US 6,845,164 B2Grant

Method and device for separating a mixture of source signals

Issue Date:2005-01-18

•11 Claims

•7 Drawing Sheets

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.

Metadata

Assignee

Telefonaktiebolaget LM Ericsson (PUBL)

Inventor

Tony Gustafsson

Application Information

Application Number:US 09/947,755

Filing Date:2001-09-07

Priority Date:2000-03-07

Art Unit:2644

Patent Drawings (7 sheets)

Description

[00002] This application is a continuation of International Application No. PCT/SE00/00451 filed on Mar. 7, 2000.

Technical Field

[00003] The present inventions relates to a metod and device for separating a mixture of source signals to regain the source signals.

Background of the Invention

[00004] In recent time several papers concerning signal separation of dynamically mixed source signals have been put forward [1-3, 8, 17, 19, 20]. In principle it is possible to separate the sources exploiting only second order statistics, cf. [8]. The blind signal separation problem with dynamic/convolutive mixtures is solved in the frequency domain in several papers presented, cf. [3, 20]. Basically, dynamic source separation in the frequency domain aims to solve a number of static/instantaneous source separation problems, one for each frequency bin in question. In order to obtain the dynamic channel system (mixing matrix), the estimates corresponding to different frequencies bins, have to be interpolated. This procedure seems to be a nontrivial task, due to scaling and permutation indeterminacies [16]. The approach in the present paper is a “time-domain approach”, see [8], which models the elements of the channel system with Finite Impulse Response (FIR) filters, thus avoiding this indeterminacies.

[00005] A quasi-maximum likelihood method for signal separation by second order statistics is presented by Pham and Garat in [11]. An algorithm is presented for static mixtures, i.e. mixing matrices without delays. Each separated signal s_ii=1, . . . , M is filtered with a Linear Time Invariant (LTI) filter h_i. The criterion used is the estimated cross-correlations for these filtered signals. The optimal choice of the filter h_i, according to [11], is the filter with frequency response inversely proportional to the spectral density of the corresponding source signal. The filters h_i, i=1, . . . , M are thus whitening filters. However, the spectral densities of the source signals are usually unknown, and perhaps time varying. One approach is to estimate these filters as done in the present paper and in the prediction error method as presented in [1]. Moreover, several aspects of the algorithm presented in [8] remained open.

Summary of the Invention

[00006] Characterizing features of the present invention, i.e. the method for separating a mixture of source signals to regain the source signals; are bringing each measured signal to a separation structure including an adaptive filter, the adaptive filter comprising filter coefficients; using a generalised criterion function for obtaining the filter coefficients, the generalised 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 generalised criterion function; and updating the adaptive filter with the filter coefficients.

[00007] Other characterizing features of the present invention, i.e. the device for separating a mixture of source signals to regain the source signals, the input to the device being based on measured signals, the device comprises: signaling links for bringing each measured signal to a separation structure including an adaptive filter, the adaptive filter comprising filter coefficients; a generalised criterion function means for obtaining the filter coefficients, the generalised criterion function means comprising cross correlation functions and a weighting matrix, the cross correlation functions being dependent on the filter coefficients; means for estimating the filter coefficients, the resulting estimates of the filter coefficients corresponding to a minimum value output of the generalised criterion function; and updating means for updating the adaptive filter with the filter coefficients.

[00008] Specific fields of application of the present invention include mobile telephone technology, data communication, hearing aids and medical measuring equipment, such as ECG. Also included is echo cancelling which can primarily occur in the telecommunications field.

Brief Description of the Drawings

[00009] FIGS. 1A-1D show the empirical and true parameter variances of a preferred embodiment of the present invention compared to a prior art signal separation algorithms.

[00010] FIGS. 2A-2D show the estimated mean value as a function of relative frequency of a preferred embodiment of the present invention compared to a prior art signal separation algorithm.

[00011] FIGS. 3A-3D show the parameter variances as a function of relative frequency of a preferred embodiment of the present invention compared to a prior art signal separation algorithm.

Description of Preferred Embodiments

[00012] 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.

[00013] 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,

\begin{matrix} x (n) = {[x_{1} (n) \dots x_{M} (n)]}^{T} = K (q) ξ (n) \\ = diag (\frac{G_{1} (q)}{F_{1} (q)}, \dots, \frac{G_{M} (q)}{F_{M} (q)}) [\begin{matrix} ξ_{1} (n) \\ ⋮ \\ ξ_{M} (n) \end{matrix}], \end{matrix}

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

[00015] A1. The generating signal ξ(n) is a realization of a stationary, white zero-mean Gaussian process:
ξ(n)εN(0, Σ), Σ=diag(σ₁², . . . σ_M²)

[00017] A2. The elements of K(q) are filters which are asymptotically stable and have minimum phase.

[00018] 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.

[00019] 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)=[y₁(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) = [\begin{matrix} 1 & B_{12} (q) & \dots & B_{1 M} (q) \\ B_{21} (q) & ⋰ & ⋮ \\ ⋮ & ⋰ & B_{(M - 1) M} (q) \\ B_{M1} (q) & \dots & B_{M (M - 1)} (q) & 1 \end{matrix}],

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, s₁(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)=[s₁(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=[d₁₁^T. . . d_MM^T]^Twhere 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].

[00025] 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.

[00026] Assuming that N samples of y₁(n) and y₂(n) are available, the criterion function proposed in [8], reads as

\overline{V} (θ) = \sum_{k = - U}^{U} {\overline{R}}_{s_{1} s_{2}}^{2} (k; θ),

where

{\overline{R}}_{s_{1} s_{2}} (k; θ) = \frac{1}{N} \sum_{n = 0}^{N - k - 1} s_{1} (n; θ) s_{2} (n + k; θ), k = 0, \dots, U .

[00028] To emphasize the dependence on θ equation (2.6) can be rewritten as

{\overline{R}}_{s_{1} s_{2}} (k; θ) = {\overline{R}}_{y_{1} y_{2}} (k) - \sum_{i} d_{12} (i) {\overline{R}}_{y_{2} y_{2}} (k - i) - \sum_{i} d_{21} (i) {\overline{R}}_{y_{1} y_{1}} (k + i) + \sum_{i} \sum_{l} d_{12} (i) d_{21} (l) {\overline{R}}_{y_{2} y_{1}} (k - i + l),

where the notation d₁₂(i) denotes the i:th coefficient of the filter D₁₂(q).

[00030] For notational simplicity, introduce the following vector as
{dot over (r)}_N(θ)=[{dot over (R)}, ₁,₂(−U, θ) . . . {dot over (R)},₁,₂(Uθ)]^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) = \frac{1}{2} {\overline{r}}_{N}^{T} (θ) W (θ) {\overline{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

{\hat{θ}}_{N} = \arg \min_{θ} V_{N} (θ) .

[00034] 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];

[00035] 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].
[00036] 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.

[00037] The purpose of the present contribution is to:

- [00038] Find the asymptotic distribution of the estimate of θ_N.
- [00039] Find the weighting matrix W(θ) that optimizes the asymptotic accuracy.
- [00040] Study an implementation of the optimal weighting scheme.

[00041] In addition to A1 and A2 the following assumptions are considered to hold throughout the description:

[00042] A3. Assume that the conditions C3-C6 in [8] are fulfilled, so that the the studied TITO system is parameter identifiable.
[00043] A4. The (minimal) value of U is defined as in Proposition 5 in [8].
[00044] A5. ∥θ∥<∞, i.e. θ₀is an interior point of a compact set D_M. Here, θ₀contains the true parameters.

[00045] 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

- [00046] 1. As N→∞, {dot over (R)},₁,₂(k, θ)→R,₁,₂(k, θ) with probability one (w.p.1). Thus
  V_N(θ)→{overscore (V)}(θ), w.p.1,
  where $\overline{V} (θ) = \frac{1}{2} r^{T} (θ) W (θ) r (θ), r (θ) = {[R_{s_{1} s_{2}} (- U, θ) \dots R_{s_{1} s_{2}} (U, θ)]}^{T} .$

[00049] The convergence in (3.1) is uniform in a set D_M, where θ is a member

\lim_{N -> \infty} \sup_{θ \in D_{M}}  V_{N} (θ) - \overline{V} (θ)  = 0, w . p .1 .

[00050] Furthermore, since the applied separation structure is of finite impulse response (FIR) type, the gradient is bounded

\max_{1 \leq i \leq nθ} {\sup_{θ \in D_{M}} \langle \frac{\partial V_{N} (θ)}{\partial θ_{i}} \rangle} \leq C w . p .1,

for N larger than some N₀<∞. In equation (3.4), C is some constant, C<∞, and nθ denotes the dimension of θ. The above discussion, together with the identifiability analysis [8] then shows the following result:

- [00052] Result 1 As N→∞,
  {overscore (θ)}_N→θ₀w/p.1.

[00054] Having established (strong) consistency, the asymptotic distribution of θ^_Nis considered. Since the θ^_Nminimizes the criterion V_N(θ), V_N(θ^_N)=0, where V_Ndenotes the gradient of V_N. By the mean value theorem,
0=V_N′({dot over (θ)}_N)=V_N′(θ₀)+V_N″(θ_ξ)({dot over (θ)}_N−θ₀),
where θ_ξ is on a line between θ^_Nand θ₀. Note, since θ^_Nis consistent, the θ^_N−θ₀and consequently, θ_ξ→θ₀, as N→∞.

[00057] Next, investigate the gradient evaluated at θ₀(for notational simplicity, let W(θ)=W)

V_{N}^{'} (θ_{0}) = {\hat{G}}^{T} W {\hat{r}}_{N} (θ_{0}) + {\frac{1}{2} [\begin{matrix} {\hat{r}}_{N}^{T} (θ_{0}) \frac{\partial W}{\partial θ_{1}} {\hat{r}}_{N} (θ_{0}) \\ ⋮ \\ {\hat{r}}_{N}^{T} (θ_{0}) \frac{\partial W}{\partial θ_{1}} {\hat{r}}_{N} (θ_{0}) \end{matrix}]}_{θ = θ_{0}} ≃ {\hat{G}}^{T} W {\hat{r}}_{N} (θ_{0}),

where

\hat{G} = \frac{\partial {\hat{τ}}_{N} (θ)}{\partial θ} ❘_{θ = θ_{0}} .

[00059] 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(θ₀) (r^_N(θ₀)). Furthermore, since r_N(θ₀)=0, the asymptotic distribution of V_N′(θ)
is identical to the asymptotic distribution of G^TWr^_N(θ₀), where

G = \frac{\partial τ (θ)}{\partial θ} ❘_{θ = θ_{0}} .

[00061] Applying for example Lemma B.3 in [15], and using the fact that both s₁(n:θ₀) and s₂(n;θ₀) stationary ARMA processes, one can show that (√N)G^TWr^_N(θ₀) converges in distribution to a Gaussian random vector, i.e.
√{square root over (N)}G^TW{dot over (r)}_N(θ₀)εAs N(0, G^TWMWG),
where

M = \lim_{N -> \infty} N E [{\dot{τ}}_{N} (θ_{0}) {\dot{τ}}_{N}^{T} (θ_{0})] .

[00064] This means that the gradient vector, is asymptotically normally distributed, with zero-mean and with a covariance matrix M.

[00065] Before presenting the main result of the current paper, the convergence of the Hessian matrix V″_Nmust be investigated. Assuming that the limit exists, define

{\overline{V}}^{′′} (θ) = \lim_{N -> \infty} V_{N}^{′′} (θ) .

[00066] To establish the convergence of V″_N(θ_ξ), the following (standard) inequality is applied

{ V_{N}^{′′} (θ_{ξ}) - {\overline{V}}^{′′} (θ_{0}) }_{F} \leq { V_{N}^{′′} (θ_{ξ}) - V_{N}^{′′} (θ_{0}) }_{F} + { V_{N}^{′′} (θ_{0}) - {\overline{V}}^{′′} (θ_{0}) }_{F},

where ∥ ∥_Fdenotes the Frobenius norm. Due to the FIR separation structure, the second order derivatives are continuous. Moreover, since θ_ξ converges w.p.1 to θ₀, 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.

[00068] It is, now, straightforward to see that the limiting Hessian V⁻″ can be written as

\begin{matrix} {\overline{V}}^{′′} (θ_{0}) = \lim_{N -> \infty} V^{′′} (θ_{0}) = G^{T} W G & w . p .1 \end{matrix}

[00069] Thus, for large N,
√{square root over (N)}({dot over (θ)}_N−θ₀)≅√{square root over (N)}(G^TWG)⁻¹G^TW{dot over (τ)}N(θ₀),
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.

[00072] Consider the signal separation method based on second order statistics, where θ^_Nis obtained from (2.10). Then the normalized estimation error, √(N)(θ^_N−θ_N), has a limiting zero-mean Gaussian distribution

[heading-00073] √{square root over (N)}({dot over (θ)}_N−θ₀)εAsN(0, P),

[heading-00074] where
P=(G^TWG)⁻¹G^TWMWG(G^TWG)⁻¹.

[00076] 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(θ₀) holds under weaker assumptions.

[00077] 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} = \sum_{p = - \infty}^{\infty} R_{s_{1} s_{1}} (p; θ_{0}) R_{s_{2} s_{2}} (p + τ; θ_{0}) .

[00078] Furthermore, introduce the following Z-transforms

Φ_{1} (z) = \sum_{k = - \infty}^{\infty} R_{s_{1} s_{1}} (k; θ_{0}) z^{- k}, Φ_{2} (z) = \sum_{k = - \infty}^{\infty} R_{s_{2} s_{2}} (k; θ_{0}) z^{- k} .

[00079] Then it follows that

\sum_{τ = - \infty}^{\infty} \sum_{p = - \infty}^{\infty} 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) .

[00080] Thus, the β_τ'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 - {\overline{B}}_{12} (z^{- 1}) {\overline{B}}_{21} (z^{- 1})) {\langle \frac{G_{1} (z)}{F_{1} (z)} \rangle}^{2} {\langle \frac{G_{2} (z)}{F_{2} (z)} \rangle}^{2} .

[00081] 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_τfor τ=0 . . . , 2U, the weighting matrix can, hence, be constructed as

W = [\begin{matrix} β_{0} & β_{1} & \dots & β_{2 U} \\ β_{1} & β_{0} & ⋰ & ⋮ \\ ⋮ & ⋰ & β_{1} \\ β_{2 U} & \dots & β_{1} & β_{0} \end{matrix}] .

[00082] 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

{\dot{x}}_{i} (n) = \frac{1}{\det {D (q, \dot{θ})}} s_{i} (n_{i} {\dot{θ}}_{N}),

as long as det{D(q, θ₀)} is minimum phase.

[00084] 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

\frac{\partial R_{s_{1} s_{2}} (k; θ)}{\partial d_{21} (i)} = - R_{y_{1} y_{1}} (k + i) + \sum_{l} d_{12} (l) R_{y_{2} y_{1}} (k - l + i)

\frac{\partial R_{s_{1} s_{2}} (k; θ)}{\partial d_{21} (i)} = - R_{y_{2} y_{2}} (k - i) + \sum_{l} d_{21} (l) R_{y_{2} y_{1}} (k + i - l) .

which are straightforward to compute.

[00086] 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
W=W₀=M⁻¹

[00088] For this choice of weighting,
P(W₀)=(G^TM⁻¹G)⁻¹

[00090] The accuracy is optimized in the sense that P(W₀)−P(W) is positive semi-definite for all positive definite weighting matrices W.

[00091] The proof follows from well-known matrix optimization results, see for example [9, Appendix 2].

[00092] 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.

[00093] 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.

[00094] Assume that the optimal weighting W₀is applied in the criterion (2.10). Let P_U(W) denote the asymptotic covariance for this case. Then
P_U(W₀)≧P_U-1(W₀)

[00096] The proof follows immediately from the calculations in [15, Complement C4.4]. Note that, when the optimal weighting, W₀is applied the matrices {P_U(W₀)} 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.

[00097] 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_Swhere F_Sis the

[00098] Here the channel system is defined by B₁₂(q)=0.3+0.1q⁻¹and B₂₁(q)=0.1+0.7q⁻¹. The source signal x₁(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.

[00099] In FIGS. 1A-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 FIGS. 1A-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.

[00100] Apparently, it gets more difficult to estimate the channel parameters when the source colors are similar. Therefore, in FIGS. 2A-2D and FIGS. 3A-3D a more careful examination of the parameter accuracy is presented. The angles of the poles are in the interval [40°, 50°]. In FIGS. 2A-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 FIGS. 3A-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 FIGS. 1A-1D, FIGS. 2A-2D, and FIGS. 3A-3D, the present invention increases the quality of the signal separation.

[00101] 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.

References

[none]

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[00117] [16] L. Tong, Y. R. Liu, V. Soon, and Y. Huang. Indeterminancy and identifiability of blind identification.
[00118] [17] S. van Gerven and D. van Compernolle. Signal Separation by symmetric adaptive decorrelation: stability, convergence and uniqueness. IEEE Trans. on Signal Processing, 43:1602-1612, 1995.
[00119] [18] M. Viberg and A. L. Swindlehurst. “Analysis of the Combined Effects of Finite Samples and Model Errors on Array Processing Performance”. IEEE Trans. on Signal Processing, SP-42: 1-12, Nov. 1994.
[00120] [19] H. Wu and J. Principe. A unifying criterion for blind source separation and decorrelation: simultaneous diagonalization of correlation matrices In Proc. of NNSP97, pages 496-505, Amelia Island, Fla., 1997.
[00121] [20] H. Wu and J. Principe. Simultaneous diagonalization in the frequency domain (sdif) for source separation. In ICA, pages 245-250, Aussois, France, 1999.

Claims

1. A method for separating a mixture of source signals to regain the source signals, the method being based on measured signals, the method comprising:

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;

said weighting matrix, being an inverse matrix of a matrix comprising an estimation of a covariant matrix for a signal;

said signal having a spectrum being a product of estimated spectrum of an incoming source-signal and the determinant of an estimated transformation function of a mixing filter;

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.

2. The method according to claim 1, wherein said spectrum of the incoming source-signal is unknown and the weighting matrix considers an assumed spectrum.

3. The method according to claim 1, wherein the weighting matrix is dependent on the filter coefficients.

4. The method according to claim 1, wherein the method is repeatedly performed on the measured signal or on fractions thereof.

5. The method according to claim 4, wherein the method is repeatedly performed according to a predetermined updating frequency.

6. The method according to claim 1, wherein the number of filter coefficients is predetermined.

7. A device for separating a mixture of source signals to regain the source signals, the input to the device being based on measured signals, the device comprising:

signaling links for bringing each measured signal to a separation structure including an adaptive filter, the adaptive filter comprising filter coefficients;

a generalized criterion function means for obtaining the filter coefficients, the generalized criterion function means comprising cross correlation functions and a weighting matrix, the cross correlation functions being dependent on the filter coefficients;

said weighting matrix, being an inverse matrix of a matrix comprising an estimation of a covariant matrix for a signal;

said signal having a spectrum being a product of estimated spectrum of an incoming source-signal and the determinant of an estimated transformation function of a mixing filter;

means for estimating the filter coefficients, the resulting estimates of the filter coefficients corresponding to a minimum value output of the generalized criterion function; and

updating means for updating the adaptive filter with the filter coefficients.

8. The device according to claim 7, wherein the weighting matrix is dependent on the filter coefficients.

9. The device according to claim 7, wherein the device is arranged to separate the measured signals or fractions thereof repeatedly.

10. The device according to claim 9, wherein the device is arranged to separate the measured signals or fractions thereof according to a predetermined updating frequency.

11. The device according to claim 7, wherein the number of filter coefficients is arranged to be predetermined.

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Abstract

Description

[00002] This application is a continuation of International Application No. PCT/SE00/00451 filed on Mar. 7, 2000.

Technical Field

[00003] The present inventions relates to a metod and device for separating a mixture of source signals to regain the source signals.

Background of the Invention

Summary of the Invention

Brief Description of the Drawings

[00009] FIGS. 1A-1D show the empirical and true parameter variances of a preferred embodiment of the present invention compared to a prior art signal separation algorithms.

[00010] FIGS. 2A-2D show the estimated mean value as a function of relative frequency of a preferred embodiment of the present invention compared to a prior art signal separation algorithm.

[00011] FIGS. 3A-3D show the parameter variances as a function of relative frequency of a preferred embodiment of the present invention compared to a prior art signal separation algorithm.

Description of Preferred Embodiments

\begin{matrix} x (n) = {[x_{1} (n) \dots x_{M} (n)]}^{T} = K (q) ξ (n) \\ = diag (\frac{G_{1} (q)}{F_{1} (q)}, \dots, \frac{G_{M} (q)}{F_{M} (q)}) [\begin{matrix} ξ_{1} (n) \\ ⋮ \\ ξ_{M} (n) \end{matrix}], \end{matrix}

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

[00015] A1. The generating signal ξ(n) is a realization of a stationary, white zero-mean Gaussian process:
ξ(n)εN(0, Σ), Σ=diag(σ₁², . . . σ_M²)

[00017] A2. The elements of K(q) are filters which are asymptotically stable and have minimum phase.

B (q) = [\begin{matrix} 1 & B_{12} (q) & \dots & B_{1 M} (q) \\ B_{21} (q) & ⋰ & ⋮ \\ ⋮ & ⋰ & B_{(M - 1) M} (q) \\ B_{M1} (q) & \dots & B_{M (M - 1)} (q) & 1 \end{matrix}],

[00026] Assuming that N samples of y₁(n) and y₂(n) are available, the criterion function proposed in [8], reads as

\overline{V} (θ) = \sum_{k = - U}^{U} {\overline{R}}_{s_{1} s_{2}}^{2} (k; θ),

where

{\overline{R}}_{s_{1} s_{2}} (k; θ) = \frac{1}{N} \sum_{n = 0}^{N - k - 1} s_{1} (n; θ) s_{2} (n + k; θ), k = 0, \dots, U .

[00028] To emphasize the dependence on θ equation (2.6) can be rewritten as

{\overline{R}}_{s_{1} s_{2}} (k; θ) = {\overline{R}}_{y_{1} y_{2}} (k) - \sum_{i} d_{12} (i) {\overline{R}}_{y_{2} y_{2}} (k - i) - \sum_{i} d_{21} (i) {\overline{R}}_{y_{1} y_{1}} (k + i) + \sum_{i} \sum_{l} d_{12} (i) d_{21} (l) {\overline{R}}_{y_{2} y_{1}} (k - i + l),

where the notation d₁₂(i) denotes the i:th coefficient of the filter D₁₂(q).

V_{N} (θ) = \frac{1}{2} {\overline{r}}_{N}^{T} (θ) W (θ) {\overline{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

{\hat{θ}}_{N} = \arg \min_{θ} V_{N} (θ) .

[00035] 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].
[00036] 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.

[00037] The purpose of the present contribution is to:

- [00038] Find the asymptotic distribution of the estimate of θ_N.
- [00039] Find the weighting matrix W(θ) that optimizes the asymptotic accuracy.
- [00040] Study an implementation of the optimal weighting scheme.

[00041] In addition to A1 and A2 the following assumptions are considered to hold throughout the description:

[00042] A3. Assume that the conditions C3-C6 in [8] are fulfilled, so that the the studied TITO system is parameter identifiable.
[00043] A4. The (minimal) value of U is defined as in Proposition 5 in [8].
[00044] A5. ∥θ∥<∞, i.e. θ₀is an interior point of a compact set D_M. Here, θ₀contains the true parameters.

- [00046] 1. As N→∞, {dot over (R)},₁,₂(k, θ)→R,₁,₂(k, θ) with probability one (w.p.1). Thus
  V_N(θ)→{overscore (V)}(θ), w.p.1,
  where $\overline{V} (θ) = \frac{1}{2} r^{T} (θ) W (θ) r (θ), r (θ) = {[R_{s_{1} s_{2}} (- U, θ) \dots R_{s_{1} s_{2}} (U, θ)]}^{T} .$

[00049] The convergence in (3.1) is uniform in a set D_M, where θ is a member

\lim_{N -> \infty} \sup_{θ \in D_{M}}  V_{N} (θ) - \overline{V} (θ)  = 0, w . p .1 .

[00050] Furthermore, since the applied separation structure is of finite impulse response (FIR) type, the gradient is bounded

\max_{1 \leq i \leq nθ} {\sup_{θ \in D_{M}} \langle \frac{\partial V_{N} (θ)}{\partial θ_{i}} \rangle} \leq C w . p .1,

- [00052] Result 1 As N→∞,
  {overscore (θ)}_N→θ₀w/p.1.

[00057] Next, investigate the gradient evaluated at θ₀(for notational simplicity, let W(θ)=W)

V_{N}^{'} (θ_{0}) = {\hat{G}}^{T} W {\hat{r}}_{N} (θ_{0}) + {\frac{1}{2} [\begin{matrix} {\hat{r}}_{N}^{T} (θ_{0}) \frac{\partial W}{\partial θ_{1}} {\hat{r}}_{N} (θ_{0}) \\ ⋮ \\ {\hat{r}}_{N}^{T} (θ_{0}) \frac{\partial W}{\partial θ_{1}} {\hat{r}}_{N} (θ_{0}) \end{matrix}]}_{θ = θ_{0}} ≃ {\hat{G}}^{T} W {\hat{r}}_{N} (θ_{0}),

where

\hat{G} = \frac{\partial {\hat{τ}}_{N} (θ)}{\partial θ} ❘_{θ = θ_{0}} .

G = \frac{\partial τ (θ)}{\partial θ} ❘_{θ = θ_{0}} .

M = \lim_{N -> \infty} N E [{\dot{τ}}_{N} (θ_{0}) {\dot{τ}}_{N}^{T} (θ_{0})] .

[00064] This means that the gradient vector, is asymptotically normally distributed, with zero-mean and with a covariance matrix M.

[00065] Before presenting the main result of the current paper, the convergence of the Hessian matrix V″_Nmust be investigated. Assuming that the limit exists, define

{\overline{V}}^{′′} (θ) = \lim_{N -> \infty} V_{N}^{′′} (θ) .

[00066] To establish the convergence of V″_N(θ_ξ), the following (standard) inequality is applied

{ V_{N}^{′′} (θ_{ξ}) - {\overline{V}}^{′′} (θ_{0}) }_{F} \leq { V_{N}^{′′} (θ_{ξ}) - V_{N}^{′′} (θ_{0}) }_{F} + { V_{N}^{′′} (θ_{0}) - {\overline{V}}^{′′} (θ_{0}) }_{F},

[00068] It is, now, straightforward to see that the limiting Hessian V⁻″ can be written as

\begin{matrix} {\overline{V}}^{′′} (θ_{0}) = \lim_{N -> \infty} V^{′′} (θ_{0}) = G^{T} W G & w . p .1 \end{matrix}

[heading-00073] √{square root over (N)}({dot over (θ)}_N−θ₀)εAsN(0, P),

[heading-00074] where
P=(G^TWG)⁻¹G^TWMWG(G^TWG)⁻¹.

[00077] 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} = \sum_{p = - \infty}^{\infty} R_{s_{1} s_{1}} (p; θ_{0}) R_{s_{2} s_{2}} (p + τ; θ_{0}) .

[00078] Furthermore, introduce the following Z-transforms

Φ_{1} (z) = \sum_{k = - \infty}^{\infty} R_{s_{1} s_{1}} (k; θ_{0}) z^{- k}, Φ_{2} (z) = \sum_{k = - \infty}^{\infty} R_{s_{2} s_{2}} (k; θ_{0}) z^{- k} .

[00079] Then it follows that

\sum_{τ = - \infty}^{\infty} \sum_{p = - \infty}^{\infty} 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) .

[00080] Thus, the β_τ'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 - {\overline{B}}_{12} (z^{- 1}) {\overline{B}}_{21} (z^{- 1})) {\langle \frac{G_{1} (z)}{F_{1} (z)} \rangle}^{2} {\langle \frac{G_{2} (z)}{F_{2} (z)} \rangle}^{2} .

W = [\begin{matrix} β_{0} & β_{1} & \dots & β_{2 U} \\ β_{1} & β_{0} & ⋰ & ⋮ \\ ⋮ & ⋰ & β_{1} \\ β_{2 U} & \dots & β_{1} & β_{0} \end{matrix}] .

{\dot{x}}_{i} (n) = \frac{1}{\det {D (q, \dot{θ})}} s_{i} (n_{i} {\dot{θ}}_{N}),

as long as det{D(q, θ₀)} is minimum phase.

[00084] 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

\frac{\partial R_{s_{1} s_{2}} (k; θ)}{\partial d_{21} (i)} = - R_{y_{1} y_{1}} (k + i) + \sum_{l} d_{12} (l) R_{y_{2} y_{1}} (k - l + i)

\frac{\partial R_{s_{1} s_{2}} (k; θ)}{\partial d_{21} (i)} = - R_{y_{2} y_{2}} (k - i) + \sum_{l} d_{21} (l) R_{y_{2} y_{1}} (k + i - l) .

which are straightforward to compute.

[00088] For this choice of weighting,
P(W₀)=(G^TM⁻¹G)⁻¹

[00090] The accuracy is optimized in the sense that P(W₀)−P(W) is positive semi-definite for all positive definite weighting matrices W.

[00091] The proof follows from well-known matrix optimization results, see for example [9, Appendix 2].

[00092] 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.

[00094] Assume that the optimal weighting W₀is applied in the criterion (2.10). Let P_U(W) denote the asymptotic covariance for this case. Then
P_U(W₀)≧P_U-1(W₀)

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