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

Abstract

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.