Fourier Analysis

Sinusoidal Functions

Fourier analysis always breaks a function down into a sum of sinusoidal functions. The reason why sinusoidal functions are used and not some other function is that the sinusoid has a unique property: it retains its shape when added to itself, provided its period (or frequency) is held constant. In other words, if two sinusoids of the same period are added, the result is always a sinusoid, regardless of relative amplitude and phase. No other periodic function has this property. For this reason, Fourier analysis is a unique solution to the problem of decomposing a complex function into a sum of simpler functions.

A pure tone is a sound whose function of air pressure against time is sinusoidal, that is, it corresponds to a sine or cosine function (also called trigonometric functions). The amplitude of the function is the maximum distance from the horizontal time axis, which in this case corresponds to ambient air pressure. In general, the greater the amplitude of a pure tone, the greater the loudness; but the loudness of a pure tone also depends strongly on its frequency (threshold of hearing, curves of equal loudness). The duration of a full cycle of the pure tone's waveform (from the time axis to the wave's crest, back to the time axis to its trough, and back to the time axis) is called its period; it is typically measured in milliseconds or ms. The number of cycles per unit time (e.g., per second) is frequency; mathematically, the period is the reciprocal of the frequency. The pitch that is perceived in a pure tone depends mainly on the frequency (the higher the frequency, the higher the pitch), but it also depends to a small extent on the amplitude: loud pure tones can have a slightly different pitch from quiet pure tones with the same frequency. Moreover, the scaling relationship between pitch and frequency is complex: doubling the frequency is not the same as doubling the perceived pitch so that it is “twice as high,” according to listeners in an experiment; nor is this relationship exactly logarithmic.

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