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.