Fourier Transforms

Summary

Introduction "Lies, Damn Lies, and Statistics." Few people appreciate statistics. But at least they seem OK with this and don't go off starting religious wars over the subject. Alas, such is not the case with Fourier transforms. In 30 years of working in acoustics, dynamic material testing, and vibrations, I have never, ever seen any topic provoke more apocalyptic levels of debate and outrage among technical types than the Fourier transform. This tool must get misused and misinterpreted by more people than anything I've ever seen. These poor souls are ready to fight to the death over their erroneous conclusions. They don't want to hear that they are adding 2 + 2 and getting 5. They don't want to see that 2 real live apples added to 2 more apples doesn't give 5. They've somehow got 5, and are absolutely certain they are correct. So herewith, my attempt to save their immortal souls.... After a brief detour into the areas of background and motivation, we begin the first stage of exorcism with.... regressions. Background Fourier transforms and spectral analysis are motivated by the fact that deformations in a tire are periodic, repeating with each and every rotation. But while the deformations are periodic, they are not simple sinusoids. They are pulses in which all the action takes place during the short period of contact with the road. This is evident in the figure below, which shows the principal strains, as well as max and secondary shear strains for a tread block rolling through contact. It is clear that these traces are not straight-forward sinusoids. Therefore, Fourier analysis of the data is not trivial. And therein lies the challenge. Regression Why talk about regression and curve-fitting on a page devoted to FFTs? Because an FFT (short for "Fast Fourier Transform") is nothing more than a curve-fit of sines and cosines to some given data. Nothing more, nothing less. This simple fact is important because people want to entrust FFTs with metaphysical abilities to...