Nonperiodic Sampling Research Articles

Nonperiodic sampling and the local three squares theorem

This paper presents an elementary proof of the following theorem: Given {r j } =1 with m=d+1, fix $$fix R \geqslant \sum\nolimits_{j = 1}^m {r_j } $$ and let Q=[−R, R]d. Then any f∈ L2(Q) is completely determined by its averages over cubes of side rj that are completely contained in Q and have edges parallel to the coordinate axes if and only if rj/rk is irrational for j≠k. Whend=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials.

Solutions to deconvolution equations using nonperiodic sampling

Explicit, compactly supported solutions, {vi, ϕ}i=1m, to the deconvolution (or Bezout) equation $$\sum\limits_{i = 1}^m {\mu _i *} \nu _{i,\Phi } = \Phi$$ (0.1) are computed where ϕ is a given function in Cc∞(Rd), and\(\mu _i = \chi _{[ - r_i ,r_j ]^d }\), i=1, ..., m for some set of positive numbers {ri}i=1m such that ri/rj is poorly approximated by rationals whenever i ≠ j. The novelty of the solution technique is that it uses new results in the theory of sampling of bandlimited functions detailed in [13] to provide simple Fourier series representations for the solutions, {vi, ϕ}i=1m, which can be easily implemented numerically.

LMS-like AR modeling in the case of missing observations

This paper presents a new recursive algorithm for the time domain reconstruction and spectral estimation of uniformly sampled signals with missing observations. An autoregressive (AR) modeling approach is adopted. The AR parameters are estimated by optimizing a mean-square error criterion. The optimum is reached by means of a gradient method adapted to the nonperiodic sampling. The time-domain reconstruction is based on the signal prediction using the estimated model. The power spectral density is obtained using the estimated AR parameters. The development of the different steps of the algorithm is discussed in detail, and several examples are presented to demonstrate the practical results that can be obtained. The spectral estimates are compared with those obtained by known AR estimators applied to the same signals sampled periodically. We note that this algorithm can also be used in the case of nonstationary signals.

Non-periodic and adaptive sampling. A tutorial review

Non-periodic and adaptive sampling. A tutorial review

Nonperiodic Sampling of Bandlimited Functions on Unions of Rectangular Lattices

It is shown that a function $f\in L^p[-R,R], 1\le p<\infty,$ is completely determined by the samples of $\hat f$ on sets $\Lambda=\cup_{i=1}^m\{n/2r_i\}_{n\in{\bf Z}}$ where $R=\sum r_i,$ and $r_i/r_j$ is irrational if $i\ne j,$ and of $\hat f^{(j)}(0) \mbox{ for } j=1,\ldots,m-1.$ If $f\in C^{m-2-k}[-R,R],$ then the samples of $\hat f$ on $\Lambda$ and only the first k derivatives of $\hat f$ at 0 are required to determine f completely. Higher dimensional analogues of these results, which apply to functions $f\in L^p[-R,R]^d$ and $C^{m-2-k}[-R,R]^d,$ are proven. The sampling results are sharp in the sense that if any condition is omitted, there exist nonzero $f\in L^p[-R,R]^d$ and $C^{m-2-k}[-R,R]^d$ satisfying the rest. It is shown that the one-dimensional sampling sets correspond to Bessel sequences of complex exponentials that are not Riesz bases for $L^2[-R,R].$ A signal processing application in which such sampling sets arise naturally is described in detail.

An autoregressive curvature model for describing cartographic boundaries

This paper introduces a new method for representing cartographic boundaries using autoregressive model parameters. An autoregressive model is presented to model a time series which describes the correlated shape variations determining the overall shape of the boundary. Such time series are obtained using an appropriate nonperiodic sampling sequence from a curvature function of the boundary. In the nonperiodic sampling scheme the sampling points are a set of highly informative and significant points. To reduce the sensitivity of curvature to the noise in boundary, the curvature values are estimated at each point using a filter with a proper degree of smoothing. The univariate AR model parameters are invariant to translation, scale, and rotation of the boundary, and as a consequence this modeling can be used to produce invariant recognition and description of cartographic boundaries.

Electronic states in GaAs/Al0.3Ga0.7As non-periodic superlattices

Transmission and photoluminescence measurements in GaAs/Al0.3As0.7As superlattices in which the layer widths vary randomly, illustrate the effects of one-dimensional non-periodic potentials on electronic energy states. The fundamental interband transition energy in the non-periodic structures is lower than that in the periodic case. A relation between this shift and the wavefunction localization length is derived. The PL emission from the non-periodic samples is ten to twenty times stronger than from the periodic samples.

Number of sampling points required for aberration correction using off-axis electron holography

The number of the sampling points required for digital aberration correction using an off-axis electron hologram is studied based on the Whittaker-Shannon sampling theorem [see, e.g., J.W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)]. The number of the sampling points in the case of a non-periodic sample depends on the width of the image-space inverse filter used for aberration correction. The minimum number estimated in this report is smaller than that reported previously [H. Lichte, Ultramicroscopy 38 (1991) 13]. Using an optimum sampling scheme we can reduce the required number of pixels for sampling the same area. For periodic specimens, the size of the inverse filter does not affect the required number of sampling points, when the sampled image distribution is made commensurate with the intrinsic periodicity of the specimen. The implication of a window function commonly used to suppress discontinuity at the sampled image boundary is also studied in the light of aberration correction. It is shown that the window function may restrict a sampling interval of an aberration function, and thus affect the required number of sampling points.

Adaptive control for non-periodic sampling using bilinear models

The synthesis of a discrete adaptive control law with simultaneous generation of a non-periodic sampling law for linear systems by means of an approximate bilinear model and standard quadratic optimization techniques is present. The usual adaptivity concept is extended to the optimally concept over a predefined planning horizon. In this way, an approximate discrete DARMA bilinear model from a state-space model is obtained by giving sufficient uniform observability requirements. The application of these bilinear models on simulated examples shows their validity in the adaptive sampling design of imperfectly known discrete linear plants under adaptive control. Global asymptotic stability of the overall (closed-loop) adaptive scheme is guaranteed under direct extensions of standard assumptions from the case of periodic sampling to the adaptive sampling case. In this way, adaptive sampling is designed in order to improve the adaptation transient performances.

Fundamental properties of linear control systems with after-effect—II: Extensions including the discrete case

The properties of linear systems with after-effect described in Part I [ Mathl Comput. Modelling 10, 473 (1988)] are extended to the discrete-time case. The use of nonperiodic sampling is also discussed.

A Lyapunov-Poincaré stability analysis for nonperiodic sampling systems

This paper relates the asymptotic and bounded-input bounded-output (BIBO) stability properties of a nonperiodic sampling system with respect to those of a nominal system obtained for a constant nominal sampling period. The stability properties depend on the nominal system state (or alternatively the current one) as does the fastest time constant of the system. Appropriate input-output, time-varying difference equations are derived under weak conditions.

On the derivation of bilinear models for optimization of the sampling periods in discrete systems

This paper deals with the development of some bilinear models for non-periodic sampling in continuous plants by modelling the sampling interval as a component of the input vector. Using such models, simple optimization techniques, which are valid for on-line computation of the sampling intervals, are derived.

Nonlinear oscillations in nonperiodic sampling systems

The existence of nonlinear oscillations in adaptive sampling systems has been studied experimentally. The interpretation of the existence of such oscillations is made from a special hysteresis nonlinear model which describes the adaptive sampling process in an exact way. A design method of the adaptive sampling criterion is proposed to improve the steady-state system response.

On the Improvement of the Adaptation Transients: An Adaptive Sampling Approach

This paper copes with the problem of improving the behaviour in adaptive control schemes during the adaptation transient. The key idea consists in the use of adaptive sampling by taking as signal to be adapted the tracking error between the reference model and the controlled system outputs. To solve the adaptive controlproblem, a recent modeling for non-periodic sampling systems is used.

A time-varying difference equation for nonperiodic sampling systems

It is shown that nonperiodic discrete-continuous sampling systems may be modeled in the time domain by a difference matrix equation which may be manipulated in the same way as in the case of periodic sampling. The output sequence may be computed by means of finite combinations of previous inputs and outputs using a finite set of time-varying parameters which are dependent on the parameters of the continuous transfer matrix and on a finite set of previous sampling periods equal to the order of the system.

Application of the non-periodic sampling to the identifiability and model matching problems in dynamic systems

The sensitivity identifiability and non-parametric model matching problems in continuous linear and time-invariant dynamic systems are studied. The point of view adopted involves solving a system of linear equations by selecting a set of output samples. For this purpose, parametrical and sampling conditions are established which, in practice, may be tested together by applying well known numerical analysis tools. The degree of arbitrariness allowed in the choice of the sampling instants is used to improve the transmission of the measuring errors towards the results

Nonperiodic sampling and model matching

It is shown that the compensator matrices in matching problems may be obtained from a system of equations using arbitrary sampling on the impulse-response matrices of both the given system and its reference model. The sampling is used to achieve an acceptable transmission of the measuring errors.

Nonperiodic sampling and identifiability

It is shown that the state transition matrix may be identified using arbitrary sampling under not very restrictive conditions. A good transmission of the relative measuring errors towards the identification results may be achieved by an adequate choice of the sampling instants using very well known numerical analysis concepts.

Effects of Low Temperature Periodic Annealing on the Deep-Level Defects in 200 KeV Proton Irradiated AlGaAs-GaAs Solar Cells

Low temperature thermal annealing (200 to 400 °C) has been shown to be effective in reducing densities of deep-level defects induced by proton irradiation in AlGaAs-GaAs solar cells. The purpose of this paper is to report our study of the effect of periodic thermal annealing on the deep level defects induced by the 200 keV proton irradiation. The AlGaAs-GaAs mesa diodes were initially irradiated at room temperature by 200 keV protons to a fluence of 1011 P/cm2, and were annealed at 200 °C, 300 °C, and 400 °C, respectively. The samples were repeatedly irradiated by the same proton fluence and then annealed at 200, 300, and 400 °C up to four irradiation and annealing cycles. The DLTS, C-V, and I-V measurements were made on these samples to determine the defect and recombination parameters as functions of proton fluence (1011 to 4×1011 P/cm2) and annealing temperature (200 to 400 °C). The results showed that densities of both electron and hole traps as well as recombination current decrease with increasing annealing temperature. For comparison, some samples were also irradiated once at a fluence of 1011, 2×1011, 3×1011, and 4×1011 P/cm2, and annealed at 200, 300, and 400 °C for six hours, respectively. No significant difference was found in defect density between the periodic and non-periodic annealed samples. The observed main electron trap was due to Ec-0.71 eV while the main hole trap was due to Ev+0.18 eV, for the 200 keV proton irradiated GaAs solar cells.

Sampling theorems and bases in a Hilbert space

A unified approach to sampling theorems for (wide sense) stationary random processes rests upon Hilbert space concepts. New results in sampling theory are obtained along the following lines: recovery of the process χ( t ) from nonperiodic samples, or when any finite number of samples are deleted; conditions for obtaining χ( t ) when only the past is sampled; a criterion for restoring χ( t ) from a finite number of consecutive samples; and a minimum mean square error estimate of χ( t ) based on any (possibly nonperiodic) set of samples. In each case, the proofs apply not only to the recovery of χ( t ), but are extended to show that (almost) arbitrary linear operations on χ( t ) can be reproduced by linear combinations of the samples. Further generality is attained by use of the spectral distribution function F (·) of χ( t ), without assuming F (·) absolutely continuous.