Generalized Spectral Coherences for Complex-Valued Harmonizable Processes

Hindberg, Heidi; Hanssen, A.

doi:10.1109/tsp.2007.893932

Cited by 8 publications

(12 citation statements)

References 26 publications

(46 reference statements)

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“…form of a second-order process whose covariance function is given by [19], [20], [23] (2) with probability one, where denotes temporal frequency, denotes time, and may be viewed as a complex-valued random measure. Equation (2) expresses a stochastic process as a summation of infinitely many randomly and infinitesimally weighted complex exponentials, in a manner analogous to how the standard Fourier transform expresses a deterministic function as a superposition of weighted complex exponentials.…”

Section: A Bifrequency Spectrummentioning

confidence: 99%

On Spatial Aliasing in Microphone Arrays

Dmochowski

Benesty

Affes

2009

IEEE Trans. Signal Process.

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Abstract-Microphone arrays sample the sound field in both space and time with the major objective being the extraction of the signal propagating from a desired direction-of-arrival (DOA). In order to reconstruct a spatial sinusoid from a set of discrete samples, the spatial sampling must occur at a rate greater than a half of the wavelength of the sinusoid. This principle has long been adapted to the microphone array context: in order to form an unambiguous beampattern, the spacing between elements in a microphone array needs to conform to this spatial Nyquist criterion. The implicit assumption behind the narrowband beampattern is that one may use linearity and Fourier analysis to describe the response of the array to an arbitrary wideband plane wave. In this paper, this assumption is analyzed. A formula for the broadband beampattern is derived. It is shown that in order to quantify the spatial filtering abilities of a broadband array, the incoming signal's bifrequency spectrum must be taken into account, particularly for nonstationary signals such as speech. Multi-dimensional Fourier analysis is then employed to derive the broadband spatial transform, which is shown to be the limiting case of the broadband beampattern as the number of sensors tends to infinity. The conditions for aliasing in broadband arrays are then determined by analyzing the effect of computing the broadband spatial transform with a discrete spatial aperture. It is revealed that the spatial Nyquist criterion has little importance for microphone arrays. Finally, simulation results show that the well-known steered response power (SRP) method is formulated with respect to stationary signals, and that modifications are necessary to properly form steered beams in nonstationary signal environments.Index Terms-Beamforming, broadband beampattern, microphone arrays, spatial aliasing, spatial sampling, wavenumber-frequency spectrum.

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Section: A Bifrequency Spectrummentioning

confidence: 99%

On Spatial Aliasing in Microphone Arrays

Dmochowski

Benesty

Affes

2009

IEEE Trans. Signal Process.

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“…We shall discuss the analysis of the covariance of a zero-mean time series X t . We shall therefore model the dual-time autocovariance function [12] of X t given by…”

Section: Second-order Modellingmentioning

confidence: 99%

“…where dZ * (f ) denotes the complex-conjugate of dZ(f ) and S(f 1 , f 2 ) is a complex-valued scalar quantity called the Loève spectrum [12]. It may appear that Equation (2) strongly resembles Equation (3) in form, but the introduction of the correlations between frequencies significantly modifies the appearance of realizations {X t }.…”

Section: A Dual-frequency Representation Of Covariancementioning

confidence: 99%

“…alpha band (8)(9)(10)(11)(12) which is present when eyes are closed and is also associated with inhibition control; (iv.) beta band (12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30) which is present during alert and active states; (v.) gamma band (30-100 Hertz) which are thought to represent the highly synchronous activity of neurons in response to a specific cognitive or motor function. To examine oscillatory properties of singlechannel brain signals, scientists use spectral analysis methods [28] which decompose the covariance of of methods for testing significance of these dependence measures.…”

Section: Eeg Examplementioning

confidence: 99%

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Modeling and Estimation of Covariance of Replicated Modulated Cyclical Time Series

Olhede

Ombao

2013

IEEE Trans. Signal Process.

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This paper introduces the novel class of modulated cyclostationary processes, a class of non-stationary processes exhibiting frequency coupling, and proposes a method of their estimation from repeated trials.Cyclostationary processes also exhibit frequency correlation but have Loève spectra whose support lies only on parallel lines in the dual-frequency plane. Such extremely sparse structure does not adequately represent many biological processes. Thus, we propose a model that, in the time domain, modulates the covariance of cyclostationary processes and consequently broadens their frequency support in the dualfrequency plane. The spectra and the cross-coherence of the proposed modulated cyclostationary process are first estimated using multitaper methods. A shrinkage procedure is then applied to each trial-specific estimate to reduce the estimation risk.Multiple trials of each series are observed. When combining information across trials, we carefully take into account the bias that may be introduced by phase misalignment and the fact that the Loève spectra and cross-coherence across replicates may only be "similar" -but not necessarily identicalacross replicates. The application of the inference methods developed for the modulated cyclostationary model to EEG data also demonstrates that the proposed model captures statistically significant crossfrequency interactions, that ought to be further examined by neuroscientists.

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“…The concept of dependence between oscillations at different frequencies has already been established under the context of harmonizable processes (see Loève, 1955;Rao, 1981Rao, , 1994Scharf et al, 1998;Rosenblatt, 2002, 2006;Hindberg and Hanssen, 2007;Hanssen et al, 2010). Formally, a time series X t belongs to the class of harmonizable processes if it admits the representation…”

Section: Introductionmentioning

confidence: 99%

Time‐Dependent Dual‐Frequency Coherence in Multivariate Non‐Stationary Time Series

Gorrostieta

Ombao

Sachs

2018

Journal Time Series Analysis

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Coherence is one common metric for cross‐dependence in multichannel signals. However, standard coherence does not sufficiently model many biological signals with complex dependence structures such as cross‐oscillatory interactions between a low‐frequency component in one signal and a high‐frequency component in another. The notion of cross‐dependence between low‐ and high‐frequency components, as defined in classical harmonizable processes, is still inadequate because it assumes time invariance and thus cannot capture cross‐frequency interactions that evolve over time. We construct a novel framework for modeling and estimating these dependencies under the replicated time series setting. Under this framework, we establish the novel concept of evolutionary dual‐frequency coherence and develop time‐localized estimators based on dual‐frequency local periodograms. The proposed nonparametric estimation procedure does not suffer from model misspecification. It uses the localized fast Fourier transform and hence is able to handle massive data. When applied to electroencephalogram data recorded in a motor intention experiment, the proposed method uncovers new and interesting cross‐oscillatory interactions that have been overlooked by the standard approaches.

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Generalized Spectral Coherences for Complex-Valued Harmonizable Processes

Cited by 8 publications

References 26 publications

On Spatial Aliasing in Microphone Arrays

On Spatial Aliasing in Microphone Arrays

Modeling and Estimation of Covariance of Replicated Modulated Cyclical Time Series

Time‐Dependent Dual‐Frequency Coherence in Multivariate Non‐Stationary Time Series

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