As an engineer, I understand the Fourier Series and all that stuff. And amongst my peers, it’s “common knowledge” that a sine wave is the most “pure” or “fundamental” wave. Musical instruments are a perfect example; the sound of each instrument is nothing more than the fundamental frequency + harmonics of various amplitudes and phases. Each component is a sine wave.
But I have to wonder… can any wave be chosen as the “fundamental” wave? Could you base all Fourier Analysis on any arbitrary wave shape, or is there something inherently special or natural about the sine wave?
It’s been years since I studied this, but isn’t the fundamental supposed to be the one with the greatest amplitude and the harmonics are of lesser and lesser amplitude as they get higher in pitch?
Amplitude of what? A sine wave? If you say “yes,” then aren’t you implying that a sine wave must be the “fundamental” wave?
My point is this: we naturally think of complex waveforms as being made of pure sine waves. Why is that? Couldn’t I invent my own waveform (something that kind of looks like a sine wave, but isn’t really a sine wave), and use it as a “fundamental” wave?
Didn’t you do this in your studies? I remember that pretty much any periodic signal can be used in much the same wave as a sine wave for this sort of analysis.
They are useful in signal processing because they move through linear filters undistorted. A triangle wave going through a generic linear filter is no longer a triangle wave.
Yes, you may expand periodic functions in terms of other sets of functions, but not just any old set you dream up. You want a complete orthogonal set. Look up the “generalized” Fourier series. Rather than try to describe it, I’ll just refer you to this article:
It mentions the alternate example of the Laplace series, and one link away, you will find it mentions the Fourier-Legendre series (using Legendre polynomials) and the Fourier-Bessel series (using Bessel functions).
Also, a sine or cosine wave is simply a relationship of the angle and radius of a point on a unit circle. If the angle changes at a constant rate, this is what we define as a sine wave.
These two qualities, constant angular velocity, and constant radius, are common to many basic harmonic functions, and the math is very simple. Thus, sine is a natural choice to use as a fundamental waveform.
You could base a system of math or wave analysis on some other non-trivial waveform, but you’d only be needlessly complicating things.
Similarly, you could write financial software that computes everything in base 7 instead of base 10 arithmetic, but it doesnt mean you should.
Yes. Basically, if an object is moving in a circle with constant speed, its x-coordinate and its y-coordinate can both be described by a sine wave, or a variation on it (such as y=3sin(t) or y = 0.75sin(t-pi/2))
But many other functions are faster to calculate. Computers don’t care about the weirdness of the formula, just how long does it take to get a certain number of bits. My long time favs are Chebyshev Polynomials which do a much better job than Sine/Cosine in interpolation for a given computational effort.
There’s also wavelet analysis, where (iirc, it’s been a while since I’ve read up on it) you can define your own base functions (within a few rules). It’s more useful for noisy data like the crash of a cymbal.
