Abstract: Summary. We prove the existence of a basis of a vector space, i.e., a set of vectors that generates the vector space and is linearly independent. We also introduce the notion of a subspace generated by a set of vectors and linear independence of set of vectors.

Abstract: Tests of large structures on-orbit will be performed with measurements at a relatively few structure points. Values for the unmeasured degrees of freedom (dofs) can be estimated based on measured dofs and analytical model dynamic information. These 'expanded' mode shapes are useful for optimal-update identification and damage location as well as test/analysis correlation. A new method of expansion for test mode shape vectors is developed from the orthogonal Procrustes problem from computational linear algebra. A subspace defined by the set of measured dofs is compared to a subspace defined by mode shapes from an analytical model of the structure. The method simultaneously expands and orthogonalizes the mode shape vectors. Two demonstration problems are used to compare the new method to current expansion techniques. One demonstration uses test data from a laboratory scale-model truss structure. Performance of the new method is comparable or superior to that of the previous expansion methods which require separate orthogonalization.

Abstract: A generalized coder that includes all types of excitation is presented. In this analysis-by-synthesis scheme, maximization formulae are the same regardless of the kind of excitation. The total excitation is expressed as a linear combination of excitation vectors. Given the number of excitation vectors or, equivalently, the bit rate of the coder, finding the vectors and their corresponding gains is a specific least-squares problem. The standard way this problem is usually solved is given, and three alternative mathematical approaches are proposed. These approaches are a Gram-Schmidt orthogonalization, a Choleski decomposition, a Householder transform. All these procedure have the same geometrical interpretation and lead to the same floating point simulation results. >

Abstract: Algorithms are presented which compute theQR factorization of a block-Toeplitz matrix inO(n) 2 block-operations, wheren is the block-order of the matrix and a block-operation is a multiplication, inversion or a set of Householder operations involving one or two blocks. The algorithms are in general analogous to those presented in the scalar Toeplitz case in a previous paper, but the basic operation is the Householder transform rather than the Givens transform, and the computation of the Householder coefficients and other working matrices requires special treatment. Two algorithms are presented-the first computes onlyR explicitly and the second computes bothQ andR.

Abstract: A real phase-only nulling algorithm for adaptive antenna arrays pattern synthesis is presented, and some computer simulation results are given. In order to increase the computation speed and the stability of the algorithm, the modified Gram-Schmidt orthogonalization procedure is used. Calculated phase weights are given, corresponding to the patterns studied in the simulation. >

Abstract: An efficient algorithm for optimizing the maximum likelihood criterion of direction-of-arrival (DOA) problems is presented. The algorithm is based on two principal theorems: the first concerns the fast Gram-Schmidt orthogonalization of a Krylov subspace, and the second provides an alternating one-dimensional maximization procedure. A combiner-lattice filter structure that facilitates highly modular and concurrent VLSI implementation for the proposed algorithm has also been developed. By investigating this structure, an instructive physical insight of the ML criterion could be revealed as the operation of notch filtering. Simulation results that demonstrate the performance of the algorithm are included. >

Abstract: Socioeconomic systems are often characterized by spatiotemporal structured data sets. If many potentially explanatory variables are available, a ranking of their relevance is desirable. A numerical procedure is presented which allows for a stepwise selective regression analysis of such variables based on least square principles. The optimal set of key socioeconomic factors is obtained from a large number of variables by an orthogonalization procedure. This orthogonalization takes into account the correlation between the variables which have been already been selected and the remaining set. Time delays of variables are also considered. As an example the key factor analysis of regional utilities of the migratory system of the Federal Republic of Germany is treated.

Abstract: A method for determining the states of the hidden units of feedforward neural networks for an arbitrary output function is proposed. The method uses properties of a vector space spanned by state vectors. The state vector used represents a set of states (input states, inner states, or output states) of a unit for many sample data. The inner state vector is the linear combination of the input state vectors and is nonlinearly transformed into the output state vector. Internal representation can be expressed by the output state vectors for the hidden units, which are the input state vectors necessary for the output unit. The problem is how to determine the appropriate internal representation in order to produce an arbitrary output function. This method, called the orthogonal complement method (OCM), is based on the orthogonality between the subspace spanned by the input state vectors and its orthogonal complement. The nonlinearity of the vector transformation is treated as a constraint with respect to the inner state vector. Unknown components of the output state vectors for the hidden units can be determined from this orthogonality, and the number of necessary hidden units can be estimated from the dimension of the subspace spanned by the input state vectors. The OCM is very useful for binary output systems, since this calculation can be simplified by using a basis of the orthogonal complement. Using a basic procedure of the OCM, a minimum network can be designed for about 70% of all four-variable Boolean functions

Abstract: Beginning with a discussion of the relationship between degrees of freedom and capacity of the system, the original work on higher order associative memories is described in three aspects, Learning, Capacity, and Generalization for pattern recognition and neural networks with the orthogonalization of binary vectors and the ternarization of weights, and their optical implementations using volume holograms are suggested for optical computing. Selection of terms is considered to satisfy the given conditions. When a simple sum of outer product learning rule is applied, higher order memories become higher order Hopfield-type memories. Their capacities are derived from SNR analyses for both nonzero diagonal and zero diagonal memories. Especially in the case of quadratic and cubic memories, optical implementations are suggested in three elegant ways due to the three-dimensional property of volume holograms. Robustness of higher order associative memories is discussed as a generalization property with consideration of dynamic range in terms of robustness of errors in input (error tolerance) and noise in the system (noise sensitivity). In the case of autoassociation or bidirectional association the energy functions are used to investigate the dynamics that provides a mechanism of escaping the local minima to find global minima. Algorithmic aspects and architectures of optical computing are discussed in terms of deterministic and random algorithms.

Abstract: Identification of multiple input output discrete time linear dynamic systems operating in open or closed loop are considered in the time invariant case. Two methods have been used for such a purpose: the recursive prediction error method on the input-output data and the successive Gram-Schmidt orthogonalization of the spectral density function of the joint input-output variable.

Abstract: Recently, for signal processing such as echo cancellation and equalization it is required to develop IIR-adaptive digital filters. This paper proposes a new-type IIR-adaptive algorithm based on the orthogonalization method, considers the convergence of the algorithm, and verifies its effectiveness by simulation. White's algorithm is one of IIR-adaptive algorithms reported thus far and Feintuch's algorithm is known as its approximation algorithm. This paper also derives an IIR-adaptive algorithm using both Feintuch's algorithm and the orthogonalization method. First, it can be shown that a generalized algorithm (IIR-LI) of IIR-learning identification method based on previous heuristic methods can be derived using the idea of Feintuch's algorithm and the conjugate gradient method. Next, a new IIR-adaptive algorithm (IIR-AACG) based on the orthogonalization principle of the conjugate gradient method is presented. Further, by applying the orthogonalization method to a past input vector sequence we derive an IIR-Affine projection algorithm (IIR-AP). Finally, these algorithms are compared by computer simulation to verify the effectiveness of the orthogonalization method.

Abstract: The objective is the order determination of non-Gaussian and nonminimum phase autoregressive moving average (ARMA) models, using higher-order cumulant statistics. The two methods developed assume knowledge of upper bounds on the ARMA orders. The first method performs a linear dependence search among the columns of a higher-order statistics matrix by means of the Gram-Schmidt orthogonalization procedure. In the second method, the order of the AR part is found as the rank of the matrix formed by the higher-order statistics sequence. For numerically robust rank determination the singular value decomposition approach is adopted. The argument principle and samples of the polyspectral phase are used to obtain the relative degree of the ARMA model, from which the order of the MA part can be determined. Statistical analysis is included for determining the correct MA order with high probability, when estimates of third-order cumulants are only available. Simulations are used to verify the performance of the methods and compare autocorrelation with cumulant-based order determination approaches. >

Abstract: The signal model presently considered is composed of a linear combination of basis signals chosen to reflect the basic nature believed to characterize the data being modeled. The basis signals are dependent on a set of real parameters selected to ensure that the signal model best approximates the data in a least-square-error (LSE) sense. In the nonlinear programming algorithms presented for computing the optimum parameter selection, the emphasis is placed on computational efficiency considerations. The development is formulated in a vector-space setting and uses such fundamental vector-space concepts as inner products, the range- and null-space matrices, orthogonal vectors, and the generalized Gramm-Schmidt orthogonalization procedure. A running set of representative signal-processing examples are presented to illustrate the theoretical concepts as well as point out the utility of LSE modeling. These examples include the modeling of empirical data as a sum of complex exponentials and sinusoids, linear prediction, linear recursive identification, and direction finding. >