How does Fourier analysis separate a sound wave into frequency components?

One complicated pressure trace can contain several regular oscillations at once. Fourier analysis describes suitable signals using sinusoidal components, each with a frequency, an amplitude and a phase. Frequency counts cycles per second, amplitude sets the size of the pressure variation, and phase locates the oscillation within its cycle.

The illustrated ideal sound-pressure signal is the sum of a 200-hertz sine wave with amplitude one pascal and a 400-hertz sine wave with amplitude half a pascal. Both begin at zero phase. The first completes one cycle in five milliseconds; the second completes two in the same interval. At each instant, their signed pressures add, rather than their maximum amplitudes simply becoming the waveform’s height everywhere.

The combined trace and the one-sided amplitude spectrum describe the same known periodic signal in different ways. The trace shows pressure variation over time. The spectrum places a line at each positive component frequency, with heights one and one-half pascal. Here 200 hertz is the fundamental frequency, and 400 hertz is its second harmonic. Over a complete fundamental period, products of different harmonics average to zero. This property, called orthogonality, lets Fourier analysis isolate each component’s coefficient, or weight.

Amplitude alone is not the full recipe. Reversing the second component’s phase changes the summed waveform while leaving these amplitude-spectrum heights unchanged. A full representation must retain phase information too. Real sounds may change with time, contain noise or require many components. Finite recording windows and measurement limits affect an estimated spectrum; this exact two-component model performs no audio recording, sampling experiment or instrument identification.