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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.

(2.13)

Figure 2.18a shows a vibration signal collected from a practical rotary spindle under the speed of 2,000 rotations per minute (rpm) with the sampling rate being 2,048 Hz. Figure 2.18b illustrates that there are three major peaks at 33.3, 66.6, and 99.9 Hz on the frequency spectrum, which correctly represent the fundamental frequency, the second harmonic frequency, and the third harmonic frequency, respectively. Some unimportant frequency components with relatively small amplitude (usually the noises) among three peaks are hard to observe in the time domain, but they are very clear to be detected and can be ignored.

Figure 2.18 A vibration signal: (a) in time‐domain; and (b) in FFT spectrum.

The set of FFT‐based SFs SF_FFT(q) can be extracted from the summation of FFT[n] values close to the qth certain frequency band delimited by a lower frequency and an upper frequency of critical characteristics, as expressed in (2.14).

(2.14)

where q=1, 2, …, Q and

 ufqqth upper frequency of the critical characteristics; and

 lfqqth lower frequency of the critical characteristics.

For stationary signals, FFT provides a good description in global frequency bandwidth without indicating the happening time of a particular frequency component and whether the resolution scale in both time and frequency domains are enough or not.

However, FFT might be limited to processing stationary signals. A highly non‐stationary signal cannot be adequately described in the frequency domain by FFT, since its frequency characteristics dynamically change over time. Thus, extracting other SFs in the time–frequency domain is necessary.