Introduction
Data acquisition for geoinformatics contains collection of spatially and/or temporally changing features, which are continuous by nature. No data acquisition can be done continuously, but by discrete sampling of the phenomenon. Discrete sampling is often then considered to be uninterrupted sequences of continuous data, which results in an underestimation of the actual signal.
Shannon’s proof (Shannon 1949, 1998) on the sampling theorem illustrates elegantly the feasibility of recovering: while sampling of a band-limited signal means a multiplication of the continuous signal with Dirac impulses in the time domain, in the frequency domain (due to the corresponding convolution) it yields repetition of the spectrum of the original function. Therefore, theoretically the original function can be recovered perfectly by filtering the sampled signals spectrum to the original bands, which means a multiplication in the frequency domain with a boxcar window of the proper size (see e.g. Chapter 3 of Marks 1991). This approach is, however, too idealized, as it assumes a band limited signal, while real signals are never exactly bandlimited, but there always an aliasing is observed. Also, in practice no ideal anti-aliasing low-pass filters can be constructed (Unser 2000). (For details on related issues of the sampling theory, the reader is referred to Marks (1991, 1993) and Unser (2000)).
In fact, in practice Shannon’s formulation is rarely used, as sinc function (the time domain equivalent of the boxcar filtering) decays slowly. For the practice, in case of an ‘appropriately’ sampled signal, intermediate values are assumed to be determined with ‘appropriate’ accuracy by linear interpolation. (Accordingly, the present study also assumes that the sampled signal is approximated by linear interpolation). Still, the relevance of Shannon’s theorem is essential, as (theoretically) any signal can be interpreted as infinite number of periodic signals as Fourier transformation provides such a decomposition.
\(f(t)=\sum_{c=0}^{\infty}{A_{c}sin(2\pi f_{c}t+\phi_{c})}\) (1)
Certainly, in practice no full equivalence of the original and the Fourier transform signals can be achieved as the transformation can make use of only finite numbers of frequencies, and also due to numerical limitations. Following, however, the idealization of Shannon, in this study, the effect of sampling is discussed for a discretization procedure with no error, and assuming no observation errors as well. Also, implicitly we assume the feasibility of (1), thus the discussion focus on periodic signals only. Note also, that in the practice of geoinformatics and Remote Sensing the discretization is often obtained by averaging over a finite segment of the data, e.g. Digital Terrain Models (DTM) are determined based on several observations referring to a certain pixel, resulting in aliasing the point-wise data by the block averaging (Földváry 2015). This study assumes perfectly point-wisely sampled data.
Nowadays, sampling algorithms are generalizing the Shannonian approach: instead of constraining the signal into a limited bandwidth before recovering it, the proper band-limited filter can be achieved by methods applying orthogonal, frequency dependent base functions, such as wavelets (Strang-Nguyen 1996; Mallat 1998), finite elements (Strang 1971; Selesnick 1999), frames (Duffin-Schaeffer 1952; Benedetto 1992), among others.
In summary, the aim of the study is to provide a mathematical tool for sampling error estimation. The sampling, by its discrete characteristics involves errors on the knowledge of the real continuous phenomenon. Without understanding the limitations of the discretization, the observed phenomenon may be interpreted falsely. In the present study, analytical formulation of the sampling error is to be provided, which embodies the characteristics of the sampling error by determining its amplitude, phase, bias and periodicity. Such information can be used for planning optimal resolution of sampling a process but cannot be used (and it is not the scope of this study) for reconstructing the original signal.
Formulation
Figure 1 displays an example of a periodic signal, which is sampled with a certain sampling period. Let \(T\) refer to the period of the signal, and \(T\) to the sampling period. If the sampling period is sufficiently fine, then the signal can efficiently be approximated by assuming linearity between two consecutive points. In practice, for sake of simplicity, such a linearity is assumed; accordingly, linear interpolation is used for approximating inner values of the signal.