Fourier Transform
Decomposing a signal into the sum of sinusoids that make it up ā turning a function of time into a function of frequency.
Faint colored waves are ingredients; the dark curve is their sum.
The Fourier transform takes a signal described as a function of time and rewrites it as a function of frequency ā revealing which sine waves, at which strengths, combine to build it. Any signal, no matter how jagged or complex, can be decomposed into (possibly infinitely many) sine and cosine waves of different frequencies and amplitudes.
This is useful because many questions are far easier to answer in the frequency domain: "what pitch is this sound?", "what part of this image is blurry detail vs. sharp edges?", "which frequencies should this filter remove?"
A musical chord played on a piano is a single, complicated pressure waveform over time. The Fourier transform decomposes it back into the individual notes (frequencies) being played ā exactly what a graphic equalizer or tuner app is doing under the hood.
If a signal is a pure 440 Hz sine wave (a single musical note, A4), what would you expect its Fourier transform to look like?
Solution
A single sharp spike at 440 Hz (and, for a real-valued signal, a mirrored spike at ā440 Hz) ā nearly all of the frequency content is concentrated at that one frequency, since the signal is just that one sine wave.
Related concepts
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