Minimization of L/sub 2/-sensitivity for state-space digital filters subject to L/sub 2/-dynamic-range scaling constraints

The problem of minimizing an L/sub 2/-sensitivity measure subject to L/sub 2/-norm dynamic-range scaling constraints for state-space digital filters is formulated. It is shown that the problem can be 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 construct the optimal state-space filter structure that minimizes the L/sub 2/-sensitivity measure subject to the scaling constraints. A numerical example is presented to illustrate the utility 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.     

Robust H/sub /spl infin// filtering for Uncertain2-D continuous systems

This paper considers the problem of robust H/sub /spl infin// filtering for uncertain two-dimensional (2-D) continuous systems described by the Roesser state-space model. The parameter uncertainties are assumed to be norm-bounded in both the state and measurement equations. The purpose is the design of a 2-D continuous filter such that for all admissible uncertainties, the error system is asymptotically stable, and the H/sub /spl infin// norm of the transfer function, from the noise signal to the estimation error, is below a prespecified level. A sufficient condition for the existence of such filters is obtained in terms of a set of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H/sub /spl infin// filter is given. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed method.     

Separate/joint optimization of error feedback and coordinate transformation for roundoff noise minimization in two-dimensional state-space digital filters

This paper is concerned with the minimization of roundoff noise subject to l/sub 2/-norm dynamic-range scaling constraints in two-dimensional (2-D) state-space digital filters. Two methods are proposed, with the first one using error feedback alone and the second one using joint error feedback and coordinate transformation optimization. In the first method, several techniques for the determination of optimal full-scale, block-diagonal, diagonal, and scalar error-feedback matrices for a given 2-D state-space digital filter are proposed. In the second method, an iterative approach for minimizing the roundoff noise under l/sub 2/-norm dynamic-range scaling constraints is developed by jointly optimizing a scalar error-feedback matrix and a coordinate transformation matrix, which may be regarded as an alternative approach to the conventional method for synthesizing the optimal 2-D filter structure with minimum roundoff noise. An analytical method for the joint optimization of a general error-feedback matrix and a coordinate transformation matrix under the scaling constraints is also proposed. A numerical example is presented to illustrate the utility of the proposed techniques.     

Separate/joint optimization of error feedback and realization for roundoff noise minimization in a class of 2-D state-space digital filters

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.     

Induced l/sub 2/ and generalized H/sub 2/ filtering for systems with repeated scalar nonlinearities

This paper provides complete results on the filtering problem for a class of nonlinear systems described by a discrete-time state equation containing a repeated scalar nonlinearity as in recurrent neural networks. Both induced l/sub 2/ and generalized H/sub 2/ indexes are introduced to evaluate the filtering performance. For a given stable discrete-time systems with repeated scalar nonlinearities, our purpose is to design a stable full-order or reduced-order filter with the same repeated scalar nonlinearities such that the filtering error system is asymptotically stable and has a guaranteed induced l/sub 2/ or generalized H/sub 2/ performance. Sufficient conditions are obtained for the existence of admissible filters. Since these conditions involve matrix equalities, the cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem subject to linear matrix inequalities, which can be readily solved by using standard numerical software. If these conditions are feasible, a desired filter can be easily constructed. These filtering results are further extended to discrete-time systems with both state delay and repeated scalar nonlinearities. The techniques used in this paper are very different from those used for previous controller synthesis problems, which enable us to circumvent the difficulty of dilating a positive diagonally dominant matrix. A numerical example is provided to show the applicability of the proposed theories.     

Roundoff Noise Minimization for 2-D State-Space Digital Filters Using Joint Optimization of Error Feedback and Realization

The joint optimization problem of error feedback and realization for two-dimensional (2-D) state-space digital filters to minimize the effects of roundoff noise at the filter output subject to$L_2$-norm dynamic-range scaling constraints is investigated. It is shown that the problem can be 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. Analytical details are given as to how the proposed technique can be applied to the cases where the error-feedback matrix is a general, block-diagonal, diagonal, or block-scalar matrix. A case study is presented to illustrate the utility of the proposed technique.     

Simple state-space formulations of 2-D frequency transformation and double bilinear transformation

In this paper, an explicit relationship between the two-dimensional (2-D) frequency transformation and the theory of linear fractional transformation (LFT) representation is shown. Based on this relationship, a simple alternative state-space formulation of 2-D frequency transformation for 2-D digital filters is derived by utilizing the well-known Redheffer star product of LFT representations. The proposed formulation is then utilized to establish a simple relationship between the state-space representations of a 2-D continuous system and a 2-D discrete system which are related by the double bilinear transformation. Moreover, the inherent relations among the proposed formulations and the existing results are discussed. It turns out that all the existing results given in the literature can be unified as special or equivalent cases by the new state-space formulation of 2-D frequency transformation in a very concise and elegant form. Numerical examples are given to illustrate the effectiveness of the proposed formulations.     

Design of two-channel low-delay IIR nonuniform-division filter banks using L/sub 1/ error criteria

The design of a two-channel nonuniform-division filter (NDF) bank with infinite impulse response (IIR) analysis/synthesis filters and low group delay in the sense of L/sub 1/ error criteria is considered. The problem formulation results in a nonlinear optimisation problem. Based on a variant of Karmarkar's algorithm, the optimisation problem is solved through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisation procedure ensures that the reflection coefficients lie in (-1, 1). Simulation results are provided for illustration and comparison.     

Sorting networks using L/sub p/ mean comparators for signal processing applications

Digital implementations of sorting networks that rely on a digital signal processor core are not as efficient as their analog counterparts. This paper builds on the L/sub p/ comparators for which efficient analog implementations exist that employ operational amplifiers. From a statistical point of view, L/sub p/ comparators are based on nonlinear means. Their probability density function and the first- and second-order moments are derived for independent uniformly distributed inputs. L/sub p/ comparators provide estimates of the minimum and maximum of their inputs. A proper approach to compensate for the estimation errors is proposed. Applications of the L/sub p/ comparators in odd-even transposition networks, median approximation networks, and min/max networks are presented.     

Genetic algorithm approach to fixed-order mixed H/sub 2//H/sub /spl infin// optimal deconvolution filter designs

The mixed H/sub 2//H/sub /spl infin// optimal deconvolution filter is proposed to achieve the H/sub 2/ optimal reconstruction and a desired robustness against the effect of uncertainties in signal processing from the H/sub /spl infin// norm perspective. However, the conventional mixed H/sub 2//H/sub /spl infin// optimal design filters are very complicated and are not practical for industrial applications. For simplicity of implementation and conservation of operation time, the fixed-order mixed H/sub 2//H/sub /spl infin// optimal deconvolution filter design is interesting for engineers in signal processing from the practical design perspective. In this study, to avoid the trap of local minima, a design method based on the genetic algorithm is introduced to treat the nonlinear optimization design problem of the fixed-order mixed H/sub 2//H/sub /spl infin// deconvolution filter. The convergence property of our design algorithm is also discussed. Finally, an example is presented to illustrate the design procedure and confirm the robustness performance of the proposed method.     

H/sub /spl infin// mode reduction for two-dimensional discrete state-delayed systems

The problem of H_infin model reduction for two-dimensional (2-D) discrete systems with delay in state is considered. The mathematical model of 2-D systems is established on the basis of the well-known Fornasini-Marchesini local state-space. First, conditions are established to guarantee the asymptotic stability and a prescribed noise attenuation level in the H_infin sense for the underlying system. For a given stable system, attention is focused on the construction of a reduced-order model, which approximates the original system well in an H_infin norm sense. Sufficient conditions are proposed for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearisation method is exploited to cast them into sequential minimisation problems subject to LMI constraints, which can be readily solved using standard numerical software. These obtained results are further extended to more general cases whose system states contain multiple delays. Two numerical examples are provided to demonstrate the effectiveness of the proposed techniques     

Analysis of 2-D state-space periodically shift-variant discrete systems

Two-dimensional (2-D) periodically shift-variant (PSV) digital filters are considered. These filters have applications in processing video signals with cyclostationary noise, scrambling digital images, and 2-D multirate signal processing. The filters are formulated in the form of a Givone-Roesser (GR) state-space model with periodic coefficients. This PSV model is then presented in block form as a shift-invariant system that also has the same GR state-space form. This block form has reduced computations and ease of analysis. An algorithm is developed that transforms any given 2-D PSV GR system to its equivalent shift-invariant model. Invertibility of this model is an important consideration, especially in image scrambling applications. A condition is established for the invertibility of the shift-invariant model of the 2-D PSV system. Also, the inverse system can be easily computed from the original. The established results are illustrated with an example.This work was supported in part by a grant from the Colorado Advanced Software Institute.     

Stability analysis of 1-D and 2-D fixed-point state-space digitalfilters using any combination of overflow and quantizationnonlinearities

A new criterion, together with its frequency-domain interpretation for the global asymptotic stability of zero-input one-dimensional (1-D) state-space digital filters under various combinations of overflow and quantization nonlinearities and for the situation where quantization occurs after summation only, is presented. A condition in closed form involving solely the parameters of the state transition matrix for the nonexistence of limit cycles in second-order digital filters is derived. Improved versions of some of the stability results due to Leclerc and Bauer (1994) are established. Finally, the approach is extended to two-dimensional (2-D) digital filters described by the Roesser and the Fornasini-Marchesini second local state-space models     

H/sub /spl infin// control of differential linear repetitive processes

Repetitive processes are a distinct class of two-dimensional (2-D) systems (i.e., information propagation in two independent directions) of both systems theoretic and applications interest. They cannot be controlled by direct extension of existing techniques from either standard [termed one-dimensional (1-D) here] or 2-D systems theory. Here, we give new results on the relatively open problem of the design of control laws using an H/sub /spl infin// setting. These results are for the sub-class of so-called differential linear repetitive processes which arise in applications.     

Acoustically actuated flextensional Si/sub x/N/sub y/ and single-crystal silicon 2-D micromachined ejector arrays

We propose using two-dimensional (2-D) micromachined droplet ejector arrays for environmentally benign deposition of photoresist and other spin-on materials, such as low-k and high-k dielectrics used in IC manufacturing. Direct deposition of these chemicals will reduce waste as well as production cost. The proposed device does not harm heat or pressure sensitive fluids and they are chemically compatible with the materials used in IC manufacturing. Each element of the 2-D ejector array consists of a flexurally vibrating circular membrane on one face of a cylindrical fluid reservoir. The membrane has an orifice at the center. A piezoelectric transducer generating ultrasonic waves, located at the open face of the reservoir, actuates the membranes. As a result of this actuation, droplets are fired through the membrane orifice. Ejector arrays were built with either Si/sub x/N/sub y/ or single-crystal silicon membranes using two different fabrication processes. We show that single-crystal silicon membranes are more uniform in their thickness and material quality than those of Si/sub x/N/sub y/ membranes. The single-crystal silicon membrane-based devices showed thickness and material uniformity across all the membranes of an array. This improvement eliminated nonuniform membrane resonance frequencies across an array as observed with Si/sub x/N/sub y/ membrane-based devices. Therefore, it should be possible to repeatably build devices and to predict their dynamic characteristics. Using the fabricated devices, we demonstrated water ejection at 470 kHz, 1.24 MHz, and 2.26 MHz. The corresponding droplet diameters were 6.5, 5, and 3.5 /spl mu/m, respectively.     

10K Gate I/sup 2/L and 1K Component Analog Compatible Bipolar VLSI Technology -- HIT-2

An advanced analog/digital bipolar VLSI technology that combines on the same chip 2-ns 10K I/sup 2/L gates with 1K analog devices is proposed. The new technology, called high-density integration technology-2 (HIT-2), is based on a new structure concept that consists of three major techniques shallow grooved-isolation, I/sup 2/L active layer etching, and I/sup 2/L current gain increase. I/sup 2/L circuits with 80-MHz maximum toggle frequency have developed compatibly with n-p-n transistors having a BV/sub CEO/of more than 10 V and an f/sub T/ of 5 GHz, and lateral p-n-p transistors having an f/sub T/ of 150 MHz.     

High-performance Pt/SrBi/sub 2/Ta/sub 2/O/sub 9//HfO/sub 2//Si structure for nondestructive readout memory

Metal-ferroelectric-insulator-semiconductor (MFIS) capacitors with 390-nm-thick SrBi/sub 2/Ta/sub 2/O/sub 9/ (SBT) ferroelectric film and 8-nm-thick hafnium oxide (HfO/sub 2/) layer on silicon substrate have been fabricated and characterized. It is demonstrated for the first time that the MFIS stack exhibits a large memory window of around 1.08 V at an operation voltage of 3.5 V. Moreover, the MFIS memory structure suffers only 18% degradation in the memory window after 10/sup 9/ switching cycles. The excellent performance is attributed to the formation of well-crystallized SBT perovskite thin film on top of the HfO/sub 2/ buffer layer, as evidenced by the distinctive sharp peaks in X-ray diffraction (XRD) spectra. In addition to its relatively high /spl kappa/ value, HfO/sub 2/ also serves as a good seed layer for SBT crystallization, making the proposed Pt/SrBi/sub 2/Ta/sub 2/O/sub 9//HfO/sub 2//Si structure ideally suitable for low-voltage and high-performance ferroelectric memories.     

A novel nonnegative decomposition method and its application to 2-D digital filter design

In designing two-dimensional (2-D) digital filters in the frequency domain, an efficient technique is to first decompose the given 2-D frequency domain design specifications into one-dimensional (1-D) ones, and then approximate the resulting 1-D magnitude specifications using the well-developed 1-D filter design techniques. Finally, by interconnecting the designed 1-D filters one can obtain a 2-D digital filter. However, since the magnitude responses of digital filters must be nonnegative, it is required that the decomposition of 2-D magnitude specifications result in nonnegative 1-D magnitude specifications. We call such a decomposition the nonnegative decomposition. This paper proposes a nonnegative decomposition method for decomposing the given 2-D magnitude specifications into 1-D ones, and then transforms the problem of designing a 2-D digital filter into that of designing 1-D filters. Consequently, the original problem of designing a 2-D filter is significantly simplified.     

Design Of 2-D Recursive Digital Filters Using Nonsymmetric Half-Plane Allpass Filters

A novel structure using recursive nonsymmetric half-plane (NSHP) digital allpass filters (DAFs) is presented for designing 2-D recursive digital filters. First, several important properties of 2-D recursive DAFs with NSHP support regions for filter coefficients are investigated. The stability of the 2-D recursive NSHP DAFs is guaranteed by using a spectral factorization-based algorithm. Then, the important characteristics regarding the proposed novel structure are discussed. The design problem of 2-D recursive digital filters using the novel structure is considered. We formulate the problem by forming an objective function consisting of the weighted sum of magnitude, group delay, and stability-related errors. A design technique using a trust-region Newton-conjugate gradient method in conjunction with the analytic derivatives of the objective function is presented to efficiently solve the resulting optimization problem. The novelty of the presented 2-D structure is that it possesses the advantage of better performance in designing a variety of 2-D recursive digital filters over existing 2-D filter structures. Finally, several design examples are provided for conducting illustration and comparison.