Cellular Automata and Fractal Evolution

1/f Noise - Power Spectrum

By combining simple sine waves of appropriate frequencies and amplitudes, electronic music and voice synthesizers can produce very complex sounds.
This process is called Fourier synthesis, after Joseph Fourier, the French mathematician and physicist who did much of the foundational work on this topic, especially as it relates to the flow of heat.
The reverse process is Fourier analysis, taking a complex sound and decomposing it into the sine waves of which it is composed.
An analog version of this is easy to produce, if you happen to be standing near a piano.
Make some loud noise - clap your hands, for instance - and you'll see some of the piano strings begin to vibrate.
Those are the piano strings whose natural frequencies are part of the sound you made.
The power spectrum is a representation of this decomposition.
First, for each sine component of frequency f, determine the amplitude A(f) of that component.
The power P(f) at frequency f is P(f) = A(f)2.
The power spectrum is a plot of P(f) as a function of f

Here are three simple examples. The power spectra are shown on the right.
A single sine wave
A single sine wave with twice the frequency and half the amplitude of the first
The sum of the first and second sine waves. Click the picture for an animation.

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