Aliasing tests and connections to superoscillation

假设我们决定通过以固定的采样率对信号进行采样来观察信号。采样率决定了在时间序列中可以观察到的信号的最大频率。这一最大频率称为奈奎斯特速率。有时信号包含比奈奎斯特速率更高的频率的内容。这样的内容不能用采样信号来表示，于是就会出现混叠现象。混叠是当高频信息在采样信号中以低于奈奎斯特速率的较低频率重新出现时，该较低频率是较高频率的“倍”。人们也可以从非常实际的角度来看待这一领域。如果一个人被给予一个固定采样频率的时间序列，那么人们通常认为没有发生混叠，并且采样时间序列中的信息内容准确地反映了信号中发生的事情。然而，这一假设从未得到检验，这在一定程度上是因为，对于平稳的时间序列，基本上不可能对其进行检验。这一点很重要，因为混叠序列可能根本不能反映真实信号，并导致不准确的估计，例如，自协方差、频谱和糟糕的预测，而且，额外的数据不会改善某些东西。然而，对于由小波过程表示的非平稳时间序列(但奇怪的是，对于时变傅立叶过程可能不是)，它开始看起来像是可以检测到可能的混叠，并且如果还跟踪信号中的频率，则该检测变得更强。最近在这方面的工作可以在埃克利和纳森(2014,2018)中找到。在埃克利和纳森的工作和物理学中的超振荡概念之间似乎也有联系，见Berry(1994)。EPSRC研究领域：统计学、信号处理、数学物理

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