Chalie Charoenlarpnopparut

Multidimensional FIR filter bank design using Grobner bases

A multivariate polynomial matrix factorization algorithm is introduced and discussed. This algorithm and another algorithm for computing a globally minimal generating matrix of the syzygy of solutions associated with a polynomial matrix are both associated with a zero-coprimeness constraint that characterizes perfect-reconstruction filter banks. Generalizations, as well as limitations of recent results which incorporate the perfect reconstruction as well as the linear-phase constraints, are discussed with several examples and counterexamples. Specifically, a Grobner basis-based proof for perfect reconstruction with linear phase is given for the case of two-band multidimensional filter banks, and the algorithm is illustrated by a nontrivial design example. Progress and bottlenecks in the multidimensional multiband case are also reported.

Gröbner Bases for Problem Solving in Multidimensional Systems

The objective here is to underscore recent usage of the algorithmic theory of Gröbner bases in multidimensional systems since that possibility was highlighted about fifteen years back. The main contribution here focuses on the constructive aspects of the solution, known to exist, of the two-band multidimensional IIR perfect reconstruction problem using Gröbner bases. Other recent research results on the subject with future prospects are also briefly cited.

Multidimensional convolutional code: progresses and bottlenecks

Multidimensional (m-D, m/spl ges/2) finite support convolutional code has become a new area of research in the multidimensional system and signal processing. While the one-dimensional convolutional code and its variants have been thoroughly understood, the m-D counterpart still lacks unified notation and practical implementation. In this document, the issues related to equivalent encoder, syndrome decoder and the realization of m-D convolutional code are discussed. The application of Grobner basis theory for finding a syndrome decoder matrix is given. In addition, the formulations of optimal realization problem and m-D convolutional code design problem is stated. Recent development and one open-problem are also reported.

Algebraic decoder of multidimensional convolutional code: constructive algorithms for determining syndrome decoder and decoder matrix based on Gröbner basis

A representation of an multidimensional (m-D) convolutional encoder is analogous to the representation of a transfer function for a MIMO m-D FIR system. The encoder matrix is usually not square and thus finding its inverse (decoder matrix) typically employs the Moore-Penrose generalized inverse. However, the result may not be FIR (polynomial matrix) even if the generator matrix is a polynomial matrix. In this paper a constructive algorithm for computing the FIR pseudo inverse, based on the usage of Gröbner basis is presented along with detailed examples. The result obtained can be parameterized to cover the class of all possible FIR inverses. In addition, by using the computation method of syzygy with the Gröbner basis module, the syndrome matrix for a given m-D convolutional encoder is shown. Furthermore, the theory of Gröbner basis is applied to solve the algebraic syndrome decoder problems using the maximum likelihood (nearest neighborhood) criteria and the procedure for 2-D convolutional code error correction is proposed. Despite the complication of the decoding process, the proposed method is the only error correcting decoder for multidimensional convolutional code available to date.

High-quality factor, double notch, IIR digital filter design using optimal pole re-position technique with controllable passband gains

Digital IIR notch filter has been employed in various communication systems to eliminated unwanted narrowband interference with known frequency. In this paper, the optimal multiple notch filter design is proposed based on the pole re-position technique. The optimality considered is based on the minimization of resulting filter magnitude response from the ideal notch magnitude response, measured in term of numerical sum of error square. Furthermore, by using Gröbner basis, the symbolic solution and design method can be generalized to the higher number of notch frequencies case. The design algorithm can be implemented directly on Matlab platform and optimal filter coefficients can be found via numerical optimization/search. The advantage of the proposed algorithm is that the designer can specify additional performance parameters such as stability margin (distant from largest pole to the nearest point on the unit circle), maximum passband gain and quality factor.

One-dimensional and multidimensional spectral factorization using Gröbner basis approach

Spectral factorization is an important step in the process of designing quadrature-mirror filter bank (QMF Bank) and other types of filter banks. In the one-dimensional case, the spectral factorization can be done effectively by finding all roots of the polynomial to be factorized and pairs up the appropriate roots to form the required factors. On the other hand, this simple approach is not general- izable to the multidimensional case since the roots of the multivariate polynomial are generally not isolated. Here, the new approach based on the usage of Grobner basis is proposed. This new approach is not only applicable to the one-dimensional case, but it is also generalizable to the multidimensional case. Furthermore, for lower order polynomial, the proposed algorithm can provide a symbolic solution. The factorization algorithm is described along with the numerical example to show the effectiveness of the proposed method.

Minimax controller design using rate feedback

A complete analytic characterization and solution construction (done either explicitly or by recursion) for the minimax control problem using optimal rate feedback is given for the case when the plant consists of a known fixed set of coupled oscillators of cardinality not exceeding three. When this is not the case, the problem appears to be analytically intractable, and suboptimal solutions based on numerical techniques are currently the only recourse.

Design of three-dimensional adaptive IIR notch filters

This paper investigates a realization of a three-dimensional (3-D) adaptive notch filter. The procedures are mainly divided into two parts: frequency-detecting and sinusoidal interference removal. The detections are based on adaptive line enhancer on infinite impulse response (IIR) lattice structure. In the interference removal part, a non-separable version of a 3-D notch filter is effectively applied. The magnitude response of a 3-D adaptive IIR notch filter is illustrated. At the end of the paper, the implementation of an IIR notch filter on a 3-D image is also conducted in order to show how to remove a sinusoidal interference superimposed on a 3-D image.

Multiple IIR notch filter design and optimization by Secant approximation

Cascading single notch filters is the usual way to construct a multiple notch filter. However, the cascading connection may introduce the uncontrollable non-unity passband gains between notch frequencies. In other words, when the pole placement of a multiple notch filter is changed, the exact notch frequencies between the notch frequencies may not be precisely controlled. To compensate this effect, the design of IIR multiple notch filters with controllable passband gain is concerned. One of the solutions was an allpass based design using the pole-reposition technique which is proposed by Thamrongmas et al. Even this technique was simple, however, it requires high number of iterations to find the optimal design parameter that makes the passband gains (between two notch frequencies) be uniformly flat. In this paper, the Secant method is applied to improve the performance of the optimal search algorithm. The result shows the significantly faster search i.e. the algorithm often converges within fewer than 10 iterations.

All-pass based IIR multiple notch filter design using Gröbner Basis

In this paper, the digital IIR multiple notch filter based on an all-pass filter design of order 2N is considered. The most important step is about the calculation of the exact notch frequencies. The previous method requires solving of non-linear polynomial equation system which can be difficult to tackle especially for the case with higher number of notch frequencies. The analytical result was derived only for the case of N = 2 not for N = 3 and higher, then the main idea proposed in this paper is focused on solving of the non-linear systems for the cases of higher number of notch frequencies, e.g. N ≥ 3, by employing “Gröbner Basis” theory which has ability to solve multi-variance polynomial equation systems by using the benefit of “lexicographical orderings” in the polynomial rings to make those systems to be “triangular systems” which can be solved by backward substitution. Although the proposed filter design is a one dimensional but the proposed technique involved multi-variance polynomials. Another advantage is that the Gröbner Basis provides symbolic solutions which allow further optimization to satisfy additional constraints such as having minimum group delay.