This paper describes the background noise estimation technique of the tensor product expansion with absolute error (TPE-AE) to estimate multiple sources. The electroencephalogram (EEG) signal produced by the saccadic eye movement is adopted to analyze relationship between a brain function and a human activity. The electrooculogram (EOG) generated by eye movements yields significant problems for the EEG analysis. The denoising of EOG artifacts is important task to perform an accurate analysis. In this paper, the two types of TPE-AE are proposed to estimates EOG and other components in EEG during eye movement. One technique estimates two outer products using median filter based TPE-AE. The another technique uses a reference signal to separate the two sources. We show that the proposed method is effective to estimate and separate two sources in EEG.

In this paper, a convolution theorem which is analogous to the theorem for Fourier transform is shown among a certain type of polynomials. We establish a fast method of the multiplication in a special class of quotient rings of multivariate polynomials over q-element finite field GF(q). The polynomial which we treat is one of expressing forms of the multiple-valued logic function from the product of the semigroups in GF(q) to GF(q). Our results can be applied to the speedup of both software and hardware concerning multiple-valued Boolean logic.

This paper proposes a background noise estimation method using an outer product expansion with non-linear filters for ELF (extremely low frequency) electromagnetic (EM) waves. We proposed a novel source separation technique that uses a tensor product expansion. This signal separation technique means that the background noise, which is observed in almost all input signals, can be estimated using a tensor product expansion (TPE) where the absolute error (AE) is used as the error function, which is thus known as TPE-AE. TPE-AE has two problems: the first is that the results of TPE-AE are strongly affected by Gaussian random noise, and the second is that the estimated signal varies widely because of the random search. To solve these problems, an outer product expansion based on a modified trimmed mean (MTM) is proposed in this paper. The results show that this novel technique separates the background noise from the signal more accurately than conventional methods.

We develop a novel construction method for n-dimensional Hilbert space-filling curves. The construction method includes four steps: block allocation, Gray permutation, coordinate transformation and recursive construction. We use the tensor product theory to formulate the method. An n-dimensional Hilbert space-filling curve of 2r elements on each dimension is specified as a permutation which rearranges 2rn data elements stored in the row major order as in C language or the column major order as in FORTRAN language to the order of traversing an n-dimensional Hilbert space-filling curve. The tensor product formulation of n-dimensional Hilbert space-filling curves uses stride permutation, reverse permutation, and Gray permutation. We present both recursive and iterative tensor product formulas of n-dimensional Hilbert space-filling curves. The tensor product formulas are directly translated into computer programs which can be used in various applications. The process of program generation is explained in the paper.

This paper proposes a novel signal estimation method that uses a tensor product expansion. When a bivariable function, which is expressed by two-dimensional matrix, is subjected to conventional tensor product expansion, two single variable functions are calculated by minimizing the mean square error between the input vector and its outer product. A tensor product expansion is useful for feature extraction and signal compression, however, it is difficult to separate global noise from other signals. This paper shows that global noise, which is observed in almost all input signals, can be estimated by using a tensor product expansion where absolute error is used as the error function.

Trellis diagrams of lattices and the Viterbi algorithm can be used for decoding. It has been known that the numbers of states and labels at every level of any finite trellis diagrams of a lattice L and its dual L* under the same coordinate system are the same. In the paper, we present concrete expressions of the numbers of distinct paths in the trellis diagrams of L and L* under the same coordinate system, which are more concrete than Theorem 2 of [1]. We also give a relation between the numbers of edges in the trellis diagrams of L and L*. Furthermore, we provide the upper bounds on the state numbers of a trellis diagram of the lattice L1L2 by the state numbers of trellis diagrams of lattices L1 and L2.

In this paper, we study trellis properties of the tensor product (product code) of two linear codes, and prove that the tensor product of the lexicographically first bases for two linear codes in minimal span form is exactly the lexicographically first basis for their product code in minimal span form, also the tensor products of characteristic generators of two linear codes are the characteristic generators of their product code.

Obtaining a linearizing feedback and a coordinate transformation map is very difficult, even though the system is feedback linearizable. It is known that finding a desired transformation map and feedback is equivalent to finding an integrating factor for an annihilating one-form for single input nonlinear systems. It is also known that such an integrating factor can be approximated using the simple C.I.R method and tensor product splines. In this paper, it is shown that m integrating factors can always be approximated whenever a nonlinear system with m inputs is feedback linearizable. Next, m zero-forms can be constructed by utilizing these m integrating factors and the same methodology in the single input case. Hence, the coordinate transformation map is obtained.

It is very difficult to obtain a linearizing feedback and a coordinate transformation map, even though the system is feedback linearizable. It is known that finding a desired transformation map and feedback is equivalent to finding an integrating factor for an annihilating one-form. In this paper we develop a numerical algorithm for an integrating factor involving a set of partial differential equations and corresponding zero-form using the C.I.R method. We employ a tensor product splines as an interpolation method to data which are resulted from the numerical algorithm in order to obtain an approximate integrating factor and a zero-form in closed forms. Next, we obtain a coordinate transformation map using the approximate integrating factor and zero-form. Finally, we construct a stabilizing controller based on a linearized system with the approximate coordinate transformation.