Rule 29 of 36 · Chapter IV — Fields & Waves

Frequency and time are one view seen twice

Why this rule exists

Any signal can be described equally well as a function of time or as a sum of sinusoids of different frequencies; the Fourier transform moves between these two views without losing information. They are not two phenomena but one object seen from two sides. This works because sinusoids form a complete basis, so any well-behaved waveform is a unique superposition of them.

In practice

Choose the domain that makes the problem simple. Filtering, resonance, and periodicity are transparent in frequency; transients and timing are clearer in time. To move over, take the Fourier transform; remember that a feature localized in one domain is spread in the other — a sharp pulse contains a broad band of frequencies, and a pure tone lasts forever.

When it doesn't apply

The clean picture assumes a linear, time-invariant system. For signals whose spectrum changes over time, a single global transform hides that; use a windowed or wavelet transform, bounded by the time–frequency resolution trade-off.