Discrete Realization Research Articles

On certain realizations of the q-deformed exterior differential calculus

We investigate two particular realizations of a q-deformed differential calculus at q being a primitive root of unity, qN = 1. Particular attention is paid to the Z3-graded case N = 3. First we construct an analogue of the exterior differential calculus on a manifold, then we introduce a discrete realization of such a calculus on generalized Clifford algebras. Finally, combining both constructions, we discuss a ZN-graded generalization of gauge theory.

Oscillation-test methodology for low-cost testing of active analog filters

The oscillation-test strategy is a low cost and robust test method for mixed-signal integrated circuits. Being a vectorless test method, it allows one to eliminate the analog test vector generator. Furthermore, as the oscillation frequency is considered to be digital, it can be precisely analyzed using pure digital circuitry and can be easily interfaced to test techniques dedicated to the digital part of the circuit under test (CUT). This paper describes the design for testability (DFT) of active analog filters based on oscillation-test methodology. Active filters are transformed to oscillators using very simple techniques. The tolerance band of the oscillation frequency is determined by a Monte Carlo analysis taking into account the nominal tolerance of all circuit under test components. Discrete practical realizations and extensive simulations based on CMOS 1.2 /spl mu/m technology parameters affirm that the test technique presented for active analog filters ensures high fault coverage and requires a negligible area overhead. Finally, the DFT techniques investigated are very suitable for automatic testable filter synthesis and can be easily integrated in the tools dedicated to automatic filter design.

Equilibria for discrete kinetic equations

We develop a systematic method of constructing equilibria for kinetic models with discrete microscopic velocities. The approach is based on a suitable entropy maximum principle. The $H$ theorem is demonstrated in the continuous and discrete space-time realizations. In addition, we discuss an extension of the Lattice Boltzmann method to irregular grids.

Quantum mechanics as a matrix symplectic geometry

The main purpose of this work is to describe the quantum analog of the usual classical symplectic geometry and then to formulate quantum mechanics as a noncommutative symplectic geometry. First, we describe a discrete Weyl-Schwinger realization of the Heisenberg group and we develop a discrete version of the Weyl-Wigner-Moyal formalism. We also study the continuous limit and the case of higher degrees of freedom. In analogy with the classical case, we present the noncommutative (quantum) symplectic geometry associated with the matrix algebraMN(C) generated by the Schwinger matrices.

The flat phase of fixed-connectivity membranes

The statistical mechanics of flexible two-dimensional surfaces (membranes) appears in a wide variety of physical settings. In this talk we discuss the simplest case of fixed-connectivity surfaces. We first review the current theoretical understanding of the remarkable flat phase of such membranes. We then summarize the results of a recent large scale Monte Carlo simulation of the simplest conceivable discrete realization of this system [1]. We verify the existence of long-range order, determine the associated critical exponents of the flat phase and compare the results to the predictions of various theoretical models.

On isotopic and discrete realizations of maps of an -dimensional sphere in Euclidean space

In this paper we consider questions of whether a compact space can be embedded in a Euclidean space. The problem of embedding an '-like' compact space in is solved affirmatively under certain restrictions on the dimension . We clarify the relations between the realization problem and areas of homotopy theory and differential topology.

Design rules for an integratable low-power amplifier/filter combination

In this paper design rules for a circuit topology in which there is an inseparable combination of an amplifier and a filter characteristic, are presented. By intentionally using the capacitance of an already present input sensor for the filtering, the total required integrated capacitance is much less than that in circuits, which have a separately designed amplifier and filter function. Consequently, it is possible to have the advantage of a better integratability. Moreover, less complexity in the design is achieved. The presented circuit shows a current-to-voltage conversion and an inherently controllable second-order low-pass filter characteristic. A discrete realization has been designed to test the circuit. This circuit operates down to a 1 V supply voltage and the transfer shows a 1.8 MΩ currentto-voltage conversion with a bandwidth of 6 kHz. Measurement results of this circuit show that a 63 dB dynamic range can be achieved with a total required integrated capacitance of only 31 pF.

Image recovery using the anisotropic diffusion equation

A new approach for image recovery using the anisotropic diffusion equation is developed which is based on the first derivative of the signal in time embedded in family of images with different scales. The diffusion coefficient is determined as a function of the gradient of the signal convolved with a symmetric exponential filter. A new discrete realization is developed for the simultaneous removal of noise and preservation of edges.

Local critical behaviour at aperiodic surface extended perturbation in the Ising quantum chain

The surface critical behaviour of the semi-infinite one-dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface-extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly, whereas the expression of the surface energy density exponent is conjectured from a finite-size scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst-van Leeuwen model.

Total error in the discrete convolution backprojection algorithm in computerized tomography

The CBP algorithm in computerized tomography (CT) is a discrete realization of a well-known tool from approximation theory; namely, the approximation of a function f in R n by its convolution with a (bandlimited) peaked kernel. The steps in a computer implementation (e.g., in medical CT scanners) of the algorithm evaluate an n-dimensional convolution by (a) interpolation of projection data (line integrals in a two-dimensional case), (b) a one-dimensional discrete convolution, and (c) interpolation of the convolved data, required in (d) a discrete backprojection (integration over a unit sphere). The total error in the algorithm is due to the discretization steps (a)–(d) and (e) the truncation error in the basic convolution approximation. In this work we augment the known error estimates for steps (b) and (b) with those for (a), (c) and (e) to arrive at a total error profile of the algorithm, which may be summarized as follows. In a discrete b-bandlimited CBP reconstruction f d b of f, under appropriate conditions in (a)–(e), the total error f − f d b is essentially of the order of ε(f, b) = sup θϵS n−1∫ [brvbar];σ[brvbar]b [brvbar]σ[brvbar] n−1[brvbar] f ̂ (σθ)[brvbar]dσ, b→∞ .

A wide-band AC-coupled current source for electrical impedance tomography

A current source suitable for application in electrical impedance tomography (EIT) is described. The first stage of the commercially available current-feedback amplifier AD844 constitutes a current-conveyor implementation and allows the construction of wide-bandwidth current sources, thus avoiding the mismatching and temperature-induced problems that arise in discrete realizations. The lack in gain accuracy of this circuit is overcome by the inclusion of its input buffer in an operational amplifier feedback loop. Saturation problems that appear when placing a DC-blocking capacitor between the source and the electrode are solved by a DC feedback that maintains DC voltage at the output near to 0 V without reducing the output impedance of the source. Two AC-coupled current sources, in both inverting and non-inverting configurations, are described and their possible applications to EIT are listed.

Realizations of regular apeirotopes

In an earlier paper, a theory of realizations of (finite) regular polytopes in euclidean spaces was developed. Here, the analogous problem of realizing regular apeirotopes (infinite polytopes) is investigated. While no complete theory is expounded, several basic results are established. Among these is the curious fact that, if a regular apeirotope has a discrete realization, then it has one with no translations in its symmetry group.

Monolithic narrow-band filter using ultrahigh-Q tunable active inductors

A tunable active inductor is presented where the novel topology enables both the inductance and series resistance to be varied. With a discrete MMIC realization of this active inductor, Q-factors in excess of 15000 have been measured over a wide range of inductance values. Applications for these active inductors include high-performance narrow-band filters, voltage controlled oscillators, and analog phase shifters. Analytical equations for the novel active inductor and a 3-resonator filter are given. The measured performance for a monolithic 2 GHz filter achieves a mean insertion loss of 0.9 dB, passband ripple of /spl plusmn/0.7 dB, with a 3 dB bandwidth of 70 MHz, and an excellent out-of-band rejection which exceeds 50 dB up to 18 GHz. >

A note on computational issues associated with restricted maximum likelihood estimation of covariance rarameters

Geostatistical investigations often require that the covariance of a Gussian random field be estimated from a single and discrete realization of the field. If this estimation problem is placed in a restricted maximum likelihood Reml framework, then teh need to construct an appropriate mean filtering matrix arises. In this paper we show that depending on the choice of the mean filtering matrix, the Reml function i may be overly sensitive to round-off errors, ii may require expensive matrix-matrix multiplications, and iii may not retain computationally exploitable structures present in the field covariance matrix. Against this background, we invoke a form of the Reml function that does not depend explicity on any filtering matrix so that the difficulties i, ii, and iii mentioned above do not arise.

On two-dimensional spectral realization

Reconstruction of a spectral density function from a finite set of covariances can be performed by maximizing an entropy functional. The method of the maximum entropy on the mean is used For computing a discrete version of this spectral density and allows one to give a new interpretation of these reconstruction methods. In fact, the authors show that the choice of the entropy is directly related to a prior distribution. In particular, they consider processes on Z/sup 2/. Steepest descent procedures permit the numerical computation of discrete realizations for a wide class of entropies. To ensure the nonnegativity of the solution related to the Burg entropy, they present a new algorithm based on a fixed-point method and the Yule-Walker equations to compute this solution. Then, the solution of the dual problem is obtained as the limit of the trajectory of an ordinary differential equation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Radar detection in K-distributed clutter

The paper compares two newly proposed detection schemes for targets in K-distributed clutter: the first implements generalised likelihood ratio test, and the second represents a discrete realisation of the classical Neymann-Pearson detector. Results indicate that the two optimisation strategies yield substantially equivalent performance, but the generalised likelihood ratio test is simpler in its structure and offers some additional advantages both at the design and at the analysis stage.

Discrete Gabor transform

A feasible algorithm for implementing the Gabor expansion, the coefficients of which are computed by the discrete Gabor transform (DGT), is presented. For a given synthesis window and sampling pattern, computing the auxiliary biorthogonal function of the DGT is nothing more than solving a linear system. The DGT presented applies for both finite as well as infinite sequences. By exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like DGT is obtained. As the discrete Fourier transform (DFT) is a discrete realization of the continuous-time Fourier transform, similarly, the DGT introduced provides a feasible vehicle to implement the useful Gabor expansion.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Note on time reversible chaos in mixmaster dynamics.

The Belinskii-Khalatnikov-Lifshitz (BKL) parametrization of the diagonal spatially homogeneous Bianchi IX vacuum (mixmaster) cosmology can be expressed as a discrete realization of a chaotic map. A well-known invertible two-dimensional version of their parametrization embodies the time reversibility of the model's Einstein equations and can be written (in both an epoch-era and era-era version) to emphasize the fact that the function that maps the first parameter is the inverse of that which maps the second. It is shown here that this can be understood in terms of the self-similar equilateral triangle minisuperspace equipotentials that describe the approximate dynamics in the anisotropy plane of minisuperspace. A new parameter is identified which has the same value in all the epochs of a given era. The BKL parameters of the two-dimensional map are different ratios of triangle scales to the new parameter. It is shown how one parameter controls the change of era (and thus the chaos) in the collapse direction while the other plays the same role for expansion.

Resistively variable capacitors using general impedance convertors

The use of general impedance convertors (GICs) for the realisation of resistively variable capacitors (RVCs) is demonstrated. Such elements are useful in analogue-adaptive circuits and for neural networks. Various design concepts have been examined with the aim of finding configurations that are suitable for discrete, and complete, or partial VLSI realisation. A number of useful circuit topologies have been found. Design guidelines are given; simulations and experimental results are compared. The resulting circuits are being considered for use in adaptive telephone hybrid networks for high-speed full-duplex data transmission.

Reduced order controller design for discrete time systems

Robust discrete control system design techniques and model reduction are discussed. A new linear quadratic guussian/loop transfer recovery procedure for discrete time systems is presented. In this technique, a full-state feedback or an output injection feedback is designed which has the desired loop shape, and then recovered by a realizable linear quadratic gaussian controller. To do this, results that show the effects of the weighting matrices (noise intensities) on linear quadratic regulator (Kalman-Bucy filter) return difference and inverse-return difference arc derived and a procedure to recover the linear quadratic regulator loop transfer function is described. The complexity of the resulting controller is then reduced without causing closed-loop instability. Two methods for model reduction are considered. The first is the discrete balanced realization and the second is P frequency weighting technique where it is possible to vary the approximation accuracy with frequency. The controller design and re...