Square the Fourier Amplitudes (Reminder!)

Do not forget to square your Fourier amplitudes! Since a power spectrum is the square of the Fourier amplitude, if one forgets to take the square operation, a 1/f2 noise may be mistaken as a 1/f noise.

It is important to remember to square the Fourier amplitudes when calculating the power spectrum of a signal. The power spectrum is a measure of the power present at different frequencies in a signal, and it is defined as the square of the Fourier amplitude. If you forget to square the Fourier amplitudes, you will not obtain the correct power spectrum of the signal.

In addition to being an important step in the calculation of the power spectrum, squaring the Fourier amplitudes can also help to distinguish between different types of noise in a signal. For example, a 1/f^2 noise spectrum, which is often observed in systems with correlated noise, will appear as a straight line on a log-log plot if the Fourier amplitudes are not squared. However, if the amplitudes are squared before plotting the spectrum, the 1/f^2 noise will appear as a parabolic curve on the log-log plot, which can help to distinguish it from a 1/f noise spectrum, which appears as a straight line on the log-log plot.

In summary, it is important to remember to square the Fourier amplitudes when calculating the power spectrum of a signal to obtain accurate results and to properly characterize the noise present in the signal.

  • 2005: Lasse Laurson, Mikko J Alava and Stefano Zapperi (2005), “Power spectra of self-organized critical sandpiles”, Journal of Statistical Mechanics, L11001
  • 2001: J Davidsen, N Luethje (2001), “Comment on ‘1/f noise in the Bak-Sneppen model’ “, Physical Review E, 63:063101.
  • 1999: P Helander, SC Chapman, RO Dendy, G Rowlands, NW Watkins (1999), “Exactly solvable sandpile with fractal avalanching”, Physical Review E, 59:6356-6360.
  • 1996: Frank Daerden, Carlo Vanderzande (1996), “1 / f noise in the Bak-Sneppen model”, Physical Review E, 53:4723-4728.
  • 1991: Hans Jacob S Feder, Jens Feder (1991), “Erratum: Self-organized criticality in a stick-slip process”, Physical Review Letters, 67(2):283.
    [text: Because of a programming error the square root of the power spectrum was used instead of the power spectrum itself. This means that the correct result is that we observe S(f) ~ 1/f2 noise (instead of 1/f noise as stated). Thus the slope in Fig. 3 is cphi = 2 +- 0.06. We conclude that 1/f2 noise is observed, as in the sandpile experiments]
  • 1990: J Kertesz, LB Kiss (Kish) (1990), “Noise spectrum in the model of self-organized criticality”, Journal of Physics A, 23:L433-L440.
  • 1989: Henrik Jeldtoft Jensen, Kim Christensen, Hans C Fogedby (1989), “1/f noise, distribution of lifetimes, and a pile of sand”, Physical Review B, 40(10):7425-7427.
    [Comment: this is the paper that pointed out that the 1/f noise claimed in Bak,Tang,Wiesenfeld (1987) paper is actually 1/f2 noise.]
  • 1987: Per Bak, Chao Tang, Kurt Wiesenfeld (1987), “Self-organized criticality: an explanation of the 1/f noise”, Physical Review Letters, 59(4):381-384.