a novel orthonormalization matrix based fast and stable dpm algorithm for principal and minor subspace tracking we note that the well known fast rayleigh s quotient based adaptive noise subspace frans frans with householder transformation hfrans and fast data projection method fdpm algorithms all inherit from the data projection method dpm algorithm but with different orthonormalization matrices starting from the dpm we analyze the orthonormalization matrices of all these algorithms and develop a novel orthonormalization matrix for our algorithm based on this novel orthonormalization matrix a fast and stable implementation of the dpm algorithm which has the merits of both the frans and fdpm approaches is investigated for principal and minor subspace tracking the proposed algorithm can switch between the principal and minor subspace tracking with a simple sign change of its step size parameter moreover it reaches the 3np lower bound of the dominant complexity and guarantees the orthonormality of the tracked subspace the numerical stability of our algorithm is established theoretically and tested numerically the strengths and weaknesses of the proposed algorithm to some existing subspace tracking algorithms are demonstrated using a de facto benchmark example simulation results are presented to demonstrate the effectiveness of the tracking algorithm advocated

spatio temporal spectral analysis by eigenstructure methods this paper presents new algorithms for estimating the spatio temporal spectrum of the signals received by a passive array the algorithms are based on the eigenstructure of the covariance and spectral density matrices of the received signals they allow partial correlation between the sources and thus are applicable to certain kinds of multipath problems simulation results that illustrate the performance of the new algorithms are presented

orthogonal eigensubspace estimation using neural networks we present a neural network nn approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum in magnitude eigenvalue this problem has its origin in the constrained minimization framework and has extensive applications in signal processing we recast this problem into the nn framework by constructing an appropriate energy function which the nn minimizes the nn is of feedback type with the neurons having sigmoidal activation function the proposed approach is analyzed to characterize the nature of the minimizers of the energy function the main result is that the matrix w is a minimizer of the energy function if and only if the columns of w are the orthogonal eigenvectors with a given norm corresponding to the smallest eigenvalue of the given matrix further all minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of differential equations which define the dynamics of the nn have also been derived results of computer simulations are presented to support our analysis

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

tracking of signal subspace projectors a new subspace tracking approach that directly operates on the projection matrix onto the signal subspace instead of tracking its unitary eigenbasis is proposed therefore the difficulties arising from introducing a calculus on the manifold of projection matrices are overcome by a proper parameterization of the projectors simulation results are presented from the field of angular frequency retrieval in array signal processing

a subspace based direction finding algorithm using fractional lower order statistics we propose several classes of fractional lower order moment flom based matrices that can be used with music to estimate the doas of independent circular signals embedded in additive s spl alpha s symmetric spl alpha stable noise e g sea clutter we run simulations with different choices of the flom parameter p for our flom based matrices and conclude that when the noise is s spl alpha s with unknown spl alpha spl ne 2 flom multiple signal classification music with p close to unity yields good performance the performance of flom music and robust covariation based roc music are similar three scenarios that contain circular signals phase modulation pm circularly symmetrical gaussian and quaternary phase shift keying qpsk and one scenario that contains noncircular signals binary phase shift keying bpsk all embedded in the same s spl alpha s noise are tested these simulation results reveal that the scenario containing bpsk signals leads to poor performance indicating that flom music is presently limited to circular signals

broad band signal subspace spatial spectrum bass ale estimation an approach to spatial spectrum estimation of broadband sources is described the approach termed broadband signal subspace spatial spectrum bass ale estimation is based on the eigenstructure of a broadband spatial temporal covariance matrix and is justified by identifying the low rank character of spatial temporal observations of broadband sources bass ale estimators are described which incorporate source focusing spatial temporal noise decorrelating a signal or noise only subspace generated from the eigenstructure of the transformed spatial temporal covariance matrix and broadband source models through discussion and simulation bass ale estimators are compared to the coherent signal subspace processor a similar subspace method and relative advantages of the two are identified

adaptive eigensubspace algorithms for direction or frequency estimation and tracking the authors present an adaptive estimator of the complete noise or signal subspace of a sample covariance matrix as well as the estimator s practical implementations the general formulation of the proposed estimator results from an asymptotic argument which shows the signal or noise subspace computation to be equivalent to a constrained gradient search procedure a highly parallel algorithm denoted the inflation method is introduced for the estimation of the noise subspace the simulation results of these adaptive estimators show that the adaptive subspace algorithms perform substantially better than p a thompson s 1980 adaptive version of v f pisarenko s technique 1973 in estimating frequencies or directions of arrival doa of plane waves for tracking nonstationary parameters the simulation results also show that the adaptive subspace algorithms are better than direct eigendecomposition methods for which computational complexity is much higher than the adaptive versions

wavelets and recursive filter banks it is shown that infinite impulse response iir filters lead to more general wavelets of infinite support than finite impulse response fir filters a complete constructive method that yields all orthogonal two channel filter banks where the filters have rational transfer functions is given and it is shown how these can be used to generate orthonormal wavelet bases a family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband butterworth filters when there is an odd number of zeros at pi it is shown that closed forms for the filters are available without need for factorization a still larger class of orthonormal wavelet bases having the same moment properties and containing the daubechies and butterworth filters as the limiting cases is presented it is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case and a constructive method is given it is also shown how compactly supported bases may be orthogonalized and bases for the spline function spaces are constructed

analysis of orthogonality error propagation for frans and hfrans algorithms in this correspondence we analyze the propagation of orthogonality error for fast rayleigh s quotient based adaptive noise subspace algorithm frans and frans with householder transformation hfrans first we examine the propagation of orthogonality error for the numerically unstable frans in the mean and in the mean square sense then an upper bound on orthogonality error is derived for the hfrans algorithm which is much more numerically stable compared to frans numerical examples are provided to corroborate the proposed error propagation models

rational filter wavelets abstract in this paper we introduce two types of rational filters for the construction of wavelets the rational filters are a natural development of the polynomial filters we use rational filters to derive a large family of the wavelets which can include daubechies wavelets and battle lemarie wavelets especially the ii type rational filter wavelets among the family have linear phases furthermore we analyze the regularity of the rational filter wavelets and estimate their regularity indices some examples are also given

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

focussing matrices for coherent signal subspace processing a class of focussing matrices is proposed for use in the coherent signal subspace method csm h wang and m kaveh ibid vol assp 33 aug 1985 when the directions of arrival of wideband sources fall into more than one group of one beamwidth each this class leads to performance substantially better than those suggested in previous studies on direction of arrival estimation new insight into the structures of various focussing matrices and their effect on the performance of csm is presented the performance of csm is compared for several classes of these matrices on the simulations as well as relative sufficiency analyses

least squares type algorithm for adaptive implementation of pisarenko s harmonic retrieval method pisarenko s harmonic retrieval method involves determining the minimum eigenvalue and the corresponding eigenvector of the covariance matrix of the observed random process recently thompson 9 suggested a constrained gradient search procedure for obtaining an adaptive version of pisarenko s method and his simulations have verified that the frequency estimates provided by his procedure were unbiased however the main cost of this technique was that the initial convergence rate could be very slow for certain poor initial conditions restating the constrained minimization as an unconstrained nonlinear problem we derived an alternative gauss newton type recursive algorithm which also used the second derivative matrix or hessian this algorithm may also be viewed as an approximate least squares algorithm simulations have been performed to compare this algorithm to a slight variant of thompson s original algorithm the most important conclusions are that the least squares type algorithm has faster convergence in the beginning while its convergence rate close to the true parameters depends on the signal to noise ratio of the input signal the approximate least squares algorithm resolves the sinusoids much faster than the gradient version

implementation of adaptive array algorithms some new efficient and numerically stable algorithms for the recursive solution of matrix problems arising in optimal beam forming and direction finding are described and analyzed the matrix problems considered are systems of linear equations and spectral decomposition while recursive solution procedures based on the matrix inversion lemma may be unstable ours are stable furthermore these algorithms are extremely fast

on data increasing in phase retrieval via quadratic inversion flattening manifold and local minima in this article we address the question of traps in phase retrieval via quadratic inversion first we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional later we recall a mathematical condition for the lack of traps afterward we illustrate how the ratio between the dimension of data space m and the number of real unknowns n r impacts on the manifold of data and consequently on the presence of local minima in particular we show that if the dimension of data space is high enough the manifold of data is sufficiently large and no traps appear in the functional in the last part of this article with reference to a particular unknown sequence we establish the minimum value of the ratio m n r such that no traps should appear in the functional in this condition phase retrieval via quadratic inversion exhibits a global convergence behavior consequently the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random

optimization based amp for phase retrieval the impact of initialization and ell_ 2 regularization we consider an ell 2 regularized non convex optimization problem for recovering signals from their noisy phaseless observations we design and study the performance of a message passing algorithm that aims to solve this optimization problem we consider the asymptotic setting m n rightarrow infty m n rightarrow delta and obtain sharp performance bounds where m is the number of measurements and n is the signal dimension we show that for complex signals the algorithm can perform accurate recovery with only m 64 pi 2 4 n approx 2 5n measurements also we provide a sharp analysis on the sensitivity of the algorithm to noise we highlight the following facts about our message passing algorithm 1 adding ell 2 regularization to the non convex loss function can be beneficial and 2 spectral initialization has a marginal impact on the performance of the algorithm the sharp analyses in this paper not only enable us to compare the performance of our method with other phase recovery schemes but also shed light on designing better iterative algorithms for other non convex optimization problems

mimo systems with limited rate differential feedback in slowly varying channels in this paper an adaptive limited feedback linear precoding technique for temporally correlated multiple input multiple output mimo channels is proposed where the receiver has perfect channel knowledge but the transmitter only receives a quantized channel direction to perform adaptation to the time correlation structure we employ a differential feedback where the amount of the perturbation added to the previous precoder is determined by the statistics of the directional variation based on random matrix quantization analysis we develop a spherical cap codebook approach where the cap is centered at the previous precoder and the radius of the cap is determined proportional to the identified directional variation if the channel is highly correlated in time it is shown that the proposed differential feedback scheme achieves significant throughput improvement in the large codebook size regime the rest of the paper is devoted to developing a systematic spherical cap codebook generation method the developed approach employs a feedback scheme that uses a differential rotation of the previously used precoder our codebook adaptation is based on generating a perturbation in euclidean space and projecting the perturbation onto the unitary space simulation results show that the proposed adaptation scheme accurately tracks the channel using only a small rate of feedback

rotational search methods for adaptive pisarenko harmonic retrieval two adaptation algorithms for adaptive pisarenko harmonic retrieval are described they are derived by considering the associated minimum eigenvalue problem as an optimization problem which seeks the minimum of a quadratic cost function given a hyperspherical constraint an iterative search procedure is used in which each search path is constrained to lie on the unit hypersphere computational complexity per iteration is approximately one third that of previous adaptive phr algorithms simulations reveal that at low snr the trial eigenvector can converge to the true minimum eigenvector of the sample covariance matrix long before this matrix is a good estimate of the true covariance matrix

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

focusing filters for wide band direction finding a time domain equivalent of the coherent signal subspace transformations cst is established for wideband direction finding in a possible multipath environment using general arrays time domain equivalents of focusing are derived based on the least squared error approach for general transformations and the taylor series expansion approach for closed form transformations the preprocessor is realized by a multichannel digital finite impulse response filter for diagonal transformations the problem reduces to implementing different delays at each sensor which in turn leads to computational simplicity various tapped delay line filters for realizing an arbitrary delay are proposed and compared simulation results reveal that low order filter structures in the time domain achieve similar performance to the frequency domain approach even at detection and resolution thresholds

on the regularity of wavelets the regularity index alpha sub n of the scaling functions sub n phi n 2 3 of multiresolution analysis introduced by i daubechies 1988 is investigated it is shown that 0 51

modal decomposition signal subspace algorithms a new method of estimating the locations of multiple wide band emitters arriving at an array of sensors is presented in this paper the wide band emitter location problem is treated as a separable multidimensional spectrum estimation problem by modal decomposition of the estimated spectral factor or spectral density matrix at the poles of the emitters this modal decomposition method extends conventional sinusoidal separation approaches to wide band emitters several methods of modal decomposition are presented and the signal subspace approach of schmidt 2 and su and morf 3 4 is extended and applied to the decomposed spectral residues for determination of emitter locations the method is asymptotically exact and provides the capability to resolve correlated or even coherent emitters

kalman based estimator for doa estimations we introduce a new kalman based noise subspace estimator combined with the root music for direction of arrival doa estimation of uncorrelated narrow band plane waves impinging on an array of sensors without both a priori knowledge of the number of sources and the inflation method presented by yang and kaveh 1988 we show that the kalman based estimator can approximately estimate the complete noise subspace with small bias for high input signal to noise ratio snr scenario the proposed algorithm needs slightly more computation operations per adaptive cycle than yang and kaveh s lms based algorithm but with much more rapid convergence simulations demonstrate the effectiveness of the proposed algorithm

on the performance of signal subspace processing part ii coherent wide band systems this is the second part of a study which deals with the performance of multiple source direction finding via signal subspace processing 1 an analytical evaluation of detection determination of the number of sources and direction estimation performance is presented for the coherent wide band signal subspace processing algorithm 2 which operates on the received array signals of large relative bandwidth the probabilities of underestimating and overestimating the number of sources are derived under asymptotic conditions and in 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 spatially white noise a scalar measure is used for the evaluation of the quality of the estimated signal subspaces it is shown that the detection and estimation performances of the practical coherent algorithm are close to those of the perfectly coherent processing and that they are relatively insensitive with respect to a preliminary estimate of the angles used to effect an approximate coherence

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data