an mdl algorithm for detecting more sources than sensors using outer products of array output 1991 2012 ieee in this paper we propose an algorithm for detecting the number m of gaussian sources received by an array of a number n n m of sensors this algorithm is based on the minimum description length mdl principle using the outer products of the array output we show that as long as the covariance matrix of the array output has the full rank n the covariance matrix of a vectorized outer product of the array output has the full rank n squared which meets a validity condition of the mdl algorithm we show by simulation that the mdl algorithm can perform substantially better than some relevant algorithms a necessary identifiability condition is also obtained for uncorrelated sources

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

robust adaptive beamforming based on the eigenstructure method in a linearly constrained beamformer with imperfect arrays the authors investigate the cause of cancelling the desired signal from a viewpoint based on the eigenstructure properties of the array covariance matrix based on this cause they propose a robust beamforming algorithm as an adaptive algorithm of the proposed beamformer the present a data domain signal subspace updating algorithm

characterizations and representations of the p q outer generalized inverse we present a new explicit expression for the p q outer generalized inverse based on the idempotent operators p and q we derive properties of the p q outer generalized inverse and analyse relations among the various p q outer generalized inverses

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

detection of coherent signals using weighted subspace smoothing a new approach to detecting the number of coherent signals incident upon a uniformly spaced linear array is presented the approach combines a modified spatial smoothing scheme and a modified mdl criterion the modified spatial smoothing scheme referred to as weighted subspace smoothing is actually a generalization of the post smoothing approach proposed by krim and proakis in 1994 it is shown that the noise eigenvalues obtained with weighted subspace smoothing are more accurate than those obtained with the original spatial smoothing we thus attempt to improve the detection performance using these more accurate eigenvalues a novel modification of the minimum description length mdl criterion is proposed to accomplish this computer simulations are presented to study the performance improvement of this new approach

automatic suppression of narrow band interference in direct sequence spread spectrum systems this paper addresses the problem of narrow band interference nbi cancellation in direct sequence spread spectrum systems the proposed procedure amounts to a preliminary nonlinear processing wherein upon projection of the received signal onto a fourier basis a number of samples having the largest modula are excluded from further processing the structure of the optimum detector operating on censored observations is obtained showing that the optimum detector performs matched filtering on the censored data the performance assessment demonstrates that this receiver is able to suppress narrow band interferers no matter what their structure provided that the censoring depth is properly chosen a blind version of such a receiver is presented also and a comparative performance assessment demonstrates that unlike other suppression procedures the proposed system allows suppression of nbi with no prior knowledge on its structure

on the equivalence between kernel quadrature rules and random feature expansions we show that kernel based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels for a particular decomposition that always exists for such kernels we provide a theoretical analysis of the number of required samples for a given approximation error leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms in particular we show that the upper bound may be obtained from independent and identically distributed samples from a specific non uniform distribution while the lower bound if valid for any set of points applying our results to kernel based quadrature while our results are fairly general we recover known upper and lower bounds for the special cases of sobolev spaces moreover our results extend to the more general problem of full function approximations beyond simply computing an integral with results in l2 and l norm that match known results for special cases applying our results to random features we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with lipshitz continuous losses

nonparametric detection of signals by information theoretic criteria performance analysis and an improved estimator determining the number of sources from observed data is a fundamental problem in many scientific fields in this paper we consider the nonparametric setting and focus on the detection performance of two popular estimators based on information theoretic criteria the akaike information criterion aic and minimum description length mdl we present three contributions on this subject first we derive a new expression for the detection performance of the mdl estimator which exhibits a much closer fit to simulations in comparison to previous formulas second we present a random matrix theory viewpoint of the performance of the aic estimator including approximate analytical formulas for its overestimation probability finally we show that a small increase in the penalty term of aic leads to an estimator with a very good detection performance and a negligible overestimation probability

on information theoretic criteria for determining the number of signals in high resolution array processing an important problem in high resolution array processing is the determination of the number of signals arriving at the array information theoretic criteria provide a means to achieve this two commonly used criteria are the akaike information criterion aic and minimum descriptive length mdl criterion while the aic tends to overestimate even at a high signal to noise ratio snr the mdl criterion tends to underestimate at low or moderate snr by excluding irrelevant parameters a new log likelihood function has been chosen utilizing this new log likelihood function gives a set of more accurate estimates of the eigenvalues and in the establishment of modified information theoretic criteria which moderate the performance of the aic and the mdl criterion computer simulations confirm that the modified criteria have superior performance

paper selecting the best linear transfer function model statistics currently used to determine the best model order based on the covariance of the parameter estimates or the instrumental product moment matrix are critically examined particularly for their applicability to situations where input and output units differ or where there is a large or small steady state gain a normalization process is advocated which extends the usefulness and reliability of the statistics in these situations a procedure for comparing determinant statistics for models of different orders is described which simplifies current methods the improved procedures and statistics are compared using simulated data and an instrumental variable estimation method confirming their worth

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

an analysis of the projection method for robust adaptive beamforming two causes of performance degradation in adaptive beamforming are perturbation error and sample covariance error these causes were discussed by feldman and griffihs see ieee trans signal processing vol 42 p 867 april 1994 which proposed an approach termed the projection method to reduce the sensitivity of the beamformer performance to both types of error this paper contains an analysis of the expected performance of the beamformer when it is implemented either with or without using the projection method based on a second order taylor series expansion of an approximate expression for the signal to interference plus noise ratio sinr at the beamformer output the analysis provides accurate results when either perturbation error or sample covariance error occurs a less accurate analysis based on a zeroth order taylor series expansion includes results presented by jablon 1986 and chang and yeh see ibid vol 40 no 11 p 1336 1992 as special cases

simple and efficient nonparametric method for estimating the number of signals without eigendecomposition inspired by the computational simplicity and numerical stability of qr decomposition a nonparametric method for estimating the number of signals without eigendecomposition mense is proposed for the coherent narrowband signals impinging on a uniform linear array ula by exploiting the array geometry and its shift invariance property to decorrelate the coherency of signals through subarray averaging the number of signals is revealed in the rank of the qr upper trapezoidal factor of the autoproduct of a combined hankel matrix formed from the cross correlations between some sensor data since the infection of additive noise is defused signal detection capability is improved a new detection criterion is then formulated in terms of the row elements of the qr upper triangular factor when finite array data are available and the number of signals is determined as a value of the running index for which this ratio criterion is maximized where the qr decomposition with column pivoting is also used to improve detection performance the statistical analysis clarifies that the mense detection criterion is asymptotically consistent furthermore the proposed mense algorithm is robust against the array uncertainties including sensor gain and phase errors and mutual coupling and against the deviations from the spatial homogeneity of noise model the effectiveness of the mense is verified through numerical examples and the simulation results show that the mense is superior in detecting closely spaced signals with a small number of snapshots and or at relatively low signal to noise ratio snr

automatic determination of the number of targets present when using the time reversal operator acoustical time reversal mirrors have been shown to provide a highly accurate means of studying and focusing on acoustical sources the dort method is a derivation of the time reversal process which allows for focusing on multiple targets an important step in this process is the determination of the number of targets or sources present this is achieved by examining the eigenvalues of the time reversal operator tro the number of significant eigenvalues is then chosen as the number of sources present however as mentioned in n mordant c prada and m fink j acoust soc am 105 2634 2642 1999 and c prada m tanter and m fink in proceedings of the ieee symposium 1997 pp 679 683 factors such as low signal to noise ratio snr small data sample array configuration and the target location may result in the eigenvalues corresponding to the targets no longer being distinguishable from the background noise eigenvalues this paper proposes a robust method of automatically determining the

direction of arrival estimation using co prime arrays a super resolution viewpoint we consider the problem of direction of arrival doa estimation using a recently proposed structure of nonuniform linear arrays referred to as co prime arrays by exploiting the second order statistical information of the received signals co prime arrays exhibit o mn degrees of freedom with only m n sensors a sparsity based recovery algorithm is proposed to fully utilize these degrees of freedom the suggested method is based on the developing theory of super resolution which considers a continuous range of possible sources instead of discretizing this range onto a grid with this approach off grid effects inherent in traditional sparse recovery can be neglected thus improving the accuracy of doa estimation we show that in the noiseless case it is theoretically possible to detect up to mn 2 sources with only 2m n sensors the noise statistics of co prime arrays are also analyzed to demonstrate the robustness of the proposed optimization scheme a source number detection method is presented based on the spectrum reconstructed from the sparse method by extensive numerical examples we show the superiority of the suggested algorithm in terms of doa estimation accuracy degrees of freedom and resolution ability over previous techniques such as music with spatial smoothing and discrete sparse recovery

a pencil music algorithm for finding two dimensional angles and polarizations using crossed dipoles the authors study the problem of finding two dimensional 2 d angles of wave arrival and wave polarizations using a uniform rectangular array of crossed dipoles the method presented effectively exploits the redundancy in this array via 2 d moving window smoothing to handle coherent sources and to achieve optimum noise sensitivity the method combines the computational advantages of the music and matrix pencil approaches the method is shown in simulation to be nearly optimum compared with the cramer rao bound

un music and un cle an application of generalized correlation analysis to the estimation of the direction of arrival of signals in unknown correlated noise a new approach is proposed for the consistent estimation of the directions of arrival doa of signals in an unknown spatially correlated noise environment the signal and noise model used is based on the assumption that the data are received by two arrays well separated so that their noise outputs are uncorrelated the generalized correlation decomposition of the cross correlation matrix between the two arrays is then introduced of particular interest is the canonical correlation decomposition the analysis of the generalized correlation leads to various interesting geometric and asymptotic properties of the eigenspace structure two algorithms un music and un cle are developed to estimate the doa of signals in unknown spatially correlated noise based on the utilization of these properties computer simulations show that these methods are superior in performance compared to conventional methods furthermore it is demonstrated that the new methods are equally effective even when only one sensor array is employed

nested arrays in two dimensions part ii application in two dimensional array processing this paper explores the practical application of a new class of two dimensional arrays namely the two dimensional nested arrays in array processing problems like two dimensional direction of arrival estimation nested arrays constitute a class of two dimensional arrays with physical sensors on lattice s whose difference co array gives rise to a virtual two dimensional array with o mn elements although the number of physical sensors used is m n the structure of the two dimensional nested array was proposed in a companion paper and several issues related to the geometrical orientation of the difference co array in the two dimensional space were addressed in this paper the practical aspects of the application of the nested array in two dimensional array processing are considered which can enable us to exploit the increased degrees of freedom offered by the co array under the constraint of a fixed total number of physical sensors the optimal structure of the 2d nested array is solved which maximizes the number of elements in the virtual co array this provides closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors it is shown that one can obtain o n2 virtual elements in the co array using only n physical sensors to exploit the increased degrees of freedom large number of virtual sensors offered by the virtual array for two dimensional doa estimation of more sources than sensors a novel algorithm based on the concept of two dimensional spatial smoothing is proposed and several issues related to identifiability in two dimensional doa estimation problem which is a fundamental issue in two dimensions are addressed the validity of the proposed methods is verified through several numerical examples

low complexity and high resolution line spectral estimation using cyclic minimization the line spectral estimation problem has applications in radar wireless communications spectroscopy and power electronics among others the signal is modeled as a sparse linear combination of complex sinusoids and the problem target is to estimate the number of sinusoids in the mixture and respective parameters of each individual sinusoid such as magnitude phase and frequency in this work we first introduce a novel formulation of the rank function which involves the solution of a multi convex optimization problem using the multi convex reformulation of the rank function an over parameterization of the line estimation problem is proposed together with a cyclic minimization procedure to obtain a solution the cyclic method iterates between the optimization of the signal subspace and null space and stops when the two are orthogonal every limit point of the sequence of iterates is shown to be a stationary point of the original problem numerical experiments show that only a small number of iterations is required for convergence the signal subspace optimization is a semidefinite program sdp and the null space optimization has a closed form solution the cost per iteration of a general purpose interior point method ipm to solve the sdp is a quartic function of the problem dimension it is shown that by exploiting the problem structure the cost per iteration of the ipm may be reduced to a cubic function of the problem dimension comparable to the cost of the alternating direction method of multipliers admm however contrarily to the latter the former achieves the precision necessary for fine frequency localization a large set of numerical experiments show the effectiveness of the proposed approach

on the performance of signal subspace processing part i narrow band systems this paper presents an analytical evaluation of detection determination of the number of sources and estimation performances of narrow band signal subspace processing for multiple source direction finding the probabilities of underestimating and overestimating the number of sources are derived under asymptotic conditions and around the threshold regions in terms of the choice of a penalty function and signal noise and array parameters for the cases of at most two closely spaced sources in the spatially white noise a scalar measure is introduced for the evaluation of the quality of the estimated signal subspace based on the statistics of this measure performance thresholds are demonstrated for the signal to noise ratio angle separation and correlation between two equipowered sources

uniform in bandwidth consistency of kernel type function estimators we introduce a general method to prove uniform in bandwidth consistency of kernel type function estimators examples include the kernel density estimator the nadaraya watson regression estimator and the conditional empirical process our results may be useful to establish uniform consistency of data driven bandwidth kernel type function estimators

statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing the performances of the akaike 1974 information criterion and the minimum descriptive length criterion methods are examined the events which lead to erroneous decisions are considered and on the basis of these events the probabilities of error for the two criteria are derived the probabilities of the first two events are derived based on the asymptotic distribution of the sample eigenvalues of an estimated hermitian matrix it is further shown that the probabilities of missing and false alarm for these two criteria can be evaluated to a close approximation although the derivation of the probabilities of error is based on an asymptotic analysis the results are confirmed to be in very close agreement with computer simulation results

detection of the number of signals a predicted eigen threshold approach a novel method for detecting the number of signals incident upon an array of sensors is described the method is based on finding upper thresholds for the observed eigenvalues of the correlation matrix of the array output the asymptotic normality of the multifold eigenvalue estimate is used to derive a prediction formula for the thresholds due to the fact that the detection error of the method can be controlled by a parameter t the performance of the method is superior to the mdl minimum description length under low snr signal to noise ratio and superior to an aic akaike information criterion under high snr the distribution functions of both missing and false alarm errors are evaluated so that the performance of the method can be analyzed and so that t can be chosen simulation results are presented to confirm the analysis

nested arrays a novel approach to array processing with enhanced degrees of freedom a new array geometry which is capable of significantly increasing the degrees of freedom of linear arrays is proposed this structure is obtained by systematically nesting two or more uniform linear arrays and can provide o n2 degrees of freedom using only n physical sensors when the second order statistics of the received data is used the concept of nesting is shown to be easily extensible to multiple stages and the structure of the optimally nested array is found analytically it is possible to provide closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors this cannot be done for existing classes of arrays like minimum redundancy arrays which have been used earlier for detecting more sources than the number of physical sensors in minimum input minimum output mimo radar the degrees of freedom are increased by constructing a longer virtual array through active sensing the method proposed here however does not require active sensing and is capable of providing increased degrees of freedom in a completely passive setting to utilize the degrees of freedom of the nested co array a novel spatial smoothing based approach to doa estimation is also proposed which does not require the inherent assumptions of the traditional techniques based on fourth order cumulants or quasi stationary signals as another potential application of the nested array a new approach to beamforming based on a nonlinear preprocessing is also introduced which can effectively utilize the degrees of freedom offered by the nested arrays the usefulness of all the proposed methods is verified through extensive computer simulations

detection of number of sources via exploitation of centro symmetry property forward backward fb averaging is a technique which can be employed in sensor array signal processing for exploiting a so called centro symmetry property of the signal subspace to improve parameter estimation accuracy although fb averaging effectively doubles the number of data samples it also introduces correlation among these data vectors this correlation complicates not only the performance analysis but also the detection of the number of sources when fb averaging is employed existing detection procedures such as sequential hypothesis sh likelihood ratio tests minimum description length mdl and an information criterion aic must be modified to correctly account for this preprocessing of the data herein proper exploitation of the centro symmetry property of the fb averaged covariance is described and a rigorous analysis is conducted it is shown that the general scheme of existing methods is still applicable to the fb averaging case but that the parameters e g degrees of freedom need to be adjusted

an operator pseudo inversion lemma the so called matrix inversion lemma is frequently used in estimation restoration and related problems this paper extends it to the singular version which is utilized in the field of image restoration