The Fourier Remodel, defined in a single sentence (Revolutions)
If, like me, you struggled to know the Fourier Transformation while you first realized about it, this succinct one-sentence colour-coded clarification from Stuart Riffle in all probability comes a number of years too late:
Stuart offers a more detailed explanation here. That is the method for the Discrete Fourier Remodel, which converts sampled alerts (like a digital sound recording) into the frequency area (what tones are represented within the sound, and at what energies?). It is the mathematical engine behind quite a lot of the expertise you employ at the moment, together with mp3 information, file compression, and even how your previous AM radio stays in tune.
The daunting method entails imaginary numbers and complicated summations, however Stuart’s concept is easy. Think about an unlimited speaker, mounted on a pole, enjoying a repeating sound. The speaker is so giant, you’ll be able to see the cone transfer forwards and backwards with the sound. Mark some extent on the cone, and now rotate the pole. Hint the purpose from an above-ground view, if the ensuing squiggly curve is off-center, then there’s frequency corresponding the pole’s rotational frequency is represented within the sound. This animated illustration (click on to see it in motion) illustrates the method:
The higher sign is make up of three frequencies (“notes”), however solely the bottom-right squiggle is generated by a rotational frequency matching one of many part frequencies of the sign.
By the way in which, no-one makes use of that method to really calculate the Discrete Fourier Remodel — use the Quick Fourier Remodel as a substitute, as applied by the fft perform in R. Because the identify suggests, it is a lot quicker.
AltDevBlog: Understanding the Fourier Transform (notice: up to date hyperlink 20 Oct 2015 with lively mirror)