Patent Application: US-94775501-A

Abstract:
device and method for separating a mixture of source signals to regain the source signals , the device and method being based on measured signals , the invention comprises : bringing each measured signal to a separation structure including an adaptive filter , the adaptive filter comprising filter coefficients ; using a generalized criterion function for obtaining the filter coefficients , the generalized criterion function comprising cross correlation functions and a weighting matrix , the cross correlation functions being dependent on the filter coefficients ; estimating the filter coefficients , the resulting estimates of the filter coefficients corresponding to a minimum value of the generalized criterion function ; and updating the adaptive filter with the filter coefficients .

Description:
in the present invention a signal separation algorithm is derived and presented . a main result of the analys is an optimal weighting matrix . the weighting matrix is used to device a practical algorithm for signal separation of dynamically mixed sources . the derived algorithm significantly improves the parameter estimates in cases where the sources have similar color . in addition the statistical analysis can be used to reveal attainable ( asymptotic ) parameter variance given a number of known parameters . the basis for the source signals , in the present paper , are m mutually uncorrelated white sequences . these white sequences are termed source generating signals and denoted by ξ k ( n ) where k = 1 , . . . , m . the source generating signals are convolved with linear time - invariant filters g k ( q )/ f k ( q ) and the outputs are , x ⁡ ( n ) = ⁢ [ x 1 ⁡ ( n ) ⁢ ⁢ … ⁢ ⁢ x m ⁡ ( n ) ] t = k ⁡ ( q ) ⁢ ξ ⁡ ( n ) = ⁢ diag ⁢ ⁢ ( g 1 ⁡ ( q ) f 1 ⁡ ( q ) , ⁢ … ⁢ , g m ⁡ ( q ) f m ⁡ ( q ) ) ⁡ [ ξ 1 ⁡ ( n ) ⋮ ξ m ⁡ ( n ) ] , referred to as the source signals and where q and t is the time shift operator and matrix transpose , respectively . the following assumptions are introduced a1 . the generating signal ξ ( n ) is a realization of a stationary , white zero - mean gaussian process : ξ ( n ) εn ( 0 , σ ), σ = diag ( σ 1 2 , . . . σ m 2 ) a2 . the elements of k ( q ) are filters which are asymptotically stable and have minimum phase . condition a1 is somewhat restrictive because of the gaussian assumption . however , it appears to be very difficult to evaluate some of the involved statistical expectations unless the gaussian assumption is invoked . the source signals x ( n ) are unmeasurable and inputs to a system , referred to as the channel system . the channel system produces m outputs collected in a vector y ( n ) y ( n )=[ y 1 ( n ) . . . y m ( n )] t = b ( q ) x ( n ). which are measurable and referred to as the observables . in the present paper the channel system , b ( q ), given in b ⁡ ( q ) = [ ⁢ 1 b 12 ⁡ ( q ) ⋯ b 1 ⁢ m ⁡ ( q ) b 21 ⁡ ( q ) ⋰ ⋮ ⋮ ⋰ b ( m - 1 ) ⁢ m ⁡ ( q ) b m1 ⁡ ( q ) ⋯ b m ⁡ ( m - 1 ) ⁡ ( q ) 1 ⁢ ] , where b ij ( q ), ij = 1 , . . . m are fir filters . the objective is to extract the source signals from the observables . the extraction can be accomplished by means of all adaptive separation structure , cf . [ 8 ]. the inputs to the separation structure are the observable signals . the output from the separation structure , s 1 ( n ), . . . , s m ( n ), depend on the adaptive filters , d ij ( q , θ ), i , j = 1 , . . . m , and can be written as s ( n , θ )=[ s 1 ( n , θ ) . . . s m ( n , θ )] t = d ( q , θ ) y ( n ), where θ is a parameter vector containing the filter coefficients of the adaptive filters . that is the parameter vector is θ =[ d 11 t . . . d mm t ] t where d ij , i , j = 1 , . . . m are vectors containing the coefficients of d ij ( q , θ ), i , j = 1 , . . . m , respectively . note , that unlike b ( q ) the separation matrix d ( q , θ ) does not contain a fixed diagonal , cf . [ 6 , 13 ]. most of the expressions and calculations in the present paper will be derived for the two - input two - output ( tito ) case , m = 2 . the main reason for using the tito case is that it has been shown to be parameter identifiable under a set of conditions , cf . [ 8 ]. however , the analysis in the current paper is applicable on the more general multiple - input multiple - output ( mimo ) case , assuming that problem to be parameter identifiable as well . assuming that n samples of y 1 ( n ) and y 2 ( n ) are available , the criterion function proposed in [ 8 ], reads as ⁢ v _ ⁢ ( θ ) = ∑ k = - u u ⁢ r _ s 1 ⁢ s 2 2 ⁢ ( k ; θ ) , r _ s 1 ⁢ s 2 ⁢ ( k ; θ ) = 1 n ⁢ ∑ n = 0 n - k - 1 ⁢ ⁢ s 1 ⁢ ( n ; θ ) ⁢ s 2 ⁢ ( n + k ; θ ) , k = 0 , ⁢ … ⁢ , u . to emphasize the dependence on θ equation ( 2 . 6 ) can be rewritten as r _ s 1 ⁢ s 2 ⁢ ( k ; θ ) = r _ y 1 ⁢ y 2 ⁢ ( k ) - ∑ i ⁢ d 12 ⁢ ( i ) ⁢ r _ y 2 ⁢ y 2 ⁢ ( k - i ) - ∑ i ⁢ d 21 ⁢ ( i ) ⁢ r _ y 1 ⁢ y 1 ⁢ ( k + i ) + ∑ i ⁢ ∑ l ⁢ d 12 ⁢ ( i ) ⁢ d 21 ⁢ ( l ) ⁢ r _ y 2 ⁢ y 1 ⁢ ( k - i + l ) , where the notation d 12 ( i ) denotes the i : th coefficient of the filter d 12 ( q ). { dot over ( r )} n ( θ )=[ { dot over ( r )}, 1 , 2 (− u , θ ) . . . { dot over ( r )}, 1 , 2 ( u θ )] t where the subscript n indicates that the estimated cross - covariances are based on n samples . furthermore , introduce a positive definite weighting matrix w ( θ ) which possibly depends on θ too . thus , the criterion , in equation ( 2 . 6 ), can be generalized as v n ⁢ ( θ ) = 1 2 ⁢ r _ n t ⁢ ( θ ) ⁢ w ⁢ ( θ ) ⁢ r _ n ⁢ ( θ ) , which will be investigated . note , the studied estimator is closely related to the type of non - linear regressions studied in [ 15 ]. the estimate of the parameters of interest are obtained as θ ^ n = arg ⁢ ⁢ min θ ⁢ v n ⁡ ( θ ) . although the signal separation based on the criterion ( 2 . 6 ) has been demonstrated to perform well in practice , see for example [ 12 ], there are a couple of open problems in the contribution [ 8 ]; 1 . it would be interesting to find the asymptotic distribution of the estimate of θ n . especially , an expression for the asymptotic covariance matrix is of interest . one reason for this interest , is that the user can investigate the performance for various mixing structures , without performing simulations . potentially , further insight could be gained into what kind of mixtures that are difficult to separate . the asymptotic covariance matrix would also allow the user to compare the performance with the cramér - rao lower bound ( crb ), primarily to investigate how far from the optimal performance of the prediction error method the investigated method is . an investigation of the crb for the mimo scenario can be found in [ 14 ]. 2 . how should the weighting matrix w ( θ ) be chosen for the best possible ( asymptotic ) accuracy ? given the best possible weighting , and the asymptotic distribution , one can further investigate in which scenarios it is worthwhile applying a weighting w ( θ )≠ i , where i denotes the identity matrix . find the asymptotic distribution of the estimate of θ n . find the weighting matrix w ( θ ) that optimizes the asymptotic accuracy . study an implementation of the optimal weighting scheme . in addition to a1 and a2 the following assumptions are considered to hold throughout the description : a3 . assume that the conditions c3 - c6 in [ 8 ] are fulfilled , so that the the studied tito system is parameter identifiable . a4 . the ( minimal ) value of u is defined as in proposition 5 in [ 8 ]. a5 . ∥ θ ∥& lt ;∞, i . e . θ 0 is an interior point of a compact set d m . here , θ 0 contains the true parameters . this section deals with the statistical analysis and it will begin with consistency . the asymptotic properties ( as n -∞) of the estimate of θ n ( θ ^ n ) is established in the following . however , first some preliminary observations are made . in [ 8 ] it was shown that 1 . as n →∞, { dot over ( r )}, 1 , 2 ( k , θ )→ r , 1 , 2 ( k , θ ) with probability one ( w . p . 1 ). thus v _ ⁢ ( θ ) = 1 2 ⁢ r t ⁢ ( θ ) ⁢ w ⁢ ( θ ) ⁢ r ⁡ ( θ ) , ⁢ r ⁡ ( θ ) = [ r s 1 ⁢ s 2 ⁡ ( - u , θ ) ⁢ ⁢ … ⁢ ⁢ r s 1 ⁢ s 2 ⁡ ( u , θ ) ] t . the convergence in ( 3 . 1 ) is uniform in a set d m , where θ is a member lim n -& gt ; ∞ ⁢ sup θ ∈ d m ⁢  v n ⁢ ( θ ) - v _ ⁢ ( θ )  = 0 , w . p ⁢ . 1 . furthermore , since the applied separation structure is of finite impulse response ( fir ) type , the gradient is bounded max 1 ≤ i ≤ nθ ⁢ { sup θ ∈ d m ⁢  ∂ v n ⁢ ( θ ) ∂ θ i  } ≤ c ⁢ ⁢ w . p ⁢ . 1 , for n larger than some n 0 & lt ;∞. in equation ( 3 . 4 ), c is some constant , c & lt ;∞, and nθ denotes the dimension of θ . the above discussion , together with the identifiability analysis [ 8 ] then shows the following result : having established ( strong ) consistency , the asymptotic distribution of θ ^ n is considered . since the θ ^ n minimizes the criterion v n ( θ ), v n ( θ ^ n )= 0 , where v n denotes the gradient of v n . by the mean value theorem , 0 = v n ′({ dot over ( θ )} n )= v n ′( θ 0 )+ v n ″( θ ξ )({ dot over ( θ )} n − θ 0 ), where θ ξ is on a line between θ ^ n and θ 0 . note , since θ ^ n is consistent , the θ ^ n − θ 0 and consequently , θ ξ → θ 0 , as n →∞. next , investigate the gradient evaluated at θ 0 ( for notational simplicity , let w ( θ )= w ) v n ′ ⁡ ( θ 0 ) = g ^ t ⁢ w ⁢ r ^ n ⁡ ( θ 0 ) + 1 2 ⁡ [ r ^ n t ⁡ ( θ 0 ) ⁢ ∂ w ∂ θ 1 ⁢ r ^ n ⁡ ( θ 0 ) ⋮ r ^ n t ⁡ ( θ 0 ) ⁢ ∂ w ∂ θ 1 ⁢ r ^ n ⁡ ( θ 0 ) ] θ = θ 0 ≃ g ^ t ⁢ w ⁢ r ^ n ⁡ ( θ 0 ) , g ^ = ∂ τ ^ n ⁢ ( θ ) ∂ θ ⁢ ❘ θ = θ 0 . note , evaluation of g is straightforward , see for example [ 8 ]. the introduced approximation does not affect the asymptotics , since the approximation error goes to zero at a faster rate than does the estimate of r n ( θ 0 ) ( r ^ n ( θ 0 )). furthermore , since r n ( θ 0 )= 0 , the asymptotic distribution of v n ′( θ ) is identical to the asymptotic distribution of g t wr ^ n ( θ 0 ), where g = ∂ τ ⁢ ( θ ) ∂ θ ⁢ ❘ θ = θ 0 . applying for example lemma b . 3 in [ 15 ], and using the fact that both s 1 ( n : θ 0 ) and s 2 ( n ; θ 0 ) stationary arma processes , one can show that (√ n ) g t wr ^ n ( θ 0 ) converges in distribution to a gaussian random vector , i . e . √{ square root over ( n )} g t w { dot over ( r )} n ( θ 0 ) εas n ( 0 , g t wmwg ), m = lim n -& gt ; ∞ ⁢ n ⁢ ⁢ e ⁡ [ τ . n ⁢ ( θ 0 ) ⁢ τ . n t ⁢ ( θ 0 ) ] . this means that the gradient vector , is asymptotically normally distributed , with zero - mean and with a covariance matrix m . before presenting the main result of the current paper , the convergence of the hessian matrix v ″ n must be investigated . assuming that the limit exists , define v _ ′′ ⁢ ( θ ) = lim n -& gt ; ∞ ⁢ v n ′′ ⁢ ( θ ) . to establish the convergence of v ″ n ( θ ξ ), the following ( standard ) inequality is applied  v n ′′ ⁢ ( θ ξ ) - v _ ′′ ⁢ ( θ 0 )  f ≤  v n ′′ ⁢ ( θ ξ ) - v n ′′ ⁢ ( θ 0 )  f +  v n ′′ ⁢ ( θ 0 ) - v _ ′′ ⁢ ( θ 0 )  f , where ∥ ∥ f denotes the frobenius norm . due to the fir separation structure , the second order derivatives are continuous . moreover , since θ ξ converges w . p . 1 to θ 0 , the first term converges to zero w . p . 1 . the second term converges also to zero w . p . 1 . this can be shown using a similar methodology that was used to show ( 3 . 3 ). note also that since the third order derivatives are bounded , the convergence is uniform in θ . it is , now , straightforward to see that the limiting hessian v − ″ can be written as v _ ′′ ⁡ ( θ 0 ) = lim n -& gt ; ∞ ⁢ v ′′ ⁡ ( θ 0 ) = g t ⁢ w ⁢ ⁢ g w . p ⁢ . 1 √{ square root over ( n )}({ dot over ( θ )} n − θ 0 )≅√{ square root over ( n )}( g t wg ) − 1 g t w { dot over ( τ )} n ( θ 0 ), assuming the inverses exists ( generically guaranteed by the identifiability conditions in a3 . here all approximation errors that goes to zero faster than o ( 1 /√( n )) have been neglected . finally , the following result can be stated . consider the signal separation method based on second order statistics , where θ ^ n is obtained from ( 2 . 10 ). then the normalized estimation error , √( n )( θ ^ n − θ n ), has a limiting zero - mean gaussian distribution obviously , the matrix m plays a central role , and it is of interest to find a more explicit expression . for simplicity we consider only the case when the generating signals are zero - mean , gaussian and white ( as stated in assumption a1 ). it seems to be difficult to find explicit expressions for the non - gaussian case . note also that this is really the place where the normality assumption in a1 is crucial . for example , the asymptotic normality of √( n ) r ^ n ( θ 0 ) holds under weaker assumptions . theorem 6 . 4 . 1 in [ 5 ] indicates precisely how the components of m can be computed . these elements are actually rather easy to compute , as the following will demonstrate . let β τ = ∑ p = - ∞ ∞ ⁢ r s 1 ⁢ s 1 ⁢ ( p ; θ 0 ) ⁢ r s 2 ⁢ s 2 ⁢ ( p + τ ; θ 0 ) . φ 1 ⁢ ( z ) = ∑ k = - ∞ ∞ ⁢ r s 1 ⁢ s 1 ⁢ ( k ; θ 0 ) ⁢ z - k , ⁢ φ 2 ⁢ ( z ) = ∑ k = - ∞ ∞ ⁢ r s 2 ⁢ s 2 ⁢ ( k ; θ 0 ) ⁢ z - k . ∑ τ = - ∞ ∞ ⁢ ∑ p = - ∞ ∞ ⁢ r s 1 ⁢ s 1 ⁢ ( p ; θ 0 ) ⁢ r s 2 ⁢ s 2 ⁢ ( p + τ ; θ 0 ) ⁢ z - τ = φ 1 ⁢ ( z - 1 ) ⁢ φ 2 ⁢ ( z ) . thus , the β τ &# 39 ; s are the covariances of an arma process with power spectrum φ 1 ⁢ ( z - 1 ) ⁢ φ 2 ⁢ ( z ) = σ 1 2 ⁢ ( σ 2 2 ⁢ ( 1 - b 12 ⁢ ( z ) ⁢ b 21 ⁢ ( z ) ) ) 2 ⁢ ( 1 - b _ 12 ⁢ ( z - 1 ) ⁢ b _ 21 ⁢ ( z - 1 ) ) ⁢  g 1 ⁢ ( z ) f 1 ⁢ ( z )  2 ⁢  g 2 ⁢ ( z ) f 2 ⁢ ( z )  2 . computation of arma covariances is a standard topic , and simple and efficient algorithms for doing this exists , see for example [ 15 , complement c7 . 7 ]. given β τ for τ = 0 . . . , 2u , the weighting matrix can , hence , be constructed as w = [ β 0 β 1 … β 2 ⁢ u β 1 β 0 ⋰ ⋮ ⋮ ⋰ β 1 β 2 ⁢ u … β 1 β 0 ] . thus , in the present problem formulation the separated signals are distorted with the determinant of the channel system the channel system determinant equals det { b ( z )}, and one may define the reconstructed signals as x . i ⁡ ( n ) = 1 det ⁢ { d ⁡ ( q , θ . ) } ⁢ s i ⁡ ( n i ⁢ θ . n ) , to complete our discussion , it is also pointed out how the matrix g can be computed . the elements of g are all obtained in the following manner . using equation ( 2 . 8 ). it follows that ∂ r s 1 ⁢ s 2 ⁢ ( k ; θ ) ∂ d 21 ⁢ ( i ) = - r y 1 ⁢ y 1 ⁢ ( k + i ) + ∑ l ⁢ d 12 ⁢ ( l ) ⁢ r y 2 ⁢ y 1 ⁢ ( k - l + i ) ∂ r s 1 ⁢ s 2 ⁢ ( k ; θ ) ∂ d 21 ⁢ ( i ) = - r y 2 ⁢ y 2 ⁢ ( k - i ) + ∑ l ⁢ d 21 ⁢ ( l ) ⁢ r y 2 ⁢ y 1 ⁢ ( k + i - l ) . next , consider the problem of choosing w . our findings are collected in the following result . the asymptotic accuracy of θ ^ n , obtained as the minimizing argument of the criterion ( 2 . 10 ), is optimized if p ( w 0 )=( g t m − 1 g ) − 1 the accuracy is optimized in the sense that p ( w 0 )− p ( w ) is positive semi - definite for all positive definite weighting matrices w . the proof follows from well - known matrix optimization results , see for example [ 9 , appendix 2 ]. the result could have been derived directly from the abc theory in [ 15 , complement c4 . 4 ]. however , the result above itself is a useful result motivating the presented analysis . before considering the actual implementation of the optimal weighting strategy , let us make a note on the selection of u . this parameter is a user - defined quantity , and it would be interesting to gain some insight into how it should be chosen . note , assumption a3 states a lower bound for u with respect to identifiability . the following result may be useful . assume that the optimal weighting w 0 is applied in the criterion ( 2 . 10 ). let p u ( w ) denote the asymptotic covariance for this case . then the proof follows immediately from the calculations in [ 15 , complement c4 . 4 ]. note that , when the optimal weighting , w 0 is applied the matrices { p u ( w 0 )} forms a non - increasing sequence . however , in practice one must be aware that a too large value of u in fact may deteriorate the performance . this phenomenon may be explained by that a large value of u means that a larger value of n is required in order for the asymptotic results to be valid . in the present section a comparison of signal separation based on an algorithm within the scope of the present invention and the algorithm in [ 8 ] will be made . the purpose for the comparison is to show the contribution of the present invention . put differently , does the weighting lead to a significant decrease of the parameter variance ? in all of our simulations , u = 6 . furthermore , the term relative frequency is used in several figures . here relative frequency corresponds to f rel = 2f / f s where f s is the here the channel system is defined by b 12 ( q )= 0 . 3 + 0 . 1q − 1 and b 21 ( q )= 0 . 1 + 0 . 7q − 1 . the source signal x 1 ( n ) is an ar ( 2 ) process with poles at radius 0 . 8 and angles π / 4 . the second source signal is , also , an ar ( 2 ) process . however , the poles are moved by adjusting the angles in the interval [ 0 , π / 2 ], while keeping the radius constant at 0 . 8 . at each angle 200 realizations have been generated and processed by the channel system and separation structure . that is to say , for each angle the resulting parameter estimates have been averaged . finally , each realization consists of 4000 samples . in fig1 a - 1d the empirical and true parameter variances are depicted . first , note the good agreement between the empirical and theoretical variances . second , observe that the proposed weighting strategy for most angles gives rise to a significant variance reduction . in fig1 a - 1d , the parameter variances as a function of relative frequency . “*” denotes empirical variance of the prior art signal separation algorithm ; “+” denotes empirical variance of the proposed weighting strategy . the solid line is the true asymptotic variance of the unweighted algorithm , and the dashed line is the true asymptotic variance for the optimally weighted algorithm . the dotted line is the crb . apparently , it gets more difficult to estimate the channel parameters when the source colors are similar . therefore , in fig2 a - 2d and fig3 a - 3d a more careful examination of the parameter accuracy is presented . the angles of the poles are in the interval [ 40 °, 50 °]. in fig2 a - 2d , the estimated mean value is depicted as a function of relative frequency . “*” denotes empirical mean value of the prior art signal separation algorithm ; “+” denotes empirical mean value of the proposed weighting strategy . the solid lines correspond to the true parameter values . note that the optimally weighted algorithm gets biased , although less biased than a prior art signal separation algorithm . this bias is probably an effect of the fact that the channel estimates of the unweighted algorithm are rather inaccurate , making the weighting matrix inaccurate as well . in fig3 a - 3d , the parameter variances is depicted as a function of relative frequency ; “*” denotes empirical variance of the prior art signal separation algorithm [ 8 ]; “+” denotes empirical variance of the proposed weighting strategy . the solid line is the true asymptotic variance of the unweighted algorithm , and the dashed line is the true asymptotic variance for the optimally weighted algorithm . the dotted line is the crb . as is indicated by the fig1 a - 1d , fig2 a - 2d , and fig3 a - 3d , the present invention increases the quality of the signal separation . further details of the present invention are that the method , according the device for separating signals , is repeatedly performed on the measured signal or on fractions thereof . also , the method may be repeatedly performed according to a predetermined updating frequency . it should be noted that the predetermined updating frequency may not be constant . further , the number of filter coefficients is predetermined . finally , the number of filter coefficients is arranged to be predetermined in the above embodiment . h . broman , u . lindgren , h . sahlin , and p . stoica . “ source separation : a tito system identification approach ”. esp , 1994 . d . c . b . chan . blind signal separation , phd thesis , university of cambridge , 1997 . f . ehlers and h . g . schuster . blind separation of convolutive mixtures and an application in automatic speech recognition in a noisy environment . ieee trans . on signal processing 45 ( 10 ): 2608 - 2612 , 1997 . m . feder , a . v . oppenheim , and e . weinstein . maximum likelihood noise cancellation using the em algorithm , ieee trans . on acoustics , speech , and signal processing , assp - 37 : 204 - 216 . february 1989 . w . a . fuller . introduction to statistical time series . john wiley & amp ; 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