Takashi Hamanaka

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. >

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.

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.

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. >

Minimum sensitivity structure of 2-D digital filters based on the Fornasini-Marchesini model

The authors treat the balanced realization and minimum sensitivity structure of two-dimensional (2-D) recursive digital filters. This structure is based on the Fornasini-Marchesini local state-space model. First, a technique is introduced for the balanced realization of 2-D recursive digital filters. Next, the sensitivities of a 2-D transfer function are investigated with respect to coefficients of the local state-space model. The overall sensitivity is then evaluated using the 2-D controllability and observability Gramians. It is also shown that the minimum sensitivity structure is easily derived from the balanced realization and under an l/sub 2/ scaling constraint it is equivalent to that with minimum roundoff noise.<<ETX>>

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

A generalized technique has been developed for the synthesis of 2-D state-space filter structures which minimize the roundoff noise under a scaling constraint. This is based on the Fornasini-Marchesini local-state-space (LSS) model. The optimal filter structure is found using a 2-D similarity transformation characterized by a nonsingular matrix which is not block-diagonal, but general. This allows one to synthesize the optimal filter structure by solving only one optimization problem. If some constraints are imposed on the Fornasini-Marchesini LSS model and the 2-D similarity transformation matrix, it is possible to easily arrive at the results based on the Roesser LSS model. The proposed theory is therefore quite general and simple.<<ETX>>