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 can-not estimate the true number of active sources, the proposed algorithm can estimate the number of active sources and their cor-responding TF values in any TF neighborhood simultaneously. An-other 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. Sim-ulation results demonstrate the validity and high performance of the proposed algorithm in both blind source separation （BSS） and DOA estimation.

Underdetermined DOA estimation via multiple time-delay covariance matrices and deep residual network

Higher-order statistics based approaches and signal sparseness based approaches have emerged in recent decades to resolve the underdetermined direction-of-arrival(DOA)estimation problem.These model-based methods face great challenges in practical applications due to high computational complexity and dependence on ideal assumptions.This paper presents an effective DOA estimation approach based on a deep residual network(DRN)for the underdetermined case.We first extract an input feature from a new matrix calculated by stacking several covariance matrices corresponding to different time delays.We then provide the input feature to the trained DRN to construct the super resolution spectrum.The DRN learns the mapping relationship between the input feature and the spatial spectrum by training.The proposed approach is superior to existing model-based estimation methods in terms of calculation efficiency,independence of source sparseness and adaptive capacity to non-ideal conditions(e.g.,low signal to noise ratio,short bit sequence).Simulations demonstrate the validity and strong performance of the proposed algorithm on both overdetermined and underdetermined cases.

Source Recovery in Underdetermined Blind Source Separation Based on Artificial Neural Network

We propose a novel source recovery algorithm for underdetermined blind source separation, which can result in better accuracy and lower computational cost. On the basis of the model of underdetermined blind source separation, the artificial neural network with single-layer perceptron is introduced into the proposed algorithm. Source signals are regarded as the weight vector of single-layer perceptron, and approximate ι~0-norm is taken into account for output error decision rule of the perceptron, which leads to the sparse recovery. Then the procedure of source recovery is adjusting the weight vector of the perceptron. What's more, the optimal learning factor is calculated and a descent sequence of smoothed parameter is used during iteration, which improves the performance and significantly decreases computational complexity of the proposed algorithm. The simulation results reveal that the algorithm proposed can recover the source signal with high precision, while it requires lower computational cost.

A robust clustering algorithm for underdetermined blind separation of sparse sources

In underdetermined blind source separation, more sources are to be estimated from less observed mixtures without knowing source signals and the mixing matrix. This paper presents a robust clustering algorithm for underdetermined blind separation of sparse sources with unknown number of sources in the presence of noise. It uses the robust competitive agglomeration （RCA） algorithm to estimate the source number and the mixing matrix, and the source signals then are recovered by using the interior point linear programming. Simulation results show good performance of the proposed algorithm for underdetermined blind sources separation （UBSS）.

Audio-Visual Underdetermined Blind Source Separation Algorithm Based on Gaussian Potential Function

Most existing algorithms for the underdetermined blind source separation(UBSS) problem are two-stage algorithm, i.e., mixing parameters estimation and sources estimation. In the mixing parameters estimation, the previously proposed traditional clustering algorithms are sensitive to the initializations of the mixing parameters. To reduce the sensitiveness to the initialization, we propose a new algorithm for the UBSS problem based on anechoic speech mixtures by employing the visual information, i.e., the interaural time difference(ITD) and the interaural level difference(ILD), as the initializations of the mixing parameters. In our algorithm, the video signals are utilized to estimate the distances between microphones and sources, and then the estimations of the ITD and ILD can be obtained. With the sparsity assumption in the time-frequency domain, the Gaussian potential function algorithm is utilized to estimate the mixing parameters by using the ITDs and ILDs as the initializations of the mixing parameters. And the time-frequency masking is used to recover the sources by evaluating the various ITDs and ILDs. Experimental results demonstrate the competitive performance of the proposed algorithm compared with the baseline algorithms.

Algorithm for source recovery in underdetermined blind source separation based on plane pursuit

In order to achieve accurate recovery signals under the underdetermined circumstance in a comparatively short time,an algorithm based on plane pursuit(PP) is proposed. The proposed algorithm selects the atoms according to the correlation between received signals and hyper planes, which are composed by column vectors of the mixing matrix, and uses these atoms to recover source signals. Simulation results demonstrate that the PP algorithm has low complexity and higher accuracy as compared with basic pursuit(BP), orthogonal matching pursuit(OMP), and adaptive sparsity matching pursuit(ASMP) algorithms.

Non-invasive foetal heartbeat rate extraction from an underdetermined single signal

Extraction of foetal heartbeat rate from a single passive sound sensor on the mother’s abdomen is demonstrated. The extraction is based on the assumption that a disjoint band of frequencies exist and foetal signal is concentrated in this band, and further that it can be represented conveniently as a set of wavelet coefficients. The algorithm has been applied to each stream of data obtained from six different channels and the detection performance is elaborated. The algorithm has also been tested on signals from non-pregnant abdomens to show successful rejection of adult heartbeat. The extraction of the desired signal is done in two stages so as to eliminate components from the maternal heart-beat.

Mixing matrix estimation of underdetermined blind source separation based on the linear aggregation characteristic of observation signals

Under the underdetermined blind sources separation(UBSS) circumstance,it is difficult to estimate the mixing matrix with high-precision because of unknown sparsity of signals.The mixing matrix estimation is proposed based on linear aggregation degree of signal scatter plot without knowing sparsity,and the linear aggregation degree evaluation of observed signals is presented which obeys generalized Gaussian distribution(GGD).Both the GGD shape parameter and the signals' correlation features affect the observation signals sparsity and further affected the directionality of time-frequency scatter plot.So a new mixing matrix estimation method is proposed for different sparsity degrees,which especially focuses on unclear directionality of scatter plot and weak linear aggregation degree.Firstly,the direction of coefficient scatter plot by time-frequency transform is improved and then the single source coefficients in the case of weak linear clustering is processed finally the improved K-means clustering is applied to achieve the estimation of mixing matrix.The proposed algorithm reduces the requirements of signals sparsity and independence,and the mixing matrix can be estimated with high accuracy.The simulation results show the feasibility and effectiveness of the algorithm.

一种基于l_p模约束的FOCUSS迭代EEG源定位新方法

如何有效地从头表记录电位中准确定位脑电源的真实活动位置是神经认知脑功能研究中的一个关键问题.本文在FOCUSS算法迭代基础上,从脑神经活动的局部稀疏性出发,提出了一种新的脑功能成像方法.在该算法中,通过把稀疏性的lp模约束加入到修改的FOCUSS算法的迭代过程中,使算法可以有效地收敛于真实的稀疏源活动位置.利用该方法对随机系统、三层球模型及真实头模型确定的稀疏欠定系统进行了求解模拟实验,结果显示了该方法在求解欠定系统及EEG源定位时具有良好的稳健性.

UBSS and blind parameters estimation algorithms for synchronous orthogonal FH signals

By using the sparsity of frequency hopping（FH） signals,an underdetermined blind source separation（UBSS） algorithm is presented. Firstly, the short time Fourier transform（STFT） is performed on the mixed signals. Then, the mixing matrix, hopping frequencies, hopping instants and the hooping rate can be estimated by the K-means clustering algorithm. With the estimated mixing matrix, the directions of arrival（DOA） of source signals can be obtained. Then, the FH signals are sorted and the FH pattern is obtained. Finally, the shortest path algorithm is adopted to recover the time domain signals. Simulation results show that the correlation coefficient between the estimated FH signal and the source signal is above 0.9 when the signal-to-noise ratio（SNR） is higher than 0 d B and hopping parameters of multiple FH signals in the synchronous orthogonal FH network can be accurately estimated and sorted under the underdetermined conditions.

Blind source separation by weighted K-means clustering

Blind separation of sparse sources （BSSS） is discussed. The BSSS method based on the conventional K-means clustering is very fast and is also easy to implement. However, the accuracy of this method is generally not satisfactory. The contribution of the vector x（t） with different modules is theoretically proved to be unequal, and a weighted K-means clustering method is proposed on this grounds. The proposed algorithm is not only as fast as the conventional K-means clustering method, but can also achieve considerably accurate results, which is demonstrated by numerical experiments.

一种新的基于稀疏贝叶斯学习的ISAR成像方法

将BP(基寻踪)算法和FOCUSS(Focal underdetermined systemsolver)算法用于ISAR成像中,由于它们具有一些缺点,在实际应用中受到很大的限制.但是SBL(稀疏贝叶斯)方法可以克服这些缺点.本文首次提出可以将SBL代替BP和FOCUSS用于ISAR成像中.真实ISAR数据的成像结果表明SBL是一种比BP和FOCUSS更有效,更有潜力的成像方法.

Investigation of the electrical conductivity beneath China using geomagnetic spatial gradient method

The data of the year 1992 from 25 geomagnetic observatories affiliated to the geomagnetic network of State Seismological Bureau of China were processed using the principle of geomagnetic spatial gradient method. Through finding out the polynomial form of optimum fitting, comparatively good C values for four harmonic components of diurnal variations were obtained. By using the inverse method of non linear underdetermined problem, the electrical conductivity structures under the observatories were investgated. It is shown that there are differences of the C values and conductivity structures in the deep underground under the south western part and northern parts and other parts of China. We studied the possibility of improving the gradient method for investigation of the deep underground conductivity structure, and it is indicated that the gradient method is hopeful in the investigation of earth′s deep conductivity structure and the applied studies concerned.

Underdetermination, Multiplicity, and Mathematical Logic

Whether a collection of scientific data can be explained only by a unique theory or whether such data can be equally explained by multiple theories is one of the more contested issues in the history and philosophy of science. This paper argues that the case for multiple explanations is strengthened by the widespread failure of models in mathematical logic to be unique, i.e., categorical. Science is taken to require replicable and explicit public knowledge; this necessitates an unambiguous language for its transmission. Mathematics has been chosen as the vehicle to transmit scientific knowledge, both because of its ＂unreasonable effectiveness＂ and because of its unambiguous nature, hence the vogue of axiomatic systems. But mathematical logic tells us that axiomatic systems need not refer to uniquely defined real structures. Hence what is accepted as science may be only one of several possibilities.

Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

It is well known that the use of Helmholtz decomposition theorem for static vector fields , when applied to the time dependent vector fields , which represent the electromagnetic field, allows us to obtain instantaneous-like solutions all along . For this reason, some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot be applied to time dependent vector fields and some modification is wanted in order to get the retarded solutions. However, the use of the Helmholtz theorem for static vector fields is correct even for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in such a way that a retarded solution can be transformed in an instantaneous one, and conversely. On this paper we want to suggest, following most of the time the mathematical formalism of Woodside in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological criterion for choosing the gauge according to the structure postulated for a global space-time, 5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is involved. So, when we relate the retarded solution to the instantaneous one what we do is to change the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time structure, and are equivalent to gauge transformations, each gauge transformation is natural for a specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically equivalent, because they can be related by means of a gauge transformation, but they are not empirically equivalent, because they have quite different observational consequences due to the different space-time structure involved.

Equivalence and Strong Equivalence Between the Sparsest and Least l1-Norm Nonnegative Solutions of Linear Systems and Their Applications

Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints,which seek the sparsest nonnegative solutions to underdetermined linear systems.Recent study indicates that l1-minimization is efficient for solving l0-minimization problems.From a mathematical point of view,however,the understanding of the relationship between l0-and l1-minimization remains incomplete.In this paper,we further address several theoretical questions associated with these two problems.We prove that the fundamental strict complementarity theorem of linear programming can yield a necessary and sufficient condition for a linear system to admit a unique least l1-norm nonnegative solution.This condition leads naturally to the so-called range space property(RSP)and the “full-column-rank”property,which altogether provide a new and broad understanding of the equivalence and the strong equivalence between l0-and l1-minimization.Motivated by these results,we introduce the concept of “RSP of order K”that turns out to be a full characterization of uniform recovery of all K-sparse nonnegative vectors.This concept also enables us to develop a nonuniform recovery theory for sparse nonnegative vectors via the so-called weak range space property.

Searching-and-averaging method of underdetermined blind speech signal separation in time domain

Underdetermined blind signal separation （BSS） （with fewer observed mixtures than sources） is discussed. A novel searching-and-averaging method in time domain （SAMTD） is proposed. It can solve a kind of problems that are very hard to solve by using sparse representation in frequency domain. Bypassing the disadvantages of traditional clustering （e.g., K-means or potential-function clustering）, the durative- sparsity of a speech signal in time domain is used. To recover the mixing matrix, our method deletes those samples, which are not in the same or inverse direction of the basis vectors. To recover the sources, an improved geometric approach to overcomplete ICA （Independent Component Analysis） is presented. Several speech signal experiments demonstrate the good performance of the proposed method.

Adaptive blind separation of underdetermined mixtures based on sparse component analysis

The independence priori is very often used in the conventional blind source separation （BSS）. Naturally, independent component analysis （ICA） is also employed to perform BSS very often. However, ICA is difficult to use in some challenging cases, such as underdetermined BSS or blind separation of dependent sources. Recently, sparse component analysis （SCA） has attained much attention because it is theoretically available for underdetermined BSS and even for blind dependent source separation sometimes. However, SCA has not been developed very sufficiently. Up to now, there are only few existing algorithms and they are also not perfect as well in practice. For example, although Lewicki-Sejnowski＇s natural gradient for SCA is superior to K-mean clustering, it is just an approximation without rigorously theoretical basis. To overcome these problems, a new natural gradient formula is proposed in this paper. This formula is derived directly from the cost function of SCA through matrix theory. Mathematically, it is more rigorous. In addition, a new and robust adaptive BSS algorithm is developed based on the new natural gradient. Simulations illustrate that this natural gradient formula is more robust and reliable than Lewicki-Sejnowski＇s gradient.

Source number estimation and separation algorithms of underdetermined blind separation

Recently,sparse component analysis (SCA) has become a hot spot in BSS re-search. Instead of independent component analysis (ICA),SCA can be used to solve underdetermined mixture efficiently. Two-step approach (TSA) is one of the typical methods to solve SCA based BSS problems. It estimates the mixing matrix before the separation of the sources. K-means clustering is often used to estimate the mixing matrix. It relies on the prior knowledge of the source number strongly. However,the estimation of the source number is an obstacle. In this paper,a fuzzy clustering method is proposed to estimate the source number and mixing matrix simultaneously. After that,the sources are recovered by the shortest path method (SPM). Simulations show the availability and robustness of the proposed method.

Improved statistical sparse decomposition principle method for underdetermined blind source signal recovery

Aiming at the statistical sparse decomposition principle(SSDP) method for underdetermined blind source signal recovery with problem of requiring the number of active signals equal to that of the observed signals, which leading to the application bound of SSDP is very finite, an improved SSDP(ISSDP) method is proposed. Based on the principle of recovering the source signals by minimizing the correlation coefficients within a fixed time interval, the selection method of mixing matrix’s column vectors used for signal recovery is modified, which enables the choose of mixing matrix’s column vectors according to the number of active source signals self-adaptively. By simulation experiments, the proposed method is validated. The proposed method is applicable to the case where the number of active signals is equal to or less than that of observed signals, which is a new way for underdetermined blind source signal recovery.