2 Nyquist frequency for irregular sampling

Let f(t) be a signal function of time t. We suppose that the power spectrum of the function f is not variable in time (it should be noted that this hypothesis is stronger than stationarity). This is the case for periodic functions. Otherwise, the Nyquist frequency is only defined very locally and loses its sense for a given irregular sampling. Moreover, a spectral analysis of an unstable signal which is undersampled may lead to wrong conclusions.

If f is sampled at time t_i we are left with (f_i,t_i) $i=1,~\ldots,N$, where N is the total number of measurements. In the frequency domain, Deeming (1975, 1976) showed how the irregular discretization affects the power spectrum $F_{N}(\nu)$: the spectrum results from the convolution between the real spectrum $F(\nu)$ and the spectral window $G_{N}(\nu)$. $G_{N}(\nu)$ is the Fourier transform of $g(t)=\sum \delta(t-t_{i})$. That is $F_{N}(\nu) = F(\nu) \otimes G_{N}(\nu)$.

We can state the result as follows. Let p be the largest value such that $\forall \, t_{i}$,$\,t_{i}= t_{1} + n_{i} p \:$, where $n_{i} \in \Bbb{N}$. p is a kind of greatest common divisor (gcd) for all (t_i-t₁). The Nyquist frequency then is:

$$\nu_{\mbox{\tiny Ny}}=\frac{1}{2p} \geq \frac{1}{2s}\cdot$$ (1)

In most practical cases $p \ll s$, therefore, even with a strong but random under-sampling, very high frequencies can be detected.