An Interactive Guide to the Fourier Transform (2012)
At the end of these types of popsci explanations you still can&#x27;t grok the math any better and you end up with a hand-wave-y misleading understanding of what&#x27;s going on.<p>It starts off by saying that the Fourier Transform &quot;measures every possible cycle&quot;. This is an <i>wrong</i> approach to thinking of the Fourier Transform. If you think in these terms you will quickly get lost. The Fourier basis is a set of sinusoids that are mutually orthogonal (ie. none can be represented as a combination of the others). They just happen to generally cover the frequency ranges you&#x27;re interested in (though this depends on your problem). If you choose some random sinusoidal signal that isn&#x27;t a harmonic of your sampling period, then the Fourier Transform gives you back a mess that&#x27;s hard to interpret (how a off-harmonic sinusoid gets &quot;smeared&quot; across the basis is actually something I frankly still don&#x27;t quite understand <a href="https:&#x2F;&#x2F;ccrma.stanford.edu&#x2F;~jos&#x2F;mdft&#x2F;Frequencies_Cracks.html" rel="nofollow">https:&#x2F;&#x2F;ccrma.stanford.edu&#x2F;~jos&#x2F;mdft&#x2F;Frequencies_Cracks.html</a> )<p>This is in contrast to the phase information that you get that <i>is</i> continuous. Each frequency has two associated basis vectors and they can give you any phase offset. The frequencies are only set harmonics, but the phase can be any value. This is the result of how the basis is setup. If anything, the phase information you get from the Fourier transform is more interesting than the frequency information you can glean.<p>The &quot;Circular Paths&quot; visualization is also highly misleading. The complex plane is a crutch that gives you pretty pictures. It pretends to be like a 2D plane but it&#x27;s actually fundamentally distinct from a Cartesian&#x2F;&quot;normal&quot; X-Y plane b&#x2F;c complex numbers can be multiplied while normal [x,y] pairs can not (b&#x2F;c the operation is simply undefined - you can&#x27;t multiply two points on a map). The product of two complex numbers is giving you another complex number.. but good luck visualizing where that point ends up! You&#x27;ll notice all complex plane examples stay on the unit circle b&#x2F;c you get a &quot;rotation&quot; but it&#x27;s an edge case. The whole 2D analogy is setting you up to be very confused