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.

Keywords

• Almost periodic sequence
• Ramanujan sums
• Ramanujan-Fourier series
• Thomae's function
• variation
• period-frequency relation