BLIND source separation is a set of techniques that is used

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1 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK A Novel Approach to Blind Separation of Delayed Sources in Linear Mixtures Jakob Ashtar, Kristoffer Møller Jørgensen, Jacob Lindvig, Lars Juul Mikkelsen, Jakob Birkedal Nielsen, Lars Sommer Søndergaard, and Dario Farina, Supervisor, Aalborg University Abstract This paper addresses the problem of blind estimation of time-delayed sources from a set of linear mixtures. The mixtures are analyzed and approximated by a linear, instantaneous signal model and the sources are recovered using second order statistical methods. An algorithm is implemented and test results are demonstrated and discussed. The proposed algorithm is referred to as SOBIDS Second Order Blind Identification of Dealyed Sources). SOBIDS is compared to another algorithm using second order statistics and test results show that SOBIDS performs better for delayed sources. I. INTRODUCTION BLIND source separation is a set of techniques that is used in many fields of engineering such as speech recognition, telecommunications, and biomedical signal processing. One of the first papers describing the problem of recovering sources was published in 985 by Herault et al. [5] These authors addressed the problem by using and extending principal component analysis. Various approaches have been proposed since then. Many of them are based on mathematical disciplines such as principal component analysis, independent component analysis, or singular value decomposition [] [2]. The classical cocktail party problem where two persons communicate in a room with loud music and background noise is an example of a situation where the human brain utilizes blind source separation. In this case the human brain focuses on a speaker, distinguishes one sound from the others, and ignores the background noise. Simulating this process with machines is, however, a very difficult task. The wide range and nature of applications further complicates the task. In some cases the sources can be assumed stationary. In other cases, e.g. speech signals, the sources are non-stationary. Signal propagation delays might also be relevant to consider. Also, the mixtures of sources may be either instantaneous and linear or convolutive. The aim of this paper is to propose a new method SOBIDS) for the separation of sources mixed with unknown, small delays in a linear, instantaneous way. The paper is organized as follows. In section II and III we provide a description of a blind identification algorithm previously proposed to separate sources from their linear, instantaneous mixtures using second order statistics []. This algorithm is referred to as SOBI Second Order Blind Identification). Section IV introduces a convolutive signal model where the sources are delayed. To solve the blind source separation problem for a convolutive model we approximate the model with a linear, instantaneous signal model. In section V we show that it is possible to blindly estimate the sources for small delays. In section VI we compare test results for SOBI and SOBIDS and discuss the performance of the new algorithm. II. LINEAR INSTANTANEOUS MIXTURES Assuming linear, instantaneous mixtures we have the following signal model. A. Signal Model Let a set of n unknown sources s t),s 2 t),...,s j t),...,s n t) be linearly combined to form m known observations y t),y 2 t),...,y i t),...,y m t). Each observation y i t) is contaminated by unknown additive noise p i t). Expressed mathematically we get the following relation: y t) = a s t)+a 2 s 2 t)+...+ a n s n t)+ p t) y 2 t) = a 2 s t)+a 22 s 2 t)+...+ a 2n s n t)+ p 2 t). y m t) = a m s t)+a m2 s 2 t)+...+ a mn s n t)+ p m t) ) The above system of linear equations can also be represented in matrix notation as a transformation T : R n R m of the source vector st): yt) = Tst)) = xt)+pt) = Ast)+pt) 2) where the observation vector yt), the signal vector xt), and the noise vector pt) are m column vectors and the source vector st) is an n column vector. The unknown m n matrix A is called the mixing matrix. B. Assumptions It is assumed that the sources are real, ergodic, zero mean stochastic processes. Since the power of the sources cannot be determined [], it is assumed without loss of generality that the sources have unit variance. Furthermore, the sources are assumed to be mutually uncorrelated spatially white). It is also assumed that the sources are uncorrelated with the additive noise. The noise process is a stationary, temporally and spatially white, zero-mean, and real random process. The noise variance is the same for all observations. The mixing matrix has full rank. Furthermore, it is assumed that there are at least as many observations as sources, that is m n.

2 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK 2 C. Derivations Based on the signal model and the above assumptions the n n covariance matrix of the sources R s τ) is given by: R s τ) = E [ st)s T t + τ) ] = diag [ R s s τ),r s2 s 2 τ),...,r sn s n τ) ] 3) where R s j s j τ) = E [ s j t)s j t +τ) ]. Since the sources are mutually uncorrelated and assumed to have unit variance, we have: R s 0) = I 4) As the noise is temporally and spatially white the m m covariance matrix R p τ) of the noise is 0 at time lags τ 0. At τ = 0 the covariance matrix R p τ) is the identity matrix scaled by the noise variance σ 2 p : R p τ) = E [ pt)p T t + τ) ] = σ 2 pδτ)i 5) where δτ) is the Kronecker delta. The m m covariance matrix R y τ) of the observations is given by: R y τ) = E [ yt)y T t + τ) ] = E [ Ast)+pt))Ast + τ)+pt + τ)) T] = E [ Ast)s T t + τ)a T + pt)p T t + τ) ] = AR s τ)a T + σ 2 pδτ)i 6) = R x τ)+σ 2 p δτ)i where R x τ) is the covariance matrix of the signal vector. III. THE SOBI ALGORITHM The method for blind separation described in [] is based on two steps: whitening and rotation. The whitening step requires an estimation of the noise variance from the observations. A. Estimation of the Noise Variance The first step is to estimate the noise variance σ 2 p. This can be done by applying 6) for τ = 0 which is the only time instant where the noise variance is a part of the equation: R y 0) = AR s 0)A T + σ 2 p δ0)i = AIA T + σ 2 pi = AA T + σ 2 p I 7) Since AA T is a symmetric matrix it is also orthogonally diagonalizable [3, pp. 445]. If we orthogonally diagonalize AA T we can rewrite 7) as: R y 0) = UDU T + σ 2 pi 8) where the column vectors of U are the orthonormal eigenvectors of the positive definite matrix AA T. The corresponding positive and sorted eigenvalues [3, pp. 456]: are in the diagonal matrix λ λ n λ m 0 9) D = diag[λ,λ 2,...,λ n,λ n+,...,λ m ] 0) Understanding that the right side of 8) is, in fact, an orthogonal diagonalization of R y 0) becomes more easy when the equation is written in a different way: R y 0) = UDU T + Uσ 2 p IUT = UD+σ 2 pi)u T = UVU T ) From ) we observe that the eigenvalues v ii in the diagonal of V contain the noise variance: v ii = λ i + σ 2 p 2) From our assumptions the mixing matrix A has full rank equal to n). Since the column vectors of AA T are linear combinations of the column vectors of A we can derive that: λ i = 0, i > n 3) This means that the last m n eigenvalues in V are: v ii = σ 2 p, i > n 4) Thus the noise variance can be estimated by calculating an average of the last m n eigenvalues v ii in V, that is: B. Whitening σ 2 p = m n m i=n+ v ii 5) From the definition of the transform in 2) we see that the n-dimensional source vector is mapped from R n to an m- dimensional signal vector in R m. Disregarding the noise vector we notice that the signal vector is a linear combination of the n linearly independent m-dimensional column vectors of A. Since there are only n linearly independent column vectors they only span R n and not all of R m. As a consequence the representation of the source vector will contain redundant information when m > n. Another problem is that the source vector is mapped from an orthogonal coordinate system to a coordinate system which is not guaranteed to be orthogonal. To eliminate the redundant information and to ensure that the source vector is mapped into an orthogonal coordinate system we introduce the whitening matrix W. The n m whitening matrix W is defined in the following way: WR x 0)W T = I WAR s 0)A T W T = I WAIA T W T = I WAA T W T = I 6) such that the transformed sources are uncorrelated at time lag τ = 0. Thus, the whitening matrix has the following property: WA)WA) T = WAA T W T = I 7) If we observe what happens when the whitening matrix is applied to the observation vector: Wyt) = Wxt)+Wpt) = WAst)+Wpt) 8)

3 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK 3 we see that the source vector is now mapped from R n to R n no redundant information) and that the transformation of the signal vector is orthogonal since WA is orthogonal). To derive an expression for the whitening matrix we continue by substituting AA T in 7) with UDU T : WUDU T W T = I 9) The last m n eigenvalues in the matrix D are zero. Thus we can reduce the size of D and U without affecting the diagonalization of AA T, that is: AA T = U r D r U T r 20) where the n n reduced eigenvalue matrix D r is: D r = diag[λ,λ 2,...,λ n ] 2) Column vector j in the m n matrix U r is the eigenvector corresponding to the eigenvalue λ j. The matrix V is reduced in a similar way, such that: We substitute UDU T in 9) with U r D r U T r and replace D r with V r σ 2 pi: V r = D r + σ 2 p I 22) and get: WU r D r U T r W T = I 23) WU r Vr σ 2 p I) U T r WT = I 24) Rearranging the equation such that it reads: WU r Vr σ 2 pi ) 2 V r σ 2 pi ) 2 U T r W T = I 25) [WU r Vr σ 2p I) 2 ][ WU r Vr σ 2 p I) 2 ] T = I 26) we see that it is possible to obtain a closed analytical expression for the whitening matrix by solving the following equation: WU r Vr σ 2 p I) 2 = I 27) under the assumption that an estimate of the noise variance is known. The matrices U r and V r can be obtained by orthogonally diagonalizing R y 0). The expression V r σ 2 pi ) 2 is calculated by taking the square root of each entry in the diagonal matrix V r σ 2 pi. As a conclusion we derive the following expression for the whitening matrix: C. Estimation of the Mixing Matrix W = V r σ 2 p I) 2 U T r 28) The final step in recovering the sources is to estimate the mixing matrix A. To get an estimate of the mixing matrix we choose to investigate the whitened covariance matrix R z τ) at different time lags τ 0: R z τ) = WR y τ)w T = WAR s τ)a T W T = WAR s τ)wa) T = QR s τ)q T, τ 0 29) and realize that the right side of the equation is an orthogonal diagonalization of R z τ). Consequently, the key idea here, since W and R y τ) can be estimated, is to solve 29) with respect to Q at K different time lags τ K > τ K > τ K 2 > τ 2 > τ > 0. The set of solutions {Q,Q 2,...,Q K,Q K } will be the basis for calculating an estimate Q of Q and hence also an estimate of the mixing matrix since: Q = ŴÂ 30) We rearrange the equation to obtain an expression for the estimate of the mixing matrix: Â = Ŵ Q 3) where Ŵ is the pseudo-inverse of Ŵ. Since it is beyond the scope of this paper to delve further into how an estimate is obtained, we refer to [] which in greater detail explains how Q is estimated using a technique called joint diagonalization. IV. MIXTURES WITH DELAYED SOURCES When propagation delays are introduced the signal model changes from a model with linear, instantaneous mixtures to a convolutive model. In the following sections we will approximate this convolutive model with a linear, instantaneous model in which the sources can be estimated using second order statistics. A. Convolutive Signal Model Let a set of n unknown sources s t),s 2 t),...,s j t),...,s n t) be linearly combined with unknown delays t i j to form m observations y t),y 2 t),...,y i t),...,y m t). Each observation y i t) is contaminated by unknown additive noise p i t). This model can be expressed mathematically in the following way: y t) = y 2 t) =. y m t) = a s t t )+...+ a n s n t t n )+ p t) a 2 s t t 2 )+...+ a 2n s n t t 2 )+ p 2 t) a m s t t m )+...+ a mn s n t t mn )+ p m t) B. Approximation of a Delayed Source 32) From the definition of the derivative: st + ε) st) ṡt) = lim 33) ε 0 ε we arrive at the following approximation when substituting ε with a small delay t i j : which shows that: t i j ṡt) st t i j ) st) 34) st t i j ) st) t i j ṡt) 35) This approximation corresponds to the Taylor expansion truncated at the first order and is used in [2] where a delay is assumed small if: t i j 36) 2 π fmax where f max denotes the maximum frequency in st).

4 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK 4 C. An Approximation of the Convolutive Signal Model From 35) it is possible to define a linear, instantaneous signal model, which is an approximation of the convolutive signal model. The approximated signal model is defined as: y i t) n j= or equivalently in matrix notation as: a i j s j t) t i j ṡ j t))+ p i t) 37) yt) Ast)+pt) 38) where the m 2n mixing matrix A is defined as: a a n a t σṡ a n t n σṡn a 2 a 2n a 2 t 2 σṡ a 2n t 2n σṡn A = a m a mn a m t m σṡ a mn t mn σṡn 39) The 2n source vector st) contains the sources and their normalized derivatives and is defined as: st) = [s t),s 2 t),...,s n t), ṡt), ṡ2t),..., ṡnt) ] T σṡ σṡ2 σṡn = [s t),s 2 t),...,s n t), s t), s 2 t),..., s n t)] T 40) where s j t) is the derivative of s j t) normalized with respect to the standard deviation σṡ j of ṡ j t). D. Assumptions The sources and noise have the same statistical properties as defined in II-B. Hence the derivatives are zero mean, second order stationary, ergodic, stochastic processes. Moreover, the derivatives are uncorrelated with the noise and also spatially white. Finally it is assumed that there are at least twice as many observations as sources, that is m 2n. The mixing matrix A has full rank equal to 2n. E. Derivations Based on the approximative signal model and the above assumptions the 2n 2n covariance matrix R s τ) is given by: [ ] Rss τ) R R s τ) = s s τ) 4) R ss τ) R s s τ) where the n n quadrant matrices R ss τ),r s s τ),r ss τ) and R s s τ) are defined as: R ss τ) = diag [ R s s τ),r s2 s 2 τ),...,r sn s n τ) ] 42) R s s τ) = diag [ R s s τ),r s2 s 2 τ),...,r sn s n τ) ] 43) R ss τ) = diag [ R s s τ),r s2 s 2 τ),...,r sn s n τ) ] 44) R s s τ) = diag [ R s s τ),r s2 s 2 τ),...,r sn s n τ) ] 45) Considering that the sources are mutually uncorrelated we have that: R ss 0) = I 46) Considering that the derivatives of the sources are mutually uncorrelated we have that: R s s 0) = I 47) From [4, pp. 65] and [4, pp. 44] we know that: and R s j ṡ j = Ṙ s j s j R s j s j τ) = Ṙs j s j τ) σṡ j R s j s j τ) = R s j s j τ) = Ṙs j s j τ) σṡ j 48) Since σ ṡ j Ṙ s j s j τ) is symmetric around τ = 0 we have that R s j s j τ) = σ ṡ j Ṙ s j s j τ) = σ ṡ j Ṙ s j s j τ) = R s j s j τ). Furthermore, we know that R s j s j τ) has a maximum for τ = 0. Based on this we derive that Ṙ s j s j 0) = 0. Combining the two findings we conclude that the covariance matrix R s τ) at τ = 0 is: R s 0) = [ ] I 0 = I 49) 0 I Due to the fact that the noise is temporally and spatially white the m m covariance matrix R p τ) of the noise is 0 at time lags τ 0. At τ = 0 the covariance matrix R p τ) is the identity matrix scaled by the noise variance σ 2 p: R p τ) = E [ pt)p T t + τ) ] = σ 2 pδτ)i 50) where δτ) is the Kronecker delta. The m m covariance matrix R y τ) of the observations is given by: R y τ) = E [ yt)y T t + τ) ] = E [ Ast)+pt))Ast + τ)+pt + τ)) T] = E [ Ast)s T t + τ)a T + pt)p T t + τ) ] = AR s τ)a T + σ 2 pδτ)i 5) = R x τ)+σ 2 pδτ)i where R x τ) is the covariance matrix of the signal vector. V. METHOD In the following text we will show that it is possible to estimate the sources using second order statistics. A. Estimation of the Whitening Matrix and the Noise Variance For τ = 0 we have: R y 0) = AR s 0)A T + σ 2 pi = AA T + σ 2 p I 52) Since 52) is similar to 7) we can estimate the whitening matrix and the noise variance using the method described in section III-A and III-B.

5 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK 5 B. Estimation of the Mixing Matrix For τ 0 the whitened covariance matrix of the observations is: WR y τ)w T = WAR s τ)a T W T 53) The covariance matrix of the sources R s τ) is in general non-diagonal. We note that the entries in 43) and 44) are the off-diagonal entries in R s τ). These entries have been shown to be related to the derivative of the autocorrelation function of the sources. The size of the off-diagonal entries compared to the size of the main diagonal entries depends on the autocorrelation functions of the sources and, thus, on the bandwidth and frequency content of each source. To estimate the mixing matrix we use the same method as described in section III-C. Due to the fact that R s τ) has to be diagonal or close to diagonal in order to use this method the performance of the SOBIDS algorithm will depend on the frequency characteristics of the sources. Normalized Amplitude Original Source Estimated Source Original Source Estimated Source Samples C. Implementation Based on our analysis we introduce a second order blind identification algorithm for delayed sources SOBIDS). SO- BIDS is implemented in the following way: ) Estimate the sample covariance R y 0) from L data samples. Denote by λ,λ 2,...,λ 2n the 2n largest eigenvalues and u,u 2,...,u 2n the corresponding eigenvectors of R y 0). 2) Calculate an estimate of the noise variance σ 2 p as the average of the m 2n smallest eigenvalues of R y 0). Calculate an estimate of the whitening matrix as: Ŵ = [ λ σ 2 p) 2 u,...,λ 2n σ 2 p) 2 ] T u 2n 3) Calculate the sample covariance matrix R z τ) of Ŵyt) for a fixed set of time lags τ {τ j j =,...,K}. 4) Calculate an estimate Q of the orthogonal matrix that is a joint diagonalizer of the set { R z τ j ) j =,...,K}. 5) Calculate an estimate of the sources and their normalized derivatives as ŝt) = Q T Ŵyt). 6) Calculate an estimate of the mixing matrix as  = Ŵ# Q. D. Delay Estimation In the following we describe a method for estimating the relative delays using the estimated source vector ŝt) and the estimated mixing matrix Â. We calculate the derivatives of the sources in ŝt) and then the variance of these derivatives to get an estimate of σṡi. Using these estimates and  it is possible to estimate the delays. Note that we have no information on the order of the entries of ŝt) or the arrangement of the columns in Â. Since ŝt) is calculated using the invers of  we do know that the order of the entries in ŝt) corresponds to the order of the columns of Â. By calculating the derivative of each entry in ŝt) we state that it is possible to do a matching of each estimated source ŝ i t) with the corresponding estimated normalized derivative s i t). This provides sufficient information to calculate the relative delay of each source. Fig. : An example of the goodness factor for two source estimates. The goodness factor is 0.93 and 0.72 for top and bottom respectively. VI. RESULTS In this section the results of our performed test of the SOBIDS and SOBI algorithms will be presented. The intention of this paper is to describe an algorithm based on principles from the SOBI algorithm but with increased performance when estimating sources with small delays. Therefore the algorithms will be tested with delayed sources and the results will be compared. These tests will be performed with noiseless and noise contaminated observations. A. Performance Index To identify and compare the performance of the two algorithms a performance index has been used. This index is based on the cross-correlation of each generated source and the corresponding estimated source and is defined as: G = n n j= max{ R s j ŝ j } 54) where max{ R s j ŝ j } is the maximum value of the normalized and absolute cross-correlation between the original source sequence s j and the estimated source sequence ŝ j. Since there are n sources we also have n maximum values and the performance parameter is therefore an average of these n maximum values. The idea is to study the behavior of G which indicates how well the algorithm performs under different circumstances, e.g. various delay values. We refer to G as the goodness factor. In fig. two different matchings of signals are shown with goodness factors of 0.72 and The maximum value of G is which corresponds to a full correlation of the matched signals.

6 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK 6 B. Simulation Environment The sources used for the simulations are generated by filtering n stationary white noise sources with second order filters with partly overlapping passbands. The observations are generated as linear mixtures of delayed sources. Noise is added for each observation. In section IV-A this mixture is shown. Preliminary tests have been made to identify the parameters which influence the goodness of the algorithms. The tests show that the goodness is almost independent of a number of parameters which therefore were decided to be kept fixed for all simulations. The tests were thus performed with: 3 sources, 4 matrices to be jointly diagonalized, and an overlap ratio of the passband filters of 0% of the filter bandwidth. The filters were created randomly with equal bandwidth. With 3 sources and 0% overlap we will produce filters with a maximum bandwidth of 0.37 f s/2. The number of observations used in the simulation differs from SOBI to SOBIDS. For SOBI the number of observations is n+2 = 5 and for SOBIDS this value is 2n+2 = 8. Tests have shown that this difference is irrelevant for the performance. The parameters essential for the performance of the algorithms are the SNR of the observations and the size of the relative delay of each source. The delay values are randomly generated and the values will be uniformly distributed in an interval from 0 to a defined maximum value. We state that the single parameter which provides the best information on how a single source is delayed in the observations is the standard deviation of the delays for the source. We therefore define the following parameter σ d as the mean of the standard deviations of the delays for each source: σ d = n n j= Vart j,...,t m j ) 55) We refer to σ d as the mean standard deviation of delay. C. Testing the Algorithms with Noiseless Observations To create an ideal environment for testing the performance when using delayed sources no noise is added to the observations when simulating. Fig. 2 shows the result of a simulation with a variation of the mean standard deviation of delay from 0 to 3. The graph for the SOBI algorithm shows that at σ d = 0 the goodness is 0.96 decreasing significantly to G 0.76 at σ d = 0.5. From this point the performance is falling slightly to a goodness of 0.7 at σ d = 3. We note that at zero delay the goodness of the SOBIDS algorithm is 0.97 decreasing to G 0.94 at σ d = 0.. At 0. σ d 0.5 the performance is approximately constant with G From that point the performance is decreasing approximately linearly to G 0.83 at σ d = 3. Comparing the SOBI and the SOBIDS algorithms we notice that with no delay the performance of SOBIDS is slightly better than SOBI. For mean standard deviation of delays of 0.2 the SOBIDS is performing approximately 5% higher than Goodness, G SOBIDS algorithm SOBI algorithm Mean standard deviation of delay, σ d Fig. 2: Result of simulation with noiseless observations with σ d 3. Goodness, G SOBIDS algorithm SOBI algorithm Mean standard deviation of delay, σ d Fig. 3: Result of simulation with noiseless observations for σ d 30. SOBI and for σ d = 0.5 the performance is approximately 22% higher. At a mean standard deviation of delay of 3 the performance of the SOBIDS algorithm is 7% higher than the SOBI. Simulations with σ d up to 30 show that the performance decreases exponentially fig. 3). D. Testing the Algorithms with Noise Contaminated Observations The performance of the SOBI and the SOBIDS is tested with noise contaminated observations to examine the performance of the algorithms in a more demanding applications. The observations have a SNR of 0dB. Fig. 4 shows the result of a simulation with a variation of the mean standard deviation of delay from 0 to 3. The performance of SOBI with noisy observations at σ d = 0 is approximately 0.82 which is 5% lower than with noiseless

7 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK 7 Goodness, G SOBIDS algorithm SOBI algorithm Mean standard deviation of delay, σ d Fig. 4: Simulation result of observations with 0dB SNR for σ d 3. Goodness, G SOBIDS algorithm SOBI algorithm E. Discussion The tests show that the proposed SOBIDS algorithm outperforms SOBI for linear mixtures of delayed sources. Adding noise has an influence on the overall performance characteristics. Since the full effect of adding noise is far from tested in details this is an interesting subject for further study concerning the robustness and effeciency of the SOBIDS algorithm in real-world cases. The performance of the algorithms with noiseless observations decreases as the relative delay increases. This can be explained from section IV-B concerning Taylor series expansion. 35) is an approximation which is only valid for small delays. The SOBIDS uses the first order Taylor series expansion and SOBI uses a zero order Taylor series expansion which explains why the sources recovered from SOBI become inaccurate at smaller delays than the SOBIDS algorithm. As expected, the performance of both algorithms decreases when noise is added. The larger the delay the smaller the difference. For 0.3 σ d 3 the performance decrease of the simulations made with noisy sources is for the SOBI algorithm between 4% and 8% and for SOBIDS between 7% and 3%. This indicates that SOBIDS is slightly less noise robust than the SOBI algorithm. The first order Taylor series expansion has a positive effect on the performance for relatively small delays. For relatively larger delays it would be interesting to recalculate the expressions for the Taylor series and expanding it to higher orders. This might result in a more constant overall performance. As a concequence more observations would be needed and there is still an unsolved problem of whitening as the second derivative of a source is not uncorrelated with the source even for time lag zero. This is left for further studies Mean standard deviation of delay, σ d Fig. 5: Simulation result of observations with 0dB SNR for σ d 30. observations. As delays become larger the difference in goodness decreases to 0.03 at σ d = 3 which corresponds to a drop in performance of 4%. Also for the SOBIDS the performance decreased when noise was added to the observations. The difference of the goodness between tests performed with noisy observations and with noiseless observations is 0. at zero delay corresponding to a performance decrease of approximately 9%. The goodness decreases to 0.79 at σ d = 0.5 which is a 9% drop in performance. At σ d = 3 the performance is 8% lower than the performance of the tests with noiseless observations. We note that there is a large decrease in performance in the interval 0 σ d 0.3. Simulations with σ d up to 30 show that the performance decreases exponentially shown in fig. 5. The goodness of the SOBIDS algorithm is approximately 0.05 higher than SOBI for σ d > 5. VII. CONCLUSION In this paper we present a novel method for blindly separating mutually uncorrelated sources from a convolutive linear mixture contaminated with white noise. The paper covers a signal model introducing an approximation of sources with small delays as a first order Taylor series expansion. The sources are estimated mainly in two steps, a whitening process in order to make the observations spatially white followed by a rotation of the whitened observations making the principal axes the estimated sources. Tests revealed that the proposed SOBIDS algorithm outperforms SOBI for linear mixtures of delayed sources. An analytical method for estimating the delays has been proposed. REFERENCES [] A. Belouchrani, K. Abed-Meraim, J. F. Cardoso and E. Moulines, "A Blind Source Separation Technique Using Second-Order Statistics", IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 2, FEBRUARY 997 [2] J. Barrère, G. Chabriel, "A Compact Sensor Array for Blind Separation of Sources", IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSŮI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 5, MAY 2002

8 GROUP 743, 7 TH SEMESTER SIGNAL PROCESSING, AUTUMN 2004, AALBORG UNIVERSITY, FREDRIK BAJERS VEJ 7A, 9220 AALBORG ØST, DENMARK 8 [3] D. Lay, "Linear Algebra and Its Applications", 2nd ed., Addison-Wesley Publishing Company, April [4] K. Sam Shanmugan, A. M. Breipohl, "Random Signals - Detection, Estimation and Data Analysis", John Wiley & Sons, 988. [5] J. Herault, C. Jutten and B Ans., "D etection de grandeurs primitives dans un message composite par une architecture de calcul neuromim etique en apprentissage non supervis e.", Proc. Gretsi pp , 985. Jakob Ashtar was born in 972. He is a student at the 7th semester Signal Processing at Aalborg University. Kristoffer Møller Jørgensen was born in 979. He is a student at the 7th semester Signal Processing at Aalborg University. Jacob Lindvig was born in 973. He is a student at the 7th semester Signal Processing at Aalborg University. Lars Juul Mikkelsen was born in 98. He is a student at the 7th semester Signal Processing at Aalborg University. Jakob Birkedal Nielsen was born in 979. He is a student at the 7th semester Signal Processing at Aalborg University. Lars Sommer Søndergaard was born in 980. He is a student at the 7th semester Signal Processing at Aalborg University. Dario Farina Supervisor) was born in 973. He Graduated from Politecnico di Torino in 998. In 200 he obtained his PhD in Electronics Engineering and Communications at Politecnico di Torino and at the Ecole Centrale de Nantes, Nantes France).