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, \(\varepsilon_{k}\).