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ref: work-0 tags: PSD FFT periodogram autocorrelation time series analysis date: 07-19-2010 18:45 gmt revision:3 [2] [1] [0] [head]

Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data Scargle, J. D.

  • The power at a given frequency as computed by a periodigram (FFT is a specific case of the periodigram) of a gaussian white noise source with uniform variance is exponentially distributed: P z(z)=P(x<Z<z+dz)=e zdzP_z(z) = P(x&lt;Z&lt;z+dz) = e^{-z}dz
    • The corresponding CDF: 1e z 1- e^{-z} or P(Z>z)=e zP(Z&gt;z) = e^{-z} which gives the probability of a large observed power at a given freq.
    • If you need to average N samples, then P(Z>z)=1(1e z) NP(Z&gt;z) = 1 - (1-e^{-z})^N where Z=max nPow(ω n)Z = max_n Pow(\omega_n)
  • Means of improving detection using a periodogram:
    • Average in time - this means that N above will be smaller, hence a spectral peak becomes more significant.
      • Cannot average too much - at some point, averaging will start to attenuate the signal!
    • Decrease the number of frequencies inspected.
  • Deals a good bit with non-periodic sampling, which i guess is more common in astronomical data (the experimenter may not take a photo every day, or the same time every day (clouds!).