Blind Separation Of Nonstationary Sources Research Articles

Bayesian Estimation in the Presence of Deterministic Nuisance Parameters—Part II: Estimation Methods

One of the fundamental issues of estimation theory is the presence of deterministic nuisance parameters. While in the Bayesian paradigm the model parameters are random, introduction of deterministic nuisance parameters into the model exceeds the Bayesian framework to the hybrid framework. In this type of scenarios, the conventional Bayesian estimators are not valid, as they assume knowledge of the deterministic nuisance parameters. This paper is the second of a two-part study of Bayesian parameter estimation in the presence of deterministic nuisance parameters. In part I, a new Cramer–Rao (CR)-type bound on the mean-square-error (MSE) for Bayesian estimation in the presence of deterministic nuisance parameters was established based on the concept of risk-unbiasedness. The proposed bound was named risk-unbiased bound (RUB). This paper presents properties of asymptotic uniform mean- and risk-unbiasedness of some Bayesian estimators: 1) the minimum MSE (MMSE) or maximum a posteriori probability (MAP) estimators with maximum likelihood (ML) estimates substituting the deterministic parameters, named MS-ML and MAP-ML, respectively, and 2) joint MAP and ML estimator, named JMAP-ML. Furthermore, an asymptotic performance analysis of the MS-ML and MAP-ML estimators is presented. These estimators are shown to asymptotically achieve the RUB, while the existing CR-type bounds can be achieved only in distinct cases. Simulations verify these results for the problem of blind separation of nonstationary sources. It is shown that unlike existing CR-type bounds, the RUB is asymptotically tight.

A new method for blind separation of nonstationary sources

The development of nonstationary signal blind separation, especially an underdetermined mixture of nonstationary sources, has become increasingly important for mechanical multi-fault diagnosis. In this paper, a novel algorithm based on the adaptive parametric time-frequency analysis theory to blindly separate the mixtures of nonstationary sources is presented. The computer simulations illustrate the applicability of our approach, and the performance of the algorithm is successfully tested on the underdetermined blind source separation of five vibration signals of a high speed train from one instantaneous mixture of these signals.

Underdetermined DOA estimation and blind separation of non-disjoint sources in time-frequency domain based on sparse representation method

This paper deals with the blind separation of nonstation-ary sources and direction-of-arrival (DOA) estimation in the under-determined case, when there are more sources than sensors. We assume the sources to be time-frequency (TF) disjoint to a certain extent. In particular, the number of sources presented at any TF neighborhood is strictly less than that of sensors. We can identify the real number of active sources and achieve separation in any TF neighborhood by the sparse representation method. Compared with the subspace-based algorithm under the same sparseness assumption, which suffers from the extra noise effect since it cannot estimate the true number of active sources, the proposed algorithm can estimate the number of active sources and their corresponding TF values in any TF neighborhood simultaneously. Another contribution of this paper is a new estimation procedure for the DOA of sources in the underdetermined case, which combines the TF sparseness of sources and the clustering technique. Simulation results demonstrate the validity and high performance of the proposed algorithm in both blind source separation (BSS) and DOA estimation.

Blind separation of non-stationary sources using continuous density hidden Markov models

Blind source separation (BSS) has attained much attention in signal processing society due to its 'blind' property and wide applications. However, there are still some open problems, such as underdetermined BSS, noise BSS. In this paper, we propose a Bayesian approach to improve the separation performance of instantaneous mixtures with non-stationary sources by taking into account the internal organization of the non-stationary sources. Gaussian mixture model (GMM) is used to model the distribution of source signals and the continuous density hidden Markov model (CDHMM) is derived to track the non-stationarity inside the source signals. Source signals can switch between several states such that the separation performance can be significantly improved. An expectation-maximization (EM) algorithm is derived to estimate the mixing coefficients, the CDHMM parameters and the noise covariance. The source signals are recovered via maximum a posteriori (MAP) approach. To ensure the convergence of the proposed algorithm, the proper prior densities, conjugate prior densities, are assigned to estimation coefficients for incorporating the prior information. The initialization scheme for the estimates is also discussed. Systematic simulations are used to illustrate the performance of the proposed algorithm. Simulation results show that the proposed algorithm has more robust separation performance in terms of similarity score in noise environments in comparison with the classical BSS algorithms in determined mixture case. Additionally, since the mixing matrix and the sources are estimated jointly, the proposed EM algorithm also works well in underdetermined case. Furthermore, the proposed algorithm converges quickly with proper initialization.

Blind Separation of Underdetermined Convolutive Mixtures Using Their Time–Frequency Representation

This paper considers the blind separation of nonstationary sources in the underdetermined convolutive mixture case. We introduce, two methods based on the sparsity assumption of the sources in the time-frequency (TF) domain. The first one assumes that the sources are disjoint in the TF domain, i.e., there is at most one source signal present at a given point in the TF domain. In the second method, we relax this assumption by allowing the sources to be TF-nondisjoint to a certain extent. In particular, the number of sources present (active) at a TF point should be strictly less than the number of sensors. In that case, the separation can be achieved thanks to subspace projection which allows us to identify the active sources and to estimate their corresponding time-frequency distribution (TFD) values. Another contribution of this paper is a new estimation procedure for the mixing channel in the underdetermined case. Finally, numerical performance evaluations and comparisons of the proposed methods are provided highlighting their effectiveness.

Underdetermined Blind Separation of Nondisjoint Sources in the Time-Frequency Domain

This paper considers the blind separation of nonstationary sources in the underdetermined case, when there are more sources than sensors. A general framework for this problem is to work on sources that are sparse in some signal representation domain. Recently, two methods have been proposed with respect to the time-frequency (TF) domain. The first uses quadratic time-frequency distributions (TFDs) and a clustering approach, and the second uses a linear TFD. Both of these methods assume that the sources are disjoint in the TF domain; i.e., there is, at most, one source present at a point in the TF domain. In this paper, we relax this assumption by allowing the sources to be TF-nondisjoint to a certain extent. In particular, the number of sources present at a point is strictly less than the number of sensors. The separation can still be achieved due to subspace projection that allows us to identify the sources present and to estimate their corresponding TFD values. In particular, we propose two subspace-based algorithms for TF-nondisjoint sources: one uses quadratic TFDs and the other a linear TFD. Another contribution of this paper is a new estimation procedure for the mixing matrix. Finally, then numerical performance of the proposed methods are provided highlighting their performance gain compared to existing ones

Blind Separation of Nonstationary Sources Based on Spatial Time-Frequency Distributions

Blind source separation (BSS) based on spatial time-frequency distributions (STFDs) provides improved performance over blind source separation methods based on second-order statistics, when dealing with signals that are localized in the time-frequency (t-f) domain. In this paper, we propose the use of STFD matrices for both whitening and recovery of the mixing matrix, which are two stages commonly required in many BSS methods, to provide robust BSS performance to noise. In addition, a simple method is proposed to select the auto- and cross-term regions of time-frequency distribution (TFD). To further improve the BSS performance, t-f grouping techniques are introduced to reduce the number of signals under consideration, and to allow the receiver array to separate more sources than the number of array sensors, provided that the sources have disjoint t-f signatures. With the use of one or more techniques proposed in this paper, improved performance of blind separation of nonstationary signals can be achieved.

Separating More Sources Than Sensors Using Time-Frequency Distributions

We examine the problem of blind separation of nonstationary sources in the underdetermined case, where there are more sources than sensors. Since time-frequency (TF) signal processing provides effective tools for dealing with nonstationary signals, we propose a new separation method that is based on time-frequency distributions (TFDs). The underlying assumption is that the original sources are disjoint in the time-frequency (TF) domain. The successful method recovers the sources by performing the following four main procedures. First, the spatial time-frequency distribution (STFD) matrices are computed from the observed mixtures. Next, the auto-source TF points are separated from cross-source TF points thanks to the special structure of these mixture STFD matrices. Then, the vectors that correspond to the selected auto-source points are clustered into different classes according to the spatial directions which differ among different sources; each class, now containing the auto-source points of only one source, gives an estimation of the TFD of this source. Finally, the source waveforms are recovered from their TFD estimates using TF synthesis. Simulated experiments indicate the success of the proposed algorithm in different scenarios. We also contribute with two other modified versions of the algorithm to better deal with auto-source point selection.

Penalty function-based joint diagonalization approach for convolutive blind separation of nonstationary sources

A new approach for convolutive blind source separation (BSS) by explicitly exploiting the second-order nonstationarity of signals and operating in the frequency domain is proposed. The algorithm accommodates a penalty function within the cross-power spectrum-based cost function and thereby converts the separation problem into a joint diagonalization problem with unconstrained optimization. This leads to a new member of the family of joint diagonalization criteria and a modification of the search direction of the gradient-based descent algorithm. Using this approach, not only can the degenerate solution induced by a null unmixing matrix and the effect of large errors within the elements of covariance matrices at low-frequency bins be automatically removed, but in addition, a unifying view to joint diagonalization with unitary or nonunitary constraint is provided. Numerical experiments are presented to verify the performance of the new method, which show that a suitable penalty function may lead the algorithm to a faster convergence and a better performance for the separation of convolved speech signals, in particular, in terms of shape preservation and amplitude ambiguity reduction, as compared with the conventional second-order based algorithms for convolutive mixtures that exploit signal nonstationarity.

Blind Separation of Nonstationary Sources

We propose a blind separation technique for nonstationary sources that exploits both auto-terms and cross-terms of the time-frequency distributions. The technique is based on the simultaneous joint diagonalization and off-diagonalization of spatial time-frequency distributions. Computer simulations demonstrate the superiority of the approach in comparison with other time-frequency based methods.

Convolutive blind separation of non-stationary sources

Acoustic signals recorded simultaneously in a reverberant environment can be described as sums of differently convolved sources. The task of source separation is to identify the multiple channels and possibly to invert those in order to obtain estimates of the underlying sources. We tackle the problem by explicitly exploiting the nonstationarity of the acoustic sources. Changing cross correlations at multiple times give a sufficient set of constraints for the unknown channels. A least squares optimization allows us to estimate a forward model, identifying thus the multipath channel. In the same manner we can find an FIR backward model, which generates well separated model sources. Furthermore, for more than three channels we have sufficient conditions to estimate underlying additive sensor noise powers. We show the good performance in a real room environments and demonstrate the algorithm's utility for automatic speech recognition.

Blind separation of nonstationary sources in noisy mixtures

A new method of blind source separation is presented that is not sensitive to additive white noise. The method exploits the nonstationarity and temporal structure of sources. The method only requires multiple time-delayed correlation matrices of the observed data at several different time-windowed data frames to estimate the mixing matrix. The implementation using joint diagonalisation (JD) is described. Simulations verify the high performance of the proposed method, especially in a low SNR environment.