16 The frequency domain

Everything so far has described a series by how each value relates to the ones before it. The frequency domain describes the same series by which cycles it is made of. Neither view adds information the other lacks (they are Fourier transforms of one another) but each makes different things obvious, and periodicity is far easier to see as a peak than as a pattern in the autocorrelations. A cycle with angular frequency \(\omega \) (radians per time unit) has period \(2\pi /\omega \): frequency \(\omega = \pi /2\) is a cycle every \(4\) observations, and \(\omega = \pi \), the highest frequency observable in discrete data, is alternation from one observation to the next.

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