Fourier Transforms

I was surprised to see such a long article on use and miuse of the Discrete Fourier Transform (DFT) with only brief and casual mention of "sidelobes" -- there wasn't formal discussion of the frequency response of the DFT window function and how it relates to transform output. The latter is typically represented evaluating the z-transform of the window function along the unit circle. While steeped in this mathematics the attendant visualizations can be helpful to impart an intuitive understanding of what the DFT is doing with its inputs. While I would recommend a text that covers the topic such as "A Digital Signal Processing Primer" by Ken Steiglitz, Gemini was able to show the response graph of a DFT window function with the following prompt: "what is the z transform of the hamming window in a discrete fourier transform context"

> This is done by choosing A and B such that the following integral is minimized

Which is an absolutely subjective choice in an of itself and immediately breaks the notion that curve-fitting done that way is going to be telling you some absolute truth about the function.

For example, you might want, in each point of the non-linear curve being fitted, throw a line perpendicular to its tangent, compute the distance to the linear fit, and sum those distances over all points of the non-linear curve.

About as intuitively correct as the "fit" proposed, yet yields a very different linear fit.

Statistics are by definition subjective unless you use a specifically demonstrated property of the particular way you decide to project your data to the simple-minded underlying statistical model.

Closest peripheral aspect is transforming integrals according to Fourier, which extricates 1 cycle in f(x)=sin(x). A synchronization may take place in transient sine coefficient, where "Mag" and lag column specifies polar coordinates.

> Because an FFT (short for "Fast Fourier Transform") is nothing more than a curve-fit of sines and cosines to some given data

That is not even wrong. A Fourier transform is a basis expansion. In particular, the full expansion is exact (not just an approximation). Of course, truncated expansions are approximations.

The actually interesting part: Why is this basis expansion so much more useful than, e.g. expanding into some eigenfunctions, Hermite polynomials, etc.? The decomposition into (complex) exponentials converts between addition and multiplication, i. e. sin(x+y), cos(x+y) you get from multiplying sin(x), cos(x), sin(y) and cos(y). This in turn has important implications such as turning derivatives into multipliers. More generally you can consider nonlinear Fourier transforms with different groups and generators other than exponentials.

TLDR: It is a transform. What you are transforming between is what makes it so useful.

Closest peripheral aspect is transforming integrals according to Fourier, which extricates 1 cycle in f(x)=sin(x). A synchronization may take place in transient sine coefficient, where "Mag" and lag column specify individual polar coordinates where the regression curve is hydrostatic.

I appreciate that others may not have got much out of this, but let me say that as someone who knows about statistical modeling, but has no physics training, this made Fourier transforms very clear.

There was another comment that referred to why we use this orthonormal basis versus another, and I think to appreciate the full reason of why this was done in the first place is important. But this presentation is a very good introduction for someone with my particular training.

FFT was such a revelation to me when I was learning about the digital and analogue interaction. The Wikipedia page is really quite about it: https://en.wikipedia.org/wiki/Fourier_series#Fourier_theorem...

I love the visualizations on that page. There were some other cool interactive visualizatiosn on bl.ocks.org, but sadly, that site has be shattered. This is the closest I could find: https://observablehq.com/@drio/visualizing-the-fourier-serie...