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A new expression of evaluating l 2 -sensitivity for generalized direct-form II structure of two-dimensional (2-D) separable-denominator digital filters is derived and analyzed. Unlike the previous work, l 2 -sensitivity is analyzed for the filter structure instead of its state-space realization. Then the resulting l 2 -sensitivity measure is compared with that deduced in a recent study of generalized direct-form II state-space realization of 2-D separable in denominator digital filters. In a numerical example, the new l 2 -sensitivity measure is minimized with respect to free parameters subject to l 2 -scaling constraints by an exhaustive search over a set of discrete values of finite cardinality, and the results are compared with those obtained by existing techniques.

In this paper, l 2 -sensitivity for the generalized direct-form II state-space realization of two-dimensional (2-D) separable-denominator digital filters is analyzed and minimized with respect to free parameters subject to l 2 -scaling constraints. Improved l 2 -sensitivity for 2-D separable-denominator digital filters is investigated by taking 0 and ±1 elements into account. An l 2 -sensitivity measure for the generalized direct-form II state-space realization of 2-D separable-denominator digital filters is deduced by applying improved l 2 -sensitivity. The l 2 -sensitivity measure is then minimized with respect to free parameters subject to l 2 -scaling constraints by employing an exhaustive search over a set of discrete values of finite cardinality. A numerical example is included to illustrate the validity and effectiveness of the proposed technique.

The paper is the first of a series of research investigations on robust digital filters which refer to filters that offer optimal performance under variations of filter parameters. We begin with quantitative characterization of performance robustness of a digital filter against parameter uncertainties. This is followed by several properties of the proposed robust performance measures and design formulations of robust FIR filters in L 2 (least-squares) and L ∞ (minimax) sense as nonsmooth convex problems. We present an accelerated subgradient algorithm for the design of L ∞ -robust FIR filters with technical details involved in implementing the proposed algorithm. A numerical example is included for illustration of the proposed design method and performance evaluation in comparison with conventional minimax FIR filters.

Based on the concept of polynomial operators, generalized direct-form II structure of two-dimensional (2-D) separable-denominator (SD) digital filters is explored. It is shown that 2-D SD digital filters can be modeled by a generalized SIMO direct- form II and a generalized MISO transposed direct-form II that are connected in cascade. Then an expression for the roundoff noise gain in the resulting structure is derived and investigated. Moreover, the roundoff noise gain is compared with that deduced in a recent study of generalized direct-form II realization of 2-D SD digital filters.

This paper presents an enhanced Steiglitz-McBride (SM) procedure for the design of stable minimax IIR digital filters. It is well known that minimax design of IIR filters is typically initiated with a nonconvex formulation, followed by a procedure to relax the original design problem to a sequence of convex sub-problems to be solved iteratively. We proposed an enhanced SM procedure that leads to improved convex relaxation relative to the conventional SM techniques, hence to improved designs. In addition, we make an observation that the state-of-the-art convex stability constraint based on strictly positive realness is equivalent to that deduced from an enhanced version of Rouche theorem from complex analysis. Design examples are presented to evaluate the new design algorithm.

In this paper, we explore generalized direct-form II realization of two-dimensional separable-denominator digital filters of order (m,n) based on the concept of polynomial operators, where we deal with a model with 3(m + n) + mn + 1 fixed parameters plus m + n free parameters. An l2-scaling method is introduced by utilizing different coupling coefficients at different branch nodes to avoid overflow. An expression for the roundoff noise gain in the realization is also examined. It is shown that the roundoff noise gain can be minimized with respect to the m + n free parameters by means of exhaustive search in a finite element space.

This paper presents a systematic study of pole and zero sensitivity minimization for state-space digital filters in several different yet related settings. First, a new weighted measure for pole and zero sensitivity for state-space digital filters is proposed and the problem of minimizing this measure is investigated. To this end, two efficient iterative techniques for minimizing this measure are developed by employing a quasi-Newton algorithm and relying on a recursive matrix equation, respectively. Furthermore, minimization of the proposed sensitivity measure subject to $l_{2}$ -scaling constraints is examined by extending the two aforementioned solution methods—one converts the constrained optimization problem at hand into an unconstrained problem and solves it using a quasi-Newton algorithm, while the other relaxes the constraints into a single constraint on matrix trace and solves the relaxed problem with an effective matrix iteration scheme. In addition, a simple yet novel method for the minimization of a zero sensitivity measure subject to minimal pole sensitivity is explored by pursuing an optimal coordinate transformation matrix. Simulation studies are presented to demonstrate the validity and effectiveness of the proposed techniques.

The joint optimization problem of high-order error feedback and realization for minimizing roundoff noise at the filter output subject to-scaling constraints for two-dimensional (2-D) separable-denominator digital filters is investigated. Linear algebraic techniques that convert the problem at hand into an unconstrained optimization problem are explored, and an efficient quasi-Newton algorithm is then applied to solve the unconstrained optimization problem iteratively. Closed-form formulas for fast and accurate gradient evaluation are derived. A numerical example is presented to demonstrate the validity and effectiveness of the proposed technique.

Joint optimization of high-order error feedback and state-space realization for minimizing roundoff noise at filter output subject to l2-scaling constraints is investigated for one- and two-dimensional state-space digital filters. Linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored, and an efficient quasi-Newton algorithm is then applied to solve the unconstrained optimization problem iteratively. In this connection, closed-form formulas are derived for fast and accurate gradient evaluation. Finally, case studies are presented to demonstrate that the high-order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state-space coordinate transformation and high-order error feedback matrices are optimized separately.

In this paper, the problem of minimizing a weighted pole and zero sensitivity measure subject to 1_2-norm dynamic range scaling constraints for state-space digital filters is investigated. First, a new measure for evaluating pole and zero sensitivity is introduced, and it is shown that the constrained optimization problem is converted into an unconstrained optimization problem by using linear algebraic techniques. Then, the unconstrained optimization problem at hand is solved iteratively by employing an efficient quasi-Newton algorithm with closed-form formulas for key gradient evaluation. A numerical example is presented to demonstrate the validity and effectiveness of the proposed iterative optimization technique.

We present a unified approach to the design of two well-known classes of computationally efficient digital filters, namely the interpolated and frequency-response-masking (FRM) FIR filters, that are optimized in minimax sense. The highly nonconvex minimax designs of these filters are carried out by jointly optimizing the subfilters involved using a convex-concave procedure (CCP) that yields an iterative and converging process where each iteration involves a convex problem of minimizing a linear function subject to convex quadratic constraints. We present an analysis to explain why CCP is well-suited for the design problems at hand. We also show that the CCP-based design framework is flexible to allow extension to the design of FRM filters that simultaneously promotes sparsity of the filter coefficients to reduce implementation complexity. Design examples with comparisons are presented to demonstrate the performance of the proposed design methods.

This paper deals with the problem of minimizing weighted p ole and zero sensitivity subject to l2-scaling constraints for state-space digital filters. A measure for pole and zero sensitivity is presented and an efficient iterative technique for minimizing this measure subject to l2-scaling constraints is developed by relaxing the constraints into a single constraint on matrix trace and solving the relaxed problem with an effective matrix iteration scheme. A numerical example is included to demonstrate the validity and effectiveness of the proposed technique.

Three-dimensional (3-D) digital filters find applications in a variety of image and video signal processing problems. This paper presents a coefficient-sensitivity analysis for a wide class of 3-D digital filters with separable denominators in local state space that leads to an analytic formulation for sensitivity minimization, and to present two solution techniques for the sensitivity minimization problem at hand. To this end, a vector-matrix-vector decomposition of a given 3-D transfer function that separates the three variables and leads to a state-space realization in a form convenient for subsequent analysis. An l2-sensitivity analysis is then performed. The result is a computationally tractable formula of the overall l2-sensitivity for 3-D digital filters. The l2-sensitivity is minimized subject to l2-scaling constraints by using one of the two solution methods proposed-one relaxes the constraints into a single trace constraint and solves the relaxed problem with an effective matrix iteration scheme; while the other converts the contained optimization problem at hand into an unconstrained problem and solves it using a quasi-Newton algorithm. A case study is presented to illustrate the validity and effectiveness of the proposed techniques.

This paper treats the minimization problem for roundoff noise in two-dimensional (2-D) recursive digital filters with an error feedback (EF). The EF is known as an available scheme for the reduction of roundoff noise, and this can be achieved by extracting a roundoff error from the quantizer and feeding it back to the adder. Since the EF circuit is an additional device of the digital filter, the bit length of the coefficients should be as short as possible for the fast arithmetic and low-cost realization. In this paper, we propose an algorithm to optimize the EF coefficients in a discrete space. The proposed algorithm is based on the branch and bound method by estimating the lower bound of the optimal solution, and the optimality of the obtained solution can be guaranteed. Finally, we give a numerical example to demonstrate the effectiveness of the proposed method, and show the optimal solution can be found without using the round-robin algorithm.

Techniques for the separate/joint optimization of error-feedback and realization are developed to minimize the roundoff noise subject to l 2-norm dynamic-range scaling constraints for a class of 2-D state-space digital filters. In the joint optimization, the problem at hand is converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained problem obtained is then solved by applying an efficient quasi-Newton algorithm. A numerical example is presented to illustrate the utility of the proposed techniques.

We study a class of composite digital filters, each has a dominating FIR component of order N and an extremely simple IIR component of order r with r-C N that are connected in parallel. We show that a constrained optimization setting known as convex-concave procedure (CCP) is naturally suited for the design of stable composite filters where the FIR and IIR components are jointly optimized in frequency-weighted minimax sense. A design algorithm based on successive CCP is presented and numerical examples are included to demonstrate that even with r =2 composite filters offer substantial performance improvement relative to their FIR-alone counterparts.

This paper investigates the problem of minimizing weighted pole and zero sensitivity for state-space digital filters. A new measure for pole and zero sensitivity is proposed and an efficient iterative technique for minimizing this measure is developed by employing a quasi-Newton algorithm. A novel and simple method is also explored for obtaining the optimal coordinate transformation matrix which minimizes a zero sensitivity measure subject to minimal pole sensitivity. A numerical example is presented to demonstrate the validity and effectiveness of the proposed technique.

The minimization problem of an L2-sensitivity measure subject to L2-norm dynamic-range scaling constraints is formulated for a class of two-dimensional (2-D) state-space digital filters. First, the problem is converted into an unconstrained optimization problem by using linear-algebraic techniques. Next, the unconstrained optimization problem is solved by applying an efficient quasi-Newton algorithm with closed-form formula for gradient evaluation. The coordinate transformation matrix obtained is then used to synthesize the optimal 2-D state-space filter structure that minimizes the L2-sensitivity measure subject to L2-norm dynamic-range scaling constraints. Finally, a numerical example is presented to illustrate the utility of the proposed technique.

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.

In the application of 2-D adaptive filters, a large amount of data must be processed and real-time processing is often required. In this paper, a new parallel-form realization of 2-D adaptive separable-denominator state-space filters suitable for high-speed processing is proposed. First, the 2-D local state-space model suitable for parallel processing is introduced. Next, the adaptive algorithm for this model is developed. This algorithm is based on the delayed least mean square (DLMS) method. In addition, the computation time required for the update of the coefficients is investigated. Finally, the proposed technique is applied to the design of 2-D digital filters in the spatial domain.

A new technique for the design of two-dimensional (2-D) separable-denominator adaptive state-space digital filters is developed using stable filter structures. First, coefficient sensitivites are related to intermediate transfer functions in order to generate gradient signals. Next, the LMS algorithm is applied to construct adaptive state-space digital filters with new systems to generate gradient signals. Finally, a numerical example is given to illustrate the validity of the proposed technique. Comparison between the proposed and conventional adaptive filters is also presented in the example.

This paper deals with the analysis and minimization of L2-sensitivity for three-dimensional state-space digital filters which are separable in the denominator. First, a novel expression for evaluating the L2-sensitivity is developed in closed form. An iterative procedure for synthesizing the optimal filter structures is then presented so as to minimize the L2-sensitivity. Finally, a numerical example is given to illustrate the utility of the proposed technique.

Since Lim's paper (see IEEE Trans. Circuits Syst., vol.33, p. 357-364, Apr. 1986) on the frequency-response-masking (FRM) technique for the design of finite-impulse response digital filters with very small transition widths, the analysis and design of FRM filters has been a subject of study. In this paper, a new optimization technique for the design of various FRM filters is proposed. Central to the new design method is a sequence of linear updates for the design variables, with each update carried out by semidefinite programming. Algorithmic details for the design of basic and multistage FRM filters are presented to show that the proposed method offers a unified design framework for a variety of FRM filters. Design simulations are included to illustrate the proposed algorithms and to evaluate the design performance in comparison with that of several existing methods.

In 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

This paper investigates the problem of minimizing roundoff noise under l/sub 2/-norm dynamic-range scaling constraints in state-space digital fibers by means of error feedback as well as joint error feedback/coordinate transformation optimization. First, several techniques, for the determination of optimal full-scale, diagonal, and scalar error-feedback matrices for a given state-space digital filter are proposed, where three realization schemes, namely, the general state-space realization, input-balanced realization, and optimal realization in the sense of Hwang-Mullis-Roberts are examined. Furthermore, an iterative approach is developed for jointly optimizing a scalar error-feedback matrix and a coordinate transformation matrix so as to minimize the roundoff noise subject to the l/sub 2/-norm dynamic-range scaling constraints. The proposed method may be regarded as an alternative, but much simpler and more general, approach to Hwang's method for synthesizing the optimal filter structure with minimum roundoff noise. A case study is included to illustrate the utility of the proposed techniques.

When multivalued images are transformed to binary images, the error diffusion method is often used because it is excellent in gray scale representation. When the error diffusion method is applied by means of a raster scan, equal-color regions with all black pixels or all white pixels occur along the edges. This phenomenon has the effect of emphasizing the edges, and the edges may be visually emphasized excessively. In this paper, in order to freely control the equal-color regions, a method that adaptively adjusts the filter coefficients in the error diffusion method based only on the difference from the points to be processed is presented. The filter coefficients are computed by using fuzzy reasoning. According to the proposed method, the width of the equal-color region near a step edge can be controlled adequately by providing appropriate values. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(2): 82–90, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.1135

In 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

Joint optimization of error feedforward, high-order error feedback and state-space realization for minimizing roundoff noise at filter output subject to l 2 -scaling constraints for state-space digital filters is investigated. Linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored, and an efficient quasi-Newton algorithm is then applied to solve the unconstrained optimization problem iteratively. In this connection, closed-form formulas are derived for fast and accurate gradient evaluation. Finally, a numerical example is presented to demonstrate that the high-order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state-space coordinate transformation as well as error feedforward and high-order error feedback matrices are optimized separately.

In this paper, a technique for designing 3-D separable-denominator state-space digital filters is developed. The design process is divided into two phases. First, the coefficient matrices related to stability are constructed for the filter to be stable by using alternating variable method. Next, the other matrices are obtained by solving linear equations. These phases are repeated until there is no significant change in the squared error function.

When a nonstationary signal containing sudden changes, such as an image signal, is degraded by an additive noise, a powerful means to recover the signal is to use a nonlinear filter. This paper uses a layered neural network, and proposes a new method to construct a nonlinear filter for restoring a signal corrupted by mixed noise (i.e., mixed noise composed of Gaussian noise and impulse noise). As the first step, a prototype filter is proposed, which is a combination of a median filter and a linear (averaging) filter. Then, it is shown that the prototype filter can be represented by a network structure. By interpreting the network representation by a layered neural network, the idea is extended to the median neural network hybrid (MNNH) filter. The MNNH filter can be trained by the back-propagation algorithm. The prototype filter already has a high mixed noise elimination performance, but the MNNH filter can further improve the performance by reflecting information on the signal to be processed (the original image and the additive noise) when such information is given. Lastly, the usefulness of the MNNH filter is demonstrated through various application examples. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(6): 52–60, 1998

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 technique for reducing roundoff noise effects subject to l2-scaling constraints is developed for two-dimensional state-space digital filters by means of high-order error feedback and state-space realization. First, the roundoff noise gain is minimized subject to l2-scaling constraints by choosing the coordinate transformation matrices appropriately. Optimal high-order error-feedback matrices are then determined so as to minimize the roundoff noise gain in the two-dimensional state-space digital filter obtained via the coordinate transformation matrices. Finally a numerical example is presented to illustrate the utility of the proposed technique.

The joint optimization problem of high-order error feedback and realization for minimizing roundoff noise at filter output subject to l2-scaling constraints is investigated for two-dimensional (2-D) state-space digital filters. Linear algebraic techniques that convert the problem at hand into an unconstrained optimization problem are explored, and an efficient quasi-Newton algorithm is then applied to solve the unconstrained optimization problem iteratively. In this connection, closed-form formulas are derived for fast and accurate gradient evaluation. Finally a numerical example is presented to illustrate the validity and effectiveness of the proposed algorithm.

Sparse digital filters are of importance as they offer improved implementation efficiency relative to their nonsparse counterparts. This paper examines the design of minimax sparse IIR filters from a sparse representation point of view. The result is a new algorithm that accomplishes a design with three phases - identification of the whereabouts of zero coefficients; optimal design subject to the sparsity constraint; and performance enhancement by further dimension reduction of the working subspace. A design example is presented for illustrating the new algorithm.

A new type of 2-D cascade-parallel adaptive infinite impulse response (IIR) filters with adaptive algorithm based on the concept of back propagation method is developed. This class of filter consists of parallel blocks each of which includes all-zero (transversal) and all-pole sections in cascade form realizations. At first a 2-D cascade-parallel filter structure is introduced, then an efficient adaptive algorithm is proposed using the concept of backpropagating the desired signal. Finally, the applications of this type of 2-D adaptive IIR filters in 2-D system identification and noise reduction in an image are illustrated with numerical examples.

This paper introduces the parallel processing into the separable-denominator 3-D adaptive state-space filter, and discusses a realization method which is suited to the high-speed processing. the LMS (least-mean-square) algorithm is used as the adaptive algorithm, and the parallel processing-oriented structure of the separable-denominator 3-D filter is used as the filter model. The parallel processing-oriented structure of the 3-D adaptive state-space filter is obtained by extending the adaptive algorithm of the 2-D state-space adaptive filter to the above 3-D structure. the proposed algorithm is evaluated in this paper in terms of the time required for the filtering and the coefficient update, as well as the number of multipliers (adders). Finally, the proposed adaptive filter is applied to the design of the 3-D state-space digital filter, and the effectiveness of the algorithm is demonstrated.

A technique for minimizing the normalized noise variance at the output of two-dimensional (2-D) recursive digital filters is developed where error spectrum shaping of the filter is performed by means of higher-order feed-back. First, the optimal solution of error feedback coefficients is derived. Then suboptimal error feedback coefficients are found in which the error feedback coefficients are symmetric. Here, both the optimal and suboptimal solutions are found analytically by solving the minimization problem of a quadratic form. Finally, a numerical example is given to check the validity of the proposed algorithm and it is shown that the proposed technique is effective even if the optimal or suboptimal coefficients are rounded to a power of 2.

A technique for the reduction of the output noise variance in state-space digital filters is developed using a high-order error feedback consisting of a finite impulse response (FIR) digital filter. First, the optimal error feedback structure using all of the elements of the coefficient matrices in the FIR digital filter is derived. Next, the suboptimal error feedback structures are shown; the elements of the coefficient matrices in the FIR digital filter are constrained to be either symmetric or antisymmetric. Finally, a numerical example is given to illustrate the utility of the proposed technique.

This paper is concerned with the reduction of roundoff noise effects subject to l 2 -scaling constraints for state-space digital filters by means of high-order error feedback and realization. First, the roundoff noise gain is mimimized subject to l 2 -scaling constraints by choosing the coordinate transformation matrix appropriately. Optimal high-order error-feedback matrices are then determined so as to minimize the roundoff noise gain in the state-space digital filter transformed by the resulting coordinate transformation matrix. Finally a numerical example is presented to illustrate the utility of the proposed technique.

For two-dimensional (2-D) state-space digital filters described by the Fornasini-Marchesini second local state-space model, the joint optimization of high-order error feedback and realization for minimizing roundoff noise at filter output subject to l 2 -scaling constraints is investigated. We present linear-algebraic techniques that convert the problem at hand into an unconstrained optimization problem, and present an efficient quasi-Newton algorithm to solve the unconstrained optimization problem iteratively, in which closed-form formulas are derived for fast and accurate gradient evaluation. A numerical example is presented to illustrate the utility and effectiveness of the proposed algorithm.

Implementing a variable fractional delay (VFD) filter in Farrow model is costly as each coefficient of a VFD filter is a polynomial rather than a numerical scalar as in a conventional digital filter. This paper presents a method for the design of VFD filters with sparse coefficients which admits efficient implementation. The design is accomplished in two phases with the first phase identifying locations in polynomial impulse response that are suitable to be set to zero and the second phase optimizing the remaining nonzero coefficients so as for the VFD filter to best approximate a desired frequency response. Performance evaluation and comparison of the proposed algorithm relative to an equivalent nonsparse counterpart are also presented.

Based on a concept of minimal decomposition and balanced model reduction, a technique is developed for designing three-dimensional (3-D) separable-denominator digital filters in the spatial domain. This is done by decomposing a desired 3-D FIR filter into three parts and by representing it by a cascade connection of three 1-D filters. The minimal decomposition treated here can be viewed as an extension of that in the 2-D separable-denominator case. A numerical example also is given to illustrate the utility of the proposed technique. The amount of calculation required here is relatively small and the stability of the resulting filter is always guaranteed.

The joint optimization problem of high-order error feedback and realization for state-space digital filters to minimize the effects of roundoff noise at the filter output subject to l 2 -scaling constraints is investigated. The problem is converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained optimization problem at hand is then solved iteratively by applying an efficient quasi-Newton algorithm with closed-form formulas for key gradient evaluation. Finally, a numerical example is presented to illustrate the utility of the proposed technique.

Coefficient sparsity of digital filters is of importance as it is directly related to implementation efficiency and cost. To date most work in the field has been focused on finite-impulse-response filters. In this paper, the design of sparse digital filters is investigated for a class of IIR filters. We propose a two-phase algorithm that promotes coefficient sparsity, maintains filter's stability, and optimizes its frequency response to approximate a desired frequency response in minimax sense. A design example is presented to illustrate the proposed algorithm and compare its performance relative to its nonsparse IIR and FIR counterparts.

An analytical technique is developed for designing multidimensional (M-D) separable-denominator recursive digital filters that meet, simultaneously, magnitude and phase specifications. A cubic symmetry that exists in the M-D frequency response is incorporated into the M-D filter structure. The denominator and the numerator of an M-D transfer function are designed separately. The former is used to approximate constant group delays and the latter is used to adjust an overall magnitude response. The filter is analytically designed by solving linear equations only. This serves to reduce the amount of calculations drastically. The stability of the resulting filter is always guaranteed. Finally, two examples are given to illustrate the utility of the proposed technique.

A two-dimensional (2-D) Lyapunov equation with constant coefficients is considered for the Fornasini-Marchesini second local state-space (LSS) model. First, a novel criterion relating to the Lyapunov equation is presented that sufficiently guarantees the asymptotic stability. A sufficiency condition that ensures the absence of limit cycles is also given. Next, the above stability condition is incorporated into the 2-D filter structure to design 2-D state-space digital filters with guaranteed asymptotic stability. An efficient method is then developed for computing the characteristic polynomial and the inverse of the system matrix. Finally, two numerical examples are given to design 2-D stable state-space digital filters. >

In this paper, we propose a fast arithmetic error feedback (EF) structure with shift operation circuits and shared multiplier. For optimization of EF network, we show the design method based on the Lagrange multiplier method for designing the shared multiplier and the branch and bound based algorithm for optimization of the shift operation circuits. The branch and bound method can reduce the calculation cost because the sub-trees can be cut based on the lower bound estimation. Finally, we present a numerical example by designing the EF with shift operation circuits and shared multiplier. From the results, we demonstrate the effectiveness of the proposed method and show the calculation time is a few seconds.

Is sparsity an issue in filter design problems? and why is it important? How a digital filter can be designed to have a sparse impulse response for efficient implementation while achieving improved performance relative to its non-sparse counterpart? In an attempt to address these questions, this paper comes up with a design technique for optimal linear-phase FIR filters with sparse impulse responses.

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 12 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.

Based on the Roesser (1975) local state-space (LSS) model, new measures are introduced to describe over a particular frequency region the sensitivities of a two-dimensional (2-D) transfer function with respect to state-space parameters. The overall sensitivity derived from these measures is called the frequency weighted sensitivity and is evaluated using 2-D generalized controllability and observability Gramians that are newly defined for 2-D state-space digital filters. Then, 2-D filter structures that minimize the frequency weighted sensitivity are synthesized for two cases of no constraint and scaling constraints on the state variables. These 2-D filter structures can also be optimized with respect to the scaling parameters in the latter case. An example is given to illustrate the utility of the proposed technique. >