Aliasing tests and connections to superoscillation

Suppose we decide to observe a signal by sampling it at a fixed sampling rate. The sampling rate determines the maximum frequency of signal that can be observed in the time series. This maximum frequency is called the Nyquist rate. Sometimes signals contain content at higher frequencies than the Nyquist rate. Such content cannot be represented by the sampled signal and then the phenomenon of aliasing occurs. Aliasing is when the higher frequency information reappears in the sampled signal at lower frequencies that are `folds' of the higher frequency, and are below the Nyquist rate. One can also look at this area from a very practical point of view. If one is given a time series at a fixed sampled frequency, then one usually assumes that no aliasing has occurred and information content in the sampled time series is an accurate reflection of what went on in the signal. However, this assumption is never tested and this is partly because, for stationary time series, it is essentially impossible to test for. This is important as an aliased series might not reflect the true signal at all and lead to inaccurate estimates of, e.g., autocovariance, the spectrum and poor forecasts and, moreover, something that extra data would not improve. However, it is beginning to look like, for non-stationary time series, represented by a wavelet process (but, strangely, probably not for a time varying Fourier process) it is possible to detect probable aliasing and this detection becomes stronger if one is also tracking frequencies in the signal. Recent work in this area can be found in Eckley and Nason (2014, 2018). There also seems to be a connection between the Eckley and Nason work and the concept of superoscillation in physics, see Berry (1994). EPSRC research area: Statistics, Signal Processing, Mathematical Physics