An adaptive algorithm for the blind separation of
periodic sources is proposed in this paper. The method uses
only the second order statistics of the data, and exploits the
periodic nature of the source signals. Simulation results show
that the proposed approach has the ability to restore statistical
independence, and its performance is comparable to that of a
well established, higher order, blind source separation method

Jonathon Chambers (1251609)

Maria G. Jafari (7207460)

English

Loughborough University Institutional Repository

   This item was submitted to Loughborough’s Institutional Repository (https://dspace.lboro.ac.uk/) by the author and is made available under the following Creative Commons Licence conditions.      For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/  A novel adaptive algorithm for the blind separation of periodic sources M. G. Jafari Centre for Digital Signal Processing Research Strand, London WC2R 2LS, U.K. J. A. Chambers Cardiff School of Engineering King's College London Cardiff University Cardiff CF24 OYF, U.K. E-mail: maria.jafari@kcl.ac.uk E-mail: chambersj @cf.ac.uk Abstract- An adaptive algorithm for the blind separation of periodic sources is proposed in this paper. The method uses only the second order statistics of the data, and exploits the periodic nature of the source signals. Simulation results show that the proposed approach has the ability to restore statistical independence, and its performance is comparable to that of a well established, higher order, blind source separation method. I. INTRODUCTION Blind source separation (BSS) addresses the problem that arises when a set of unobservable source signals must be recovered from a set of measurements, generated when the original signals are mixed by an unknown medium. The blind source separation problem is typically solved by exploiting the second or higher order statistics of the data, with higher order methods generally representing the preferred approach, since they require few assumptions other than the statistical indevendence of the sources. For this reason they have 111. The performance of the proposed method is shown by simulation in Section IV, and conclusions are drawn in Section V. A. Pmblem statement The aim of BSS methods is to recover the source signals s ( k )  E C", from the observed mixtures x ( k )  E C", which are generated by the mixing model x ( k )  = As(k) (1) where A E Cnx" is an unknown, full column rank, mix- ing matrix, and k is the discrete time index. The original sources, assumed to be zero mean, unit variance, and mutually statistically uncorrelated, are then estimated according to the separating model Y ( k )  = W ( k ) x ( k )  (2) resulted in a large number of algorithms (e.g. [I], [21, [31, [4], [5]). Conversely, second order methods require shorter data records, and do not limit the number of Gaussian sources that can be separated to one, but their derivation is usually based on additional assumptions made on the nature of the original signals, such as statistical non-stationarity where y ( k )  E C" represents the recovered sources, and W ( k )  E C"'" is the separating matrix. In this paper, it is also assumed that the sources are temporally periodic, such that the i-th source, with fundamental period can be written as of the sources [6], [7], [SI, presence of time correlations in s ; ( k )  = S i ( k  + nT,) (3) where n E Z, and Z denotes the field of integers. Periodicity of the source implies that its autocorrelation function, defined as a function of lag 7, ( T )  = (si (k) sf(k + T ) ) ,  is also periodic with period Ti, that is stationaty signals [9], [IO], [ I l l ,  or cyclostationarity 1121, [131, [141. A novel adaptive algorithm, for the blind separation of periodic source signals, based only on second order statistical information is proposed in this paper. Source separation is performed by diagonalising the output autocorrelation P ~ , ( T )  = pS, (7  + nT,) = (S, (k) s:(k + 7 + Q,)) (4) matrix at a lag T corresponding to the fundamental period of periodicity of one of the source signals. Although the method requires that the delay T is equal to the fundamental period of one of the sources, it will successfully separate the the exact period. Thus, we begin by stating the blind source separation the proposed algorithm is based is described in Section method in Section I-A. The identification principle on which 11, while the sequential algorithm is introduced in Section 0-7803-8359-1/04/%20.00 02004 IEEE 59 where (.)' and (.) represent, respectively, complex conju- gation and statistical expectation. Thus, the autocorrelation matrix of the source signals, at a lag 7 = T, becomes sources even when the selected lag does not correspond to Rs(T,) = ( s ( k ) s H ( k + T ; ) )  ( 5 )  where ( . ) H  denotes the Hermitian transpose operator. From (4). the elements of the matrix in (5) are given by Qj # i Qj = a ,  Tj = nT4 (6) ( s , ( k ) s ; ( k t T j ) )  = p$,(T!) ,  Qj = i, Tj # nT' Authorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on December 11, 2009 at 05:06 from IEEE Xplore.  Restrictions apply. where n E Z. Thus, the source autocorrelation matrix at lag T; is given by where 1 . 1  denotes the absolute value. Differentiation of the first term on the right hand side of (11) with respect to the Rs(T;) =&(Ti) ( 7 )  separating matrix W ( k )  leads to where &(Ti) is the eigenvalue matrix of R,(T,), whose ele- ments are defined in (6). Note that the source autocorrelation a l o d d e t ( W  (k) W H  (k))) = 2(w (k) W~ ( k ) ) - ~ W  (k) a w ( k )  matrix is not guaranteed to be positive definite. In this paper, distinct and non-zero. to (12) we do assume, for algorithm development, that the eigenvalues of the = T, are Following the approach in [ I 5 1 9  it can be shown that differen- tiation of the second term on the right hand side of (I 1) leads autocorrelation matrix at the delay 11. IDENTIFICATION PRINCIPLE a ( y i  (k )y f (k  dWij  (k) *)) = (yf (k + T ) Z j  (k)) + (yi ( k ) $ ( k  + T ) ) ,  V i , j  = 1, ..., m (13) The output autocorrelation matrix evaluated at a lag T = T, is given by where it has been assumed that RY(kTi) = ( Y ( k ) Y H ( k  +Ti)) = ( W ( k ) A s ( k ) s H ( k  + T ; ) A H W H ( k  + T;)) WI3 (k) s w;j (k + T )  ( 14) = P(k) ( s ( k ) s H ( k  + T;))  P H ( k  +Ti) = P (k) A,(T,)PH (k +Ti) In matrix form, (13) becomes (8) a F (IY, (k) Y f  (k + a) = ( Y ( k + T ) X H ( k )  i=l aw (k) where P(k) = W ( k ) A  is the global mixing-separating ma- trix. The above expression allows the derivation of a periodic tively performing blind Source separation on the basis of only second order statistical information. To see this, let R; (T,) denote the output autocorrelation matrix at convergence, that is (9) decorrelation algorithm, which has also the ability of effec- + y ( k )  X H  ( k  T ) )  (15) Employing the natural gradient descent method1, the gradient of (11) is obtained from (12) and (15) as V J  (W ( k ) )  = a J ( W ( k ) ) W H ( k ) W ( k )  = -  I - - ( y ( k + r ) y H ( k )  [ T lim Ity (k,T;) = R; (T;) k-m By definition, the whitening operation implies that R:(T') = P ( k ) A . , ( T , ) P H ( k )  =D(T , )  (IO) + Y ( k ) Y H ( k + T ) ) ] W ( k )  (17) venience, the approximation in (14) is not explicitly shown. In terms of the output periodic autocorrelation function ( S ) ,  (I7) leads Io the following learning rule where D(T,) is a diagonal matrix and, since the system (10) where the assumption (14) has again been used. and for con- approaches steady.state, p (k) = p ( k  + T ~ ) .  ~~~~~i~~ (10) implies that p ( k )  is a ,,,,ita,,, matrix which can diagonalise the output autocorrelation matrix provided that its eigenvalues are distinct and non-zero. Moreover, the columns of P (k) are the eigenvectors of RE (T;). W ( k + l )  = W ( k ) + q  W(k) (18) 1 The matrix R, ( k , T , )  converges to D(T;) rather than &(Ti) because of the indeterminacies of scaling and permutation, which are due to the non-uniqueness of the eigenvectors contained in P (k). Thus, D(4)  estimates a matrix whose diagonal elements are the eigenvalues of %(k, E )  within a scaling and permutation ambiguity. Also, it  should be noted that P ( k )  is the eigenvector matrix of Ity (k,T;) and not Ra (7';). 111. PERIODIC ALGORITHM A sequential algorithm for the decorrelation of periodic sig- nals can be derived by minimising the following information theoretic criterion [ 151 1 J ( W  ( k ) )  = - ;; l o g ( d e t ( W  ( k )  W H  ( k ) ) )  where q is a step-size parameter. However, the algorithm (18) will not separate the sources when % ( k , T , )  is not positive definite, since the direction of descent may vary during the adaptive procedure, causing some of the elements of the estimated separating matrix to oscillate between positive and negative values. Thus, following the approach in [151, we replace R, (k,Ti) in (18) with R y ( k , T , ) S ( y ( k , T , ) ) ,  where 'The Atick-Redlich formula can also be used here. In lhis case, the same gradient is obtained using [I51 60 Authorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on December 11, 2009 at 05:06 from IEEE Xplore.  Restrictions apply. Hence, (18) becomes W ( k + l )  = W ( k ) + q  1-5 [R,H(k,Ti)S(y(k,T,)) { l  +R, ( k ,  Ta) s (Y ( k ,  Ta))lI w ( k )  (20) where it can be shown that the matrix R, ( k ,  T,) S (y ( k ,  T,)) is guaranteed to be positive definite. It should be noted that the matrix Rf ( k ,  T,) + R, ( k ,  Ti) is Hermitian, which implies that it does not offer enough degrees of freedom for source separation 1161. Conversely, Hermitian, because both ELy ( k ,  Ti) and R; ( k ,  Ti) are multiplied by S (y ( k ,  Ti)), thus ensuring that a sufficient number of degrees of freedom is preserved for the solution of the BSS problem. For the practical implementation of (ZO), the matrices ELy (k,T,)  and S(y(k,T,)) are replaced by their instantaneous estimate, given by Rf(kT,)S(y(k,T,)) + R,(k,Ti)S(y(k,T,)) is not f b ( k > T d  =Y(k)YH(k+Td (21) and where so that the algorithm (20) becomes W ( k + l )  = W ( k ) + v  I-- Rf(k,T,)S(y(k,T,)) { 3" +R,(k,T,)S(y(k,Ti))])W(k) (23) The algorithm proposed here has the ability to separate source signals for any arbitrary value of the lag r .  Thus, although the method has been derived for the case of temporally periodic sources, provided that the source autocomelation function is non-zero, and that the sources have different spectral charac- teristics, the assumption that the lag equals the fundamental period of periodicity can be relaxed. IV. SIMULATION RESULTS The performance of the proposed method was compared to that of the well established EASI algorithm, given by [SI W(k + 1) = w  ( k )  + P {I - Y ( k )  YT ( k )  + Y ( k )  fT (Y ( k ) )  - f ( Y  (k))YT(k)}W(k) (24) where p is a positive step-size parameter, and f (y ( k ) )  is an odd non-linear function of the output y(k). Firstly, a sinusoidal source s l ( k )  with fundamental period TI = 100 samples and a signal s ~ ( k )  uniformly distributed in [-1,1], both shown in Fig. I ,  were mixed by a time-invariant 2 x 2 mixing matrix, so that the mixtures q ( k )  and z 2 ( k )  were generated. Separation was performed with (23), when 2 -- 0 - r .. . .- -20 l""" 7""" wnn 0 1wO Zoo0 WO 4000 MW 5 Sample number Fig. 1 .  Original sour~e signals (SI ( k )  and sz(k)). and mixed signals ( z l ( k )  and T Z ( ~ ) ) .  0 1WO 20W 3WO 40W MW Sample number Fig. 2. Signals recovered with the EASI algorithm (yf(k)  and y F ( k ) ) .  and with the algorithm (23) ($(e)  and y f ( k ) ) .  q = 0.003, Ti = 100, and with the EASI algorithm (24), when p = 0.003, and f,(yi(k)) = y:(k). In Fig. 2 yf(k) and yf(k) represent the signals recovered with the EASI algorithm, and y f ( k )  and yT(k) are the outputs of the proposed algorithm. The results illustrate that the proposed method recovers the original components, and preserves waveform similarity in a similar fashion as the EASI algorithm. The two upper plots in  Fig. 3 depict the support of the joint probability density function (pdf) of the source signals (upper-left plot), and the mixtures (upper-right). The support of the joint pdf of the sources separated with the proposed periodic BSS algorithm and with EASI, following initial 61 Authorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on December 11, 2009 at 05:06 from IEEE Xplore.  Restrictions apply. 3, 3, Fig. 3. Scatter plots of the source signals (upper-left plot), the mixtures (upper-right plat). and the outputs of the proposed algorithm (23) ( y f ( k )  and yF(k)) (lower-left plot), and EASI (yf(k) and v f ( k ) )  (lower-right plot). Fig. 4. EASI algorithm (24). for a single realisalion. Performance indices obtained with thc algorithm (23) and with the convergence of the algorithm (i.e., for 500 5 k 5 5000). are depicted,respectively, in the lower-left and lower-right plots. The shape of the joint distribution of the source signals clearly indicates that they are statistically independent, since the knowledge of the value of one does not convey any information about the value of the other. The signals generated by the mixing procedure, 21 and 22, are no longer independent, and the joint pdf now has the form of a parallelogram. The results depicted in the lower plots of Fig. 1, show that the periodic decorrelation algorithm proposed here has the necessary degrees of freedom to allow it to rotate the joint pdf of the recovered sources, as well as decorrelate the signals, leading to the restoration of statistical independence. The performance of the periodic BSS algorithm (23) and EASI (24) was compared in terms of the performance index (PI), as defined in [17], where generally a low PI indicates better performance. The performance indices for the two methods are compared in Fig. 4, which illustrates that the proposed algorithm is capable of separating the source signals, with a performance comparable to the behaviour of EASI. 2, I 1000 2000 3wo 4m sow -2' 0 I I 0 low 2m 3000 4" 5ooo I I 0 tow 2wo 3000 4m 5ooo Sample number Fig. 5. Original sourcc signals s i ( k )  . S Z ( L ) .  and s g ( k )  - EASI l a  Penmlc BSS 1 6  w Fig. 6. algorithm and EASI, averaged over I00 independent trials. Performance index obtained with thc proposed periodic BSS Next, the average performance of the proposed algorithm is investigated. Fig. 5 shows a signal uniformly distributed in [-1,1], a sign and an amplitude-modulated signal, given by s , ( k )  = s g n ( c o s ( I r r f ~ t , k ) )  S z ( k )  = Sin(2?rfit,k)Sin(2lif3t~k) where t ,  = 9 x fl = 155Hz, fi = 9Hz,  and f3 = 300Hi. The sources were mixed by a stationary 3 x 3 mixing matrix, and noise was added such that the signal to noise ratio was 5dB. Separation was performed with (23). when I) = 0.003, T, = (f2 + f3)-' where (f2 + A )  is the upper 62 Authorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on December 11, 2009 at 05:06 from IEEE Xplore.  Restrictions apply. side band of source s1 (k), and with the EASI algorithm (24). when f i  = 0.003. The PI resulting from the application of the two methods, and averaged over 100 independent trials, is shown in Fig. 6. It illustrates that the behaviour of the proposed method is similar to that of EASI, both during and following initial convergence. V. CONCLUSIONS In this paper a novel sequential algorithm for the blind separation of periodic source signals is introduced, which exploits the temporal periodic nature of the source signals, and thus performs separation based only on second order statistics. Computer simulations have shown that the method allows the recovery of the original sources, and restoration of statistical independence. Moreover, its behaviour is comparable to that of EASI, during and following initial convergence. REFERENCES [I] S .  Amari. A. Cichocki, and H. H. Yang. “A new learning algorithm for blind signal separation,” in  Advances in Neurol Informarion Processing Systems. vol. 8. 1996, pp. 752-763. [2] C. Jutten and J. Herault. “Blind separation of sources, Pan1 An adaptive algorithm baaed on a neummimetic architecture,*’ Signrrl Processing. vol. 24, pp. 1-10, 1991. [3] P. Camon. “Independent component analysis, a new concept?” Signol Pmcesring. vol. 36. pp. 287-314, 1994. [4] A. J. Bell and T. I. Sejnowski. “An infarmation-maximiration approach to blind separation and blind deeonvalution,” Neuml Computation. vol. 7, pp. 1129-1159. 1995. [SI 1. F, Cardoso and B. Laheld, “Equivariant adaptive source separation,” IEEE Tram. on Signor Processing. vol. 44. pp. 3017-3030. 1996. [6] K. Matsuoka, M. Ohya. and M. Kawamoto, “A neural net for blind separation of nonstationary signals,” Neural Networks, vol. 8. pp. 41 1- 419. 1995. [7] S .  Chai and 0. Y. Lee, “Nonstationary source separation,” in Pmc. of the IEEE Region 10 Cvnference, vol. 1, 1999, pp. 670-673. [8] D. L. Jones. “A new method for blind sou~ccc separation of nonstationary signals:’ in Pmc. ofthe IEEE Inr. Conf: on Acoustic, Speech, und Sifinol Pmcessing, vol. 5, 1999, pp. 2893-2896. [9] A. Belouchrani, K. Abed-Meraim, J. Cardoso, and E. Moulines, “ A  blind source separation technique using second-order statistics; IEEE Trorrms. on Signal Processing, vol. 45, pp. 434-444, 1997. [IO] S .  Choi and A. Cichoek. “Blind separation of nonstationary sources in noisy mixtures,” Elecrmnics Letters. vol. 36, pp. 848-849. 2000. [I l l  A. Belouchrani and A. Cichocki. “A robust procedure in blind Source separation context,” Electronics Letters, vol. 36, pp. 2050-2051, 2000. [I21 A. Ferreol and P. Chevalier, “On the behaviour of current second and higher order blind source separation methods for cyclostationary sources.” IEEE T r m .  on Sifinol Processing, vol. 48, pp. 1712-1725, 2000. [I31 K. Abed-Meraim. Y. Xiang, J. H. Manton, and Y. Hua, “Blind source separation using second-order cyclostatianary statistics; IEEE Trans. on Sigml Processing. vol. 49, pp. 694-701, 2001 [I41 M. G. Jafari and J. A. Chambers, “A new natural gradient algorithm for cyclostatianary sources,” To appear: E E  Proceedings Vision, Image and Signal F’roees:iing, 2002. [ 151 A. Cichocki and S. Amari. Adnprive Blind Signal nndlmge Processing: Learning Alfiorithins and Applicntionr. [I61 I. E Cardoso, ‘‘Blind signal separation: statistical principles,” Proceed- ings of the IEEE, vol. 86, pp. 2009-2025. 1998. [17] A. Cichocki, B. 83rsier. A. Back. and S. Amari. “On-line adaptive algorithms in non..stationary environmenls using a modified conjugate gradient approach,” in PNC. ofrhe IEEE Workshop on Neural Nemorks for Signal Procesring. 1997, pp. 316325.  John Wiley, 2002. 63 Authorized licensed use limited to: LOUGHBOROUGH UNIVERSITY. Downloaded on December 11, 2009 at 05:06 from IEEE Xplore.  Restrictions apply. 

A novel adaptive algorithm for the blind separation of periodic sources

An adaptive algorithm for the blind separation of periodic sources is proposed in this paper. The method uses only the second order statistics of the data, and exploits the periodic nature of the source signals. Simulation results show that the proposed approach has the ability to restore statistical independence, and its performance is comparable to that of a well established, higher order, blind source separation method

Jafari, MG

Chambers, JA

Surrey Research Insight

https://repository.lboro.ac.uk/articles/A_novel_adaptive_algorithm_for_the_blind_separation_of_periodic_sources/9555689/files/17187572.pdf

A novel adaptive algorithm for the blind separation of periodic sources

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