**3. Piezoelectric Diaphragms**

The piezoelectric diaphragms have a very simple construction and are available in different sizes. This type of transducer consists of a circular piezoelectric ceramic (active element typically of barium titanate or PZT), usually ranging from 0.1 to 2 mm in thickness, mounted on a circular metal plate (diaphragm available in brass, nickel alloy, or stainless steel). The ceramic is coated with a thin metallic film (usually silver) that serves as an electrode. Figure 1 shows a typical PZT diaphragm and its parts [18]. Piezoelectric transducers can operate both as sensors and actuators due to the piezoelectric effect [46]. The piezoelectric effect consists of the generation of an electric dipole in a material that is subjected to a mechanical force, which results in an electrical output voltage. The polarization produced by the voltage creates charges and therefore an electric field. In the reverse effect, the application of an electric voltage in the piezoelectric material causes a mechanical deformation [47].

**Figure 1.** Murata piezoelectric sensor 7BB-35-3 used in the tests.

According to Castro et al. [22], in piezoelectric material, there is an electromechanical coupling, i.e., an electric field applied to the material generates a mechanical deformation while a mechanical change generates an electric load. Thus, piezoelectric transducers are capable of generating an electrical voltage when altered by a mechanical stress, which is generated by an acoustic wave. The application of piezoelectric sensors for different purposes can be found in the specific literature [48,49].

### **4. Signal Processing**

The discrete Fourier transform (DFT) is a method for the analysis of frequency spectra in digital signal processing, usually implemented by the fast Fourier transform (FFT). However, the FFT is inadequate for identifying non-stationary transient information because it has no time resolution [50]. In this context, an alternative approach is to segment the sequence into a set of short subsequences,

with each subsequence centered on uniform time intervals and its DFT calculated separately, thereby obtaining the short-time Fourier transform (STFT) [51]. Thus, the STFT is defined by Kim et al. [52]

$$STFT\left(t,\omega\right) = \int\_{-\infty}^{\infty} h(u)f(t+u)e^{-j\omega u} \, du\tag{1}$$

where *f*(*t*) is a given signal in the time domain, *t* is the time, ω the frequency, and *h*(*u*) the temporal window function such as rectangular, Gaussian, Blackman, Hanning, Hamming, Kaiser, etc.

The result obtained by the STFT is a two-dimensional representation of the signal in time and frequency. However, the limitation imposed by Heisenberg's uncertainty principle requires a relation between the resolutions (time–frequency). The accuracy in time and frequency are mainly determined through the window length, which is constant for all frequencies. In this way, a longer window results in a better resolution in the frequency domain. However, to obtain a more accurate time resolution, a window of smaller length is used. Therefore, it is necessary to know the resolution in time and frequency to obtain all relevant information in both domains [53]. The resolution in time and frequency can be defined by Equations (2) and (3), respectively.

$$
\Delta t = N \ast T\_s \tag{2}
$$

$$
\Delta F = m \frac{F\_s}{N} \tag{3}
$$

where *N* is the window length; *Ts* the sampling period; *Fs* the sampling frequency; and *m* the window coefficient, and where the sampling period is equal to the inverse of the sampling frequency and the window coefficient depends on the type of window used, e.g., *m* = 2 for a rectangular window.

Many statistical parameters are applied to the signals collected during the monitoring of manufacturing processes. The most-used statistic is the root mean square (RMS) [54], however, other statistics are also applied to these signals and are able to diagnose events that occur in the monitored processes. In this context, the application of counts [7], DPO (power deviation) [55], and ROP [56] is highlighted.

According to Lin et al. [57], the ratio of power (ROP) is a statistical method for analyzing the ratio of a given frequency band with the total power spectrum. The ROP statistic can be defined as

$$ROP = \frac{\sum\_{k=n1}^{n2} |\mathbf{x}\_k|^2}{\sum\_{k=0}^{N-1} |\mathbf{x}\_k|^2} \tag{4}$$

where *N* is the block data size; *n*1 and *n*2 define the frequency range for the analysis; and *xk* is the *k*th discrete Fourier transform [58].

The study of similarity between two or more sensor signals, statistics, or parameters is of vital importance in the validation of new techniques. The correlation study between a widely known variable in relation to a new variable in the test phase is done through several statistics and indices. In this sense, the application of correlation coefficient [59], wavelet coherence [60], and magnitude-squared coherence [61] is highlighted

According to Scannell et al. [62], the magnitude-squared coherence (MSC) lists common frequencies between two signals and evaluates their similarity. The MSC between the time domain signals *x* and *y* can be calculated by Equation (5). Results in the interval between 0 and 1 indicate the level of spectral similarity between both time-domain signals.

$$\mathcal{C}\_{xy} = \frac{\left|P\_{xy}(f)\right|^2}{P\_{xx}(f)P\_{yy}(f)}\tag{5}$$

where *Pxx*(*f*) and *Pyy*(*f*) are the power spectral densities of *x* and *y* and *Pxy*(*f*) is the cross power spectral density at frequency *f*.
