Analysis of Frequency Domain Data Generated by a Quartz Crystal

Background

There are different methods for frequency counting that include conventional counters, reciprocal counting, interpolating reciprocal counting, time-stamping counting, method for measurement of absolute values, method for measurement of relative values, and universal method of dependent count (Johansson, 2005; Kalisz, 2004; Kirianaki et al., 2001). In the recent years, the principle of rational approximations has been proposed for measuring the frequency of a desired signal. This method requires to compare a signal to measure with other whose frequency is known. Both signals are required to go through a signal conditioning process, where the signals are converted into streams of pulses with a rectangular shape, which appear at regular intervals defined by the frequency of original signals (Hernández Balbuena et al., 2009). These conditioned signals are compared using an AND condition, and a third signal is generated: the signal of coincident pulses. As a result, a coincidence appears where a pulse of the desired signal overlaps in time with a pulse of the reference signal. After the first coincidence, the counting of pulses in three signals starts; when there is another coincidence, an approximation to the desired frequency is obtained. As any other time-frequency measurement technique, when the measurement time increases, the accuracy of measurement increases and uncertainty decreases. The principle of rational approximations has plenty of advantages over other measurement techniques, namely: continuous measurement without dead time, uncertainty limited by the accuracy of reference frequency, approximations to desired frequency are obtained in very short time, and ease of implementation (Avalos-Gonzalez, Sergiyenko, et al., 2018; Murrieta-Rico, Sergiyenko, et al., 2016; Murrieta-Rico et al., 2017, 2018; Murrieta-Rico, Petranovskii, Galván, et al., 2021).

A continuous measurement generates a vast amount of information, and in most cases, these datasets are difficult to interpret. Among the tools available for data analysis, the principal component analysis (PCA) is a method that is widely spread, and it allows to obtain information regarding the relationship of measured variables. In the PCA, the dimensionality of input datasets is reduced, the interpretability is increased and loss of information is avoided; this is done after the creation of new and uncorrelated variables, which successively maximize variance (Jolliffe & Cadima, 2016). Although PCA has been widely developed from statistics point of view, it finds quite diverse applications. Some of them include, dimension reduction on non-Euclidean manifolds with PCA (Mardia et al., 2022), diagnosis of dental pathologies (Nouir et al., 2022), geographically and temporally weighted analysis (Han et al., 2022), etc.