Separable-denominator Digital Filters Research Articles

Invariance of Second-Order Modes under Frequency Transformation in 2-D Separable Denominator Digital Filters

This paper discusses the invariance of the second-order modes of 2-D separable denominator digital filters under frequency transformation. This paper first derives a state-space description for 2-D digital filters obtained by 2-D frequency transformation and then represents the controllability Gramians and the observability Gramians of the transformed 2-D digital filters. This description proves that the second-order modes of 2-D separable denominator digital filters are not invariant under all frequency transformations, but invariant under specific frequency transformations of which transfer functions are strictly proper. These frequency transformations include lowpass-bandpass and lowpass-bandstop frequency transformations keeping the same bandwidth as that of the prototype lowpass filter. It is also shown that the horizontal second-order modes are invariant when only horizontal transformation is applied, and the vertical second-order modes are invariant when only vertical transformation is applied. This paper further remarks significance brought by the invariance of the second-order modes under frequency transformation.

A New Structure of Error Feedback in 2-D Separable-Denominator Digital Filters

In this paper, we propose a new error feedback (EF) structure for 2-D separable-denominator digital filters described by a rational transfer function. In implementing two-dimensional separable-denominator digital filters, the minimum delay elements structures are common. In the proposed structure, the filter feedback-loop corresponding to denominator polynomial is placed at a different location compared to the commonly used structures. The proposed structure can minimize the roundoff noise more than the previous structure though the number of multipliers is less than that of previous one. Finally, we present a numerical example by designing the EF on the proposed structure and demonstrate the effectiveness of the proposed method.

Design of 2D separable‐denominator state‐space digital filters using orthogonal projection

AbstractIn this paper, a design method is proposed for a two‐dimensional separable‐denominator state‐space digital filter based on the projection method. First, the square error function is expressed in terms of the coefficient matrix of the local state‐space model. Next, the given FIR filter is realized by a local state‐space model. An equivalent transform based on the eigenvalue–eigenvector decomposition of the impulse response Grammian is applied to this filter. The obtained model is approximated at lower order by orthogonal projection. The projection parameters are optimized by an iterative algorithm based on the error function, deriving a lower dimensional model with a higher design accuracy. The conventional lower dimensional approximation algorithm based on rounding down of the coefficient matrix is used to find the initial values in the present method. The stability of the filter obtained by the present method is always guaranteed. The effectiveness of the present method is confirmed by numerical examples. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(6): 10–18, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10066

Schur Decomposition and Model Reduction in the Design of Two-Dimensional Separable-Denominator Digital Filters

A new and numerically efficient technique for designing two-dimensional (2-D) separable-denominator digital filters using Schur decomposition (SD) and the diagonal symmetry of the 2-D impulse response specifications. This technique is based on two steps: first, the 2-D impulse response specifications are decomposed into two cascaded specifications, representing SIMO and MISO 1-D digital filters. By using the symmetry property of the 2-D impulse response matrix and that the left and right eigenspaces obtained by SD are a transpose of each other, the design problem of the two 1-D digital filters is reduced to the design problem of only one 1-D digital filter. Thus, the computational effort is reduced by approximately half. Further computational reduction is obtained by applying the SD to only part of the 2-D impulse response matrix, the leading principal minor submatrix, and from the symmetry property of the class of filters that are considered here, the decomposition of the impulse response matrix is obtained. In the second step, a model reduction algorithm is used to design a filter that approximates the 1-D specifications obtained from the first step. Two design examples illustrate the advantages of the proposed technique.

L/sub 2/-sensitivity analysis and minimization of 2-D separable-denominator state-space digital filters

For two-dimensional (2-D) state-space digital filters that are separable in the denominator, the coefficient sensitivity is analyzed by using a pure L/sub 2/-norm, and then, the problem of minimizing the L/sub 2/-sensitivity is considered. First, a novel expression is developed in closed form for the evaluation of the L/sub 2/-sensitivity. Next, an iterative procedure is presented for synthesizing the optimal filter structures that minimize the L/sub 2/-sensitivity. Finally, a numerical example is given to illustrate the utility of the proposed technique.

Error Spectrum Shaping in Two‐Dimensional Separable‐Denominator Digital Filters

AbstractIn the implementation of recursive digital filters, the characteristics might be degraded due to finite word length effects. The degradation generates noise at the filter output which is caused by roundoff errors in the product quantization. A technique called error spectrum shaping (ESS) can be applied to suppress the noise. ESS usually relies on the use of feedback of roundoff errors and sometimes their feedforward. In this paper, a technique is developed for ESS in two‐dimensional separable‐denominator digital filters by applying both feedback and feedforward of roundoff errors. A two‐dimensional separable denominator is implemented with the minimum number of delay elements and includes two quantizers. A genetic algorithm is also applied to obtain the suboptimal discrete solution of feedback–feedforward coefficients for the purpose of cost reduction. Finally, a numerical example is given to illustrate the utility of the proposed technique. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(4): 10–19, 2002

Synthesis of two-dimensional IIR digital filters with desired spatial domain specifications and no overflow oscillations

A design technique of two-dimensional (2D) recursive infinite-area impulse response (IIR) digital filters (DFs) which approximates desired spatial-domain specifications and guarantees no overflow oscillations is proposed in this paper. First, it is shown that the balanced approximation is not optimal for l2 model reduction. From this point of view, a new design technique of 2D separable-denominator (SD) digital filters is studied using reduced-dimensional decomposition and the l2 model reduction methods based on balanced gains. This result is used as a starting point of a design algorithm for 2D non-separable ND DFs. Second, a design problem of 2D ND DFs is formulated as a nonlinear minimization problem with equality and inequality constraints, and the Broyden-Fletcher-Goldfarb-Shanno algorithm is applied to solve it. Finally, two examples are given to show the effectiveness of the proposed design technique.

Synthesis of two-dimensional IIR digital filters with desired spatial domain specifications and no overflow oscillations

A design technique of two-dimensional (2D) recursive infinite-area impulse response (IIR) digital filters (DFs) which approximates desired spatial-domain specifications and guarantees no overflow oscillations is proposed in this paper. First, it is shown that the balanced approximation is not optimal for l2 model reduction. From this point of view, a new design technique of 2D separable-denominator (SD) digital filters is studied using reduced-dimensional decomposition and the l2 model reduction methods based on balanced gains. This result is used as a starting point of a design algorithm for 2D non-separable ND DFs. Second, a design problem of 2D ND DFs is formulated as a nonlinear minimization problem with equality and inequality constraints, and the Broyden-Fletcher-Goldfarb-Shanno algorithm is applied to solve it. Finally, two examples are given to show the effectiveness of the proposed design technique.

Design of Multidimensional Separable-Denominator DigitalFilters in the Spatial Domain

Based on a concept of minimal decomposition and balanced realization, a novel technique is developed for designing multidimensional (M-D) separable-denominator digital filters in the spatial domain. This is done by decomposing a desired M-D FIR filter into M parts and by representing it by a cascade connection of M 1-D filters. The minimal decomposition treated here can be viewed as an extension of that in 2-D separable-denominator case in some sense. However, the proposed technique differs from the existing one even in 2-D case and is simpler. Two numerical examples are also given to illustrate the utility of the proposed technique. The amount of calculation required here is relatively small and the stability of the filter is always guaranteed.

A lattice realization for 2-D separable-denominator digital filters

New properties useful in connection with 1-D and 2-D lattice realizations are developed. These properties enable one to represent given, complicated 2-D separable-denominator digital filters in terms of simpler, more elemental building blocks which consist of two 1-D lattice realizations having dynamics in different directions and connected in a cascade form. The matrix-relationship between a 2-D discrete Schwarz form and a controller-observer canonical form is also derived. A notable property of the proposed 2-D lattice realization is that the impulse response energy of a 2-D separable-denominator digital filter can be readily obtained from the reflection coefficients and input/output tapped coefficients of the realization.

Synthesis of 2-D separable-denominator digital filters with low sensitivity

This paper treats the coefficient sensitivity of two-dimensional (2-D) separable-denominator digital filters using the Roesser local state-space model. Two techniques suitable for 2-D separable-denominator digital filters are developed for synthesizing the filter structure with low sensitivity, one free from 1 2 scaling constraints on the state variables, and the other under the scaling constraints. In the paper, it is clarified that the filter structures with low sensitivity can be easily derived from the balanced realization. Finally, an example is given to illustrate the utility of the proposed techniques.

Design of 2-D separable-denominator digital filters using impulse response grammians

A technique is developed for designing 2-D separable-denominator digital filters in the spatial domain. First, a given 2-D FIR digital filter is realized in a canonic form by the Roesser model whose transfer function is separable in the denominator. Then, the reduced-order model is obtained using eigenvalue-eigenvector decomposition on the impulse response Grammians derived from the realization. The proposed technique utilizes the impulse response Grammians considered here for 2-D separable-denominator digital filters, which are extensions of those proposed for single-input/single-output 1-D linear systems. The stability of the resulting filter is always guaranteed. Finally, two numerical examples are given to illustrate the utility of the proposed technique.

Realization of 2-D separable-denominator digital filters for real-time and parallel processing

This paper considers the 2-D separable-denominator digital filter, and discusses its real-time and parallel-processing-oriented realization. The aim of this paper is to employ a large number of processors concurrently, and to realize a real-time processing minimizing the data throughput delay by applying pipeline processing. More precisely, the case where the poles of the 2-D separable-denominator transfer function are composed of real and complex poles is considered, and two new methods of realization are presented based on the Roesser-type local state-space model, which is suited to the real-time processing. It is shown that the data throughput delay in those methods is equal to the time for two multiplications and one addition, or the time for one multiplication and two additions. The processor factor and the efficiency in each realization are investigated, and the effectiveness of the realization methods are verified from the viewpoints of real-time processing and parallel processing.

Efficient design of two-dimensional separable denominator digital filters using symmetries

The authors propose a technique for reduction of computational effort in 2-D separable-denominator digital filter design. The technique exploits diagonal symmetry or antisymmetry of 2-D impulse response specifications. It is known that a 2-D separable-denominator digital filter can be decomposed into a cascade of two 1-D digital filters with 1-input/multi-output and multi-input/1-output, respectively. Such a decomposition is called the reduced-dimensional decomposition. It is shown that the design problems of two 1-D digital filters can be reduced to the design problem of a single 1-D digital filter by exploiting diagonal symmetry or antisymmetry. As a result, the computing time in the proposed design procedure is 40-50% shorter than in a design method that does not use symmetries. >

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

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.

Design of 2-D separable-denominator digital filters based on the reduced-dimensional decomposition

The paper proposes a technique for the design of 2-D separable-denominator digital filters. The technique is based on decomposing both 2-D separable-denominator state-space models and the given 2-D impulse response specifications into 1-D ones. It is proved that the given 2-D specifications can be optimally decomposed into 1-D specifications via singular value decompositions. Then, 1-D digital filters are designed to approximate the decomposed 1-D specifications. By using balanced approximation as the 1-D design algorithm, a practical algorithm is presented to design stable 2-D separable-denominator digital filters. The algorithm is computationally efficient and numerically stable.

Synthesis of three–dimensional separable–denominator digital filters with minimum roundoff noise

AbstractA roundoff noise analysis is presented for three‐dimensional (3‐D) local state‐space digital filters with separable denominator. It is clarified that the output‐noise power and the signal scaling condition are closely related to three, and three positive‐definite matrices, respectively. A procedure is also developed for the synthesis of an optimal 3‐D local state‐space digital filter with separable denominator which minimizes the output‐noise due to roundoff subject to a scaling condition on the state variables. The optimal state‐space realization can be obtained through the 3‐D similarity transformation by individually solving three 1‐D optimization problems.

Synthesis of 2-D separable denominator digital filters with minimum roundoff noise and no overflow oscillations

This paper presents a new expression of the variance of roundoff noise in 2-D separable denominator digital filters described by Roesser's local state-space model. The covariance matrices and noise matrices necessary to analyze the variance of roundoff noise under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling are obtained as the solutions of Liapunov equations. The synthesis problem of 2-D separable denominator digital filters with minimum roundoff noise are formulated under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling. The synthesis method of 2-D minimum noise realizations under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling is proposed by applying the minimization technique of roundoff noise in the 1-D case. Furthermore, it is shown that 2-D minimum noise realizations are rotated and scaled balanced realizations. Using this property, 2-D minimum noise realizations are proved to be free of overflow oscillations under zero input conditions, if their second-order modes are distinct.