Tutorial: Sub-Nyquist artifacts and sampling moiré effects

Sampling moiré effects are well known in signal processing. They occur when a continuous periodic signal g(x) is sampled using a sampling frequency fs that does not respect the Nyquist condition, and the signal frequency f folds-over and gives a new, false low frequency in the sampled signal. However, some visible beating artifacts may also occur in the sampled signal when g(x) is sampled using a sampling frequency fs which fully respects the Nyquist condition. We call these phenomena sub-Nyquist artifacts. Although these beating effects have already been reported in the literature, their detailed mathematical behaviour is not widely known. In this paper we study the behaviour of these phenomena and compare it with analogous results from the moiré theory. We show that both sampling moirés and sub-Nyquist artifacts obey the same basic mathematical rules, in spite of the differences between them. This leads us to a unified approach that explains all of these phenomena, and puts them under the same roof. In particular, it turns out that all of these phenomena occur when the signal frequency f and the sampling frequency fs satisfy f ≈ (m/n)fs with integer m,n, where m/n is a reduced integer ratio; cases with n = 1 correspond to true sampling moiré effects.