scholarly journals Using Variability Related to Families of Spectral Estimators for Mixed Random Processes

2001 ◽  
Vol 123 (4) ◽  
pp. 572-584
Author(s):  
Li Wen ◽  
Changxue Wang ◽  
Peter Sherman
Keyword(s):  
Point Spectrum ◽  
Random Processes ◽  
Stationary Processes ◽  
Wide Sense ◽  
Dirac Delta ◽  
Harmonic Content ◽  
Spectral Estimators ◽  
Delta Functions ◽  
Spectral Estimator

Traditionally, characterization of spectral information for wide sense stationary processes has been addressed by identifying a single best spectral estimator from a given family. If one were to observe significant variability in neighboring spectral estimators then the level of confidence in the chosen estimator would naturally be lessened. Such variability naturally occurs in the case of a mixed random process, since the influence of the point spectrum in a spectral density characterization arises in the form of approximations of Dirac delta functions. In this work we investigate the nature of the variability of the point spectrum related to three families of spectral estimators: Fourier transform of the truncated unbiased correlation estimator, the truncated periodogram, and the autoregressive estimator. We show that tones are a significant source of bias and variability. This is done in the context of Dirichlet and Fejer kernels, and with respect to order rates. We offer some expressions for estimating statistical and arithmetic variability. Finally, we include an example concerning helicopter vibration. These results are especially pertinent to mechanical systems settings wherein harmonic content is prevalent.

On random processes that are almost strict sense stationary

1976 ◽  
Vol 8 (04) ◽  
pp. 820-830
Author(s):  
Dag Tj⊘stheim
Keyword(s):  
Stationary Process ◽  
Random Processes ◽  
Stationary Processes ◽  
Strict Sense ◽  
Wide Sense ◽  
Original Process ◽  
Extended Class ◽  
Gaussian Case

An extension of the class of strict sense stationary processes is studied. The extended class represents the strict sense analogy of an extension of wide sense stationary processes considered in an earlier paper [9]. The relations between the various types of processes defined are investigated in the general and in the Gaussian case, and some examples are given. It is shown that associated with a process belonging to the extended class there is a strict sense stationary process. The associated strict sense stationary process is unique iff the original process is ergodic.

On random processes that are almost strict sense stationary

1976 ◽  
Vol 8 (4) ◽  
pp. 820-830 ◽  
Author(s):  
Dag Tj⊘stheim
Keyword(s):  
Stationary Process ◽  
Random Processes ◽  
Stationary Processes ◽  
Strict Sense ◽  
Wide Sense ◽  
Original Process ◽  
Extended Class ◽  
Gaussian Case

An extension of the class of strict sense stationary processes is studied. The extended class represents the strict sense analogy of an extension of wide sense stationary processes considered in an earlier paper [9]. The relations between the various types of processes defined are investigated in the general and in the Gaussian case, and some examples are given. It is shown that associated with a process belonging to the extended class there is a strict sense stationary process. The associated strict sense stationary process is unique iff the original process is ergodic.

Dirac Delta Functions in the Laplace‐Type Expansion of rnYlm(θ, φ)

1969 ◽  
Vol 51 (6) ◽  
pp. 2359-2362 ◽  
Author(s):  
Kenneth G. Kay ◽  
H. David Todd ◽  
Harris J. Silverstone
Keyword(s):  
Dirac Delta ◽  
Delta Functions ◽  
Laplace Type

scholarly journals Electrostatic and magnetostatic fields of point dipoles revisited

2019 ◽  
Vol 65 (1) ◽  
pp. 71 ◽  
Author(s):  
Y. Muniz ◽  
Anderson José Fonseca ◽  
C. Farina
Keyword(s):  
Magnetic Dipole ◽  
Electric Dipole ◽  
Alternative Procedure ◽  
Dirac Delta ◽  
Delta Functions ◽  
Point Electric Dipole

After reviewing how the Dirac delta contributions to the electrostatic and magnetostatic fields of a point electric dipole and a point magnetic dipole are usually introduced, we present an alternative procedure for obtaining these terms based on a regularization prescription similar to that used in the computation of the transverse and longitudinal delta functions. We think this method may be useful for the students in other analogous calculations.

Review of basic quantum physics

Quantum 20/20 ◽  
2019 ◽  
pp. 1-20
Author(s):  
Ian R. Kenyon
Keyword(s):  
Quantum Physics ◽  
Particle Density ◽  
Electromagnetic Coupling ◽  
Black Body ◽  
Black Body Radiation ◽  
Gauge Transformations ◽  
Dirac Delta ◽  
Wave Particle Duality ◽  
Delta Functions ◽  
Lorentz Invariants

Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (01) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

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