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

CHINESE JOURNAL OF MECHANICAL ENGINEERING 88 Vol.20, No.4, 2007LI Ning SHI TielinSchool of Mechanical Scienceand Engineering,Huazhong University of Scienceand Technology,Wuhan 430074, ChinaESTIMATION 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 difficult 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 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. Key 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 considering 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. 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 be 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 proposed 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 comparing 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 the linear instantaneous combinations of sources, that is0)x(t) = As(t)where 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 with full-row rank and its column vectors are not proportional. The element s/,(t) of the source signal s(t) can be given by1 This project is supported by National Natural Science Foundation of China (No. 50675076). Received September 9, 20O6; received in revised form April 2,2007; accepted April 16, 2007J» (0 = 2A exP( J <°J) + 2X exp( j &Jt)(2)where a>M denotes a non-commor. jreqviency in sh(t), and non-common frequencies diffrr in vaiue from each other; a>ydenotes a common frequency of aH sources, and common frequencies 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*,(') = Zfl* 2A exP(Jo>«0 + Zc*v expOOYjLa A exp( jaj) + ££a*c*v exp(jd>,0 =*=l 1.1tel 1T.|I,ZaA exp(jav) + IX «P(Jfi>,')(3)where aih is an element of ith row and Ath column in AThus, there is the following modelx(t) = AM,5,(t) + AM2s2(t) = AM&(t) + Ds2(t) (5)where s,(t) is an N-dimensional (N = Nt + 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. The 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 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 by© 1994-2007 Chin Acade c Journal Electron c ublishing use. A rights res rtp /c k .nCHINESE JOURNAL OF MECHANICAL ENGINEERING 89 J?;(r) = £x,(f+r )*;(/) =jJm 7J0I Ll>A exp( jat + r) +>Xexp(ja>v(' + r)ixwi)(r-l s=lp=)fIX V|UVllWlVi9 CUV 11V111 U1V MU11V OVIUVV VI UUl, U1VU 1 Ml* IBUV9 CUVlimf |XXSai»ayf*M«exP(J°'«(' + r)-J£0')+ not re'ated t0 * me column vectors in A are not proportional,r_>" Iwi-ir-ii-ithe ratio of PSD in Fn H T> is not ennnl to that in F/i (W(6)ZZ2XAA exp( - coJ + )wv(t + r) +r»1 J> 1=1Z2£fl A<*# «p( j«k(f + r) - j<y) +W i-l pA V VI2Xrf* exp(jdJv(t + f) - &/) dtnl p*lNote that the non-common frequencies 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) isR;t) = Lim - f I i£a*fli exp(jowr) + '-*° 1 v k- 1-1IX*",exp(jtt>r) W = 22fl*BA x exp( ioiT) + £«/,</ exp(jtt>vr) And the power spectral density P(a>) isP'jico) - 2itj2 .RJ(r)exv(-ja>r)dr =From Eq. (13), it can be found that, whether the non-common frequencies are from the same source or not, their PSD ratios areNow suppose that a>v is a non-common frequency of another source s(0- According to Eqs. (9)(12), the ratio between Pu("°,) (i'=lm)and Pa>) ate»visthe ratio of PSD in Eq. (12) is not equal to that in Eq. (13).(14)Now, for all m observation signals, if there are N non-common frequencies, using Eq. (13), the PSD matrix Pi at the N non-common frequencies can be obtainedP'M)PiM)*;,(«»)p;m)p;m)pN)>,)pimjpti(fl>)KM)p;mK)p;m)p;m)PU(o>)p;m)p;m)K)*(«.> (*.)p;,(<°2)1l(»»)J' 111ujK)VM) lAl(»W)-W®.)W»2) " ujK)(8)/W) uji) " W,2.2 PSD matrix at the non-common frequencyAs is known, in Eq. (13), (i(a>)(Jk = l,2,-,m) is not relatedSuppose wq is a non-common frequency of source st), then to k, and the m values of Afcu(a») is equal. But in fact, there arepx, )-2na a b2(9) alwavs some no*se influence and calculation errors when sam-'' '* * "pling and processing signals. As a result, usually the m values ofFor simplicity, the special case with i,j=, 2 is firstly considered, 4u,i(to) is not absolutely equal and must be replaced by the filterand we define two ratios at to,value Xu(m) which could be the mean of 4ui(°>) or the mean1 to)K)_2*<yV£_a2P*mK -' /*(«,) 2natpapb2 alpafter removing maximum and minimum, and so on. Hence, Eq.(10)= -*(ID/>2>,) 2na2palpb2n alpP =(15)From Eqs. (9)(11)(14) can be simplified as follows1 11V,) .,K) :K)KM) iK) KMn)= -2t(12)2.lK) =jya(«t)_2« a» (<*>,) 2napbM a<P/».i(*».) K) " -Ma*),Suppose that in the TV non-common frequencies, oi-oj*, arewhere, *-lm. Eq. (12) shows that the ratio between /£(») frequency components of *,(/), and «»,-«», are of those 52(t),the rest may be deduced by analogy, w/,l+lcoi,n are the frequencyand '(to) at any non-common frequency to, is always aap and not related to k or any others. Namely, the m values of Akz,a)q) is equal.components of s(f). Eq. (IS) can be rewritten as =(*.! P» - PJ© 1 4 2007 Chin cade c ourna E ctron ub ishing se.iserv d./ w k* 90 *LI Ning, et al: Estimation of the number of correlated sources with common frequencies based on power spectral density(16)(18)>/«)KM*)>,«)«-,KM*.,«) KM*)».-/i = l,2,"-,n =0*0 fl».=a»vFrom 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 column 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 frequencies 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.13 PSD matrix at the common frequencySuppose <ba is a common frequency in all sources, the powerspectral density PJ(a>) is/?(».) = 2*dJja(17)As in section 2.2, the ratio between /£(©) k, i-lm) andEq. (18) shows that the ratio between/>4*(<5(i)andP,'(<»(,)at any common frequency 5>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 frequency can be expressed as follows1 1l*iM) KM) KMv),(<y,) KM) K(&y)KM,) KM) KMv) i i l d»*n4,<*»dndw d,v(19)dml dm2 Vi dl2d* dFrom 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 zero, 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 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 obtained0=*,=tf, PoCwfinon frequencyJ|(0du1 1d* da,.(20)0,.Finally, from Ru the basic principle for estimating the number cf correlated 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 different 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 are 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 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 number 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 simulation 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 conditions'61. It must be pointed out that because of the calculation error, sampling error, the influence of noise, and so on, if the relative error of two or more column vectors in simulation© 19 0 7 hi a cic Jou n 1 ctrh gh / w ki tCHINESE JOURNAL OF MECHANICAL ENGINEERING91 (21)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 the equal.Experiment 1: This is a simulation experiment free of noise, and all sources have only one common frequency. The details are as followsj, (/) = 1.23sin(2jr90/ + 5) +1.46sin(2*60.) s2(t) = 1.68sin(27t40/ + 23) +1.75sin(27t800 +2.13sin(27tl20f + 12) j3(0 = 1.643sin(2rc30r + 12)l + 0.94sin(2*40/) s,(t) = 1.83sin27il00r + .12sin(27c30t)The mixing matrix A is(1.250 1.73 0 2.04 1.86 1.10 1.680.932.08And the power spectrums of sources and mixing signals are respectively given in Fig. 1 and Fig. 2.200010000g 4000r_lHIll . Ill-3- 2 000-« 01 i I i I iI 2000 3 1000I 400 200-ul_iIIlJ1IIJ' » I1_LI II s Iiiii20 40 60 80 100 120 140160 180 200 Frequency /HzU60004000E 2 000f °u 10000I 5000I0lOOOOp5 000-0-Fig. 1 Power spectmn; of Ac so.irces jhli 111' i i >i i i i_i_L_iiiiIAl.uJIII20 40 60 80 100 120 140160 180 200