Lambert, R. H. (1996). Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. Ph.D. thesis, USC.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the data is whitened prior to measuring sparseness. The connection between (6) and the normalized cumulant based kurtosis # # 4 is the following # # 4 (x) E x white 3. 7) Instead of the standard normalized kurtosis, we can use the generalized normalized kurtosis (or Gray s variable norm) [5, 8], defined as # p,q (x) E x E q x p q c pq , 8) where c pq is a positive constant, such that, for the Gaussian distribution # p,q =0and p, q are chosen suitably positive (typically, q =2and p =1, 3, 4, 6) In the special case for p =4, q =2and c pq =3, the generalized kurtosis ....

R. H. Lambert. Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures. Ph.D. dissertation, University of Southern California, 1996.

....with respect to leads to (5) If we divide (5) by (4) and flip the sign we get (6) 1045 9227 01 10.00 2001 IEEE IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 12, NO. 3, MAY 2001 619 For unit variance, we can find from the general expression for the th order moment of a generalized Gaussian signal [4] (7) is the gamma function given by .For , 7) gives (8) or, for unit variance, we have (9) Inserting this value for into (6) yields the activation function (10) Apart from a scaling constant, this nonlinearity has also been derived in [5] Using (see, for example, 6] leads to (11) Both ....

R. H. Lambert, "Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. dissertation, Univ. Southern California, Los Angeles, 1996.

....that has delayed taps, enough large to 1 take into account the di erences of the delay of arrival. Then the task of multichannel signal separation is to recover sources from their convolutive mixtures. This task can be done in either time domain [17, 3, 7, 2, 18, 5, 4] or in frequency domain [10, 11, 15]. Although the frequency domain approach enjoys the computational power of FFT, it requires the block processing that causes system latency. Moreover di erent permutations occur in each frequency bin, which might result in severe performance degradation. Some methods [13, 14] were proposed to ....

R. H. Lambert. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, University of Southern California, May 1996.

....to be applied in eq. 9) 3 Circular Filtering The formulation of the FIR filtering as a matrix multiplication is not the only one possible. Therefore, we will now adopt a different argumentation and will show that eq. 9) is correct only for minimum phase filters. As have been shown by Lambert [6] and [3] the deconvolution task can also be formulated using so called quadratic circular matrices 1 B instead of the Toeplitz matrices used so far. Compared to the earlier work on CM the following conduction gives a new interpretation of the objective function with respect to the filter ....

R. H. Lambert. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, University of Southern California, Dep. of Electrical Eng., 1996.

....MCBD can be either approached from the BSS side by replacing the scalars in the matrices by filter polynomials or from the BD side by replacing the single filter polynomial by a matrix of filter polynomials. Merging the two equations (6) and (8) and borrowing the FIR matrix algebra notations from [8] we get W k 1 =W k 1 I (1 )1 u H k y k 1 y k u H k W k . 17) denotes an FIR matrix or vector [8] whose elements are filter vectors in the frequency domain, e.g. W] il = w il . 1 is a 11 FIR matrix containing all ones, i.e. 1 . 1] and I = ....

....the single filter polynomial by a matrix of filter polynomials. Merging the two equations (6) and (8) and borrowing the FIR matrix algebra notations from [8] we get W k 1 =W k 1 I (1 )1 u H k y k 1 y k u H k W k . 17) denotes an FIR matrix or vector [8], whose elements are filter vectors in the frequency domain, e.g. W] il = w il . 1 is a 11 FIR matrix containing all ones, i.e. 1 . 1] and I = diag(1, 1) 0 50 100 0.1 0 0.1 0.2 0.3 p 11 (q) 0 50 100 0.1 0 0.1 0.2 0.3 p 12 (q) 0 50 100 0.1 0 0.1 0.2 0.3 taps p 21 (q) 0 50 ....

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R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996.

.... e jsj : 52) If we divide (52) by (6) and flip the sign we get g(s) p 0 S (s) pS (s) jsj 1 sign(s) jsj 1 sign(s) 53) For unit variance, we can find from the general expression for the nth order moment of a generalized Gaussian signal [16] E(jXj n ) n 1 ) 1 ) n : 54) For n = 2, Eq. 54) gives E(jXj 2 ) 3 ) 1 ) 2 : 55) or, for unit variance, we have = s ( 1 ) 3 ) 56) Inserting (56) into (53) yields (7) Using (x) 1 x) sin( x) from [17] leads to g(s) ....

R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996.

....and reverberation. Thus it is reasonable to model the propagation characteristics as an multivariate FIR lter. Then the task of multichannel signal separation is to recover sources from their convolutive mixtures. This task can be done in either time domain [9, 2, 1, 3] or in frequency domain [5]. Although frequency domain approach enjoys the computational power of FFT, it requires the block processing that causes system latency. Moreover di erent permutation occurs in each frequency bin, which might results in severe degradation of performance without special case [7] In this paper we ....

R. H. Lambert. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, University of Southern California, May 1996.

....is maintained [22] Moreover, since no other processing is required, 7) and (8) are ideal for extension to adaptive paraunitary systems. To develop procedures similar to (7) and (8) to solve (1) 3) we rely on recent works relating instantaneous blind source separation and blind deconvolution [23] [25] These works have indicated that spatial only adaptive algorithms can be extended to adaptive single and multichannel linear systems if the following three rules are followed: 1. Multiplication of two matrices in the spatial only case is equivalent to convolution of their associated ....

R.H. Lambert, "Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. thesis, University of Southern California, Los Angeles, CA, May 1996.

.... although other (y) are not guaranteed to work [9] di erent (y) can provide more ecient convergence behavior in some cases [10] Several authors have noted that ICA and blind deconvolution are related problems, to the extent that ICA methods can be translated to the blind deconvolution task [11] [13] In this case, x(k) is prewhitened by a prewhitening lter with impulse response p j , 0 j N , to produce the signal v(k) such that Efv(k)v (k j)g j , N j N . Then, y(k) is computed as y(k) L X l=0 w l (k)v(k l) 9) and each w l (k) is adjusted over time to satisfy ....

....2 y Figure 2: Evolution of EfISI(i)g for the various block update algorithms. 10] S.C. Douglas and S. Y. Kung, Design of estimation de ation approaches to independent component analysis, Proc. 32nd Asilomar Conf. Signals, Syst. Comput. Paci c Grove, CA, vol. 1, pp. 707 711, Nov. 1998. [11] R.H. Lambert, Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures, Ph.D. dissertation, Univ. Southern California, Los Angeles, CA, May 1996. 12] S.C. Douglas and S. Haykin, On the relationship between blind deconvolution and blind source separation, ....

R.H. Lambert, \Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. dissertation, Univ. Southern California, Los Angeles, CA, May 1996.

....time domain becomes [4] We may keep the same form of the equation 4 for the separation filter by moving into the frequency domain representation where the elements of the above matrices are filter coefficients. Then the multiplication operation replaces the convolution property in (4) Lambert [5] showed that FIR polynomial matrix algebra can be used as an efficient tool to solve problems easily in the frequency domain. From the above, the learning rule of equation 4 can be reformulated as follows: We must note that in the above formula, the non linear function operates in the time domain ....

Lambert R.: "Multi channel blind deconvolution: FIR matrix algebra and separation of multipath mixtures". PhD Thesis, University of Southern California, Dept. Of Electrical Engineering, (1996).

....universal method in [4] as Efjy(k)j 2 g = jjw(k)jj 2 (14) for prewhitened signal mixtures. In fact, several authors have noted that blind deconvolution and blind source separation are closely related, to the extent that ICA methods can be e ectively translated to the blind deconvolution task [19] [23] In this case, x(k) is prewhitened by a prewhitening lter with impulse response p i (k) 0 i N , to produce the prewhitened signal v(k) such that Efv(k)v (k l)g = l ; N l N: 15) 2 Then, the coecients of an FIR equalizer given by y(k) L X l=0 w l (k)v(k l) 16) are ....

R.H. Lambert, \Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. dissertation, Univ. Southern California, Los Angeles, CA, May 1996.

....in Eq. 3) but by the natural gradient [2] resulting in the modified learning rule [3] W k 1 = W k Gamma I Gamma OE(u)u T Delta W k : 5) Based on the update equation (5) the solution to the multichannel blind deconvolution task can be obtained following the rules given by Lambert [4]: replace scalar weights by appropriate polynomials in the complex variable z, multiplications by convolutions, and transpose by hermitian . This directly yields the learning rule [5] W k 1 (z) W k (z) Gamma I Gamma h(u(z) u T (z Gamma1 ) Delta W k (z) 6) with h i (u i (z) ....

Russell H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of South California, May 1996.

....k (dotted) for a blind source separation problem with four binary distributed sources, OE(u) sgn(u) batch size B = 100, and step size = 0:01. In the notation for SCBD problems we denote convolution of sequences by multiplication of polynomials in z. The standard algorithm for SCBD is given by [10] w k 1 (z) w k (z) k B X t Gamma 1 Gamma OE(u t (z) u t (z Gamma1 ) Delta w k (z) 70) and can be examined for stability and steady state error with the same method as was done above for BSS. The equivariance property which guarantees that the satisfactory level for convergence ....

....MCBD problem, the observation signal vector x is a result of mixing and convolving the source signals. The sources have to be non Gaussian and spatially as well as temporally indepen dent. Based on the update equation (9) the solution to the MCBD task is obtained by following the rules of Lambert [10]: ireplace scalar weights by appropriate polynomials in the complex variable z, multiplications by convolutions and the transpose operation by the hermitian onej. This directly yields the learning rule [ W k 1 (z) W k (z) k B X t Gamma I Gamma OE(u t (z) u T t (z Gamma1 ) ....

Russell H. Lambert. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, University of South California, May 1996.

.... tFu v jxw y (52) If we divide (52) by (6) and flip the sign we get B = m 9 MUP = m M N = m g g p t j W X L Y = m n p g zg t j W X LTY = m i (53) For unit variance, we can find p from the general expression for the th order moment of a generalized Gaussian signal [16] 2 = g g U F q t q = t p (54) For o , Eq. 54) gives 2 = g g x q = t q = t p (55) or, for unit variance, we have p q = q = 56) Inserting (56) into (53) yields (7) Using q = q = S 9 D W X Y = D from [17] leads to B = ....

R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996.

....party problem, wireless communication) the scalar elements of the mixing matrix A have to be re placed by polynomials describing the appropriate convolutions. The solution to this convolutive prob lem can be obtained from the instantaneous mixing case (5) following the rules given by Lambert [4]: replace scalar weights by appropriate polynomials in the complex variable z, multiplications by convo lutions, and transpose by hermitian . This directly yields the learning rule for blind source separation of convolutive mixtures, also known as blind multi channel deconvolution [5] W k 1 (z) ....

Russell H. Lambert, Multichannel Blind Decon volution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of South California, May 1996.

.... If we divide (52) by (6) and flip the sign we get g(s) Gamma p 0 S (s) pS (s) ff jsj fi ff Gamma1 sign(s) fi = ff fi ff ffjsj ff Gamma1 sign(s) 53) For unit variance, we can find fi from the general expression for the nth order moment of a generalized Gaussian signal [16] E(jXj n ) Gamma( n 1 ff ) Gamma( 1 ff ) fi n : 54) For n = 2, Eq. 54) gives E(jXj 2 ) Gamma( 3 ff ) Gamma( 1 ff ) fi 2 : 55) or, for unit variance, we have fi = s Gamma( 1 ff ) Gamma( 3 ff ) 56) Inserting (56) into (53) yields (7) Using Gamma(x) ....

R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996.

....similar in nature for a given probability distribution of the signals to separate. In fact, the exact curve of the nonlinearity might not matter [7] Whereas the minimization of the mutual information leads to a pdf independent polynomial with several terms [5] both Infomax and Bussgang technique [8] lead to g(u i ) # log p S (u i ) #u i = p 0 S (u i ) p S (u i ) 3) where p S (u ) and p S (u i ) are the pdf and its derivative, respectively, of the source signals. 3) is referred to as the score function of a certain pdf p S ( We assume the same probabilistic model for all source ....

R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996.

....(1998) Lewicki and Olshausen (1999) or by variational methods (Jordan et al. 1999) It would be interesting to compare these possibility to the other methods presented in this paper. Another important direction is towards the problem of simultaneous blind deconvolution and separation, as in Lambert (1996). In this case the matrices A and W will have linear NC 2079, Zibulevsky, Sparse Decomposition 14 Cardoso s JADE Fast ICA BS Infomax SpectInfomax SpectBFGS (8.8 ) 8.6 ) 7.1 ) 0.9 ) 0.67 ) Figure 11: Percent relative error of separation of seven musical sources recovered by (1) ....

Lambert, R. H. (1996). Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, USC.

....or by a matching Gaussian, in the spirit of [21, 20] or by variational methods [17] It would be interesting to compare these possibility to the other methods presented in this paper. Another important direction is towards the problem of simultaneous blind deconvolution and separation, as in [18]. In this case the matrices A and W will have linear lters as an elements, and multiplication by an element corresponds to convolution. Even in this matrix of lters context, most of the formulae in this paper remain valid. 8 Conclusions We showed that the use of sparse decomposition in a ....

R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, PhD thesis, USC, 1996.

....the source signals are unknown. The available microphone signals only contain a mixture of these. Several researchers believe that Higher Order Statistics (HOS) are necessary for signal separatio(e.g. 1] 2] 3] so many existing algorithms are quite complex because of using HOS explicitly [1][4] or implicitly [5] 6] Another e#ective method to separate signals is beamforming technique [7] which is based on taking advantage of di#erent location information of di#erent sources. Unfortunately, in many applications, when the speakers are in a typical room, like an o#ce, a beamforming ....

R.H.Lambert, Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures,Univer- sity of Southern California, Ph.D. Thesis, May, 1996.

....g(s) ps(s) s) ps(s) Equalization algorithms are derived by finding a filter that minimizes the difference between the output and the true source estimated through g. LMS is typically the criterion. Extensions for multichannel blind deconvolution and separation were presented by Lambert [42, 43, 45]. Coupled with FIR matrix algebra [46, 43] efficient separation methods seem to result. See also [106, 107] for application of these methods for the overdetermined mixing case. It is notable that the nonlinearity has the same exact form as in the entropy maximization, and maximum likelihood ....

....are derived by finding a filter that minimizes the difference between the output and the true source estimated through g. LMS is typically the criterion. Extensions for multichannel blind deconvolution and separation were presented by Lambert [42, 43, 45] Coupled with FIR matrix algebra [46, 43] efficient separation methods seem to result. See also [106, 107] for application of these methods for the overdetermined mixing case. It is notable that the nonlinearity has the same exact form as in the entropy maximization, and maximum likelihood approaches, leading to similar separation ....

R. H. Lambert. Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures. PhD dissertation, University of Southern California, Department of Electrical Engineering, May 1996.

....is an important research topic with a vast literature. We shall here describe only a special case of the problem that is closely connected to ours. In blind deconvolution, a convolved version x(t) of a scalar signal s(t) is observed, without knowing the signal s(t) or the convolution kernel [126, 42, 54, 53, 92, 129, 130, 148]. The problem is then to nd a separating lter h so that s(t) h(t) x(t) An illustration can be found in Fig. 3. The equalizer h(t) is assumed to be a FIR lter of suOEcient length, so that the truncation eoeects can be ignored. A special case of blind deconvolution that is especially ....

....formulation is only approximative, the exact formulation using linear lters would lead to essentially the same algorithms and convergence proofs. Also blind separation of several convolved signals ( multi channel deconvolution ) can be represented combining these two approaches, see, for example, [41, 123, 137, 149, 150, 134, 92]. 3.5.4 Other applications Due to the close connection between ICA and projection pursuit on the one hand, and between ICA and factor analysis on the other, it should be possible to use ICA on many of the applications where projection pursuit and factor analysis are used. These include ....

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R. H. Lambert. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, Univ. of Southern California, 1996.

....signal separation in the frequency domain. The FDODF can be derived from the FT of the TDODF adaptation equations, which results in: 2 1 21 1 2 12 f f f f H H U U W U U W D D (2) where H denotes conjugate transposition and # is the learning rate. Lambert [7] showed that FIR polynomial matrix algebra can be used as an efficient tool to solve problems easily in the frequency domain where the convolution operations of the time domain are replaced by matrix multiplications. It can be observed that the above learning rule is in the same form of the least ....

Lambert R.: "Multi channel blind deconvolution: FIR matrix algebra and separation of multipath mixtures". PhD Thesis, University of Southern California, Dept. Of Electrical Engineering, (1996).

....of the filter operation in Eq. 2) an approximate solution to the minimum entropy deconvolution can be obtained [2] In that paper a triangular Toeplitz matrix has been used to express the filter operation. Later a different matrix formulation based on circular matrices has been proposed [7, 4]. However, the relation between both methods and the implications of the different approximations remain unclear. In the following investigation we will show that the matrix expression of Eq. 2) that is based on a Toeplitz matrix is only suitable if the unknown filter fa k g is constrained to be ....

....to be applied in Eq. 10) 3. CIRCULAR FILTERING The formulation of the FIR filtering as a matrix multiplication is not the only one possible. Therefore, we will now adopt a different argumentation and will show that Eq. 10) is correct only for minimum phase filters. As have been shown by Lambert [7] and [4] the deconvolution task can also be formulated using so called quadratic circular matrices 1 B instead of the Toeplitz matrices used so far. Compared to the earlier work on CM the following conduction gives a new interpretation of the objective function with respect to the filter ....

R. H. Lambert. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, University of Southern California, Department of Electrical Engineering, 1996.

....exactly to the separating solutions. In experiments, local optimization of J 1 usually provides the separated sources. 3. 2 Relationship with Bussgang criterion and entropy based contrasts The form Efky Gamma g(y)k 2 g is quite similar to the Bussgang cost function used in blind equalization [19, 9, 10]. We use Lambert s notation and approach [19] where the nonlinearity is chosen to be g(y) GammaEfjyj 2 gp 0 y (y) p y (y) 13) In (13) p y (y) is the density of y and p 0 y (y) its derivative. In particular, Lambert [19] has given various algorithms for minimizing the cost function ....

....local optimization of J 1 usually provides the separated sources. 3. 2 Relationship with Bussgang criterion and entropy based contrasts The form Efky Gamma g(y)k 2 g is quite similar to the Bussgang cost function used in blind equalization [19, 9, 10] We use Lambert s notation and approach [19], where the nonlinearity is chosen to be g(y) GammaEfjyj 2 gp 0 y (y) p y (y) 13) In (13) p y (y) is the density of y and p 0 y (y) its derivative. In particular, Lambert [19] has given various algorithms for minimizing the cost function Efky Gamma g(y)k 2 g. However, Lambert s ....

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R. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. Ph.D. dissertation, Univ. of Southern California, Dept. of Electrical Eng., May 1996.

....(8) one can easily see that in this case the criterion J 1 (W) actually depends on sixth order (and higher) statistics. 3.3 Relationships to Other Approaches 3.3. 1 Blind Equalization Using Bussgang Approaches The form Efky Gamma g(y)k 2 g is similar to the Bussgang blind equalization cost [18, 39], in which the nonlinearity is chosen to be g(x) GammaEfjxj 2 gp 0 x (x) p x (x) 11) where p x is the probability density function of x. Since the mixtures are whitened, the expectation Efjxj 2 g = 1, and for p x (x) 1= cosh(x) we get g(x) sinh(x) cosh(x) cosh 2 (x) tanh(x) ....

....nonlinear PCA learning rule. The reasons for this are discussed in [40] Although in the Bussgang blind equalization cost it is assumed that the density p x is known, it has been shown that it is not necessary to exactly match the nonlinearity with the source density [36] Based on Lambert s work [39], we have also shown in [40] that the minimization of the cost Efky Gamma g(y)k 2 g can be interpreted as nding an extremal point of the sum of negentropies J = X i Eflog p y i (y i ) log pG (y i )g Here p y i is the probability density function of the i:th output signal y i and pG is the ....

R. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. Ph.D. dissertation, Univ. of Southern California, Dept. of Electrical Eng., May 1996.

.... characteristics of a room can be modeled as a finite impulse response (FIR) filter and convolved with the original sound source to simulate the signal recorded by a microphone [11] Several researchers have extended blind source separation algorithms to cope with delayed and convolved sources [15, 8, 9, 4]; most of these algorithms have only implemented NxN configurations, using N sensors to separate out N sources. Lambert [8] implemented an MxN example with more sensors than sources. Researchers often use beamforming microphone arrays when recording sounds in a reverberant environment. ....

....simulate the signal recorded by a microphone [11] Several researchers have extended blind source separation algorithms to cope with delayed and convolved sources [15, 8, 9, 4] most of these algorithms have only implemented NxN configurations, using N sensors to separate out N sources. Lambert [8] implemented an MxN example with more sensors than sources. Researchers often use beamforming microphone arrays when recording sounds in a reverberant environment. Beamforming arrays target their sound capture toward a desired spatial area, improving upon the signal to noise ratio (SNR) of the ....

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Lambert, Russell H. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD Thesis, University of Southern California, 1996.

....approach, the other an equalizer structure. Either structure speeds up the separation of the remaining unknown sources considerably. Both approaches can be extended to the multichannel blind deconvolution case where the elements of matrix A are filter polynomials by using FIR matrix algebra [8]. In this case the mixing matrix A(z)aswellasthe separation matrix W(z) contain filter polynomials as their matrix elements. ACKNOWLEDGEMENT The authors would like to thank Russell Lambert and Te Won Lee for helpful discussions. 6. ....

R.H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures,Ph.D.the- sis, University of Southern California, 1996.

....filtering imposed on the sources by the room environment, propagation delays and wall reflections. Under these circumstances we have to use a more general mixing scenario known as convolutive mixture. To adequately express the mixing phase we make use of the FIR Matrix Algebra proposed by Lambert [10]. Using its notation, we can express the convolved mixing case in the form: t t s A x = 2) where A is the mixing matrix with each element being a FIR filter. For example, for A 2x2 , equation (2) gives: 2 22 1 21 2 2 12 1 11 1 t s t a t s t a ....

Lambert R.: "Multi channel blind deconvolution: FIR matrix algebra and separation of multipath mixtures". PhD Thesis, University of Southern California, Dept. Of Electrical Engineering , 1996.

....MCBD can be either approached from the BSS side by replacing the scalars in the matrices by filter polynomials or from the BD side by replacing the single filter polynomial by a matrix of filter polynomials. Merging the two equations (6) and (8) and borrowing the FIR matrix algebra notations from [8] we get W k 1 =W k 1 # I # (1 )1 u H k y k # 1 y k u H k # W k . 17) denotes an FIR matrix or vector [8] whose elements are filter vectors in the frequency domain, e.g. W] il = w il .1is a 1 1FIR matrix containing all ones, i.e. 1. 1] and I = ....

....the single filter polynomial by a matrix of filter polynomials. Merging the two equations (6) and (8) and borrowing the FIR matrix algebra notations from [8] we get W k 1 =W k 1 # I # (1 )1 u H k y k # 1 y k u H k # W k . 17) denotes an FIR matrix or vector [8], whose elements are filter vectors in the frequency domain, e.g. W] il = w il .1is a 1 1FIR matrix containing all ones, i.e. 1. 1] and I = diag(1, 1) 0 50 100 0.1 0 0.1 0.2 0.3 p 11 (q) 0 50 100 0.1 0 0.1 0.2 0.3 p 12 (q) 0 50 100 0.1 0 0.1 0.2 0.3 taps p 21 (q) 0 50 100 0.1 0 ....

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R.H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures,Ph.D.the- sis, University of Southern California, 1996.

....for example, mixtures of white sources can be separated only by exploiting higher orders as in DCA I. Torkkola (1996) proposed the ordinary gradient rule corresponding to the frequency domain version of (39) and its relative gradient form was described by Lee, Bell and Lambert (1997) see also Lambert 1996); in the absence of a spatio temporal spectral error function, both relied on information maximization considerations in the frequency domain. A rule similar to (39) appeared in (Cochocki et al. 1996) Methods that use cumulant information in the frequency domain (i.e. polyspectra) were suggested ....

Lambert, R. (1996). Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures. PhD Thesis, University of Southern California.

....matrix the harder the separation task is for algorithms that do not exhibit the equivariant behaviour [1, 3] In the presence of noise the task becomes harder also for equivariant algorithms. The level of difficulty can be controlled by adjusting the eigenvalue spread of the mixing filter matrix [11]. 2. There is a continuum from instantaneous mixing to delayed mixing, i.e. convolutive mixing with only one nonzero coefficient per filter. This can be used to measure the ability of an algorithm to deal with simple convolutive mixing. 3. There is also a continuum from delayed mixing to real ....

Russell H. Lambert. Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures. PhD dissertation, University of Southen California, Department of Electrical Engineering, May 1996.

....and speech processing area due to their prospective application in speech recognition, telecommunications and medical signal processing. The blind source separation problem has been studied by researchers in the field of neural networks [1, 2, 5, 9, 10, 16, 17] and statistical signal processing [3, 7, 11, 15, 19]. Comon [7] defines the concept of independent component analysis (ICA) which measures the degree of independence among outputs using contrast functions approximated by the Edgeworth expansion of the Kullback Leibler divergence. The higher order statistics is approximated by cummulants up to 4th ....

....the weight update. Another way of dealing with filters in a multichannel representation is to transform the learning rules into the frequency domain where the convolution and deconvolution property becomes a multiplication and division operation. In particular, the use of FIR polynomial techniques [11] present an efficient tool to solve true phase inverse systems allowing a simple implementation of noncausal filter solutions. The significance of the methods is shown by the successful separation of two voices and separating a voice that has been recorded with loud music in the background. We ....

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R. Lambert. Multichannel blind deconvolution: Fir matrix algebra and separation of multipath mixtures. Thesis, University of Southern California, Department of Electrical Engineering, May 1996.

....this approach to a feedback system with only cross filters. A full filter feedback system is presented in [11] and [4] Since feedback systems are limited to minimum phase mixing systems, the general assumption of non minimum phase systems can be overcome by using a feedforward unmixing system [10, 12, 8, 5]. The infomax algorithm has been used to separate voices recorded in real environments [10, 11, 6] A simple time delayed decorrelation (TDD) algorithm [14] has been shown to be highly effective under the minimumphase constraint. The TDD algorithm can in some circumstances achieve the same ....

.... in [11] and [4] Since feedback systems are limited to minimum phase mixing systems, the general assumption of non minimum phase systems can be overcome by using a feedforward unmixing system [10, 12, 8, 5] The infomax algorithm has been used to separate voices recorded in real environments [10, 11, 6]. A simple time delayed decorrelation (TDD) algorithm [14] has been shown to be highly effective under the minimumphase constraint. The TDD algorithm can in some circumstances achieve the same separation quality much faster which is important for online implementations. The convergence of the ....

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Lambert, R. (1996). Multichannel blind deconvolution: Fir matrix algebra and separation of multipath mixtures. Thesis, University of Southern California, Department of Electrical Engineering.

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Lambert, R. H. (1996). Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. Ph.D. thesis, USC.

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R. H. Lambert, "Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. dissertation, University of Southern California, 1996.

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R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996. -2 0 2 0 5 10 -2 0 2 0 5 10 -2 0 2 0 5 10 -2 0

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R.H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures,Ph.D.the- sis, University of Southern California, 1996. 142

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R. H. Lambert, "Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. dissertation, University of Southern California, 1996.

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R.H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures,Ph.D.the- sis, University of Southern California, 1996.

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R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996.

No context found.

R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California, 1996.

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R. H. Lambert. Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. PhD thesis, Univ. of Southern California, 1996.

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R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of Southern California,

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R. H. Lambert, "Multichannel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. dissertation, Univ. Southern Calif., Los Angeles, 1996.

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R. H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures,Ph.D.the- sis, University of Southern California, 1996.

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Russell H. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures, Ph.D. thesis, University of South California, May 1996.

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R. Lambert, Multichannel Blind Deconvolution: FIR Matrix Algebra and Separation of Multipath Mixtures. Ph.D. dissertation, Univ. of Southern California, Dept. of Electrical Eng., May 1996.

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R. H. Lambert, "Multi-channel blind deconvolution: FIR matrix algebra and separation of multipath mixtures," Ph.D. dissertation, Dept. Elec. Eng., Univ. Southern California, Los Angeles, 1996.

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Lambert, R. ~1996!. Multichannel blind deconvolution: FIR matrix algebra 176 T.-P. Jung et al. and separation of multipath mixtures. Thesis, University of Southern California, Department of Electrical Engineering.

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