| Electronic music synthesizers are quite common now. |
| By combining simple sine waves of appropriate frequencies and amplitudes, very complex sounds can be produced. |
| 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 that make it up. |
| 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 frequency f sine component in the signal, determine the amplitude A(f) of the frequency f sine 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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