ESTIMATION OF THE NUMBER OF CORRELATED SOURCES WITH COMMON FREQUENCIES BASED ON POWER SPECTRAL DENSITY (2).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 m 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 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(jo0 + 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 lla exp( jat + r) +xexp(jav( + r)ixwi)(r-l s=lp=)fix v|uvllwlvi9 cuv 11v111 u1v mu11v oviuvv vi uul, u1vu 1 ml* ibuv9 cuvlimf |xxsaiayf*mexp(j( + r)-j0)+ not reated 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) +r1 j 1=1z2fl a*# p( jk(f + r) - jr) w = 22fl*ba x exp( ioit) + /,vr) and the power spectral density p(a) ispjico) - 2itj2 .rj(r)exv(-jar)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 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).(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 obtainedpm)pim)*;,()p;m)p;m)pn),)pimjpti(fl)km)p;mk)p;m)p;m)pu(o)p;m)p;m)k)*(. (*.)p;,()(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*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/)km*),)-,km*.,) km*).-/i = l,2,-,n =0*0 fl.=avfrom 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.(1) 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.(1) 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 ) 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,(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,i i i i_i_l_iiiiial.ujiii20 40 60 80 100 120 140160 180 200 frequency /hz fig. 2 power spectrum of the mixing signalsfig. 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 source; 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 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. 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-bound of the number of sources is5. the true value is inside of the bound.table 1 ratio of psd at different frequenciesratiofrequency/7hz1030406070*2,i(u)0.6480.620001.6221.0213,|()00000.864003.463ratiofrequency j7hz80901001201302,l(fi)001.607000000*3,i(a)00000000*u()00.622000kl()000000a2j(b)0.889001.7750.8951.630note: a(,/) 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 can be seen that there are 3 groups of column vectors that can be regarded as being close, and the 3 groups are