# Frequency probabilities from Fourier transform

Fourier transform is an estimator of frequency. Because its an estimator, there is always some uncertainty associated with my coefficients as described by the Fourier/Gabor limit. I'm wondering how to generate a probability density of frequency from the transformed data. My intuition is that I like what they do in physics with wave function probability amplitudes: modulus square the coefficients and normalize.

Edit: Fourier transform is NOT an estimator of frequency when integrating across time from -infinity to infinity. When you do not do this, you do not get distinct poles, but gaussian-like curves. First question, does it make sense or can you think of a time-limited fourier-transform in terms of probability? Given my time-limited fourier transform, what is the probability that frequency f is precsent?

• The Fourier transform is not an estimator of frequency. It is a transform from one domain to the related frequency domain (e.g. from time to frequency). There is no estimation.
– Peter K.
Oct 1, 2022 at 19:44
• As Peter said, we need to be a bit more nit-picky here (sorry!) as it's really important to understand what you want to estimate when discussing the limits of accuracy/precision of that estimate. Is it the frequency of a tone (or a discrete set of tones) or is it a spectrum? The PSD (what you would be most tempted to call "spectrum") inherently is a stochastic description of a random process in frequency domain. But it's not clear what you would mean with "probability density of frequency". For that you would first need to define "frequency of what", specifically! Oct 2, 2022 at 0:49
• If this concerns finite measurements, staring at DFT won't do much except for pure tones. Prefer time-frequency analysis. Oct 2, 2022 at 11:17
• Appreciate the nit-pickiness actually. I am looking for probability density of tone frequencies with time-limited discrete measurements. If I normalized PSD so that area under curve = 1, would that be what I'm looking for? Oct 2, 2022 at 15:54
• I guess since my signal is transient I would do ESD instead? Oct 2, 2022 at 17:40

The Power Spectral Density can be estimated using a number of methods.

• The most straightforward, like you said, is taking the magnitude squared of the frequency coefficients: $$\mathbf{PSD} = |X(k)|^2$$
• A better estimator can be obtained through Welch's method, which uses frequency averaging to increase the estimate's SNR.
• Here is a great resource for proper PSD scaling.
• I've never known it as Welch's method. But this power spectrum estimation, in general, is not new. Sliding window, pick a good window (I've been using Gaussian of late but have also used Kaiser and Hamming and Hann), apply DFT to the windowed signal, magnitude square, pick out discrete peaks at integer index locations, apply quadratic interpolation to get a precise peak location. Precision is not the same as accuracy, but the expected accuracy can be determined in analysis, and there are different ways of doing that analysis to get an idea of how off some frequency estimate is. Oct 1, 2022 at 21:31
• Are you sure you want to "reduce the estimate's SNR"? Most of us have spent a lifetime struggling to increase SNR! :-) Oct 7, 2022 at 18:47
• Ha, I ALWAYS make that mistake, thank you for catching! Edited
– Jdip
Oct 7, 2022 at 19:01