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Mathematical Analysis of Random Noise-Conclusion
About: This article is published in Bell System Technical Journal.The article was published on 1945-01-01 and is currently open access. It has received 807 citations till now.
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TL;DR: In this paper, upper and lower bounds are obtained for the probability that a separable random process X ( t ) will not be confined within the time-dependent barriers - λ 1 ( t ), and λ 2 ( t ).
15 citations
TL;DR: In this paper, the authors proposed a quasi-dynamic unidirectional asymmetric circular crack model, where the slip is specified by employing Eshelby's static solution at every instant in time.
Abstract: We propose a unidirectional asymmetrical circular crack model, and we investigate the far-field body-wave radiation. The model is a quasi-dynamic one, as the slip is specified by employing Eshelby's static solution at every instant in time. However, unlike previously proposed symmetrical circular crack models that nucleate at the center of a circular fault and spread radially, symmetrically in all directions, the proposed model exhibits unidirectional propagation. We derive compact closed form expressions for the far-field pulse radiated by the source model. As expected, the radiation of the model exhibits strong azimuthal variation. We investigate the stopping phases of the model that are emitted when the rupture front stumbles on the circular barrier at the edge of the fault, and we examine their radiation pattern. We study the seismic energy and seismic efficiency of the model and compare them with those of the symmetrical circular crack model. We find that the seismic efficiency of the unidirectional asymmetrical circular fault model is smaller by a factor of approximately 2-3 as compared to the seismic efficiency of the symmetrical circular crack model. This is explained by the fact that the average rupture velocity of the asymmetrical model is smaller by a similar factor as compared to that of the symmetrical model. Finally, we employ various objective definitions of the corner frequency, and we investigate how their predictions compare with graphically picked corner frequencies.
Manuscript received 25 October 2000.
15 citations
TL;DR: A new package dti for R is presented, which provides functions for the analysis of diffusion weighted data within the diffusion Tensor model, including smoothing by a recently proposed structural adaptive smoothing procedure based on the propagation-separation approach in the context of the widely used diffusion tensor model.
Abstract: Diffusion weighted imaging has become and will certainly continue to be an important tool in medical research and diagnostics. Data obtained with diffusion weighted imaging are characterized by a high noise level. Thus, estimation of quantities like anisotropy indices or the main diffusion direction may be significantly compromised by noise in clinical or neuroscience applications.
Here, we present a new package dti for R , which provides functions for the analysis of diffusion weighted data within the diffusion tensor model. This includes smoothing by a recently proposed structural adaptive smoothing procedure based on the propagation-separation approach in the context of the widely used diffusion tensor model. We extend the procedure and show, how a correction for Rician bias can be incorporated. We use a heteroscedastic nonlinear regression model to estimate the diffusion tensor. The smoothing procedure naturally adapts to different structures of different size and thus avoids oversmoothing edges and fine structures.
We illustrate the usage and capabilities of the package through some examples.
15 citations
Cites background or methods from "Mathematical Analysis of Random Noi..."
...After fast Fourier transform into image space the resulting observed signal follows a Rician distribution (Rice 1945) with parameters ζ and σ....
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...In the case of single receiver coil systems and additive complex Gaussian noise in k-space, referred to as thermal noise below, this results in image grey values that follow a Rician distribution Rice(θ, σ) (Rice 1945)....
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...After fast Fourier transform into image space the resulting observed signal follows a Rician distribution (Rice 1945) with parameters ζ and σ. ζ is the signal of interest while σ corresponds to the standard deviation of the errors in k-space....
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TL;DR: In this paper, the first passage density in the presence of a second absorbing barrier has been obtained for the case of a continuous differentiable Gaussian vector process, which is a special case of the problem of first passage time problems.
Abstract: : Durbin has presented a compact formula for the first passage density of a Gaussian process, which is locally like Brownian motion, to a smooth barrier. In previous works, we have extended the formula to the case of processes which are smooth functions of a continuously differentiable Gaussian vector process and to more general kinds of first passage time problems, so called marked crossings. In the present paper we obtain similar results for the first passage density in presence of a second absorbing barrier and use it to construct upper and lower bounds for the first passage, rainflow cycle amplitude, zerocrossing wave-length and amplitude densities. Numerical examples illustrate the results.
15 citations
TL;DR: In this article, an objective and simple method has been developed to estimate the background continuum in Ge gamma ray spectra, which requires no special procedures, and can be readily automated, based upon the inherent statistical properties of the experimental data itself, nodes which reflect background samples are located and used to produce an estimate of the continuum.
15 citations