Partial source separation from unknown correlation mixture for eliminating unknown periodic disturbances from random measured signals

Separating and eliminating periodic disturbances from measured signals are a key problem to obtain original responses used for further system identification and evaluation. Actual periodic disturbances are partial unknown sources in measured signals and have certain correlation with random noise sources in time domain. In this paper, a separation problem on partial unknown sources such as periodic sources correlated with random noises is introduced. A partial unknown source separation technique is proposed by combining signal eigenspace transformation, covariance joint diagonalization and decorrelation of correlation sources. The partial source separation procedure has two main stages: obtain uncorrelated sources by eigenspace transformation and joint diagonalization; and obtain partial periodic sources correlated with random noises from the uncorrelated sources by decorrelation. The proposed partial source separation technique is supported by several theorems. Under given assumptions, the separation technique will result in accurate partial sources. The separation technique has main features such as partial unknown sources separated from measured signals, separated periodic sources correlated with random noise sources, and being suitable for dominant random noises and non-dominant periodic disturbance sources in measured signals. Numerical results are presented to illustrate the effectiveness of the separation technique.


Introduction
Structural health monitoring or damage diagnosis is an important problem in engineering. Using structural dynamic responses for identifying modal feature variation is a common way to solve the problem [1][2][3][4]. A complex large-scale structure system in complicated dynamic surroundings has undeterminable model with excitation and only responses are usable. Disturbed or distorted response signals used will result in unreliable feature identification. For example, vibration response signals measured from a maglev train are inevitably disturbed by intensive electromagnetic fields which provide power and support. The electromagnetic interference is generally periodic with high frequency. Hence, separating and eliminating the periodic disturbances from the measured signals are a key problem to obtain original responses used for further identification. The periodic disturbances are partial unknown sources in the measured signals. The response signals commonly contain strong random noise sources which have certain correlation with deterministic sources in time domain. Therefore, that is a separation problem on partial unknown sources where there is correlation between noises and determinacy.
Conventional blind source separation problem has been studied and several separation techniques were presented as well as image recognition [5][6][7][8][9][10][11][12][13]. For example, the blind source separation technique using signal second-order statistics can effectively extract simultaneous uncorrelated sources and mixture [7]. The separation procedure is composed of the singular value decomposition of signal covariance matrix and the joint diagonalization of transformed covariance matrices with various time lags. In this technique, it is assumed that all sources are fully uncorrelated and the intensity of white noises is relatively small for removal. The blind source separation technique has been applied to structural mode identification and anomaly detection based on extracted mixture by using response second-order statistics [14][15][16][17][18][19][20]. In those researches, all sources were separated except for small noises removed and particularly the sources were assumed as fully uncorrelated. However, actual structural vibration responses commonly contain strong random noises which characterize system dynamics. The random noises have certain correlation with deterministic sources such as periodic in time domain. These sources in measured response signals are dependent or correlative. Thus, the uncorrelated sources obtained by the blind source separation technique will be a combination of the correlated sources. That is each uncorrelated source may contain certain components of the correlated sources. A further decorrelation procedure is necessary, but all correlated sources are difficult to obtain without other conditions. Nevertheless, the separation of partial unknown correlated sources is possible, where deterministic sources such as periodic are uncorrelated and they have certain correlation with random noise sources. The periodic sources are expected to separate and eliminate from the measured signals.
In the present paper, a separation problem on partial unknown sources is introduced in which random noises have certain correlation with deterministic such as periodic sources. A new partial unknown source separation technique is proposed by combining signal eigenspace transformation, covariance joint diagonalization and decorrelation of correlation sources. The partial source separation procedure has two main stages: first, obtain uncorrelated sources by eigenspace transformation and joint diagonalization; and second, obtain partial periodic sources correlated with random noises from the uncorrelated sources by decorrelation. The proposed partial source separation technique is supported by several theorems. In section 2, the partial separation problem is described and basic assumptions are presented. In section 3, analytical expressions on signal eigenspace transformation and covariance joint diagonalization are presented. Several theorems are proved to support the expressions. A condition on the number of time lags in the joint diagonalization is explained. Section 4 proposes the decorrelation analysis including separating periodic disturbance sources correlated with random noises and extracting corresponding mixture matrix. The original response signals are recovered by eliminating the periodic disturbances from the measured signals based on relevant theorems. The effect of estimated correlation functions between periodic disturbances and random noises on the separation results is analyzed. The indices of relative differences and similarity for separation results are presented. Numerical results illustrate the effectiveness of the proposed separation technique presented in section 5.

Problem description
2.1. Partial source separation Consider measured signal x i (t) (i = 1, 2,K, n), which is a mixture of system vibration response source e sj (t) ( j = 1, 2,K, N s ) and periodic disturbance source e pj (t) ( j = 1, 2,K, N p ) as follows: where t is time and , ..., , , , ...,  Signal X contains generally periodic w j (t), oscillatorily decayed w k (t) and random noise w l (t). Denote a correlation function in time domain by where τ is a time lag and T R is time length. If a process has zero mean, its correlation function is equal to covariance. For an ergodic random process, its statistics in time domain will tend towards those over event space. The auto-correlation functions of the periodic, decayed and zero-mean random processes can be obtained respectively as where a j and ω j are amplitude and frequency, respectively, and σ l 2 is covariance. For a pure random process, its correlation time is equal to zero and hence, σ l = 0 for τ ≠ 0. If the periodic w j (t) and decayed w k (t) have different frequencies and the random w l (t) is uncorrelated with the others, the cross-correlation functions obtained are As the system response can contain periodic and non-periodic sources, N s + N p sources are divided into N I periodic and N J non-periodic uncorrelated sources (N s + N p = N I +N J =N). The number n of signals is determined by number of sensors in measurement. The number N s of response sources is corresponding to number of excitation sources and N p is number of other disturbance sources such as measurement noises and electromagnetic interference. As the analysis with equations (3)-(8), two source processes are uncorrelated if their correlation function in time domain (3) is equal to zero. For example, periodic sources with different frequencies and infinite time lengths are uncorrelated. N is corresponding to number of the uncorrelated sources. However, the sources and mixed signals used have actually finite time lengths with non-ideal factors and thus the sources have certain correlation. To separate sources from measured signals, many sensors are used and then the number n of signals is usually not smaller than the number N of sources.

Assumptions
For an actual random process, it has generally certain correlation with deterministic processes in time domain. Their cross-correlation functions are unequal to zero. However, a partial source separation procedure can be two stages: first, to obtain uncorrelated sources E j (t), and second, to obtain partial periodic sources F j (t) correlated with random noises. It is assumed for the first stage that: (I) all sources are uncorrelated with each other based on equations (7) and (8); (II) disturbances to system response are periodic sources; (III) number n of measurement signals is not smaller than number N of uncorrelated sources; (IV) mixture matrix in equation (1) has full column rank. For the second stage, the correlation between random and periodic sources will be considered to replace the first assumption. The assumptions are based on many practical problems including dynamic modelling under multiple periodic excitations with random noises and signal processing such as electromagnetic interference elimination of random vibration signals measured from maglev trains.

Signal eigenspace transformation and covariance joint diagonalization
The blind source separation technique using signal second-order statistics has been proposed. However, as a part of the partial source separation procedure, different expressions on signal eigenspace transformation and covariance joint diagonalization based on the assumptions (I)-(IV) are given in the following.

Signal eigenspace transformation
For time lag τ = 0, the correlation function matrix of measured signals is expressed as where subscripts I and J denote periodic and non-periodic sources, respectively, and , , 10 in which N is number of sources, N I and N J are numbers of periodic and non-periodic sources, respectively. By singular value decomposition, the correlation function matrix is expressed as where U o is unitary matrix and Λ o is diagonal matrix with eigenvalues. The following proposition can be inferred.
Proposition 1. Based on the assumptions (I)-(IV ), letting Λ o be an eigenvalue matrix defined by singular value decomposition of the correlation function matrix R XX (0) (11) and R EE (0) be the correlation function matrix of sources E(t), then the rank of Λ o is equal to the rank of R EE (0).
Proof. By combining equations (9) with (11), the correlation function of X is U o is a non-singular unitary matrix. Then the rank of R XX (0) is equal to that of Λ o . The mixture matrix H has full column rank. Then the rank of R XX (0) is equal to that of R EE (0). Therefore, the rank of Λ o is equal to the rank of R EE (0). From proposition 1, the number of non-zero elements or eigenvalues in Λ o is equal to that in R EE (0), which is N I +N J .
Rearrange Λ o and U o as where subscripts Z and A denote zero and non-zero parts of eigenvalues in Λ o . The singular value decomposition (11) becomes where E n (t) is normalized sources, H n is corresponding mixture matrix and Introduce a transformation matrix Its generalized inverse is denoted by T # . There is the following proposition.
Proposition 2. Let Λ A be non-zero eigenvalue diagonal matrix defined by singular value decomposition of the correlation function matrix R XX (0), U A be corresponding part of unitary matrix U o , R E n E n (0) be diagonal correlation function matrix of normalized sources E n and H n be corresponding mixture matrix. The transformation matrix T is defined by equation (17). Then T # H n is a unitary matrix, i.e., (I N is identity matrix) where superscript # denotes generalized inverse. Based on equations (12)- (17), T # H n is a square matrix. By using equations (16), (12) and (14),

T H T H T H H T T HR H T T R T T U U T T TT T I
Matrix T can be used to transform signal X into its eigenspace, but sources are not separated yet. Mixture matrix H n cannot be determined only by the unitary matrix T # H n . It is necessary to use properties of the correlation function matrix R XX (τ) with time lag τ.

Covariance joint diagonalization
The measured signal X are firstly transformed into its eigenspace as For a time lag τ > 0, the following proposition and theorems can be inferred.
Proposition 3. Based on the assumptions (I)-(IV), letting R ZZ (τ) be the correlation function matrix of transformed signal Z (21), then R ZZ (τ) has the singular value decomposition with unitary matrix T # H n , where H n is the mixture matrix (16) corresponding to normalized sources E n (t).
Proof. Based on equation (21), the correlation function of Z is As given by expression (10), the correlation function of E n is From proposition 2, T # H n is a unitary matrix. Thus, the right side of equation (22) is the singular value decomposition of R ZZ (τ) with unitary matrix T # H n . The eigenvalue of the correlation function matrix R ZZ (τ) generally varies with time lag τ, and it is 1 for τ = 0. Based on proposition 3, T # H n can be determined by the condition of unitary matrix for all time lags. However, T # H n is invariant to various time lags while R ZZ (τ) varies with time lag. It may result in numerical indeterminacy that the unitary matrix is calculated directly by all singular value decomposition of R ZZ (τ) with various time lags. The joint diagonalization of R ZZ (τ) with various τ is an effective approach to determine T # H n .
The joint diagonalization of R ZZ (τ) is to find a unitary matrix V which makes VR ZZ (τ)V T diagonal or nondiagonal elements of VR ZZ (τ)V T close to zero. It can be implemented by minimizing the absolute nondiagonal elements of VR ZZ The joint diagonalization with infinite various time lags is unpractical, and finite time lags τ = τ i , i = 1, 2,K, N d are considered. Under certain conditions, the joint diagonalization with adequate time lags is sufficient for determining the unitary matrix V. The diagonalization of R ZZ (τ) is expressed as be defined by the joint diagonalization (25) and R E n E n (τ i ) be the diagonal correlation function matrix of normalized sources E n (t). The two diagonal matrices have corresponding diagonal elements otherwise only the row order of the unitary matrix is adjusted. Then the two diagonal matrices are identical, i.e.,   As the mixture matrix H n has been extracted as equation (39) by the signal eigenspace transformation and covariance joint diagonalization, the corresponding normalized sources can be separated from the signal X using the generalized inverse of H n , but the separated sources are fully uncorrelated as the assumption (I). However, actual random noises have certain correlation with deterministic or periodic sources and therefore, the expected original sources correlative are not the separated uncorrelated sources. The extracted mixture and separated sources are unexpected practical the original and further decorrelation analysis is necessary.

Decorrelation analysis of correlation sources and partial unknown source separation
In the present stage, the correlation between random and periodic sources is considered to replace the assumption (I). The mixture matrix H n (39) is just for the uncorrelated separated sources E n (t). However, the expected original sources denoted by F(t) are correlative between random and deterministic sources. Then the uncorrelated sources E n (t) are a combination of the original sources F(t), i.e., E n (t) = BF(t), where B is a combination coefficient matrix. The B will be estimated by the following decorrelation.

Decorrelation analysis
Based on equation (1) and disturbance E pn (t) chosen by practical problem, the uncorrelated separated sources are expressed as where subscripts s and p denote response sources and periodic disturbance sources, respectively, and Each uncorrelated separated source in E n (t) has been normalized and then is suitable to further determine the original sources F(t). To obtain periodic disturbance sources F p (t), the uncorrelation condition is used.
The periodic disturbance sources with different frequencies in F p (t) are uncorrelated with each other, but they have certain correlation with random noise sources in F s (t). Let the correlation function matrix be Then the uncorrelation condition is expressed as where A E is a diagonal matrix representing periodic source amplitudes with Introduce the following functions with respect to frequency ω where y i (t) is the ith element of E n (t). The amplitude and initial phase of the periodic sources F pj (t) are determined by where a j is the jth diagonal element of A E and B p,ij is the (i × j)th element of B p . By maximization of a j (47), the frequency ω j of the periodic sources is obtained. The amplitude a j of the periodic sources is calculated by expression (45). The combination coefficient B p,ij is calculated by expression (47) and amended by expression (44) with the correlation function (42). In general, the correlation function can be determined by prior statistics in the frequency band around ω j or by reference signals without the periodic disturbance with frequency ω j . Then the combination coefficient matrix B p with H n can be used for separating the periodic disturbance sources F p (t) from E n (t) or X(t).
From theorem 3, F p (t) is the expected original periodic disturbance sources which have certain correlation with random noises, and the corresponding mixture matrix is H n B p . By eliminating the periodic disturbances from the measured signals, the original response signals are recovered as Partial source separation procedure Based on the above analysis and results on signal eigenspace transformation, covariance joint diagonalization and decorrelation of correlation sources, a partial unknown source separation procedure for eliminating periodic disturbances is proposed as follows: (I) calculate the correlation function matrix R XX (0) of signal X(t) and the singular value decomposition of R XX (0) to determine the transformation matrix T; (II) transform signal X(t) into its eigenspace as Z(t), and implement the joint diagonalization of R ZZ (τ) with various time lags to determine the unitary matrix V or mixture matrix H n = TV T for the uncorrelated sources E n (t); (III) transform signal X(t) into E n (t), make its maximization analysis in frequency domain to determine the periodic disturbances F p (t), and estimate the combination coefficient matrix B p by decorrelation; (IV) eliminate the periodic disturbances from the measured signals to obtain the recovered response signals as correlated with the disturbances. In the above procedure, the separation calculation is mainly singular value decomposition, joint diagonalization and decorrelation. For data statistics, only the correlation function matrices R XX and R ZZ in time domain are calculated and the other is basic matrix operation. The statistics and estimation based on probability densities are avoided. As the number of signals increases, the dimension of matrices increases and the calculation time in the singular value decomposition will increase accordingly. However, the calculation time in the joint diagonalization and decorrelation will mainly depend on the number of sources. Then the calculation time increases nonlinearly with the size of separation problem and the number of signals used is determined mainly by practical problems and also constrained by the algorithm such as singular value decomposition.
The random noises may have a large intensity and certain correlation with the periodic disturbances. Under the given calculation accuracy, errors may be from the smaller signal time length, lower sampling frequency and inaccurate estimation of the correlation function between the periodic disturbances and random noises. Based on equation (40), the ith uncorrelated separated source is expressed as where subscripts I and J denote periodic and non-periodic (or random noise) sources, respectively. E Ij and E Jk are uncorrelated sources, and then for the periodic source F pj correlated with non-periodic sources, the mixture H I,ij of the corresponding periodic source E Ij contains a part å a k ijk which belongs to the non-periodic sources. The equation for coefficient α ijk and the correlation function between the periodic (E I ) and non-periodic (E J ) sources can be derived from equation (51) as When the recovered signals are equal to the original signals, the relative difference ε i = 0 and similarity γ i = 1. Then smaller ε i and γ i closer to 1 indicate that the recovered signals are more accurate.

Numerical example
To examine the proposed partial unknown source separation technique, consider five response signals (x i , i = 1, 2,K, 5) as shown in figures 1(a)-5(a) which are used for separating and eliminating periodic disturbances. The sampling frequency is 500 Hz and time length is 4s. The signals are the mixture of system responses and disturbances, where the responses include a random noise (e s1 ) and an oscillatorily decayed source (e s2 ) and the disturbances have two periodic sources (e p1 , e p2 ) with different frequencies (100 Hz, 120 Hz) (number of periodic sources N I =2 and number of non-periodic sources N J =2). The mixture matrix corresponding to normalized sources (e p1 , e p2 , e s2 , e s1 ) is It is seen from equation (56) that the non-diagonal elements of cross-correlation are small compared with the diagonal elements of auto-correlation, but the random noise has certain correlation with periodic disturbances. The random noise has the absolute correlation function with time lag smaller than 0.06 and then it has a little correlation time or wide-band spectrum. By the singular value decomposition of the correlation function matrix R XX (0) of the signals, the eigenspace transformation using T and the joint diagonalization of the correlation function matrix R ZZ (τ) of the transformed signals, the normalized uncorrelated sources E n (t) and corresponding mixture matrix H n are obtained. The extracted mixture matrix is The total error (ε i ) of the response signals by eliminating periodic disturbances is small and however, the response signals in time domain have certain difference from the original. Further by the decorrelation with the maximization analysis of the normalized uncorrelated sources E n (t) in frequency domain and the estimation of the combination coefficient matrix B p , the periodic disturbances F p (t) correlated with the random noise and the corresponding mixture matrix H n B p are obtained. The mixture matrix is The mixture matrix (59) has a better accuracy than that of equation (57) by comparing with the first two columns of equation (55) for the periodic disturbances correlated with the random noise. Based on the results, the two As inaccurate estimation of the correlation function between periodic disturbances and random noises will result in certain errors of the separation, for example, extracted mixture matrix, the effect of the estimation errors on the mixture matrix is discussed based on equations (51) and (52). Figure 6 shows the relative differences (ε hep ) of extracted and original mixture coefficients corresponding to the periodic source (e p1 ) in signals (x i ) versus the relative difference (ε Rpr ) of estimated and original correlation functions between the periodic source and random noise. When the relative difference of the correlation function is smaller than 3%, it has very slight effect on the extracted mixture coefficients. When the relative difference of the correlation function is smaller than 20%, the relative differences of the extracted mixture coefficients are smaller than 4.5%. Thus the proposed separation technique is insensitive to the estimation errors of the correlation function between the periodic source and random noise.

Conclusion
Separating and eliminating periodic disturbances from measured signals are a key problem to obtain original responses used for further system identification and evaluation. Actual periodic disturbances are partial unknown sources in measured signals and have certain correlation with random noise sources. In this paper, a separation problem on partial unknown sources is introduced in which random noises have certain correlation with deterministic such as periodic sources. A new partial unknown source separation technique is proposed by combining signal eigenspace transformation, covariance joint diagonalization and decorrelation of correlation sources. The partial source separation procedure has two main stages: first, obtain uncorrelated sources by eigenspace transformation and joint diagonalization; and second, obtain partial periodic sources correlated with random noises from the uncorrelated sources by decorrelation. The proposed partial source separation technique is supported by several theorems. A condition on the number of time lags in the joint diagonalization is presented. Under given assumptions, the separation technique will result in accurate partial sources. Calculation errors may be from smaller signal time length, lower sampling frequency and inaccurate estimation of correlation functions between periodic disturbances and random noises. The effect of estimated correlation functions on the separation results is analyzed. The proposed separation technique has main features as follows: (I) partial unknown sources are separated from measured signals; (II) separated periodic sources are correlated with random noise sources; (III) random noises can be dominant in measured signals or have a large intensity; (IV) periodic disturbance sources can be non-dominant in measured signals; and (V) only correlation functions of measured signals with estimated correlation functions between periodic disturbances and random noises are used in calculation of statistics. Numerical results show that the original response signals can be recovered well (in terms of mixture coefficients, relative differences, similarity) in time domain from the measured signals in which periodic disturbances are correlated with random noises. The separation results are insensitive to the estimation errors of correlation functions between periodic disturbances and random noises. The proposed separation technique is applicable to, for example, signal processing such as electromagnetic interference elimination of random vibration signals measured from maglev trains and dynamic modelling under multiple periodic excitations with random noises.