Sadao Maekawa

Synthesis of 2-D state-space digital filters with low sensitivity based on the Fornasini-Marchesini model

The balanced realization and low-sensitivity structure of two-dimensional recursive digital filters are considered in the framework of the Fornasini-Marchesini local state-space model. A procedure is introduced for the balanced realization of 2-D recursive digital filters. The sensitivities of a 2-D transfer function are investigated with respect to the coefficients in the local state-space model. The overall sensitivity is evaluated using the 2-D controllability and observability Gramians. The filter structure reducing the overall sensitivity is synthesized for two cases: one free from l/sub 2/ scaling constraints on the state variables and the other under the scaling constraints. These filter structures are shown to be closely related to the balanced realization. An example is given to illustrate the utility of the proposed technique. >

Separable-denominator 2-D rational approximation via 1-D based algorithm

In this paper, an efficient technique is developed for the rational approximation of multi-input multi-output one-dimensional (1-D) digital filters. Then, after showing that a causal 2-D recursive digital filter with separable denominator can be characterized by two multi-output 1-D recursive digital filters, the 1-D technique is applied to approximate a 2-D digital filter by one being separable in the denominator. The parameters of the approximating filter can be obtained by solving linear equations by means of an efficient recursive algorithm. Moreover, the stability of the approximating filter is always ensured. Three examples are presented to illustrate the utility of the proposed technique.

Design of two-dimensional recursive digital filters using mirror-image polynomials

In this paper, a computationally efficient technique is developed for the design of two-dimensional (2-D) recursive digital filters that meet simultaneously magnitude and group delay specifications. The denominator in the filter is chosen structurally stable so that the filter stability is always guaranteed. Alternatively, the numerator is expressed in terms of a 2-D mirror-image polynomial. This makes it possible to design the denominator and the numerator separately. As a result, the amount of calculations can be reduced to some extent and the convergence can also be improved. Finally, two examples are given to illustrate the utility of the proposed technique.

Canonic Form State Space Realization of Two Dimensional Transfer Functions having Separable Denominator

In this paper a procedure is developed for the computation of a state space realization from two dimensional transfer functions with separable denominator. The procedure relies on a canonic form for the state space realization.

A generalized on the study synthesis of 2-D state-space digital filters with minimum roundoff noise

The Fornasini-Marchesini local state-space (LSS) model is used as the basis for a novel expression for the output-noise variance due to roundoff together with an I/sub 2/ scaling on the state variables. An optimal Fornasini-Marchesii LSS model structure is then synthesized that minimizes the output noise due to roundoff, subject to an l/sub 2/ scaling constraint. The synthesis utilizes a 2-D similarity transformation matrix that is not block-diagonal, but general. This requires solving only one optimization problem. Some constraints imposed on the Fornasini-Marchesini LSS model and the 2-D similarity transformation yield the results obtained with the Roesser LSS model. The proposed synthesis theory is therefore quite general and simple. An example is given to illustrate its utility. >

Approximation and minimum roundoff noise synthesis of 3-d separable-denominator recursive digital filters

Abstract In respect to a state-space model for causal three-dimensional (3-D) recursive digital filters, a necessary and sufficient condition is derived for a 3-D separable-denominator recursive model to be separately locally controllable and separately locally observable. To approximate a given 3-D digital filter by a 3-D separable-denominator recursive model, a technique is developed via singular value decomposition of three finite Hankel matrices. The resulting 3-D separable-denominator recursive model is advantageous to the computational speed and implementation cost in addition to the ease of stability testing. A roundoff noise analysis and a scaling on state variables are discussed for 3-D separable-denominator recursive models. A technique is then presented for the synthesis of an optimal 3-D separable-denominator recursive model which minimizes the output noise due to roundoff subject to a scaling condition on the state variables. Finally, two examples are given to illustrate the utility of the proposed techniques.

The capacity of associative memories with malfunctioning neurons

Hopfield associative memories with alphan malfunctioning neurons are considered. Using some facts from exchangeable events theory, the asymptotic storage capacity of such a network is derived as a function of the parameter alpha under stability and attractivity requirements. It is shown that the asymptotic storage capacity is (1-alpha)(2)/n(4 log n) under stability and (1-alpha)(2)(1-2rho)(2)n/(4 log n) under attractivity requirements, respectively. Comparing these capacities with their maximum values corresponding to the case when there is no malfunctioning neurons, alpha=0, shows the robustness of the retrieval mechanism of Hopfield associative memories with respect to the existence of malfunctioning neurons. This result also supports the claim that neural networks are fault tolerant.

Rational approximation of three-dimensional digital filters

Abstract Two techniques are developed for approximating a given three-dimensional (3-D) digital filter by a 3-D rational function. The one is by a general 3-D rational function and the other by a 3-D rational function which is separable in the denominator. Each technique relies on the use of mixed first and second information, in the form of a finite portion of the impulse response and its autocorrelation sequence. The approximation is performed by solving a set of linear equations. The separable-denominator approximation is more advantageous due to the guaranteed stability and reduced amount of calculations.

Balanced realization and model reduction of 3-D separable-denominator transfer functions

Abstract After showing a necessary and sufficient condition to be separately locally controllable and separately locally observable, controllability and observability Gramians are introduced for a three-dimensional (3-D) separable-denominator stat-space model. A technique is then developed for realizing a 3-D minimal state-space model in balanced form from a given 3-D rational transfer function with separable denominator. This technique is applicable to the model reduction of 3-D separable-denominator digital filters.

Relationship between two forms in the 2-D unit noise matrix

Two forms defining the 2-D unit noise matrix proposed by W.-S. Lu and A. Antonious (ibid., vol.CAS-33, p.965-73 Oct., 1986) are investigated using R.P. Roessers (1975) local state-space model, and their relationship is clarified. It is shown that although these forms are not identical they are compatible in the optimal 2-D digital filter synthesis. >