Projection Approximation Subspace Tracking Research Articles

Direction of Arrival Tracking by Adaptive Subspace Methods

The radar tracking system prefers to have an estimated covariance matrix in a single snapshot in order to estimate the next position of a moving target. Recent research provides the inflation method and the projection approximation subspace tracking (PAST) method to estimate the DOA based on successive snap shots where an objective function, such as minimize the DOA estimation error, is defined and minimized. In this paper we provide complete analysis in addition tocomputer simulations for the mentioned recently published DOA estimation methods. Their behaviors are compared with the traditional, parametric, Eigen decomposition and spatial smoothing methods. Although the spatial smoothing method outperforms the other traditional method, that it could estimate the DOA of both coherent and non coherent signals at higher SNR. The recent DOA estimation method provided the capability of tracking a moving source as well as estimating the DOA of fixed location source. PAST method, can track moving signal sources, which change their direction.

Fast Subspace Tracking Algorithm Based on the Constrained Projection Approximation

We present a new algorithm for tracking the signal subspace recursively. It is based on an interpretation of the signal subspace as the solution of a constrained minimization task. This algorithm, referred to as the constrained projection approximation subspace tracking (CPAST) algorithm, guarantees the orthonormality of the estimated signal subspace basis at each iteration. Thus, the proposed algorithm avoids orthonormalization process after each update for postprocessing algorithms which need an orthonormal basis for the signal subspace. To reduce the computational complexity, the fast CPAST algorithm is introduced which has complexity. In addition, for tracking the signal sources with abrupt change in their parameters, an alternative implementation of the algorithm with truncated window is proposed. Furthermore, a signal subspace rank estimator is employed to track the number of sources. Various simulation results show good performance of the proposed algorithms.

Moving Sound Source Localization Based on Sequential Subspace Estimation in Actual Room Environments

This paper presents a novel method for moving sound source localization and describes its performance evaluation in actual room environments. The method is based on MUSIC (MUltiple SIgnal Classification) which is one of the highest-resolution localization methods. When using MUSIC, computation of eigenvectors of correlation matrix is required for estimation, which often results in a high computational cost. In particular, in the case of a moving source, this is a crucial drawback, because estimation must be performed at every observation time. Furthermore, since the correlation matrix varies its characteristics due to spatial-temporal nonstationarity, the matrix must be estimated using only a few observed samples, which degrades the estimation accuracy. In this paper, PAST (Projection Approximation Subspace Tracking) is applied for sequential estimation of the eigenvectors spanning the subspace. In PAST, eigen-decomposition is not required, and therefore it is possible to reduce the computational costs. Several experimental results in actual room environments are presented in order to demonstrate the superior performance of the proposed method. © 2011 Wiley Periodicals, Inc. Electron Comm Jpn, 94(7): 17–26, 2011; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/ecj.10333

DOA Estimation in Mobile Communications System using Subspace Tracking Methods

The radio environments in mobile communications are complicated and time-varing in general; therefore, we need high resolution DoA (direction of arrival) estimation methods that can follow quickly the change of radio environments. High resolution DoA estimation methods have been proposed which are based on the eigen decomposition of the correlation (covariance) matrix of an array input. MUSIC (Multiple Signal Classification) is one of a typical of such methods. However, these methods must normally repeat high-load computation involving the eigen decomposition of a correlation matrix every time a snapshot is taken. Therefore, it takes a very long time to obtain the estimated DoA when the number of array elements is too large. In addition, it is quite inefficient in the case that the DoA estimation is carried out continuously. To solve the above problems, Bi-SVD (Bi-Iteration Singular Value Decomposition) and PAST (Projection Approximation Subspace Tracking) have been proposed and investigated, which are typical methods of successively updating (tracking) eigenvectors in the signal subspace of correlation matrix. The radio environments in mobile communications are complicated and time-varing in general; therefore, we need high resolution DoA (direction of arrival) estimation methods that can follow quickly the change of radio environments. High resolution DoA estimation methods have been proposed which are based on the eigen decomposition of the correlation (covariance) matrix of an array input. MUSIC (Multiple Signal Classification) is one of a typical of such methods. However, these methods must normally repeat high-load computation involving the eigen decomposition of a correlation matrix every time a snapshot is taken. Therefore, it takes a very long time to obtain the estimated DoA when the number of array elements is too large. In addition, it is quite inefficient in the case that the DoA estimation is carried out continuously. To solve the above problems, Bi-SV

Blind multiuser detection using the subspace-based linearly constrained LSCMA

The least squares constant modulus algorithm (LSCMA) is a popular constant modulus algorithm (CMA) because of its global convergence and stability. But the performance will degrade when it is affected by the problem of interference capture in the MC-CDMA system that has several constant modulus signals. In order to overcome this shortage, a linearly constrained LSCMA multiuser detection algorithm is proposed by using the spreading code of the desired user to impose linear constraint on the LSCMA. To further enhance the performance, we project the weight vector obtained by the proposed linearly constrained LSCMA algorithm onto the signal subspace and propose a subspace-based linearly constrained LSCMA multiuser detection algorithm. The proposed algorithm ensures the algorithm convergence to the desired user and suppresses the noise subspace in the weight vector. Thus the performance of the system is improved. Moreover, to reduce the computational complexity, an improved projection approximation subspace tracking with deflation (PASTd) algorithm is proposed for adaptive signal subspace estimation. The simulation results demonstrate that the proposed algorithm achieves better output signal-to-interference-plus-noise ratio (SINR) and bit error rate (BER) performance than the traditional LSCMA algorithm, linearly constrained LSCMA algorithm and subspace-based MMSE algorithm.

Decision Directed Channel Estimation Aided OFDM Employing Sample-Spaced and Fractionally-Spaced CIR Estimators

In this letter we characterize the substantial difference between two channel estimation approaches, namely the sample-spaced (SS) and the fractionally-spaced (FS) channel impulse response (CIR) estimators. The achievable performance of decision-directed channel estimation (DDCE) methods employing both the SS- and the FS-CIR estimators is analyzed in the context of an OFDM system. The performance of the two estimation methods is compared and it is shown that the DDCE scheme employing the projection approximation subspace tracking (PAST)-aided FS-CIR estimator outperforms its SS-CIR estimator-based counterpart

Multivariable Adaptive Control of Synchronous Machines in a Multimachine Power System

A multivariable self-tuning adaptive control scheme used to enhance power system stability in a multimachine environment is presented. The controller is implemented locally for individual generators with supplementary stabilizing signals through the AVR and governor. A discrete multivariable state space model is developed to represent the generator. The recursive subspace identification method based on the projection approximation subspace tracking approach is employed to update generator model parameters online. A generalized predictive control strategy with constraints on the control signals is used. Simulations studies carried out on a five-machine power system without infinite bus show that the proposed multivariable adaptive controller is effective in damping both local and interarea mode oscillations under small as well as large disturbances. The self-coordinating ability of the adaptive controller with the existing conventional controllers is also demonstrated.

The modified natural power method for principal component computation

A modified version of the natural power method (NP) for fast estimation and tracking of the principal eigenvectors of a vector sequence is Presented. It is an extension of the natural power method because it is a solution to obtain the principal eigenvectors and not only for tracking of the principal subspace. As compared with some power-based methods such as Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method, the modified natural power method (MNP) has the fastest convergence rate and can be easily implemented with only O(np) flops of computation at each iteration, where n is the dimension of the vector sequence and p is the dimension of the principal subspace or the number of the principal eigenvectors. Furthermore, it is guaranteed to be globally and exponentially convergent in contrast with some non-power-based methods such as MALASE and OPERA.

RLS-based adaptive algorithms for generalized eigen-decomposition

The aim of this paper is to develop efficient online adaptive algorithms for the generalized eigen-decomposition problem which arises in a variety of modern signal processing applications. First, we reinterpret the generalized eigen-decomposition problem as an unconstrained minimization problem by constructing a novel cost function. Second, by applying projection approximation method and recursive least-square (RLS) technique to the cost function, a parallel adaptive algorithm for a basis for the r-dimensional (r>0) dominant generalized eigen-subspace and a sequential algorithm based on deflation technique for the first r-dominant generalized eigenvectors are derived. These algorithms can be viewed as counterparts of the extended projection approximation subspace tracking (PAST) and PASTd algorithms, respectively. Furthermore, we modify the parallel algorithm to explicitly estimate the first r-generalized eigenvectors in parallel, not the generalized eigen-subspace. More important, the modified parallel algorithm can be used to extract multiple generalized eigenvectors of two nonstationary sequences, while the proposed sequential algorithm lacks this ability because of slow convergence of minor generalized eigenvectors due to error propagation of the deflation technique. Third, following convergence analysis methods for PAST and PASTd, we prove the asymptotic convergence properties of the proposed algorithms. Finally, computer simulations are performed to investigate the accuracy and the speed advantages of the proposed algorithms.

Dyes in dermatology

The rotation of spacecraft flexible appendage may cause changes in modal parameters. For this time-varying system, the computation cost of the frequently-used singular value decomposition (SVD) identification method is high. Some control problems, such as the self-adaptive control, need the latest modal parameters to update the controller parameters in time. In this paper, the projection approximation subspace tracking (PAST) recursive algorithm is applied as an alternative method to identify the time-varying modal parameters. This method avoids the SVD by signal subspace projection and improves the computational efficiency. To verify the ability of this recursive algorithm in spacecraft modal parameters identification, a spacecraft model with rapid rotational appendage, Soil Moisture Active/Passive (SMAP) satellite, is established, and the time-varying modal parameters of the satellite are identified recursively by designing the input and output signals. The results illustrate that this recursive algorithm can obtain the modal parameters in the high signal noise ratio (SNR) and it has better computational efficiency than the SVD method. Moreover, to improve the identification precision of this recursive algorithm in the low SNR, the wavelet de-noising technology is used to decrease the effect of noises.

Direction-of-Arrival Tracking Scheme for DS/CDMA Systems: Direction Lock Loop

In this paper, a new direction-of-arrival (DOA) tracking algorithm, direction lock loop (DiLL), is proposed for wireless direct-sequence code-division multiple-access systems. It has a similar concept to the delay lock loop that is used for timing synchronization. It may track the DOA of sources by iterations. Its computational requirements are NK+O(K) for coherent DiLL and 2NK+O(K) for noncoherent DiLL, where N is the number of antenna elements and K is the number of signal sources, which is less than that of the projection approximation subspace tracking with deflation algorithm. The DOA tracking accuracy is demonstrated by analysis and computer simulations.

On a Class of Orthonormal Algorithms for Principal and Minor Subspace Tracking

This paper elaborates on a new class of orthonormal power-based algorithms for fast estimation and tracking of the principal or minor subspace of a vector sequence. The proposed algorithms are closely related to the natural power method that has the fastest convergence rate among many power-based methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method. A common feature of the proposed algorithms is the exact orthonormality of the weight matrix at each iteration. The orthonormality is implemented in a most efficient way. Besides the property of orthonormality, the new algorithms offer, as compared to other power based algorithms, a better numerical stability and a linear computational complexity.

A novel subspace tracking using correlation-based projection approximation

In this paper, we propose a novel subspace estimation technique, which is called correlation-based projection approximation subspace tracking (COPAST). The COPAST utilizes the projection approximation approach onto the correlation matrix to develop the subspace tracking algorithm. With the projection approximation, the RLS-based COPAST and the sequential-based COPAST algorithms are presented. The RLS-based COPAST algorithm has the better performance but the higher computational complexity than the recently developed PAST method. On the other hand, the sequential-based COPAST has reduced the computational complexity to nearly that of the PAST. Besides, the sequential-based COPAST has faster initial convergence speed than the PAST, while both nearly converge to the same value.

Spatio-temporal blind adaptive multiuser detection

We propose blind multiuser detection schemes with antenna arrays, which is based on signal subspace estimation. They are a multichannel extension of the decorrelating and minimum mean-square-error detectors, and therefore they share their immunity to near-far effects. The blind scheme may be seen as an extension of the results in of Wang and Poor (see IEEE Trans. Inform. Theory, vol.44, p.677-90, 1998). However, it is seen that compared with the latter results when spatial diversity is considered, the proposed spatio-temporal detectors offer, with little attendant increase in computational complexity, a better performance. A blind adaptive implementation based on a new orthogonal PAST (projection approximation subspace tracking) algorithm, which is shown to be efficient for subspace tracking, is proposed. Also, we develop a blind estimation of the spatial signature based on the orthogonality between noise and signal subspaces. It is seen that the blind adaptive multiuser detection and blind spatio-temporal signature estimation can he integrated jointly.

A New Look at the Power Method for Fast Subspace Tracking

A class of fast subspace tracking methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method can be viewed as power-based methods. Unlike many non-power-based methods such as the Given's rotation based URV updating method and the operator restriction algorithm, the power-based methods with arbitrary initial conditions are convergent to the principal subspace of a vector sequence under a mild assumption. This paper elaborates on a natural version of the power method. The natural power method is shown to have the fastest convergence rate among the power-based methods. Three types of implementations of the natural power method are presented in detail, which require respectively O(n2p), O(np2), and O(np) flops of computation at each iteration (update), where n is the dimension of the vector sequence and p is the dimension of the principal subspace. All of the three implementations are shown to be globally convergent under a mild assumption. The O(np) implementation of the natural power method is shown to be superior to the O(np) equivalent of the Oja, PAST, and NIC methods. Like all power-based methods, the natural power method can be easily modified via subspace deflation to track the principal components and, hence, the rank of the principal subspace.

An efficient PASTd-algorithm implementation for multiple direction of arrival tracking

We propose a simple direction-of-arrival (DOA) tracking scheme involving the projection approximation subspace tracking with deflation (PASTd)-algorithm and a Kalman filter. The DOAs are obtained via a Newton-type method initialized with the DOAs predicted by the Kalman filter. The data association is then avoided. The tracking capability of the proposed algorithm is demonstrated by computer simulations in scenarios involving targets with crossing trajectories.

A new adaptive algorithm for minor component analysis

In this paper, a new adaptive algorithm for minor component analysis is proposed. The (PASTd) projection approximation subspace tracking with deflation algorithm for principal component analysis by Yang is modified by putting the minus sign to the adaptive gain with orthonormalization. Using the ODE (ordinary differential equation) approach, the convergent point of the algorithm is shown to be the desired one. Simulation results show that the proposed algorithm has better estimation accuracies than the existing adaptive algorithm.

Instrumental variable subspace tracking using projection approximation

Subspace estimation plays an important role in, for example, sensor array signal processing. Recursive methods for subspace tracking with application to nonstationary environments have also drawn considerable interest. In this paper, instrumental variable (IV) extensions of the projection approximation subspace tracking (PAST) algorithm are presented. The IV approach is motivated by the fact that PAST gives biased estimates when the noise is not spatially white. The proposed algorithms are based on a projection like unconstrained criterion, with a resulting computational complexity, of 3ml+O(mn), where m is the dimension of the measurement vector; l is the dimension of the IV vector; and n is the subspace dimension. In addition, an extension to a "second order" IV algorithm is proposed, which in certain scenarios is demonstrated to have better tracking properties than the basic IV-PAST algorithms. The performance of the algorithms is demonstrated with a simulation study of a time-varying array processing scenario.

Two Improved Algorithms for Adaptive Subspace Filtering

In this paper, two improved algorithms with less computational complexities for adaptive subspace filtering are proposed. First, for signal subspace filtering, the basic projection approximation subspace tracking (PAST) algorithm by Yang is modified to update only the diagonal elements of the covariance matrix. Second, for noise subspace filtering, especially the extraction of the eigenvector associated with the minimum eigenvalue, a modified PAST algorithm by putting the minus sign to the adaptive gain with the normalization is proposed. Using the ODE approach, the convergent points of both algorithms are shown to be the desired ones. Some simulation results show that both algorithms have better numerical and convergent properties than the existing algorithms.

Asymptotic convergence analysis of the projection approximation subspace tracking algorithms

Subspace tracking plays an important role in a variety of adaptive subspace methods. In this paper, we present a theoretical convergence analysis of two recently proposed projection approximation subspace tracking algorithms (PAST and PASTd). By invoking Ljung's ordinary differential equation approach, we derive a pair of coupled matrix differential equations, whose trajectories describe the asymptotic convergence behavior of the subspace tracking algorithms. We discuss properties of the matrix differential equations and determine their asymptotically stable equilibrium states and domain of attraction. It turns out that, under weak conditions, both PAST and PASTd globally converge to the desired signal subspace or signal eigenvectors and eigenvalues with probability one. Numerical examples are also included to illustrate the asymptotic convergence rate of the algorithms.