Source separation in post-nonlinear mixtures

We address the problem of separation of mutually independent sources in nonlinear mixtures. First, we propose theoretical results and prove that in the general case, it is not possible to separate the sources without nonlinear distortion. Therefore, we focus our work on specific nonlinear mixtures known as post-nonlinear mixtures. These mixtures constituted by a linear instantaneous mixture (linear memoryless channel) followed by an unknown and invertible memoryless nonlinear distortion, are realistic models in many situations and emphasize interesting properties i.e., in such nonlinear mixtures, sources can be estimated with the same indeterminacies as in instantaneous linear mixtures. The separation structure of nonlinear mixtures is a two-stage system, namely, a nonlinear stage followed by a linear stage, the parameters of which are updated to minimize an output independence criterion expressed as a mutual information criterion. The minimization of this criterion requires knowledge or estimation of source densities or of their log-derivatives. A first algorithm based on a Gram-Charlier expansion of densities is proposed. Unfortunately, it fails for hard nonlinear mixtures. A second algorithm based on an adaptive estimation of the log-derivative of densities leads to very good performance, even with hard nonlinearities. Experiments are proposed to illustrate these results     

Spatial analysis in passive listening using adaptive techniques

The various known methods intended for continuous analysis of uncorrelated source fields (MEM-AR, Capon, Borgiotti-Lagunas, Bienvenu-Schmidt) are shown to be based upon a unique conventional diagram, namely one of a spatial linear multiport filter followed by a variance estimator. This filter has K inputs corresponding to K sensors and a single output, and acts as a scalar product system. A fundamental property is then the possibility of having zero responses for the signals coming from K - 1 directions. Using the classical matched filter principle for minimization of undesirable signals (when looking in a given direction), a general class of data-adaptive spatial filters, from which all the above-mentioned methods can be derived with specific choices of parameters, is defined. These filters can be further submitted to normalizing constraints; two examples of this are discussed. This new interpretation leads to the consideration of the above-mentioned methods as special applications of the well-known adaptive array techniques to the estimation problem. Moreover, the asymptotic behaviors are examined when a noise subtraction technique based upon the reduction of the smallest cross-spectral matrix eigenvalue is applied. The convergence toward the Pisarenko solution is ascertained for the estimated directions of the sources, whereas it is not for their estimated intensities (except in the case of the Capon method). In the case of partially correlated sources, the above-mentioned methods are still valid for the localization of sources, but not for their intensities. Other appropriate techniques, such as those of Schmidt, Wax, Shan, and Kailath, must then be considered.     

A Novel Online Mutual Coupling Compensation Algorithm for Uniform and Linear Arrays

In this paper, a novel online mutual coupling compensation algorithm especially tailored to uniform and linear arrays is presented. It is conceived to simultaneously compensate for mutual coupling and estimate the direction-of-arrivals (DOAs) of signals impinging on the array since the estimated calibration matrix can be embedded within any classical super-resolution direction-finding method. An alternating minimization procedure based on closed-form solutions is performed to estimate the mutual coupling matrix in the field of complex symmetric Toeplitz matrices. Unlike many existing array calibration methods, it requires neither the presence of calibration sources nor previous calibration information as initialization. Computer simulations show the effectiveness of the proposed technique and prove that the nice statistical properties of classical super-resolution DOA estimation algorithms can be restored despite the presence of mutual coupling     

Does independent component analysis play a role in unmixing hyperspectral data?

Independent component analysis (ICA) has recently been proposed as a tool to unmix hyperspectral data. ICA is founded on two assumptions: 1) the observed spectrum vector is a linear mixture of the constituent spectra (endmember spectra) weighted by the correspondent abundance fractions (sources); 2)sources are statistically independent. Independent factor analysis (IFA) extends ICA to linear mixtures of independent sources immersed in noise. Concerning hyperspectral data, the first assumption is valid whenever the multiple scattering among the distinct constituent substances (endmembers) is negligible, and the surface is partitioned according to the fractional abundances. The second assumption, however, is violated, since the sum of abundance fractions associated to each pixel is constant due to physical constraints in the data acquisition process. Thus, sources cannot be statistically independent, this compromising the performance of ICA/IFA algorithms in hyperspectral unmixing. This paper studies the impact of hyperspectral source statistical dependence on ICA and IFA performances. We conclude that the accuracy of these methods tends to improve with the increase of the signature variability, of the number of endmembers, and of the signal-to-noise ratio. In any case, there are always endmembers incorrectly unmixed. We arrive to this conclusion by minimizing the mutual information of simulated and real hyperspectral mixtures. The computation of mutual information is based on fitting mixtures of Gaussians to the observed data. A method to sort ICA and IFA estimates in terms of the likelihood of being correctly unmixed is proposed.     

Blind separation of instantaneous mixtures of nonstationary sources

Most source separation algorithms are based on a model of stationary sources. However, it is a simple matter to take advantage of possible nonstationarities of the sources to achieve separation. This paper develops novel approaches in this direction based on the principles of maximum likelihood and minimum mutual information. These principles are exploited by efficient algorithms in both the off-line case (via a new joint diagonalization procedure) and in the on-line case (via a Newton-like procedure). Some experiments showing the good performance of our algorithms and evidencing an interesting feature of our methods are presented: their ability to achieve a kind of super-efficiency. The paper concludes with a discussion contrasting separating methods for non-Gaussian and nonstationary models and emphasizing that, as a matter of fact, “what makes the algorithms work” is-strictly speaking-not the nonstationarity itself but rather the property that each realization of the source signals has a time-varying envelope     

Using the Penalized Mutual Information Criterion in the Multivariate Edgeworth-Expanded Gaussian Mixture Density for Blind Separation of Convolutive Post-Nonlinear Mixtures

This paper proposes the blind separation of convolutive post-nonlinear (CPNL) mixtures based on the minimization of the penalized mutual information criterion. The proposed algorithm is based on the estimation score function difference (SFD) and the Newton optimization. Compared with the blind source separation of a linear mixture, the separation performance of a nonlinear mixture is strongly related to the accuracy of the score function estimation. Under this framework, the multivariate Edgeworth-expanded Gaussian mixture density is adopted to estimate the SFD, which preserves the higher-order statistical structure of the data as compared to the nonparametric density estimation. Also, the Newton optimization converges faster than the steepest descent gradient. In order to calculate the Hessian matrix, the Taylor expansion of the penalized mutual information criterion is extended to second order. The minimization of the penalized mutual information criterion ensures a priori normalization of the estimated sources, thus avoiding scale indeterminacy. The proposed algorithm has a better performance, and at the same time it speeds up the convergence. Simulations with computer-generated data and synthetic real-world data show the effectiveness of the proposed algorithm.     

An NMF algorithm for blind separation of convolutive mixed source signals with least correlation constrains

Most of the existing algorithms for blind sources separation have a limitation that sources are statistically independent. However, in many practical applications, the source signals are non- negative and mutual statistically dependent signals. When the observations are nonnegative linear combinations of nonnegative sources, the correlation coefficients of the observations are larger than these of source signals. In this letter, a novel Nonnegative Matrix Factorization (NMF) algorithm with least correlated ...     

Localization of brain electrical activity via linearly constrainedminimum variance spatial filtering

A spatial filtering method for localizing sources of brain electrical activity from surface recordings is described and analyzed. The spatial filters are implemented as a weighted sum of the data recorded at different sites. The weights are chosen to minimize the filter output power subject to a linear constraint. The linear constraint forces the filter to pass brain electrical activity from a specified location, while the power minimization attenuates activity originating at other locations. The estimated output power as a function of location is normalized by the estimated noise power as a function of location to obtain a neural activity index map. Locations of source activity correspond to maxima in the neural activity index map. The method does not require any prior assumptions about the number of active sources of their geometry because it exploits the spatial covariance of the source electrical activity. This paper presents a development and analysis of the method and explores its sensitivity to deviations between actual and assumed data models. The effect on the algorithm of covariance matrix estimation, correlation between sources, and choice of reference is discussed. Simulated and measured data is used to illustrate the efficacy of the approach     

Convolutive blind source separation by minimizing mutual information between segments of signals

A method to perform convolutive blind source separation of super-Gaussian sources by minimizing the mutual information between segments of output signals is presented. The proposed approach is essentially an implementation of an idea previously proposed by Pham. The formulation of mutual information in the proposed criterion makes use of a nonparametric estimator of Renyi's /spl alpha/-entropy, which becomes Shannon's entropy in the limit as /spl alpha/ approaches 1. Since /spl alpha/ can be any number greater than 0, this produces a family of criteria having an infinite number of members. Interestingly, it appears that Shannon's entropy cannot be used for convolutive source separation with this type of estimator. In fact, only one value of /spl alpha/ appears to be appropriate, namely /spl alpha/=2, which corresponds to Renyi's quadratic entropy. Four experiments are included to show the efficacy of the proposed criterion.     

离散稳恒信号的多重分形谱的计算及其应用

对于未知信号而言,一般将其视为稳恒信源的输出。因而,利用统计的方法计算信源输出信号的多重分形谱,与理论上计算的结果加以比较,据此就可以判断信源模型参数估计的合理性。该文给出了计算信号多重分形谱的一般方法,并且探讨了计算过程中的相关问题。并将该方法应用于染色体中碱基序列的分析中,实验结果表明,在某种程度上,碱基序列可视为某个离散稳恒信源的输出。。     

On the mutual information of memoryless channels

Letx_1, cdots ,x_nandy_1, cdots, y_nbe input and output sequences of a channel. In the case of memoryless input sources, the following inequality on mutual information is well known: begin{equation} I((x_1, cdots ,x_n),(y_1, cdots ,y_n)) geq sum I(x_i, y_i). end{equation} It is straightforward to show that the inequality sign is reversed if the channel instead of the input source is memoryless. In this paper we establish these inequalities when the input and output are functions instead of sequences.     

Gradient of mutual information in linear vector Gaussian channels

This paper considers a general linear vector Gaussian channel with arbitrary signaling and pursues two closely related goals: i) closed-form expressions for the gradient of the mutual information with respect to arbitrary parameters of the system, and ii) fundamental connections between information theory and estimation theory. Generalizing the fundamental relationship recently unveiled by Guo, Shamai, and Verdu/spl acute/, we show that the gradient of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the best estimate of the input given the output. Gradients and derivatives with respect to other parameters are then found via the differentiation chain rule.     

Sufficient conditions for the local convergence of constant modulusalgorithms

The constant modulus (CM) criterion has become popular in the design of blind linear estimators of sub-Gaussian i.i.d. processes transmitted through unknown linear channels in the presence of unknown additive interference. The existence of multiple CM minima, however, makes it difficult for CM-minimizing schemes to generate estimates of the desired source (as opposed to an interferer) in multiuser environments. In this paper, we present three separate sufficient conditions under which gradient descent (GD) minimization of CM cost will locally converge to an estimator of the desired source at a particular delay. The sufficient conditions are expressed in terms of statistical properties of the initial estimates, specifically, CM cost, kurtosis, and signal-to-interference-plus-noise ratio (SINR). Implications on CM-GD initialization methods are also discussed     

Globally convergent blind source separation based on a multiuser kurtosis maximization criterion

We consider the problem of recovering blindly (i.e., without the use of training sequences) a number of independent and identically distributed source (user) signals that are transmitted simultaneously through a linear instantaneous mixing channel. The received signals are, hence, corrupted by interuser interference (IUI), and we can model them as the outputs of a linear multiple-input-multiple-output (MIMO) memoryless system. Assuming the transmitted signals to be mutually independent, i.i.d., and to share the same non-Gaussian distribution, a set of necessary and sufficient conditions for the perfect blind recovery (up to scalar phase ambiguities) of all the signals exists and involves the kurtosis as well as the covariance of the output signals. We focus on a straightforward blind constrained criterion stemming from these conditions. From this criterion, we derive an adaptive algorithm for blind source separation, which we call the multiuser kurtosis (MUK) algorithm. At each iteration, the algorithm combines a stochastic gradient update and a Gram-Schmidt orthogonalization procedure in order to satisfy the criterion's whiteness constraints. A performance analysis of its stationary points reveals that the MUK algorithm is free of any stable undesired local stationary points for any number of sources; hence, it is globally convergent to a setting that recovers them all.     

On the Use of Spatial Time Frequency Distributions for Signal Extraction

This paper deals with the extraction of signals from their instantaneous linear mixtures using time-frequency distributions. Fundamentally, this problem is a signal synthesis from the time-frequency (t-f) plane. However with the incorporation of the spatial information provided by a multisensor array, the problem can be posed as special case of blind source separation. So far, the blind source separation has been solved using only statistical information available on the source signals. Herein, we propose to solve the aforementioned problem using time-frequency signal representations and the spatial array aperture. The proposed approach relies on the difference in the t-f signatures of the sources to be separated. It is based on the diagonalization of a combined set of spatial time-frequency distribution matrices. A numerical example is provided to illustrate the effectiveness of our method.     

Source Extraction by Maximizing the Variance in the Conditional Distribution Tails

This paper presents a method for signal extraction based on conditional second-order moments of the output of the extraction filter. The estimator of the filter is derived from an approximate maximum likelihood criterion conditioned on a presence indicator of the source of interest. The conditional moment is shown to be a contrast function under the conditions that 1) all cross-moments of the same order between the source signal of interest and the other source signals are null and 2) that the source of interest has the largest conditional moment among all sources. For the two-source two-observation case, this allows us to derive the theoretical recovery bounds of the contrast when the conditional cross-moment does not vanish. A comparison with empirical results confirms these bounds. Simulations show that the estimator is quite robust to additive Gaussian distributed noise. Also through simulations, we show that the error level induced by a rough approximation of the presence indicator shows a strong similarity with that of additive noise. The robustness, with respect both to noise and to inaccuracies in the prior information about the source presence, guarantees a wide applicability of the proposed method.     

互耦条件下米波雷达角度估计的自校正算法

针对互耦条件下米波雷达目标低角估计问题,提出了一种波达方向估计的自校正算法。分析了米波雷达在互耦条件下目标回波存在多径时的信号形式;利用均匀线阵互耦矩阵的特点对接收数据协方差矩阵的信号参数形式进行变换;在未知互耦校正信息的情况下,基于子空间原理得到目标波达方向的角度搜索函数,还推导出表面反射系数和阵列互耦矩阵的计算表达式。与同类校正算法相比,该算法在存在多径相干信号和未知互耦矩阵的情况下,不损失阵列孔径,不需要辅助阵元和校准源,且不需要迭代运算。仿真结果验证了该算法的有效性。     

Multidimensional Blind Separation Using Higher-Order Statistics: Application to Non-Cooperative STBC Systems

Blindly separating the intercepted signals is a challenging problem in non-cooperative multiple input multiple output systems in association with space–time block code (STBC) where channel state information and coding matrix are unavailable. To our knowledge, there is no report on dealing with this problem in literature. In this paper, the STBC systems are represented with an independent component analysis (ICA) model by merging the channel and coding matrices as virtual channel matrix. Analysis shows that the source signals are of group-wise independence and the condition of mutual independence can not be satisfied for ordinary ICA algorithms when specific modulations are employed. A new multidimensional ICA algorithm is proposed to separate the intercepted signals in this case by jointly block-diagonalizing (JBD) the cumulant matrices. In this paper, JBD is achieved by a 2-step optimization algorithm and a contrast function is derived from the JBD criterion to remove the additional permutation ambiguity with explicit mathematical explanations. The convergence of the new method is guaranteed. Compared with the ICA-based channel estimation methods, simulations show that the new algorithm, which does not introduce additional ambiguities, achieves better performance with faster convergence in a non-cooperative scenario.     

Second-Order Approximation for DOA Estimation of Near-Field Sources

The direction-of-arrival estimation of near-field sources can be formulated as a multidimensional nonlinear optimization problem, where a performance index is minimized with respect to azimuth, range, and source power. For the single source case, under the assumption that the range is relatively larger than the interelement distance, we use the second-order approximation to derive a simpler performance index parameterized by azimuth only. The minimization of the new index is easier than that of the original one parameterized by azimuth, range, and source power. Moreover, the proposed method considers the degradation of signal powers, giving more accurate estimation results. Also for the multiple source case, an efficient computation method is developed by using the second-order approximation.