Signals and Fourier analysis

Fourier's result is that any reasonable signal can be written as a sum of sinusoids of different frequencies, amplitudes and phases. The list of those components is the signal's spectrum, and the key point is that it is not a summary: the transform is invertible, so the spectrum and the time trace contain exactly the same information in two different presentations. Choosing which one to work in is a matter of convenience, and problems that are awkward in time — filtering out a hum, identifying a periodicity — are often trivial in frequency.

The two domains are linked by a reciprocal relationship that governs much of practice. A signal confined to a short time must be built from a wide range of frequencies, and a signal with a sharply defined frequency must last a long time. This is why a brief laser pulse is spectrally broad, why a musical note must be held to have a definite pitch, and — in the same mathematics — why position and momentum obey an uncertainty relation.

Sampling is where this becomes an engineering constraint. Recording a continuous signal as discrete measurements can be lossless, but only if the sampling rate exceeds twice the highest frequency present: the Nyquist criterion. Sample any slower and high frequencies masquerade as low ones — aliasing — which is irreversible once it has happened, since nothing in the recorded data distinguishes the impostor from the real thing. This is why audio is sampled at 44.1 kHz for a 20 kHz hearing range, and why an anti-aliasing filter must remove the too-high frequencies before the sampler, not after.