seen that there are 3 groups of column vectors that can be regarded as being close, and the 3 groups are corresponded to by the following frequenciesaccording to the proposed method in the paper, the lower-bound of the number of sources is 3. in table 3, the vectors not close to any others are

42、those 860 hz, 1 295 hz and 1 860 hz correspond to. they may be common frequencies or non-common frequencies with only one frequency. but, it is certain that the upper-bound of the number of sources is 6. at fault condition (the progressed pitting in both gears) are shown in fig. 4, and ratios of psd at 17 frequency pulses are given in table 4.in table 4, similarly by comparing eve