About Fourier Transforms

There are variants of the Fourier transform, involving either cos(ω t) or cos(2π ω t); the normalization of the inverse transform is affected. As McComb (The Physics of Fluid Turbulence, Oxford U.P., 1992) put it: `it is all about where you put the factor 2p’. Here, we place the factor in the exponential, so the normalization factors for transform and inverse are 1; (alternative simplifies the manipulation of derivatives, as in Lesieur’s Turbulence or McComb’s Turbulence).

The definitions are well known, and the Parseval theorem leads to the definition of power spectra: the total energy E of the signal is recovered by adding (integral sum) the spectral energy E(k) at all frequencies. I do not like the mathematicians’ term, that energy is “conserved”, a thermodynamic concept in my mind; in my view, the Parseval formula “accounts for” the energy in the frequency domain.

Fourier transforms are calculated using the FFT approach. Note that Matlab provides FFT and inverse routines, in 1 and multiple dimensions; the frequency vector (2-sided) needs to be calculated (see Matlab help files on this topic).

My unorthodox streak manifests itself in the presentation of the spectra. Click here for the Matlab code, the result is shown on the figure:

There are two good independent reasons to use log-log coordinates (right column on the figure): the range of the variables covers several orders of magnitude, and/or we expect power-law behavior (which is displayed as a straight line on log-log). We see that an approximate slope -1 fits the sample data (which is not 1/f noise, obviously!). But if the purpose is to show the distribution of energy in the spectrum, the formulae Energy = ∫ E(f) df = ∫ f.E(f) d(log f) should be a guide. The log scale for E(f) distorts the relative contributions to total energy, so both log-log plots are misleading (we are trained to make the corrections); the linear-linear plot for E(f) vs f (bottom left) is a correct rendition of the formula, but is not very popular for obvious reasons; the log-linear plot for f.E(f) vs log(f) (top left) is also correct, and I have a strong preference for it: the energy scale (linear) shows correct relationships across the spectrum, and the use of log(frequency) allows the identification of dominant frequencies. I believe this was known in the turbulence community many years ago as the `Marseille’ plot (was it used at the IMST? Did the conformism latent in peer review demand its demise? I need to look it up some dusty shelves, or maybe a reader will let me know (jlewalle@syr.edu --- Thanks!!!)) And it turns out that the top-left version also fits my formulae for wavelet transforms and corresponding spectra (why use a linear scale for frequency?? All frequencies are logarithmic in nature.)

So, I’ll use f.E(f) vs log(f) when physical insight is the primary concern, log(f.E(f)) vs log(f) if power-law scaling is expected or assumed. The fact that both fit naturally in my version of continuous wavelet transforms (formulae and plotting) makes this sensible. I’ll argue elsewhere that the same might apply to the orthogonal wavelet spectra (where the log(f) scale occurs naturally!). See examples and the comparison of spectra in the mexhat pages.