Implementation of 2-D digital filters by iterative methods

A 2-D digital filter with a rational frequency response can be expanded into an infinite sequence of filtering operations. Each filtering operation can be implemented by convolution with a low-order 2-D finite-extent impulse response. This sequence of filtering operations can be viewed as an iteration where a new estimate of the output signal is formed from the previous estimate. If a convergence constraint is satisfied, the sequence of estimates will approach the desired output signal. In practice, the number of iterations is finite. Consequently, the frequence response that is actually realized by the iterative implementation is an approximation to the desired rational frequency response.

A simplified nonlinear transient analysis method for gas bearings was recently published by the authors (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” J. Tribol., 134(1), 011704). The method uses the fact that linearized dynamic characteristics of gas bearings, namely the impedances, can be approximated by rational transfer functions. The method gave good results if the rational transfer function approach approximated the linearized dynamic characteristics well. Indeed, each of the four complex impedances Zαβ,α,β={x,y} had one or two poles depending on the order of the rational function that were used. These poles appear as supplementary eigenvalues of the extended matrix of the homogeneous system of first order differential equations describing the model of the rotor. They govern the stability of the dynamic model in the same way as the original eigenvalues do and therefore they impose non-negligible constraints on the rational function approximation of the impedances of gas bearings. The present improvement of the method overrides this problem. The basic idea is to impose the same set of poles for Zxx, Zxy, Zyx, and Zyy. By imposing this constraint, the poles are stable and the introduction of artificial instability or erratic eigenvalues is avoided. Campbell and stability diagrams naturally taking into account the variation of the dynamic coefficients with the excitation frequency can now be easily plotted. For example, the method is used for analyzing the stability of rigid and flexible rotors supported by two identical gas bearings modeled with second order rational transfer functions. The method can be applied to any bearing or seal whose impedance is approximated by rational transfer functions.

Uncertainty principle and sparse reconstruction in pairs of orthonormal rational function bases

The transfer functions G(s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G^(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n=1,2,…,N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G^(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G^(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G(s). The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection.

Mapping from the s-domain to the z-domain via the magnitude-invariance method

This paper describes a simple procedure for synthesizing an active distributed RC network which, by using dominant poles and zeros, realizes a very accurate approximation of an arbitrary stable rational transfer function. The network uses a single uniformly distributed RC line with taps spaced along its length. A linear combination of tap voltages is added to the input signal to form the driving voltage for the RC line; the output signal is also a linear combination of the tap voltages. The network offers a number of significant advantages. Since it realizes a nearly rational transfer function, the approximation problem can be conveniently solved using readily available results on rational function approximation. Also, the network uses only one uniform RC line, the transfer function can be changed simply by changing resistor values, and the frequency can be scaled by minor connection changes. Thus one standard network with minor modification is useful for a wide variety of applications. This paper develops the design procedure and derives the various sensitivity functions of importance. Two example designs are carried out: an approximation to a second-order low-pass transfer function and an approximation to a second-order band-pass transfer function with a Q of 100. The sensitivities for the examples are very reasonable and the measurements made on laboratory models indicate excellent agreement with theoretical predictions.

1 IntroductionIn the analysis of large amounts of data to obtain BLUP, large sets of mixed model equations must be solved iteratively, which can involve considerable computing time. In real life, solutions are required only periodically for breeders to choose the best individuals, so that computing time is not usually a limiting factor. In simulation studies involving evaluation of individuals by BLUP, many rounds of evaluation are required for each simulated population. Since several or many replicates are usually required to obtain an accurate result from stochastic simulations, computing time can become a limiting factor. One of the factors that can drastically affect computing time in iterative methods is the criterion for ceasing iteration, or convergence criterion (CC). With iterative methods, a disadvantage can be that the rate of convergence can be slow, or under certain circumstances not converge at all. Nevertheless, when the system converges, the more stringent the CC, the more accurate the solutions. The more stringent the CC the more iterations and hence more computing time is required. The objectives of this study were to investigate how much response to selection is affected by the stringency of the CC and how much reranking occurs among selected individuals at different levels of the convergence criteria. These explorations provided a profile analysis of the computing time spent for each of the major subroutines in the BBSIM program.

Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce a general technique to transform a PDE problem on an unbounded domain to a PDE problem on a bounded domain. Our method uses the Kelvin Transform, which essentially inverts the distance from the origin. However, naive application of this coordinate mapping can still result in a singularity at the origin in the transformed domain. We show that by factoring the desired solution into the product of an analytically known (asymptotic) component and another function to solve for, the problem can be made continuous and compact, with solutions significantly more efficient and well-conditioned than traditional finite element and Monte Carlo numerical PDE methods on stretched coordinates. Specifically, we show that every Poisson or Laplace equation on an infinite domain is transformed to another Poisson (Laplace) equation on a compact region. In other words, any existing Poisson solver on a bounded domain is readily an infinite domain Poisson solver after being wrapped by our transformation. We demonstrate the integration of our method with finite difference and Monte Carlo PDE solvers, with applications in the fluid pressure solve and simulating electromagnetism, including visualizations of the solar magnetic field. Our transformation technique also applies to the Helmholtz equation whose solutions oscillate out to infinity. After the transformation, the Helmholtz equation becomes a tractable equation on a bounded domain without infinite oscillation. To our knowledge, this is the first time that the Helmholtz equation on an infinite domain is solved on a bounded grid without requiring an artificial absorbing boundary condition.

The accuracy of iterative and non-iterative strain-gage balance load calculation methods was compared using data from the calibration of a force balance. Two iterative and one non-iterative method were investigated. In addition, transformations were applied to balance loads in order to process the calibration data in both direct read and force balance format. NASA's regression model optimization tool BALFIT was used to generate optimized regression models of the calibration data for each of the three load calculation methods. This approach made sure that the selected regression models met strict statistical quality requirements. The comparison of the standard deviation of the load residuals showed that the first iterative method may be applied to data in both the direct read and force balance format. The second iterative method, on the other hand, implicitly assumes that the primary gage sensitivities of all balance gages exist. Therefore, the second iterative method only works if the given balance data is processed in force balance format. The calibration data set was also processed using the non-iterative method. Standard deviations of the load residuals for the three load calculation methods were compared. Overall, the standard deviations show very good agreement. The load prediction accuracies of the three methods appear to be compatible as long as regression models used to analyze the calibration data meet strict statistical quality requirements. Recent improvements of the regression model optimization tool BALFIT are also discussed in the paper.

Basilar-membrane responses to white Gaussian noise were recorded using laser velocimetry at basal sites of the chinchilla cochlea with characteristic frequencies near 10 kHz and first-order Wiener kernels were computed by cross correlation of the stimuli and the responses. The presence or absence of minimum-phase behavior was explored by fitting the kernels with discrete linear filters with rational transfer functions. Excellent fits to the kernels were obtained with filters with transfer functions including zeroes located outside the unit circle, implying nonminimum-phase behavior. These filters accurately predicted basilar-membrane responses to other noise stimuli presented at the same level as the stimulus for the kernel computation. Fits with all-pole and other minimum-phase discrete filters were inferior to fits with nonminimum-phase filters. Minimum-phase functions predicted from the amplitude functions of the Wiener kernels by Hilbert transforms were different from the measured phase curves. These results, which suggest that basilar-membrane responses do not have the minimum-phase property, challenge the validity of models of cochlear processing, which incorporate minimum-phase behavior.

Rational transfer function models for nitrifying trickling filters

We revisit the problem of approximating a multiple-input multiple-output $p\times m$ rational transfer function $H(s)$ of high degree by another $p\times m$ rational transfer function $\widehat{H}(s)$ of much smaller degree, so that the $\mathcal{H}_2$-norm of the approximation error is minimized. We show that in the general case of higher-order poles in the reduced-order model, called the defective case, the stationary points of the $\mathcal{H}_2$-norm of the approximation error can still be characterized by tangential interpolation conditions. We also indicate that the sensitivity of the solution of this problem depends on the parameterization used.

The injection of water at substantial distances from the oil-water contact may cause only 10 to 50 percent of the injected water actually to advance the oil-water contact. The remainder of the water serves to increase the pressure in the aquifer, and its effect is dissipated in the pore volume-compressibility term. Introduction It is fairly common practice to reinject water into the aquifer near the oil-water interface in water-drive reservoirs. There have been many studies of aquifer behavior without reinjection, but apparently no analytical studies have been made to indicate performance when injection wells are present in the performance when injection wells are present in the aquifer. The diffusivity equation may be used to describe the pressure behavior in an aquifer. When sources are present it can be written as present it can be written as(1) A general solution of Eq. 1 may be obtained as the sum of two solutions. One is a general solution of the homogeneous equation, obtained by dropping the source term in Eq. 1 and still satisfying the required boundary conditions. Another is a general solution of Eq. 1 satisfying homogeneous boundary conditions. The first solution, which corresponds to the aquifer without reinjection, has been given quantitative and qualitative treatment in the literature. The latter solution will be treated here. Solutions with variable injection rates are obtained by superimposing solutions with constant rates, so we have considered only the latter. Five cases have been treated here:A linear aquifer of finite width and extent. Water is injected into the aquifer at point xo, yo at constant rate. The pressure along the line x = 0 is maintained at zero pressure. (See Fig. 1.) Equations showing the pressure distribution, rates, velocity and cumulative pressure distribution, rates, velocity and cumulative influx have been developed. Certain numerical results are shown.A linear aquifer of finite width and extent. Water is injected into the aquifer at point xo, yo at constant rate. The rate of production along the oil-water water contact is maintained constant. Equations showing the pressure distribution inside the aquifer are presented. pressure distribution inside the aquifer are presented.A linear aquifer of finite width and extent. Constant pressures are maintained at opposite ends of the aquifer. Water is injected at constant rates at point xo, yo. Equations showing the pressure distribution are shown.A linear aquifer of finite width and infinite extent. The pressure is maintained constant at the oil-water contact. Water is injected at a constant rate at point xo, yo.A linear aquifer of finite width and finite extent. The pressure at the oil-water contact is maintained constant. Water is injected at constant rate per unit length along a fine xo. The Linear Aquifer The linear closed aquifer, performing at constant terminal pressure was selected as the first case to be analyzed. The development of the equations is based on uniform injection per unit of thickness, and gravity effects are neglected. JPT P. 1210

Systematic methods are used to find all possible conservation laws of a given type for certain systems of partial differential equations, including some from fluid mechanics. The necessary and sufficient conditions for a vector to be divergence-free are found in the form of a system of first order, linear, homogeneous partial differential equations, usually overdetermined. Incompressible, inviscid fluid flow is treated in the unsteady two-dimensional and steady three-dimensional cases. A theorem about the degrees of freedom of partial differential equations, needed for finding conservation laws, is proven. Derivatives of the dependent variables are then included in the divergence-free vectors. Conservation laws for Laplace's equation are found with the aid of complex variables, used also to treat the two-dimensional steady flow case when first derivatives are included in the vectors. Conservation laws, depending on an arbitrary number of derivatives, are found for a general first order quasi-linear equation in two independent variables, using two differential operators, which are associated with the derivatives with respect to the independent variables. Linear totally hyperbolic systems are then treated using an obvious generalization of the above operators.

Complex coefficient digital filters, with applications for processing real sequences, are examined. The method is quite general, and allows any real rational transfer function to be expressed in terms of a complex rational transfer function of reduced order. When implemented in complex hardware form, the reduction of filter order can provide an increase in computational efficiency and speed. Conventional filter structures, such as parallel, cascade, lattice, and state-space forms, are extended to the complex domain. Illustrative examples of complex coefficient filter synthesis are included, along with coefficient sensitivity comparisons between the complex coefficient filters and their real coefficient counterparts.