> [!tldr] > The **Nyquist frequency** of a discrete time series $(X_{t})$ with interval $\Delta t$ between each observation is given by $\omega_{N}:= \pi / \Delta t$. It is the highest frequency at which a signal can be captured by the time series. > > If the time series has length $N$, the **fundamental (Fourier) frequency** is given by $\omega_{1}:= 2\pi / (N \Delta t)$, which is the lowest frequency at which periodic signals can be identified in the time series. Note that the Nyquist frequency is determined by the measurement interval $\Delta t$ only, and the fundamental frequency by the length $N$ only. - With too large a $\Delta t$, high-frequency signals might complete cycles before being measured again. - With too small an $N$, it's impossible to tell long cycles apart from a global trend. - For example, a daily temperature measurement has $\Delta t=1$, so the Nyquist frequency is $\pi$, and the time series fails to capture the intraday temperature patterns, which have frequency $2\pi$. - If there are $364$ data points, then the fundamental frequency is $2\pi / 364$, would fail to capture the yearly patterns of frequency $2\pi / 365$. A similar notion is a lower bound on the frequency of signals that the time series can capture, namely the **fundamental Fourier frequency** $\omega_{1}=2\pi / (N \Delta t)$, where $N$ is the length of the (realized) time series. - For example, with only $6$ months of data, we can't tell if winter is colder than summer because of a seasonal pattern, or because of a permanent downward trend. Therefore it is common to evaluate the Fourier representation at the **harmonics** of the fundamental frequency, i.e.: $\omega_{p}:= p\omega_{0}=\frac{2p\pi }{N \Delta t},\,p=1,\dots,N.$As a result, it's helpful to select $N$ so that the harmonics contain interpretable/interesting frequencies: e.g. for weekly data, choosing $N=52$ allows $\omega_{4}$ to reflect monthly patterns, and $\omega_{13}$ quarterly ones.