Random Sampling Estimation for Almost Periodically Correlated Processes

Dehay, Dominique; Monsan, Vincent

doi:10.1111/j.1467-9892.1996.tb00286.x

Cited by 23 publications

(39 citation statements)

References 9 publications

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“…This result was observed for univariate continuous time PC processes in [9] and was more thoroughly investigated in [11]. The proof can also be found in the survey paper [4]. The proof here is omitted due to similarity with the univariate continuous time case.…”

Section: Strongly Periodically Correlated Fieldssupporting

confidence: 51%

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Correlation and spectral theory for periodically correlated random fields indexed on Z2

Hurd

Kallianpur

Farshidi

2004

Journal of Multivariate Analysis

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We show that a field X ðm; nÞ is strongly periodically correlated with period ðM; NÞ if and only if there exist commuting unitary operators, U 1 and U 2 that shift the field unitarily by M and N along the respective coordinates. This is equivalent to a field whose shifts on a subgroup are unitary. We also define weakly PC fields in terms of other subgroups of the index set over which the field shifts unitarily. We show that every strongly PC field can be represented as X ðm; nÞ ¼Ũ m 1Ũ n 2 Pðm; nÞ whereŨ 1 andŨ 2 are unitary and Pðm; nÞ is a doubly periodic vectorvalued sequence. This leads to the Gladyshev representations of the field and to strong harmonizability. The 2-and 4-fold Wold decompositions are expressed for weakly commuting strongly PC fields. When the field is strongly commuting, a one-point innovation can be defined. For this case, we give necessary and sufficient conditions for a strongly commuting field to be PC and strongly regular, although possibly of deficient rank, in terms of periodicity and summability of the southwest moving average coefficients. r 2004 Elsevier Inc. All rights reserved.

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Section: Strongly Periodically Correlated Fieldssupporting

confidence: 51%

“…In the communications context, periodically correlated processes are also called cyclostationary [7]. For a survey of univariate PC and almost PC processes, see Dehay and Hurd [4].…”

Section: Introductionmentioning

confidence: 99%

Correlation and spectral theory for periodically correlated random fields indexed on Z2

Hurd

Kallianpur

Farshidi

2004

Journal of Multivariate Analysis

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“…Conversely, if iscyclostationary then (19) and therefore we have (20) thus proving that is ( )-DSI. A first immediate consequence, using (14), is the general form for the correlation function of ( )-DSI processes:…”

Section: B Dsi and Cyclostationary Processesmentioning

confidence: 77%

“…A nonparametric estimation of can be characterized in this manner. 3) For ( )-DSI, when is known, estimators for or adapted from those in the cyclostationary case [20] can be constructed. This will generalize the estimators for self-similar processes, given in [13], to the DSI problem.…”

Section: Toward Mellin-based Tools For Estimationmentioning

confidence: 99%

Stochastic discrete scale invariance

Borgnat

Flandrin

Amblard

2002

IEEE Signal Process. Lett.

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Abstract-A definition of stochastic discrete scale invariance (DSI) is proposed and its properties studied. It is shown how the Lamperti transformation, which transforms stationarity in self-similarity, is also a means to connect processes deviating from stationarity and processes which are not exactly scale invariant: in particular we interpret DSI as the image of cyclostationarity. This theoretical result is employed to introduce a multiplicative spectral representation of DSI processes based on the Mellin transform, and preliminary remarks are given about estimation issues.

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“…If the process X t = X(t) is stationary, the spectral mass is concentrated on the diagonal line dH (u, v) = S v -Ui oxdF(u). If the process X t is periodically or almost periodically correlated, then the spectral mass is concentrated on at most countably many lines which are parallel to the diagonal [Gladyshev (1963) and Dehay and Hurd (1993)…”

Section: Allmentioning

confidence: 99%

Spectral Analysis for Harmonizable Processes

Lii

Rosenblatt

2011

Selected Works of Murray Rosenblatt

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Spectral estimation of nonstationary but harmonizable processes is considered. Given a single realization of the process, periodogram-like and consistent estimators are proposed for spectral mass estimation when the spectral support of the process consists of lines. Such a process can arise in signals of a moving source from array data or multipath signals with Doppler stretch from a single receiver. Such processes also include periodically correlated (or cyclostationary) and almost periodically correlated processes as special cases. We give detailed analysis on aliasing, bias and covariances of various estimators. It is shown that dividing a single long realization of the process into nonoverlapping subsections and then averaging periodogramlike estimates formed from each subsection will not yield meaningful results if one is estimating spectral mass with support on lines with slope not equal to 1. If the slope of a spectral support line is irrational, then spectral masses do not fold on top of each other in estimation even if the data are equally spaced. Simulation examples are given to illustrate various theoretical results.

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Random Sampling Estimation for Almost Periodically Correlated Processes

Cited by 23 publications

References 9 publications

Correlation and spectral theory for periodically correlated random fields indexed on Z2

Correlation and spectral theory for periodically correlated random fields indexed on Z2

Stochastic discrete scale invariance

Spectral Analysis for Harmonizable Processes

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