Figure 1. Sampled signal is approximated by linear
interpolation. (The periodic signal is represented by the blue curve,
the sampled values are red circles, and the linearly interpolated signal
is the red dashed line).
Assume that the periodic signal with a period of \(T\) (equivalently can
be described by its frequency, \(\omega=1/T\) ) is sampled regularly
by a constant sampling rate, \(T=1/f_{s}\), where \(f_{s}\) is the
sampling frequency. The dimension of the signal may range at different
scales, even the content of the signal can be different, e.g. time in
seconds, hours, years, ages or length in mm, m, km, AU. Therefore,
instead of discussing actual reliable scales and magnitudes, in the
present analysis the periodic function is characterized by its
amplitude, \(A\) and frequency, \(\omega\), while the sampling is
defined by the sampling rate, \(T\), or equivalently by the ratio of the
frequency of the signal, \(\omega\) and the observation, \(f_{s}\)
\(N=\frac{\ \omega}{f_{s}}=\frac{T}{T}\) (2).
The amplitude, \(A\) is set arbitrarily to a unit, and error estimation
due to the sampling is performed by considering two parameters: the
frequency, \(\omega\) of the signal and the ratio \(N\). The error
estimates are provided in percentage of the amplitude, \(A\). Beyond\(A\), \(\omega\) and \(N\), also the phase of the signal, \(\phi\) is
used for defining the periodic signal, however the calculus later is
performed independently of this variable. All in all, the periodic
function is defined as
\(f(x)=Asin(2\pi\omega\ x+\phi)\) (3),
where x is the independent variable. The sampling affects the
knowledge of the observed signal between the sampling epochs. Therefore,
sampling error is modelled here as the difference of real and the
(linearly) interpolated values. According to Figure 2, when the function
value at an arbitrary epoch, xk falling into the
interval of [xi ,xi+1 ], the error due to the sampling becomes
the difference of the real, f(xk) and the
interpolated, fint(xk) function
values, \(\varepsilon_{k}\).