the Fourier transform tells you how much of each ingredient “note” (sine wave or circle) contributes to the overall wave. Here’s why Fourier’s trick is useful. Imagine you were talking to your friend over the phone and you wanted to get them to draw this squarish wave. The tedious way to do this would be to read out a long list of numbers that represent the height of the wave at every instant in time. With all these numbers, your friend could patiently stitch together the original wave. This is essentially how old audio formats like WAV files worked. But if your friend knew Fourier’s trick, you could do something pretty slick: You could just tell them a handful of numbers—the sizes of the different circles in the picture above. They can then use this circle picture to reconstruct the original wave.

2022-11-14: The nuclear origins of Fast Fourier Transforms

And this trick works even if the signal is composed of a bunch of different frequencies. If the sine waves frequency is one of the components of the signal it will correlate with the signal producing a non-0 area. And the size of this area tells you the relative amplitude of that frequency sine wave in the signal. Repeat this process for all frequencies of sine waves and you get the frequency spectrum. Essentially which frequencies are present and in what proportions. If the signal is a cosine wave, then even if you multiply it by a sine wave of the exact same frequency, the area under the curve will be 0. For each frequency, we need to multiply by a sine wave and a cosine wave and find the amplitudes for each. The ratio of these amplitudes indicates the phase of the signal that is how much it’s shifted to the left or to the right. You can use Euler’s formula so you only need to multiply your signal by one exponential term. Then the real part of the sum is the cosine amplitude and the imaginary part is the sine amplitude.