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Power Spectral Density Function, supplied by MathWorks Inc, used in various techniques. Bioz Stars score: 90/100, based on 1 PubMed citations. ZERO BIAS - scores, article reviews, protocol conditions and more
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Power Spectral Density Function, supplied by MathWorks Inc, used in various techniques. Bioz Stars score: 90/100, based on 1 PubMed citations. ZERO BIAS - scores, article reviews, protocol conditions and more
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![Examples of the source dataset. Survey data of the EMFISIS Waves instrument on Van Allen Probe A. The time interval corresponds to one orbital period of the spacecraft, from perigee to perigee. ( a ) Local plasma <t>density</t> from the upper hybrid frequency for a geomagnetically disturbed orbit of 28 September 2017; the red and blue horizontal line on the top indicates the position within the plasmapause and outside of it, respectively, according to an empirical model . ( b ) Trace of the <t>power</t> <t>spectral</t> density <t>matrix</t> of the three magnetic field components as a function of frequency and time. ( c ) Signed magnetic field ellipticity plotted when the magnetic power spectral density is above the P 0 = 10 −7 <t>noise</t> threshold. ( d – f ) Plasma density, trace of the magnetic power spectral density matrix, and signed ellipticity for a geomagnetically calm orbit of 1 January 2015. Four types of whistler mode emissions are shown by numbers and arrows: 1—plasmaspheric hiss, 2a–f—lower band chorus and exohiss, 3—upper band chorus, 4—equatorial noise, 5—lightning generated whistlers. Overplotted dashed and solid lines show the frequency interval of lower-band exohiss/chorus between 0.1 and 0.5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{{ce}0}$$\end{document} f ce 0 , where the equatorial electron cyclotron frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{{ce}0}$$\end{document} f ce 0 is estimated from the locally measured magnetic field strength using Eq. . Dotted lines show high density estimates of the local lower hybrid frequency in a proton-electron plasma. Coordinates of the spacecraft are given at the bottom of each ellipticity plot: time UTC, magnetic latitude ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda }_{m}$$\end{document} λ m ) in degrees, magnetic local time (MLT) in hours, and the L parameter in the dipole approximation.](https://pub-med-central-images-cdn.bioz.com/pub_med_central_ids_ending_with_4541/pmc12274541/pmc12274541__41597_2025_5531_Fig1_HTML.jpg)
![Examples of high-resolution spectrograms of lower-band chorus / exohiss. Panels a–f show the trace of the power spectral density matrix of the three magnetic field components as a function of frequency and time, obtained from 6 seconds long continuous burst mode intervals of the EMFISIS Waves instrument on Van Allen Probe A, for events marked 2a–f in Fig. . Spacecraft position and locally measured electron cyclotron frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{{ce}}$$\end{document} f ce are given on the top of each panel. Overplotted dashed and solid lines show the frequency interval of lower-band exohiss/chorus between 0.1 and 0.5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{{ce}0}$$\end{document} f ce 0 , where the equatorial electron cyclotron frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{{ce}0}$$\end{document} f ce 0 is estimated using Eq. . The waveforms, from which panels a–f have been obtained were transformed into stereo wave files (stored on 10.6084/m9.figshare.29433323.v1) using two magnetic field components perpendicular to the local magnetic field line.](https://pub-med-central-images-cdn.bioz.com/pub_med_central_ids_ending_with_4541/pmc12274541/pmc12274541__41597_2025_5531_Fig2_HTML.jpg)
![Probability density function of the simulated trace of the noise power spectral density matrix. ( a ) for n l = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4$$\end{document} 4 averaged independent spectral estimates; ( b ) for n l = 32; ( c ) for n l = 256, relevant for the three groups of frequency channels of the Cluster STAFF-SA instruments. Red line: probability density estimated by a histogram from a Monte Carlo simulation of the sensor noise. Blue line: scaled \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }^{2}$$\end{document} χ 2 distribution model according to Eq. . Vertical black dotted line: detection threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S}_{0}$$\end{document} S 0 (Eq. ) for the trace of the power spectral density matrix to reach above \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S}_{0}$$\end{document} S 0 with a probability P 0 = 10 −7 .](https://pub-med-central-images-cdn.bioz.com/pub_med_central_ids_ending_with_4541/pmc12274541/pmc12274541__41597_2025_5531_Fig4_HTML.jpg)
![Correction factors and detection thresholds for the simulated trace of the power spectral density matrix obtained from a windowed noise waveform. ( a ) Correction factors from Eqs. and , relevant to the Survey dataset of the EMFISIS Waves instrument on Van Allen Probes. Black dots: average values of the correction factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{c\nu }$$\end{document} C c ν for the number of degrees of freedom of the core distribution model; Red dots: average values of the correction factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{{cs}}$$\end{document} C cs for the scale of the core distribution model; Black line with error bars: average values and standard deviations of the correction factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{t\nu }$$\end{document} C t ν for the number of degrees of freedom of the tail distribution model; Red line with error bars: average values and standard deviations of the correction factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{{ts}}$$\end{document} C ts for the scale of the tail distribution model. The results are given as a function of the binary logarithm of the number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n}_{l}$$\end{document} n l of averaged spectral estimates; ( b ) Detection thresholds at P 0 = 10 −7 , for the trace of the magnetic spectral density matrix in n a = 3 dimensions, as a function of the binary logarithm of the number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n}_{l}$$\end{document} n l of averaged spectral estimates, assuming a constant sensor noise level of 100 fT 2 /Hz. Red diamonds: thresholds based on the tail distribution model from Eq. , which are used in our procedure, black dots: comparison with thresholds obtained from the core distribution model from Eq. , black solid line: comparison with thresholds based on the noise waveform without a window function, according to Eq. .](https://pub-med-central-images-cdn.bioz.com/pub_med_central_ids_ending_with_4541/pmc12274541/pmc12274541__41597_2025_5531_Fig6_HTML.jpg)
![Probability density function (PDF) of the trace of the magnetic power spectral density matrix. Normalized histograms were obtained from the Survey data measured by the EMFISIS Waves instrument onboard Van Allen Probe A between 14:00 and 17:00 UT on January 1, 2015: ( a ) channel number l = 2, central frequency f l = 4 Hz, number of averaged power spectra n l = 1; ( b ) l = 36, f l = 397 Hz, n l = 22; ( c ) l = 58, f l = 5011 Hz, n l = 271, and from the STAFF-SA instrument onboard the Cluster 1 spacecraft between 18:00 and 21:00 UT on March 8, 2002: ( d ) the 6 th frequency channel with a central frequency of 28 Hz and n l = 4 averaged spectra; ( e ) the 13 th channel at 141 Hz and with n l = 32 averaged spectra; ( f ) the 24 th channel at 1794 Hz, with n l = 256 averaged spectra. Blue curves show results of the models based on the scaled \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }^{2}$$\end{document} χ 2 distributions .](https://pub-med-central-images-cdn.bioz.com/pub_med_central_ids_ending_with_4541/pmc12274541/pmc12274541__41597_2025_5531_Fig7_HTML.jpg)
![Distribution of Survey data search coil noise for the Van Allen Probe A EMFISIS instrument. Percentiles of the probability distribution of the trace of the magnetic power spectral density matrix as a function of frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{l}$$\end{document} f l , obtained from all 65 frequency channels \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l$$\end{document} l of Survey data measured by the EMFISIS Waves instrument onboard Van Allen Probe A between 14:00 and 17:00 UT on January 1, 2015, and corresponding to examples from Fig. . The colored dotted lines show the theoretical cumulative probabilities of the core model distribution according to Eq. , with the root-mean-square noise level \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{l}^{2}$$\end{document} σ l determined using the observed median value. The black dotted line corresponds to the P 0 = 10 −7 detection threshold from the tail model distribution according to Eqs. and , with the same \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{l}^{2}$$\end{document} σ l .](https://pub-med-central-images-cdn.bioz.com/pub_med_central_ids_ending_with_4541/pmc12274541/pmc12274541__41597_2025_5531_Fig8_HTML.jpg)
![Distribution of search coil noise for Cluster 1 STAFF-SA instrument. Percentiles of the probability distribution of the trace of the magnetic power spectral density matrix of the instrumental noise as a function of frequency \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{l}$$\end{document} f l obtained from all 27 frequency channels \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l$$\end{document} l of the STAFF-SA instrument onboard Cluster 1. The data were acquired during a calm interval on March 8, 2002 from 18:00 to 21:00 UT. Dotted lines show theoretical cumulative probabilities of the scaled \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }^{2}$$\end{document} χ 2 distribution according to Eq. , with the root-mean-square noise level \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{l}^{2}$$\end{document} σ l 2 determined using the observed median value, and with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =2\,{n}_{l}\,{n}_{a}$$\end{document} ν = 2 n l n a degrees of freedom, where n a = 3 is the number of search coil antennas, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n}_{l}$$\end{document} n l is the number of onboard averaged power spectra. n l = 4 for l = 1…9, n l = 32 for l = 10…18, n l = 256 for l = 19…27. The black dotted line corresponds to P 0 = 10 −7 from Eqs. , .](https://pub-med-central-images-cdn.bioz.com/pub_med_central_ids_ending_with_4541/pmc12274541/pmc12274541__41597_2025_5531_Fig9_HTML.jpg)