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| '''SigSpec''' is an acronym of "SIGnificance SPECtrum" and addresses a statistical technique to provide the reliability of periodicities in a measured (noisy and not necessarily equidistant) [[time series]].<ref>{{cite journal | author = P. Reegen | title = SigSpec - I. Frequency- and phase-resolved significance in Fourier space | journal = Astronomy and Astrophysics | volume = 467 | pages = 1353–1371 | year = 2007 | doi = 10.1051/0004-6361:20066597 | bibcode=2007A&A...467.1353R|arxiv = physics/0703160 }}</ref> It relies on the amplitude [[Spectral density|spectrum]] obtained by the [[Discrete Fourier transform]] (DFT) and assigns a quantity called the spectral significance (frequently abbreviated by “'''sig'''”) to each [[amplitude]]. This quantity is a [[logarithm]]ic measure of the [[probability]] that the given amplitude level is due to [[white noise]], in the sense of a [[type I error]]. It represents the answer to the question, “What would be the chance to obtain an amplitude like the measured one or higher, if the analysed time series were [[random]]?”
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| SigSpec may be considered a formal extension to the [[Least-squares spectral analysis|Lomb-Scargle periodogram]],<ref>{{cite journal | author = N. R. Lomb | title = Least-squares frequency analysis of unequally spaced data | journal = Astrophysics and Space Science | volume = 39 | pages = 447–462 | year = 1976|bibcode = 1976Ap&SS..39..447L |doi = 10.1007/BF00648343 }}</ref><ref>{{cite journal | author = J. D. Scargle | title = Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data | doi=10.1086/160554 | journal = The Astrophysical Journal | volume = 263 | pages = 835–853 | year = 1982 | bibcode=1982ApJ...263..835S}}</ref> appropriately incorporating a time series to be averaged to zero before applying the DFT, which is done in many practical applications. When a zero-mean corrected dataset has to be statistically compared to a [[random sample]], the [[Sample mean and sample covariance|sample mean]] (rather than the [[Mean|population mean]] only) has to be zero.
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| == Probability density function (pdf) of white noise in Fourier space ==
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| Considering a time series to be represented by a set of <math>K</math> pairs <math>(t_k,x_k)</math>, the amplitude [[Probability density function|pdf]] of white noise in [[Fourier space]], depending on [[frequency]] and [[Phase (waves)|phase]] angle may be described in terms of three parameters, <math>\alpha_0</math>, <math>\beta_0</math>, <math>\theta_0</math>, defining the “sampling profile”, according to
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| :<math>\tan 2\theta_0 = \frac{K\sum_{k=0}^{K-1}\sin 2\omega t_k - 2\left(\sum_{k=0}^{K-1}\cos\omega t_k\right)\left(\sum_{k=0}^{K-1}\sin\omega t_k\right)}{K\sum_{k=0}^{K-1}\cos 2\omega t_k - \big(\sum_{k=0}^{K-1}\cos\omega t_k\big)^2 + \big(\sum_{k=0}^{K-1}\sin\omega t_k\big)^2},</math> | |
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| :<math>\alpha_0 = \sqrt{\frac{2}{K^2}\left( K\sum_{k=0}^{K-1}\cos ^2\left(\omega t_k-\theta_0\right) -\left[\sum_{l=0}^{K-1}\cos\left(\omega t_k-\theta_0\right)\right]^2\right)},</math>
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| :<math>\beta_0 = \sqrt{\frac{2}{K^2}\left( K\sum_{k=0}^{K-1}\sin ^2\left(\omega t_k-\theta_0\right) -\left[\sum_{l=0}^{K-1}\sin\left(\omega t_k-\theta_0\right)\right]^2\right)}.</math>
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| In terms of the phase angle in Fourier space, <math>\theta</math>, with
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| :<math>\tan\theta = \frac{\sum_{k=0}^{K-1}\sin\omega t_k}{\sum_{k=0}^{K-1}\cos\omega t_k},</math>
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| the probability density of amplitudes is given by | |
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| :<math>\phi (A) = \frac{KA\cdot\operatorname{sock}}{2<x^2>}\exp\left(-\frac{KA^2}{4<x^2>}\cdot\operatorname{sock}\right),</math>
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| where the sock function is defined by
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| :<math>\operatorname{sock}(\omega ,\theta) = \left[\frac{\cos^2\left(\theta - \theta_0\right)}{\alpha_0^2}+\frac{\sin^2\left(\theta - \theta_0\right)}{\beta_0^2}\right]</math>
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| and <math><x^2></math> denotes the [[variance]] of the [[Dependent and independent variables|dependent variable]] <math>x_k</math>.
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| == False-alarm probability and spectral significance ==
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| Integration of the pdf yields the false-alarm probability that white noise in the [[time domain]] produces an amplitude of at least <math>A</math>,
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| :<math>\Phi_\operatorname{FA}(A) = \exp\left(-\frac{KA^2}{4<x^2>}\cdot\operatorname{sock}\right).</math> | |
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| The sig is defined as the negative logarithm of the false-alarm probability and evaluates to
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| :<math>\operatorname{sig}(A) = \frac{KA^2\log e}{4<x^2>}\cdot\operatorname{sock}.</math>
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| It returns the number of random time series one would have to examine to obtain one amplitude exceeding <math>A</math> at the given frequency and phase.
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| == Applications ==
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| SigSpec is primarily used in [[asteroseismology]] to identify [[variable star]]s and to classify stellar pulsation (see references below). The fact that this method incorporates the properties of the time-domain sampling appropriately makes it a valuable tool for typical astronomical measurements containing data gaps.
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| == References ==
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| <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
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| {{Reflist}}
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| * {{cite journal
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| | author = M. Breger, S. M. Rucinski, P. Reegen
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| | title = The Pulsation of EE Camelopardalis
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| | journal = The Astronomical Journal | |
| | volume = 134 | pages = 1994–1998 | year = 2007
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| | doi = 10.1086/522795
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| | bibcode=2007AJ....134.1994B|arxiv = 0709.3393 }}
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| * {{cite journal
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| | author = M. Gruberbauer, K. Kolenberg, J. F. Rowe, D. Huber, J. M. Matthews, P. Reegen, R. Kuschnig, C. Cameron, T. Kallinger, W. W. Weiss, D. B. Guenther, A. F. J. Moffat, S. M. Rucinski, D. Sasselov, G. A. H. Walker | |
| | title = MOST photometry of the RRdLyrae variable AQLeo: two radial modes, 32 combination frequencies and beyond
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| | journal = Monthly Notices of the Royal Astronomical Society | |
| | volume = 379 | pages = 1498–1506 | year = 2007
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| | doi = 10.1111/j.1365-2966.2007.12042.x
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| | bibcode=2007MNRAS.379.1498G|arxiv = 0705.4603 }}
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| * {{cite journal
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| | author = M. Gruberbauer, H. Saio, D. Huber, T. Kallinger, W. W. Weiss, D. B. Guenther, R. Kuschnig, J. M. Matthews, A. F. J. Moffat, S. M. Rucinski, D. Sasselov, G. A. H. Walker
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| | title = MOST photometry and modeling of the rapidly oscillating (roAp) star γ Equulei
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| | journal = Astronomy and Astrophysics
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| | volume = 480 | pages = 223–232 | year = 2008
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| | doi = 10.1051/0004-6361:20078830
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| | bibcode=2008A&A...480..223G|arxiv = 0801.0863 }}
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| * {{cite journal
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| | author = D. B. Guenther, T. Kallinger, P. Reegen, W. W. Weiss, J. M. Matthews, R. Kuschnig, A. F. J. Moffat, S. M. Rucinski, D. Sasselov, G. A. H. Walker
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| | title = Searching for p-modes in η Bootis & Procyon using MOST satellite data
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| | journal = Communications in Asteroseismology
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| | volume = 151 | pages = 5–25 | year = 2007
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| | doi = 10.1553/cia151s5
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| |bibcode = 2007CoAst.151....5G }}
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| * {{cite journal
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| | author = D. B. Guenther, T. Kallinger, K. Zwintz, W. W. Weiss, J. Tanner
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| | title = Seismology of Pre-Main-Sequence Stars in NGC 6530
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| | journal = The Astrophysical Journal
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| | volume = 671 | pages = 581–591 | year = 2007
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| | doi = 10.1086/522880
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| | bibcode=2007ApJ...671..581G}}
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| * {{cite journal
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| | author = D. Huber, H. Saio, M. Gruberbauer, W. W. Weiss, J. F. Rowe, M. Hareter, T. Kallinger, P. Reegen, J. M. Matthews, R. Kuschnig, D. B. Guenther, A. F. J. Moffat, S. M. Rucinski, D. Sasselov, G. A. H. Walker
| |
| | title = MOST photometry of the roAp star 10 Aquilae
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| | journal = Astronomy and Astrophysics
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| | volume = 483 | pages = 239–248 | year = 2008
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| | doi = 10.1051/0004-6361:20079220
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| | bibcode=2008A&A...483..239H|arxiv = 0803.1721 }}
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| * {{cite journal
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| | author = T. Kallinger, D. B. Guenther, J. M. Matthews, W. W. Weiss, D. Huber, R. Kuschnig, A. F. J. Moffat, S. M. Rucinski, D. Sasselov
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| | title = Nonradial p-modes in the G9.5 giant ε Ophiuchi? Pulsation model fits to MOST photometry
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| | journal = Astronomy and Astrophysics
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| | volume = 478 | pages = 497–505 | year = 2008
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| | doi = 10.1051/0004-6361:20078171
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| | bibcode=2008A&A...478..497K|arxiv = 0711.0837 }}
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| * {{cite journal
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| | author = T. Kallinger, P. Reegen, W. W. Weiss
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| | title = A heuristic derivation of the uncertainty for frequency determination in time series data
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| | journal = Astronomy and Astrophysics
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| | volume = 481 | pages = 571–574 | year = 2008
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| | doi = 10.1051/0004-6361:20077559
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| | bibcode=2008A&A...481..571K|arxiv = 0801.0683 }}
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| * {{cite book
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| | author = P. Reegen
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| | title = "SigSpec - reliable computation of significance in Fourier space", in ''The A-Star Puzzle, Proceedings IAU Symp. 224'', eds. J. Zverko, J. Ziznovsky, S.J. Adelman, W.W. Weiss
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| | work = The A-Star Puzzle, Proceedings of IAU Symposium 224
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| | location = Cambridge, UK
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| | publisher = Cambridge University Press
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| | year = 2005
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| | pages = 791–798
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| | isbn = 0-521-85018-5
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| }}
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| * {{cite journal
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| | author = P. Reegen, M. Gruberbauer, L. Schneider, W. W. Weiss
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| | title = Cinderella - Comparison of INDEpendent RELative Least-squares Amplitudes
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| | journal = Astronomy and Astrophysics
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| | volume = 484 | pages = 601–608 | year = 2008
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| | doi = 10.1051/0004-6361:20078855
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| | bibcode=2008A&A...484..601R}}
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| * {{cite journal
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| | author = C. Schoenaers, A. E. Lynas-Gray
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| | title = A new slowly pulsating subdwarf-B star: HD 4539
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| | journal = Communications in Asteroseismology
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| | volume = 151 | pages = 67–76 | year = 2007
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| | doi = 10.1553/cia151s67
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| |bibcode = 2007CoAst.151...67S }}
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| * {{cite journal
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| | author = M. Zechmeister, M. Kuerster
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| | title = The gemeralised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms
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| | journal = Astronomy and Astrophysics
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| | volume = 496
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| | pages = 577–584
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| | year = 2009
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| | doi = 10.1051/0004-6361:200811296
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| | bibcode=2009A&A...496..577Z
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| |arxiv = 0901.2573 }}
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| * {{cite journal
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| | author = K. Zwintz, T. Kallinger, D. B. Guenther, M. Gruberbauer, D. Huber, J. Rowe, R. Kuschnig, W. W. Weiss, J. M. Matthews, A. F. J. Moffat, S. M. Rucinski, D. Sasselov, G. A. H. Walker, M. P. Casey
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| | title = MOST photometry of the enigmatic PMS pulsator HD 142666
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| | journal = Astronomy and Astrophysics
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| | volume = 494
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| | pages = 1031–1040
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| | year = 2009
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| | doi = 10.1051/0004-6361:200811116
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| | bibcode=2009A&A...494.1031Z
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| |arxiv = 0812.1960 }}
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| * {{cite journal
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| | author = K. Zwintz, M. Hareter, R. Kuschnig, P. J. Amado, N. Nesvacil, E. Rodriguez, D. Diaz-Fraile, W. W. Weiss, T. Pribulla, D. B. Guenther, J. M. Matthews, A. F. J. Moffat, S. M. Rucinski, D. Sasselov, G. A. H. Walker
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| | title = MOST observations of the young open cluster NGC 2264
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| | journal = Astronomy and Astrophysics
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| | volume = 502
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| | pages = 1239–252
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| | year = 2009
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| | doi = 10.1051/0004-6361:200911863
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| }}
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| == External links ==
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| * [http://www.sigspec.org Website with further information on SigSpec calculation, etc.]
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| [[Category:Time series analysis]]
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| [[Category:Fourier analysis]]
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| [[Category:Digital signal processing]]
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