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Note: This question has been marked as a duplicate of What do the X and Y axis stand for in the Fourier transform domain?. While the answer will likely have an overlap, so the question should be linked, I see the context and background as sufficiently different to warrant a separate question for anyone who is having a similar issue to what I'm having; and it has been noted that some additional information may be useful in answering this question.

I'm trying to find the periods of time series and was directed toward Fourier analysis. I am having a hard time understanding what is going on in ref/Fourier > Applications > Frequency Identification. Would anyone be able to explain, or direct me to an explanation somewhere? Thank you!

EDIT: Here is some sample data. It is a list representing a 55-year time series with one measurement per year, for a total of 56 points. What I have been doing is subtracting off the seventh-order polynomial regression and running the absolute value of the fourier on both the raw data and the data with the overall trend removed.

{0.,0.,0.,0.00118064,0.,0.00437956,0.,0.,0.0035461,0.00107875,0.00715015,0.00501505,0.0040568,0.00294985,0.00514933,0.00609137,0.0047619,0.00100806,0.00196464,0.000882613,0.00187793,0.00282486,0.00172563,0.000807754,0.00318218,0.00233645,0.0015528,0.00149365,0.000776398,0.000688705,0.00198151,0.00236593,0.00225904,0.00149589,0.,0.00332889,0.00179748,0.00126743,0.00289352,0.00218103,0.00265675,0.00104275,0.00509495,0.00292887,0.00161616,0.00327869,0.00413793,0.00223928,0.00502067,0.00434075,0.00392465,0.00469373,0.00595745,0.00521147,0.00567752,0.00293255}

Note, for the Fourier graphs below, the data have not been padded to the right; should I do that?

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    $\begingroup$ related $\endgroup$
    – andre314
    Commented May 14, 2017 at 19:50
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    $\begingroup$ I have put some basic information on Fourier here. Let me know if this helps of if you need more. $\endgroup$
    – Hugh
    Commented May 14, 2017 at 21:02
  • $\begingroup$ @Hugh: Thank you, that's some very helpful information; though one thing I'm actually still not clear on is, what do each of the axes represent with standard Fourier in Mathematica? $\endgroup$ Commented May 15, 2017 at 22:13
  • $\begingroup$ @andre: That looks like exactly what I'm looking for. Is it necessary to pad to the right? And I'm not clear on how you converted the position of the peak to the period. Thank you! $\endgroup$ Commented May 15, 2017 at 22:14
  • $\begingroup$ I have tried to add some extra information to the article here. However, I think you need to describe your difficulty more precisely for me to help you. Try to give us an example of your problem. $\endgroup$
    – Hugh
    Commented May 16, 2017 at 11:30

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