On the Relation Between Fourier Frequency and Period for Discrete Signals, and Series of Discrete Periodic Complex Exponentials

Discrete complex exponentials are almost periodic signals, not always periodic; when periodic, the frequency determines the period, but not viceversa, the period being a chaotic function of the frequency, expressible in terms of
Thomae's function
. The absolute value of the frequency is an increasing function of the subadditive functional of
average variation
. For discrete signals that are either sums or series of periodic complex exponentials, the decomposition into their periodic, additive components allows for their
filtering according to period
. Likewise, their
period-frequency spectrum
makes predictable the effects on period of convolution filtering. Ramanujan-Fourier series are a particular case of the signal class of
series of periodic complex exponentials
, a broad class of signals on which a transform, discrete both in time and in frequency, called the
DFDT Transform
, is defined.