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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

      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.

      2.3.3.3 Time–Frequency Domain

      The time‐frequency analysis describes a nonstationary signal in both the time and frequency domains simultaneously, using various time‐frequency representations. The advantage is the ability to focus on local details compared to other traditional frequency‐domain techniques.

      Although short‐time Fourier transform (STFT) method is proposed to retrieve both frequency and time information from a signal afterward, the deficiency is still yet to be overcome completely. STFT calculates FT components of a fixed time‐length window, which slides over the original signal along the time axis.

STFT adopts an unchanged resolution in both time and frequency domains, as shown in Figure 2.19. Heisenberg uncertainty principle [15] states that it is impractical to use good resolution in both time and frequency axes since the product of the two axes is a constant. A longer window has better time resolution but worse frequency resolution, and vice versa. In general, nonstationary components often appear in high frequency and only happen in a very short period of time, but this unchanged window length makes resolution in high frequency unclear.

Figure 2.19 Unchanged resolution of STFT time‐frequency plane.

      One representative technique to solve the FT‐related issues is the wavelet packet transform (WPT) decomposition [10, 11, 16]. WPT not only dynamically changes resolutions both in time and frequency scales but also has more options to change its convolution function depending on characteristics of the signal.

In regards to the resolution of Figure 2.20, it is assumed that low frequencies last for the entire duration of the signal, whereas high frequencies appear from time to time as short bursts. This is often the case in practical applications.

Figure 2.20 Dynamic window of WPT time‐frequency plane.

      In this section, WPT serves as the major time‐frequency analysis method to extract useful SFs for various machinery applications. WPT is a generalization of DWT to provide a richer information and it can be implemented by DWT‐based MRA as introduced in Section 2.3.2.2.

      As illustrated in Figure 2.16, although DWT provides flexible time‐frequency resolution, it suffers from a relatively low resolution in the high‐frequency region since only the approximation coefficients can be sent to the next level and split into approximation and detail coefficients repeatedly. Thus, some transient elements existing in the high‐frequency region are difficult to be captured and differentiated. By these procedures, any detail and approximation of the signal can be obtained at each resolution level depending on the analysis requirements.

Figure 2.21 illustrates a WPT‐based fully binary decomposition tree. In WPT, the decomposition occurs in both approximation and detail coefficients. Then the same signal as illustrated in Figure 2.16 can be successively decomposed into different levels using a series low‐pass g[k] (scaling function) and high‐pass h[k] (wavelet function) filters that divide spectrums into one low‐frequency band and one high‐frequency band, which can be represented by approximation (gray blocks) and detail (white blocks) coefficients, respectively.

Figure 2.21 WPT decomposition binary tree.

      Note that, even detail coefficients in the high‐frequency region can be decomposed into higher level with a better resolution. Finally, a three‐level WPT produces a total of eight frequency sub‐bands in the third level, with each frequency sub‐band covering one‐eighth of the signal frequency spectrum.

Thus, for a discrete signal with length N, and given WPT coefficients at the final level L as defined in Section

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