Disclaimer: I am a geophysicist working with time-domain data which needs to be converted to the frequency domain evaluated at a specific set of logarithmically-spaced frequencies. The textbooks I am reading go through the following workflow but seem to skip over a certain aspect:

Suppose you have some random, wide-sense stationary time series $A(t)$ with length $N$ and sampling rate of $\Delta t$ which you want to convert to the frequency domain. A brute force Fourier transform of the time series will not work so instead a windowed method (e.g. Welch's method) is used instead.

My understanding of Welch's method is that you divide $A(t)$ into $K$ segments (or batches) each with length $L$ with some shift, $S$ between each segment. You may also choose to taper each segment with a Hanning window or cosine bell or something like this. Next, you compute the Fourier transform of each windowed, tapered segment which results in $L$ complex Fourier coefficients.

Each segment has a maximum frequency of $f_{MAX} = 1/(2\Delta t)$ (from Nyquist) with a total of $L$ linearly-spaced frequencies. Suppose that you do not need all $L$ Fourier coefficients at all $L$ frequencies. Suppose you only want 6 frequencies which are logarithmically-spaced (e.g. at 0.01, 0.1, 1, 10, 100 and 1000 Hz).

Now, here is where I start getting confused. How do you get the 6 Fourier coefficients at the evaluation frequencies? Various (geophysics) textbooks I have looked at seem to skim over this. One book briefly mentions something about Parzen windows and smoothing. The other seems to skip suddenly from a frequency-space with several hundred linearily-spaced frequencies to one with only 10 without any description of how these are pulled from the spectrum.

I don't think it is possible to just pull the values directly from the spectrum, is it? You must have to do some sort of smoothing or averaging over multiple frequencies (e.g. using a Parzen window)?

Question: How do you apply a Parzen window to extract a specific evaluation frequency from a spectrum? Is this the correct method or is there something better or more well-suited?


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