Origin of Rice's formula

The digital record of the voltage in a radio telescope receiver, after frequency conversion and sampling at a finite rate, is not a perfect representation of the original analog signal. To detect and characterise a transient event with a duration comparable to the inverse bandwidth it is necessary to compensate for these effects, altering the statistical properties of the signal and making it difficult to determine the significance of a potential detection. We present an analysis of these modified statistics and demonstrate them with experimental results from Bedlam, a new digital backend for the Parkes radio telescope.

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“…An early form of this result appears in Rice (1945, eqn 3.3-5); refer to Rainal (1988) for a detailed discussion. P( f , f ) can be rewritten as…”

Section: Interpolated Signalmentioning

confidence: 99%

Noise statistics in a fast digital radio receiver: the Bedlam backend for the Parkes radio telescope

2013

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“…The history of this formula is given in [14], and the theory has expanded into a large literature with the surveys [15,16] which include 133 and 82 references, respectively. The problem is to adapt these works to the frequency domain to study level crossing rates of spectrum estimates.…”

Section: Crossing Problems In the Time Domainmentioning

confidence: 99%

Spacing and shape of random peaks in non-parametric spectrum estimates

Thomson

Haley

2014

Proc. R. Soc. A.

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In this paper, expressions are derived for the expected number of spurious peaks in a spectrum estimate, that is, crossings above a given significance level per frequency unit, as well as the expected width of these peaks. In numerous scientific applications, spectrum estimates are used for the purpose of identifying sinusoidal or modal components, often thinning large sets of candidate frequencies with coincidence detection. Because one always expects numerous false peaks in a spectrum estimate, knowing the expected rate of false peaks helps to decide whether the number observed is abnormal and hence determine the true nature of the process. An example using solar wind data from the Advanced Composition Explorer is given where spectra display pathological numbers of significant peaks, while temporally permuted versions of the data possess spectra with the number expected for a white, Gaussian process. The permutation test is a valuable diagnostic for processes suspected to contain many line components.

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“…In the vein of (Kac, 1943), Rice presents the formula for the expected number of solutions to the equation F (x) = 0 as the integral of the crossing intensity ν 0 (x), which obviously may depend on x. Rainal (1988) gives an interesting and personal history of the origin of Rice's formula.…”

Section: )mentioning

confidence: 99%

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

Lindgren¹

2019

Statist. Sci.

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We describe and compare how methods based on the classical Rice's formula for the expected number, and higher moments, of level crossings by a Gaussian process stand up to contemporary numerical methods to accurately compute the statistical distribution of the maximum over a xed interval, the length of excursions above a level, and the geometry of oscillations, and other characteristics of the sample paths.We illustrate the relative merits in accuracy and computing time of the Rice moment methods and the exact numerical method, developed since the late 1990s, on three groups of distribution problems, the maximum over a nite interval and the waiting time to rst crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.We also treat the notoriously dicult problem of dependence between successive zero crossing distances. The exact solution has been known since at least 2000, but it has remained largely unnoticed outside the ocean science community.Extensive simulation studies illustrate the accuracy of the numerical methods. As a historical introduction an attempt is made to illustrate the relation between Rice's original formulation and arguments and the exact numerical methods.MSC 2010 subject classications: Primary 60G17; secondary 60-08, 60G10, 60G70, 65C50.

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Origin of Rice's formula

Cited by 20 publications

References 18 publications

Noise statistics in a fast digital radio receiver: the Bedlam backend for the Parkes radio telescope

Noise statistics in a fast digital radio receiver: the Bedlam backend for the Parkes radio telescope

Spacing and shape of random peaks in non-parametric spectrum estimates

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

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