Abstract: Orthogonalization techniques are disclosed which can be used, for example, to improve subtration of CDMA signals from a composite spread spectrum signal or to improve detection of CDMA signals from the composite spread spectrum signal. According to exemplary embodiments, a Gram-Schmidt orthogonalization process is used to modify signature sequences which are each associated with a particular CDMA signal in the spread spectrum composite signal. These modified signature sequences can then be used to spread correlations of the original signature sequences in the subtraction process or in the detection process to improve performance. Pre-orthogonalization according to the present invention, eliminates or mitigates multiple access interference.

Abstract: The paper presents an adaptive Volterra filter that employs an orthogonalization procedure of Gaussian signals for Volterra system identification. The algorithm is capable of handling arbitrary orders of nonlinearity P as well as arbitrary lengths of memory N for the system model. The adaptive filter consists of a linear lattice predictor of order N, a set of Gram-Schmidt orthogonalizers for N vectors of size P+1 elements each, and a joint process estimator in which each coefficient is adapted individually. The complexity of implementing this adaptive filter is comparable to the complexity of the system model when N is much larger than P, a condition that is true in many practical situations. Experimental results demonstrating the capabilities of the algorithm are also presented in the paper.

Abstract: A principle of ‘joint-space orthogonalization’ is proposed as an extended notion of hybrid (force and position) control for robot manipulators under geometric constraints. The principle realizes the hybrid control in a strict sense by letting position feedback signals be orthogonal in joint space to the contact force vector whose components exert at corresponding joints. This orthogonalization is executed via a projection matrix computed in real-time from a Jacobian matrix of the constraint equation in joint coordinates. To show the important role of the principle in control of robot manipulators, two basic set-point control problems are analysed. One is a hybrid PID control problem for robot manipulators under geometric endpoint constraint and another is a coordinated control problem of two arms. It is shown that passivity properties of residual dynamics of robots follow from the introduction of a quasi-natural potential and the joint-space orthogonalization. Various stability problems of PID-type feedback control schemes without compensating for the gravity force and with or without use of a force sensor are discussed from passivity properties of robot dynamics with the aid of the hyper-stability theory.

Abstract: We give an explicit formula for the solution to the initial value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szeg\"{o}, and is also interpreted as a consequence of the QR factorization method of Symes \cite{symes}. The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridiagonal formulae are given for the case of matrices with $2M+1$ nonzero diagonals.

Abstract: This paper concerns a class of infinite block matrices that are analogous to finite block Toeplitz matrices. Also studied are corresponding matrix-valued functions that are orthogonal for a matrixvalued inner product. An appendix presents basic results on orthogonalization in a Hilbert module.

Abstract: The incomplete orthogonalization method (IOM) proposed by Saad for computing a few eigenpairs of large nonsymmetric matrices is generalized into a block incomplete orthogonalization method (BIOM). It is studied how the departure from symmetry ‖A − A H ‖ affects the conditioning of the block basis vectors generated by BIOM, and some relationships are established between the approximate eigenpairs obtained by BIOM and Ritz pairs. It is proved that BIOM behaves much like generalized block Lanczos methods if the basis vectors of the block Krylov subspace generated by it are strongly linearly independent. However, it is shown that BIOM may generate a nearly linearly dependent basis for a general nonsymmetric matrix. Numerical experiments illustrate the convergence behavior of BIOM.

Abstract: The parallel performance of a dense, standard and generalized, real, symmetric eigensolver based on bisection for eigenvalues and repeated inverse iteration and reorthogonalization for eigenvectors is described. The performance of this solver, called PeIGS, is given for two test problems and for three ``real-world`` quantum chemistry applications: SCF-Hartree-Fock, density functional theory,and Moeller-Plesset theory. The distinguishing feature of the repeated inverse iteration and orthogonalization method used by PEIGS is that orthogonalization may be performed across multiple processors as dictated by the spectrum. For each problem we describe the spectrum and the clustering of the eigenvalues, the most important factor in determining the execution time. For a spectrum that is well spaced, there is essentially no orthogonalization time. Most of the time is consumed in the Householder reduction to tridiagonal form. For large clusters, almost all of the time is consumed in the Householder reduction and in orthogonalization. Performance results from the Intel Paragon, and Kendall Square Research KSR-2 are reported.

Abstract: In this paper, we propose an alternate non-parametric statistically optimal method of null filtering. One of the important features of this method lies in its ability to process signals of short record lengths. The optimality criteria for maximum output SNR and the minimum mean-square error are combined to generate the new approach. The method is first designed for the coherent case (where the desired signal shape is a priori known) and later extended to include the non-coherent case based on orthogonal signal expansion. To deal with a non-orthogonal signal expansion, we propose a sliding Gram-Schmidt orthogonalization. An application to separate two closely spaced damped sinusoids is considered. Simulation results are presented comparing the proposed methods with the conventional one based on Constrained Notch Filtering.

Abstract: Modified programmable canonical correlation analyzers (PCCA) are developed to exploit recursion and feedback for improved blind adaptive spatial filtering. Specific implementations are developed utilizing an alternating block power method with a generalized Gram-Schmidt orthogonalization procedure. Several realization of these new recursive/feedback PCCAs are developed for exploitation of cyclostationarity and constant modulus signal properties. The performance of the proposed techniques is evaluated empirically and characterized in terms of output SINR and convergence behavior.

Abstract: Certain vector sequences in Hermitian or in Hilbert spaces, can be orthogonalized by a Fourier transform. In the finite-dimensional case, the discrete Fourier transform (DFT) accomplishes the orthogonalization. The property of a vector sequence which allows the orthogonalization of the sequence by the DFT, called circular stationarity (CS), is discussed in this paper. Applying the DFT to a given CS vector sequence results in an orthogonal vector sequence, which has the same span as the original one. In order to obtain coefficients of the decomposition of a vector upon a particular nonorthogonal CS vector sequence, the decomposition is first found upon the equivalent DFT-orthogonalized one and then the required coefficients are found through the DFT. It is shown that the sequence of discrete Gabor (1946) basis functions with periodic kernel and with a certain inner product on the space of N-periodic discrete functions, satisfies the CS condition. The theory of decomposition upon CS vector sequences is then applied to the Gabor basis functions to produce a fast algorithm for calculation of the Gabor coefficients. >

Abstract: In this paper, we study the effects of orthogonalization on sequential, multisensor, and multispectral satellite images for CFAR point target detection incorporting fusion techniques. Although the K-L orthogonalization offers the best CFAR detection performance, it requires central fusion. A version of the G-S orthogonalization method, which preprocesses data in a pipeline form, offers a comparable CFAR detection to that of the K-L method. Point target CFAR detection is carried out by employing various fusion approaches on the orthogonal data. Sensor level fusion with quality information is shown to be preferable when the proposed sequential G-S orhtogonaltization is applied. The proposed CFAR approach is applied to dissimilar sensors and avoids overloading the communication channel transmitting only in the case of target detection. Trade-off studies and experimental results on real and simulated data are presented.

Abstract: This paper addresses the use of a fast orthogonalization process to find the nodes of an RBF network which produce the best match to a target function Several applications of RBF networks using this fast orthogonal search technique have been investigated and a classification problem is presented The problem involves classification of human chromosomes, which is a highly complicated 30 dimensional and 24 class problem Experimental results show that the fast orthogonal search technique not only outperforms the traditional technique, but it also uses much less time and effort

Abstract: Gram-Schmidt orthogonalization algorithm is an interesting theme in the field of adaptive beam-forming and filtering as a fast algorithm. However, a key problem associated with this algorithm is that the number of orthogonalization, namely, the dimensions of interference subspace, is required to know prior. In this paper we derive a threshold and adopt it to detect the number of orthogonalization in the procedure of Gram-Schmidt (GS) orthogonalization decomposition, and this detection approach is simpler and faster than the approach based on eigenanalysis. Finally, computer simulation results were presented too.

Abstract: We present a new deterministic algorithm which gives a complete factorization ofxn− 1 over finite fields of characteristic 2. To do so we adapt a well-known procedure of orthogonalization in order to compute primitive idempotents over extensions instead of over GF(2).

Abstract: Very successful multiple video ghost cancellation simulations have been obtained with the application of a designed learning algorithm, based on the orthogonal least square method [OLS], to a channel identification process based on a FIR system model. The algorithm can be visualized by assuming the existence of a data matrix, at the input of the equalizer, which is created with a time shifting process, in order to generate "M" column vectors. The designed algorithm processes these vectors and, with the aid of an orthogonalization method, calculates a set of the most representative one, with respect to a desired output signal. The delays and amplitudes of the ghosts were obtained with the aid of a forward regressor method. The process has shown to be very effective for the accurate calculation of the ghost parameters, even in the presence of considerable noise levels, and is also used to train RBF approximation networks for the systematic selection of its centroids. In all the tests performed in the paper, the proposed technique has given much better results than using conventional algorithms. >

Abstract: Summary and conclusions In this paper a least-squares polynomial preconditioner of low degree for use with the GMRES algorithm is discussed. The parameters of the polynomial preconditioner are computed from eigenvalue estimates that can be obtained from the GMRES iterative process. Since we have considered polynomials of low degree only stability is not a major problem. The algorithm is simple and therefore will applicable in real life. Two different methods for obtaining eigenvalue estimates are discussed. A simple and stable approach is to calculate the Ritz values with Arnoldi’s method. In literature it was reported that the zeroes of the GMRES residual polynomial may yield more robust eigenvalue estimates. We describe in this paper a stable strategy for computing these zeroes. An experiment where polynomial preconditioners based on these two eigenvalue estimates are compared renders insufficient information to draw soundly based conclusions, and more research on this topic will.be necessary. Experiments indicate that a considerable reduction of the CPU-time can be reached if the polynomial preconditioner is applied in combination with GMRES with restarts after cycles of iterations. The experiments do not indicate that a significantly greater reduction of the CPU-time can be achieved if a polynomial preconditioner of high degree is applied. We observed only a marginal reduction of the CPU-time if a polynomial preconditioner of degree higher than two was applied. For full GMRES no reduction of the number of matrix-vector products can be achieved. However, since the orthogonalization of the basis for the Krylov subspace becomes increasingly expensive an important reduction of the CPU-time can be achieved by applying the polynomial preconditioner. Moreover, since the number of iterations is decreased if the polynomial preconditioner is applied, far less basis vectors for the Krylov subspace need to be stored. We have

Abstract: A technique is introduced to whiten the inputs of an adaptive filter in such a way as to improve the convergence of the normalized least mean-squares (NLMS) adaptation algorithm. This approach, based on the orthogonalization of successive input vectors, is shown to provide a better conditioned input while introducing some added misadjustment. It is shown, however, that in some applications the gains achieved are considerably more than the losses incurred.

Abstract: Adaptive beamforming using signal cyclostationarity can preserve the desired signal and cancel the interferers without prior information of the steering vector. We consider the Cross-SCORE processor which is one of this class of beamformers and uses time-consuming eigenvalue decomposition (EVD) to compute the weight vectors. Thus, this processor is not suitable for real-time processing. We apply a modular Gram-Schmidt orthogonalization (GSO) structure in conjunction with a power normalization scheme to the Cross-SCORE processor and propose a LMS based adaptive algorithm to update the weight vectors. Due to the pipeline and parallel properties of the modular GSO structure, our approach is very suitable for real-time processing and the required computing time for the array to process an output is O(N), where N is the number of array elements.

Abstract: It has been known for decades that sub-integrated elements not only exhibit higher accuracy but also consume less cpu time However, they are plagued by the spurious zero energy modes Techniques for suppressing the spurious modes become a popular area of finite element research In this paper, hybrid stress formulation is used to devise the stabilization vectors for the sub-integrated 9-node Lagrangian Co shell element The assumed stress shape functions are contravariant in nature and modified from the spurious zero energy modes of the sub-integrated element The so-formed leverage vectors can effectively stabilize the sub-integrated element Most importantly, the vectors can be derived explicitly and constructed without resorting to numerical integration nor the Gram-Schmidt orthogonalization scheme Benchmark tests show that the element is promising compared to other state-of-the-art finite element models

Abstract: Image and video signal processing applications often require filters with unknown or time-varying characteristics. Two-dimensional adaptive filters have been examined recently as a proposed solution to these problems. The following system considerations have driven research on cost-effective acceleration algorithms for 2-D adaptive filters. First the high data rates in digital video processing demand computational efficiency, and second, the nonstationary signal properties of images require optimized convergence speed. We present an overview of structures and algorithms developed to achieve an improved rate of convergence with reduced computational complexity. These include 2-D Newton-type adaptive filters and 2-D transform domain adaptive filters. The results are benchmarked against simple 2-D LMS and RLS adaptive filters.

Abstract: The concept of a W-matrix is used to give an elementary interpretation of a biorthogonal wavelet decomposition of signals The authors also give a method to modify the decomposition to give an orthogonal projection on the space spanned by the scaling vectors Roughly speaking, their treatment is a finite-length analog of the well-known theory of multiresolution analysis of Meyer and Mallat Their approach differs in that it deals directly with the discrete case, it takes care of the boundary elements without explicit padding,and it uses a notion similar to that of semiorthogonality introduced by Chui Their algorithm has flexibility in the choice of filter coefficients The decomposition, orthogonalization, and restoration algorithms are computationally fast

Abstract: Very successful multiple video ghost cancellation simulations have been obtained with the application of a designed learning algorithm, based on the orthogonal least square method, to a channel identification process based on a FIR system model. The algorithm can be visualized by assuming the existence of a data matrix, at the input of the equalizer, which is created with a time shifting process, in order to generate "M" column vectors. The designed algorithm processes these vectors and, with the aid of an orthogonalization method, calculates a set of the most representative one, with respect to a desired output signal. The ghost parameters were obtained with the aid of a forward regressor method. The process has shown to be very effective for the accurate calculation of the ghost parameters, even in the presence of considerable noise levels, and is also used to train RBF approximation networks for the systematic selection of its centroids. In all the tests performed in this work, the proposed technique has given much better results than using conventional algorithms.