Input Data Matrix Research Articles

An efficient algorithm for the 2-D discrete cosine transform

In this paper, we present an efficient algorithm for computing the 2-D discrete cosine transform (DCT). First, we arrange the input data matrix in a certain order. Second, we formulate the output data matrix to a block-structured matrix-vector multiplication, and show that the block-structured matrix has the same properties as the 1-D DCT kernel matrix. Therefore, the 2-D DCT problem is reduced to the 1-D DCT problem. We thus can provide a procedure for computing the 2-D DCT which is similar to that of the 1 -D DCT. The primary appeal of this algorithm lies not only in its low computational cost but also in its regular structure as the transform size increases.

QRD-based multichannel adaptive lattice algorithms for the parameter identification problem

A pair of multichannel recursive least squares (RLS) adaptive lattice algorithms based on the order recrusive of lattice filters and the superior numerical properties of Givens algorithms is derived in this paper. The derivation of the first algorithm is based on QR decomposition of the input data matrix directly, and the Givens rotations approach is used to compute the QR decomposition. Using first a prerotation of the input data matrix and then a repetition of the single channel Givens lattice algorithm, the second algorithm can be obtained. Both algorithms have superior numerical properties, particularly the robustness to wordlength limitations. The parameter vector to be estimated can be extracted directly from internal variables in the present algorithms without a backsolve operation with an extra triangular array. The results of computer simulation of the parameter identification of a two-channel system are presented to confirm efficiently the derivation.

Multilayer neural networks for reduced-rank approximation

This paper is developed in two parts. First, the authors formulate the solution to the general reduced-rank linear approximation problem relaxing the invertibility assumption of the input autocorrelation matrix used by previous authors. The authors' treatment unifies linear regression, Wiener filtering, full rank approximation, auto-association networks, SVD and principal component analysis (PCA) as special cases. The authors' analysis also shows that two-layer linear neural networks with reduced number of hidden units, trained with the least-squares error criterion, produce weights that correspond to the generalized singular value decomposition of the input-teacher cross-correlation matrix and the input data matrix. As a corollary the linear two-layer backpropagation model with reduced hidden layer extracts an arbitrary linear combination of the generalized singular vector components. Second, the authors investigate artificial neural network models for the solution of the related generalized eigenvalue problem. By introducing and utilizing the extended concept of deflation (originally proposed for the standard eigenvalue problem) the authors are able to find that a sequential version of linear BP can extract the exact generalized eigenvector components. The advantage of this approach is that it's easier to update the model structure by adding one more unit or pruning one or more units when the application requires it. An alternative approach for extracting the exact components is to use a set of lateral connections among the hidden units trained in such a way as to enforce orthogonality among the upper- and lower-layer weights. The authors call this the lateral orthogonalization network (LON) and show via theoretical analysis-and verify via simulation-that the network extracts the desired components. The advantage of the LON-based model is that it can be applied in a parallel fashion so that the components are extracted concurrently. Finally, the authors show the application of their results to the solution of the identification problem of systems whose excitation has a non-invertible autocorrelation matrix. Previous identification methods usually rely on the invertibility assumption of the input autocorrelation, therefore they can not be applied to this case.

Multipath suppression performance of an adaptive array antenna with a systolic array processor

AbstractThis paper describes an adaptive array antenna using a signal processing system which is implemented by a systolic array processor (SAP), SAP‐AA. Computer simulations show the behavior of the SAP‐AA against multipath fading. The SAP realizes a parallel pipelined processing by regularly placing many SAP cells with identical functions and structures so that it achieves the high throughput rate in the data processing. Also, the SAP has many desirable features for very large‐scale integration (VLSI) system design and enables miniaturization, weight reduction, and mass production of the processing equipment.In this paper, first, the SAP‐AA is described which realizes the least squares (LS) algorithm based on the QR decomposition for the input data matrix using the Givens rotations. In addition, the fundamental performances of the SAP‐AA are studied including both the transient response and the steady‐state behavior in multipath signal environments. To date, there has been no report which treats the multipath signal suppression performance of the SAP‐AA. Finally, it is concluded from the discussion that the SAP‐AA is effective for unwanted signal suppression in a multipath signal environment.

Substantive nonadditivity in social science research A note on induced collinearity and measurement and testing of effects

Theories employed to explain regularities in social behavior often contain reference (explicit or implicit) to the presence of nonlinear and/or nonadditive (i.e., multiplicative) relationships among germane variables. While such nonadditive features are theoretically important, the inclusion of quadratic or multiplicative terms in structural equations to model such features can cause significant methodological problems. This paper estimates a set of equations and formally examines how the inclusion of quadratic terms and multiplicative interaction terms contribute to the level of collinearity or ill-conditioning of the input data matrix and the precision of the parameter estimates. Subsequently we examine how effects of explanatory variables in nonadditive models can be measured and tested for statistical significance. The results indicate that collinearity may not be as big a problem for linear structural social science models as is often believed. Further, although collinearity is increased by adding quadratic and/or multiplicative terms, the effects of the collinearity tend to be localized and entail only variables with a common base. The findings suggest the substantive insight gained from including theoretically appropriate nonlinear and nonadditive terms outweigh the methodological problems they create.

Performance Analysis of FFT Algorithms on Multiprocessor Systems

A decimation-in-time radix-2 fast Fourier transform (FFT) algorithm is considered here for implementation in multiprocessors with shared bus, multistage interconnection network (MIN), and in mesh connected computers. Results are derived for data allocation, interprocessor communication, approximate computation time, and speedup of an N point FFT on any P available processing elements (PE's). Further generalization is obtained for a radix-r FFT algorithm. An N X N point two-dimensional discrete Fourier transform (DFT) implementation is also considered when one or more rows of the input data matrix are allocated to each PE.

An APL program for minimum-error Guttman scaling

Computer programs which perform minimum-error Guttman scaling are not widely available. The most accessible program to perform any type of Guttman scaling is the SPSS (Nie, Hull, Jenkins, Steinbrenner, & Bent, 1975) program, which uses the Goodenough (1944) procedure for counting errors. Unfortunately, according to McConaghy (1976), the statistical tests used to evaluate the scales produced by SPSS are based on the minimum-error rather than the Goodenough procedure. This discrepancy produces an overly conservative statistical test. A program written to perform minimum-error Guttman scaling uses an error-counting technique and statistical tests that are consistent with each other. The program permits ambiguous score patterns that have minimum Guttman errors at two or more true score values to be treated in any of four different ways. These treatments include random selection among the possible score values, selection of the middle score, and selection of the highest or the lowest of the possible scores. The variety of alternatives illustrates the continuing nature of the controversy which still surrounds the scaling procedure (Dotson & Summers, 1970; Henry, 1952; Wimberley, 1976). Computer and Language. The program is written in the APL language for use at a computer terminal. The language and the program combine to permit a high degree of interaction between the user and the computer. The algorithms can be translated into FORTRAN, but would result in a program more than five times as long ae the APL program. The program will function on any computer with an APL system. Input, A matrix containing the data to be analyzed must be prepared before the program is executed. Each row of the matrix represents a case; each column represents the responses to an item forming the scale. Responses must be coded in binary form, with 1 indicating than an item was passed and 0 indicating that an item was failed. Other input is requested during the execution of the program: specification of processing, statistics and output options, and the insertion of a title for each analysis. Computational Procedure. The entire matrix (or any subset of columns selected by the user) is rearranged according to the number of cases passing each item. The response pattern of each case is then compared to a matrix of perfect-scale types. The number of Guttman errors associated with the comparison to each perfect scale is stored in an error vector. Unambiguous cases are assigned the score corresponding to the perfect-scale type which resulted in the fewest Guttman errors. Ambiguous cases are handled in one of four ways, depending on the user's preference. Statistics are computed after each case is assigned a true Guttman score. (See Appendix A for an example of perfect data; Appendix B shows random data.) Output. The following information may be printed at the terminal or stored on an APL file for later printing on a high-speed line printer: (1) number of cases passing each item, (2) coefficient of reproducibility, (3) minimum marginal reproducibility, (4) percent improvement, (5) coefficient of scalability, (6) descriptive statistics and frequency distributions for true scores and for case-bycase Guttman errors, and (7) complete listing of scale score, number of errors, and response pattern for all cases. Restrictions. The input data matrix should be limited to no more than 10,000 elements (e.g., 500 cases and 20 items). Availability. Copies of this paper and a source listing with complete documentation may be obtained without charge from Marshall H. Segall, Interdisciplinary Social Science Program, Syracuse University, Syracuse, New York 13210.