selected here to demonstrate how to choose essential and concise SFs. By the same token, the essential and concise SFs of other sensors, such as dynamometers, acoustic‐emission sensors, and thermometers, can also be identified. In summary, the six vibration SFs selected include: max, RMS, avg, skew, kurt, and std; while the four current SFs chosen are RMS, avg, max, and CF.
Cross‐Correlation SFs
Generally, the Pearson product‐moment correlation coefficient [13] measures the similarity degree between two signals in the same time without any time lag. Given that x(t) and y(t) are two continuous‐time signals with time T, and their correlation coefficient value crxy, ranging from negative one to positive one, is defined in (2.7).
The cross‐correlation is similar to correlation coefficient, but it takes time lag into consideration. One signal is allowed to be time‐shifted and slide over the other to compare the similarity of two independent signals at each stride. It helps to find out where the two waveforms match the best at a certain time.
For random signals, the cross‐correlation CRxy between x(t) and y(t) is expressed in (2.8):
where γxy is the covariance expressed in (2.9):
Thus, cross‐correlation can be simplified by the ratio in (2.10):
For deterministic signals, the cross‐correlation of two continuous and periodic signals x(t) and y(t) can be defined by the integration from +∞ to −∞ as in (2.11), where notation * denotes the complex conjugate; x(t) is fixed and y(t) is shifted forward/backward by m, which is the displacement, or the so‐called lag.
For discrete signals x[n] and y[n] with length N, the cross‐correlation is defined as in (2.12), the products of two signals and the integration are replaced by any interval of period T at each point.
Cross‐correlation repeats to successively slide one signal along the x‐axis and compare with the other until the maximized correlation value is found. The reason is that two signals with the same sign (both positive or negative) tend to have a large correlation. Especially, when both peaks or troughs are aligned, it must be the best correlation. On the other hand, when signals have opposing signs at a certain time, its correlation or integral area must be small.
Cross‐correlation is very useful in the pattern recognition within a signal or between two signals. It is widely used to check the stability of sensor data and remove noise in a mass production environment. Note that, each CRxy can serve as a critical SF in a set, which can be expressed as SFCR(xy) = CRxy.
Autocorrelation SFs
Autocorrelation, or the so‐called serial correlation, performs the same cross‐correlation procedure of a signal with the time‐shifted form of itself. Thus, all autocorrelation has to do is to replace y(t) with x(t) from (2.8) to (2.12). Note that, the maximized value of the autocorrelated signal always exists at the displacement zero, which means that two signals are totally overlapped. Autocorrelation is widely used in signal processing for recognizing some repeating patterns, such as detecting the missing frequencies or presence of critical information in a periodic signal.
2.3.3.2 Frequency Domain
Frequency‐domain SFs can reflect the signal’s power distribution over a range of frequencies. Theoretically, periodic signals are composed of many sinusoidal signals with different frequencies, such as the triangle signal, which is actually composed of infinite sinusoidal signal (fundamental and odd harmonics frequencies).
Thus, the frequency spectrum of the periodic signal can be obtained by the projection of these sine and cosine signals in the frequency axis by the Fourier transform (FT) technique [10], which is probably the most widely used method for signal processing. Since then, a signal can be represented by the spectrum of frequency components in the frequency domain.
As the conversion of time and frequency domain shown in Figure 2.17, one time‐domain signal composed of two different waveforms with frequency is converted into the frequency domain. Two magnitudes of corresponding sine or cosine signals are represented at the specific location on the frequency spectrum.
Figure 2.17 View of the time and frequency domains.
One drawback is that the calculation and execution are very time‐consuming when dealing with a large amount of datasets. Thus, fast Fourier transform (FFT) based on FT is implemented to deal with nonperiodic functions and discrete time‐domain signals [10]. FFT can reduce the complexity of computing FT and rapidly compute the global information of the frequency distribution from any signal. The famous mathematician Gilbert Strang also described that FFT is “the most important numerical algorithm of our lifetime” in 1994 [14].
FFT directly decomposes any discrete signal x[n] into the frequency spectrum by the orthogonal trigonometric basis functions as in (2.13), where l = 1, 2, …, N.