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

• Main Authors: Alfredo Restrepo, Julian Quiroga, Jairo Hurtado
• Format: Article
• Language: English
• Published: IEEE 2021-01-01
• Series: IEEE Open Journal of Signal Processing
• Subjects: Almost periodic sequence; Ramanujan sums; Ramanujan-Fourier series; Thomae's function; variation; period-frequency relation
• Online Access: https://ieeexplore.ieee.org/document/9373991/

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 <italic>Thomae&#x0027;s function</italic>. The absolute value of the frequency is an increasing function of the subadditive functional of <italic>average variation</italic>. For discrete signals that are either sums or series of periodic complex exponentials, the decomposition into their periodic, additive components allows for their <italic>filtering according to period</italic>. Likewise, their <italic>period-frequency spectrum</italic> makes predictable the effects on period of convolution filtering. Ramanujan-Fourier series are a particular case of the signal class of <italic>series of periodic complex exponentials</italic>, a broad class of signals on which a transform, discrete both in time and in frequency, called the <italic>DFDT Transform</italic>, is defined.