periodiciteter

Periodiciteter, or periodicities, are the properties of a quantity that repeats at regular intervals in time or space. In continuous time, a function f(t) is periodic with period T > 0 if f(t+T) = f(t) for all t. The smallest positive T is the fundamental period; the repetition frequency is f = 1/T (angular frequency ω = 2π/T). In discrete time, a sequence x[n] is periodic with period N if x[n+N] = x[n] for all n, with N the smallest positive integer that satisfies this condition.

Periodicity can be exact or approximate; many real-world signals exhibit quasi-periodic or cyclic patterns with varying or multiple periods. When multiple periods are present, a signal can be described as a sum of harmonics at integer multiples of a fundamental frequency or as a superposition of several periodic components.

Detection and analysis methods include the Fourier transform and periodograms to identify dominant frequencies, autocorrelation functions to reveal repeating lags, and Lomb-Scargle methods for unevenly sampled data. Wavelet analysis can track time-localized or changing periodicities.

Examples include a pure sine wave, which is perfectly periodic; biological signals like heartbeats or circadian rhythms, which are quasi-periodic; and seasonal environmental patterns or annual economic cycles that show yearly periods.

Applications span science and engineering: physics and astronomy (pulsations and orbital periods), biology (circadian rhythms), meteorology and economics (seasonality), engineering (signal processing), and music (rhythmic structures).

Challenges include noise, non-stationarity, irregular sampling, aliasing, and the presence of multiple interacting periods. Properly identifying periodicities supports modeling, forecasting, and understanding the processes that generate observed data.