Estimation of the Bispectrum

Rosenblatt, Murray; Ness, John W. Van

doi:10.1214/aoms/1177699987

Cited by 184 publications

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“…Murray's interest in higher-order spectra began more than four decades ago [34], which ultimately led to the deconvolution problem [35]. Murray's fundamental contributions in both areas have had a long lasting impact on many aspects of statistical probems and applications as described in this brief article.…”

Section: If the Output Process {X(t)} Is Observed With An Independentmentioning

confidence: 99%

“…Murray Rosenblatt's interest and insight in this problem may have stemmed from his interest in the higher order spectra, especially higher-order cumulant spectra which is fundamental in dealing with nonGaussian processes [34,4,5,19]. The kth-order cumulant spectral density of {x(t)} is given by In the paper [35], Murray laid out the basic idea on how the phase function h(X) can be identified up to a linear shift of cX using the phase of the bispectra of the process.…”

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confidence: 99%

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Rosenblatt’s Contributions to Deconvolution

Davis

Lii

Politis

2011

Selected Works of Murray Rosenblatt

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A simple model of deconvolution can be described as observing {x(t)} which is a convolution of a signal {s(t)} with a filter {f(j)},x = s*f. More specifically, we haveThe problem of deconvolution is to recover {s(t)} based on the output process {x(t)}. If the filter {f(j)} is known then the problem is fairly straightforward. The blind deconvolution, in signal processing terminology, is to recover {s(t)} based solely on {x(t)} without knowing {/(j)}. Statisticians may be more interested in the estimation of {f(j)} under certain conditions on {s(t)} and {/(j)}-This problem and its many variations have very broad applications in signal processing, image restoration, geo-exploration, seismology, radio astronomy among others [11,33,38,39].Assume that the signal random variables {s(t)} are independent and identically distributed with mean 0 and variance 1. Let the filter {f(j)} be a sequence of real constants such that oo

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Section: If the Output Process {X(t)} Is Observed With An Independentmentioning

confidence: 99%

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confidence: 99%

Rosenblatt’s Contributions to Deconvolution

Davis

Lii

Politis

2011

Selected Works of Murray Rosenblatt

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“…PROOF. The proof follows using the integrability properties of r 2 , r z and £ 4 and (6.1) and (6.2) (see [18] and [19]). …”

Section: And (C) Y(n) Is a Real Function That Is O(n) Thenmentioning

confidence: 99%

“…It is not necessary in proving the following theorems, but is included merely to permit the statement and proof of Theorem 5 to be written more compactly. A discussion of the nonsymmetric case including a statement of the theorems is in [19].…”

Section: Estimation Of the Bispectral Densitymentioning

confidence: 99%

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Estimation of the Bispectrum

Davis

Lii

Politis

2011

Selected Works of Murray Rosenblatt

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1. Summary. Recently interest has arisen in statistical applications of the bispectrum of stationary random processes. (The bispectrum can be thought of as the Fourier transform of the third-order moment function of the process.) The principal area of statistical harmonic analysis to receive attention previous to this time has been second-order (i.e. spectral) theory on which there is a vast literature. However, the spectrum is most useful in problems of a "linear nature" (see discussion beginning on p. viii of Blackman and Tukey [2]) and provides insufficient information in nonlinear problems.A desire to study phenomena of a nonlinear character has attracted attention to the higher order theory. Such was the case, for example, in a recent study by Hasselmann, Munk and MacDonald [8] where the bispectrum is used in connec tion with oceanographic problems, among which, as the authors state, a number of interesting phenomena such as surf beats, wave breaking, and the energy transfer between wave components can be explained only by the nonlinearity of the wave motion.The bispectrum therefore provides a first glimpse at the nonlinear effects. It is the purpose of the present paper to discuss estimating the weighted and un weighted bispectral density given a set of observations of the process. The rele vant properties (consistency and asymptotic unbiasedness) of the estimates are derived for certain general classes of processes.2. Introduction. The indexed set of random variables, {X t }, is called a con tinuous parameter random process if the index set, T, is the real line and it is called discrete parameter if T is the positive and negative integers. For our purposes we take X t to have mean zero and finite sixth-order moments and to be real-valued sixth-order weakly stationary all of which implies that for all t, Ex t = 0,Consider r(v) and r z {vi, *> 2 ) to be in L x (or k) and in the continuous parameter case to be continuous. It is well known from second-order harmonic theory that

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Bispectral Analysis of 2‐D Signals

Caillec¹,

Garello²

2008

Two‐Dimensional Signal Analysis

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Estimation of the Bispectrum

Cited by 184 publications

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Rosenblatt’s Contributions to Deconvolution

Rosenblatt’s Contributions to Deconvolution

Estimation of the Bispectrum

Bispectral Analysis of 2‐D Signals

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