**3. Envelope analysis**

Bearing faults can be classified as localized (single-point) or extended. Incipient localized faults produce sharp impulses that cover a large bandwidth. These faults, in general, are associated with small pits or spalls. On the other hand, extended faults effect is not so apparent or highlighted in the spectrum and its bandwidth is limited. Brinelling and corrosion are examples of extended bearing faults. It is also possible that a small localized fault becomes an extended fault as the fault evolves over time. Regardless of the type of fault, in general, bearing failure can be detected using envelope analysis [3].

It is also important to observe that signals produced by bearing faults (localized or extended) are typically nonstationary, i.e., signals whose statistical parameters vary in time. More specifically, localized bearing faults signals can be modelled as cyclostationary or pseudocyclostationary [8, 24].

Over the years, the envelope analysis or high frequency resonance demodulation has been widely used for identifying localized faults in rolling bearings. Each time a bearing component strikes the fault surface, a mechanical shock occurs. Consequently, an impulse is generated and structural resonances of the system are excited by it. In addition, these impulses are modulated in amplitude. This way, through the envelope analysis, it is possible to obtain demodulated signals, which are directly related to the bearing condition [8].

The following steps perform envelope analysis. First, digital bandpass filtering of acquired signal in a suitable frequency band, in general, around the machine mechanical resonance is performed. Following, the filtered signal is demodulated. Finally, the resulting signal frequency spectrum is estimated, resulting in the envelope spectrum, whereby it is possible to identify the periodic components associated with a fault in a bearing component [16, 25]. In other words, it is possible to identify the repetition frequency of the impulses caused by a fault simply analyzing the envelope signal spectrum, which, in general, it is not possible by using the raw spectrum [17]. Fourier transform is applied in order to obtain the envelope spectrum.

One of the most used tool for demodulation or envelope extraction is Hilbert transform [26, 27]. First, the acquired signal is bandpass filtered around a machine resonance frequency, and then Hilbert transform is applied. This digital technique reduces the data length and allows flexibility for bandpass filter specification [28].

However, it is important to observe that a suitable frequency band to filter the signal must contain impulses generated by the fault and amplified by machine mechanical or structural resonances [8]. Therefore, one of the main difficulties in using envelope analysis is undoubtedly the choice of an appropriate frequency band for filtering the signal. In order to circumvent this drawback, algorithms based on spectral kurtosis have been successfully applied, which is discussed later in the chapter.

### **3.1. Hilbert transform**

the signal. In this case, the original signal [*y*(*t*)]*<sup>t</sup>*=1

ing failure can be detected using envelope analysis [3].

allows flexibility for bandpass filter specification [28].

**3. Envelope analysis**

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stationary [8, 24].

that, an average of *L* = *N*/*K* periodograms is computed. Welch method can be seen as evolution of Bartlett method; since the estimation is performed considering that the signal segments are

Bearing faults can be classified as localized (single-point) or extended. Incipient localized faults produce sharp impulses that cover a large bandwidth. These faults, in general, are associated with small pits or spalls. On the other hand, extended faults effect is not so apparent or highlighted in the spectrum and its bandwidth is limited. Brinelling and corrosion are examples of extended bearing faults. It is also possible that a small localized fault becomes an extended fault as the fault evolves over time. Regardless of the type of fault, in general, bear-

It is also important to observe that signals produced by bearing faults (localized or extended) are typically nonstationary, i.e., signals whose statistical parameters vary in time. More specifically, localized bearing faults signals can be modelled as cyclostationary or pseudocyclo-

Over the years, the envelope analysis or high frequency resonance demodulation has been widely used for identifying localized faults in rolling bearings. Each time a bearing component strikes the fault surface, a mechanical shock occurs. Consequently, an impulse is generated and structural resonances of the system are excited by it. In addition, these impulses are modulated in amplitude. This way, through the envelope analysis, it is possible to obtain

The following steps perform envelope analysis. First, digital bandpass filtering of acquired signal in a suitable frequency band, in general, around the machine mechanical resonance is performed. Following, the filtered signal is demodulated. Finally, the resulting signal frequency spectrum is estimated, resulting in the envelope spectrum, whereby it is possible to identify the periodic components associated with a fault in a bearing component [16, 25]. In other words, it is possible to identify the repetition frequency of the impulses caused by a fault simply analyzing the envelope signal spectrum, which, in general, it is not possible by using the raw spec-

One of the most used tool for demodulation or envelope extraction is Hilbert transform [26, 27]. First, the acquired signal is bandpass filtered around a machine resonance frequency, and then Hilbert transform is applied. This digital technique reduces the data length and

However, it is important to observe that a suitable frequency band to filter the signal must contain impulses generated by the fault and amplified by machine mechanical or structural resonances [8]. Therefore, one of the main difficulties in using envelope analysis is undoubtedly the choice of an appropriate frequency band for filtering the signal. In order

demodulated signals, which are directly related to the bearing condition [8].

trum [17]. Fourier transform is applied in order to obtain the envelope spectrum.

overlapped and windowed. Thus, variance is reduced, but also the resolution [23].

*<sup>N</sup>* with *N* samples is split into *K* segments, such

As mentioned before, bearing fault signals can be seen as amplitude modulated signal, such that, carrier frequency, represented by high frequency resonances are modulated by bearing characteristic frequencies. Hilbert transform can be used for the demodulation process in envelope analysis when modulated signal is proved to be analytic [8].

When envelope analysis is performed based on Hilbert transform, the frequency band to be demodulated can be properly separated from adjacent components that could interfere with the analysis. Impulse response function produced by bearing faults has real and imaginary parts of its corresponding frequency function related by Hilbert transform [8].

In general, signal-to-noise ratio is used as an indication of the frequency band where the modulated signal should be filtered. After filtering, selected frequency band is shifted at low frequencies in the spectrum and padded with zeros to double the length in order to obtain a one-side spectrum. When computing the inverse Fourier transform of this one-side spectrum, an analytic signal is obtained, such that, its imaginary part is the Hilbert transform of the real part. In this way, envelope corresponds to the modulus of real and imaginary parts. However, it is more interesting to analyze the squared envelope, since it can improve signal-to-noise ratio by removing extraneous components in practical situations [28].

#### **3.2. Kurtogram**

A rolling bearing fault excites high frequency resonances in the rotating machine, which can produce modulations at bearing characteristic frequencies. Therefore, characteristic frequency components should be demodulated using an optimal selection of frequency and bandwidth (*f*, *Bw*) for bearing fault identification based on envelope analysis. In this sense, spectral kurtosis based algorithms, such as kurtogram, aims to find this combination in a computationally efficient way [25].

Initially, spectral kurtosis (SK) was defined based on short-time Fourier transform (STFT) for impulsivity measurement as a function of frequency, and it was mainly applied to sonar signal analysis [17]. Some years ago, SK was also considered and applied for bearing fault analysis [29].

Thus, spectral kurtosis of a signal *x*(*t*), i.e., kurtosis value for each frequency (*f*), can be computed based on the STFT (*X*(*t*, *f*)) of this signal, such that [8, 30]:

$$\text{SK}\{f\} = \frac{\langle \text{X}^{\text{4}}(t,f) \rangle}{\langle \text{X}^{\text{2}}(t,f) \rangle^{2}} - 2 \tag{10}$$

where *X*(*t*, *f*) corresponds to the envelope as a time-frequency function; *X*<sup>2</sup> (*t*, *f*) represents the power spectrum values calculated for each time (*t*); and the average of all these power spectral values (〈*X*<sup>2</sup> (*t*, *f* )〉) corresponds to the power spectrum of the analyzed signal as a whole. In addition, the constant factor 2 is subtracted, so that, for Gaussian signal, Eq. (2) turns to zero [8]. In this sense, spectral kurtosis can be understood as a filter so that its value is maximum in the frequency bands containing impulsive signals and zero for that frequency bands dominated by stationary signals [29].

Since using short-time Fourier transform, parameters, such as window length, can directly affect the spectral kurtosis calculation; therefore, considering an impulsive signal, the window shorter than the spacing between two consecutive pulses and longer than an individual pulse shall provide a maximum kurtosis value. A detailed investigation about the relation between spectral kurtosis value and window length was conducted in Ref. [28]. Additionally, in Ref. [29], it was depicted that the square root of the spectral kurtosis is equivalent to the optimum Wiener filter and it demonstrated a close relation between optimum matched filter and spectral kurtosis value. For envelope analysis, in order to obtain an optimum result, it is of utmost importance to specify properly filter center frequency and bandwidth. For this purpose, the concept of kurtogram emerges as a tool to find the optimum filter for envelope analysis based on spectral kurtosis values. Kurtogram displays the spectral kurtosis values as a function of frequency and windows length, which define the spectral resolution. Experiments showed that the filter set from kurtogram was more efficient for outer race fault detection, when compared with Wiener and matched filters [28].

Fast kurtogram algorithm was developed as an extension of the kurtogram, especially considering that it was costly and inefficient to analyze all possible combinations of frequency and windows length. Fast kurtogram computes spectral kurtosis using digital filters, instead of short-time Fourier transform, following a dyad-decomposition so-called 1/3-binary tree. This decomposition is similar to discrete wavelet packet transform, where frequency bands are divided into bands with one half of their previous width, but here, divisions by 1/3 are also included [30].

As an alternative for fast kurtogram, the wavelet kurtogram algorithm was developed. In this case, nonorthogonal complex Morlet wavelets are used for signal decomposition and it is considered that the optimum combination center frequency and bandwidth for envelope analysis could be found based on a 1/nth-octave wavelet analysis. In general, the sequence 1/1, 1/2, 1/3, 1/4, 1/8, 1/12, …, 1/*n*th-octave is used, although, any sequence could be applied. Besides, before wavelet decomposition, the original signal power spectral density is prewhitened by an autoregressive model in order to enhance the fault detection into the envelope spectrum. Additionally, applied complex Morlet wavelet was optimized, since several filter banks are tested and the selected for envelope analysis is the one that maximizes the SK. The scheme of signal decomposition by means of filter bank for SK optimization is similar to that one used in kurtogram [17].

Wavelets are used because they present an impulse response with a constant damping ratio, which is more suitable for impulsive signals analysis in comparison with STFT. Besides, complex Morlet wavelet is analytic; therefore, its Fourier transform presents only positive frequencies. Thus, SK for each wavelet filter can be calculated considering that the product of the Morlet wavelet coefficients and their complex conjugate corresponds to the squared envelope of the filtered signal [17]. Here, it is also important to notice that using the quadratic envelope has been more advantageous for bearing signal analysis [28], which will be discussed in the next section.

The SK calculation could be enhanced by prewhitening the spectrum of the signal to be analyzed. Through the prewhitening, signal spectrum becomes almost constant, similar to the white noise spectrum. This process reduces variations that could occur in transient signals spectrum, which can lead to inaccurate SK calculations [17]. An autoregressive model can be used for signal spectrum prewhitening. In this case, the model error corresponds to the noise, but especially to the nonstationary part of the signal, which contains information related to bearing fault. In other words, it is possible to say that a digital filter (linear prediction filter), which is designed based on an autoregressive signal model, predicts the deterministic part of the signal; and the prediction error, which contains an impulsive signal that will be used for machine condition analysis [31].

$$\begin{aligned} \text{An autoregressive model (AR) of order } p \text{ can be represented by [32]:}\\AR(\mathbf{k}) &= -\sum\_{l=1}^{p} a(\mathbf{i}) \mathbf{x}(\mathbf{i}+\mathbf{k}) \leftarrow \text{error(k)} \end{aligned} \tag{11}$$

where *a*(*i*), *i* = 1, 2, 3, …, *p*, corresponds to the linear prediction filter weighting coefficients; *error* (*k*) is a whitened signal, which is the difference between the original and the predicted signals. Minimum least square error is used to find the coefficients of the linear predictor. Model order (*p*) will be one that maximizes the kurtosis of the *error*(*k*), such that, this residual signal will contain fault related impulse signals. Besides, (*p*) must be smaller (in number of samples) than that the space between two consecutive bearing faults impulses [17].

#### **3.3. Squared envelope analysis**

frequency bands containing impulsive signals and zero for that frequency bands dominated by

Since using short-time Fourier transform, parameters, such as window length, can directly affect the spectral kurtosis calculation; therefore, considering an impulsive signal, the window shorter than the spacing between two consecutive pulses and longer than an individual pulse shall provide a maximum kurtosis value. A detailed investigation about the relation between spectral kurtosis value and window length was conducted in Ref. [28]. Additionally, in Ref. [29], it was depicted that the square root of the spectral kurtosis is equivalent to the optimum Wiener filter and it demonstrated a close relation between optimum matched filter and spectral kurtosis value. For envelope analysis, in order to obtain an optimum result, it is of utmost importance to specify properly filter center frequency and bandwidth. For this purpose, the concept of kurtogram emerges as a tool to find the optimum filter for envelope analysis based on spectral kurtosis values. Kurtogram displays the spectral kurtosis values as a function of frequency and windows length, which define the spectral resolution. Experiments showed that the filter set from kurtogram was more efficient for outer race fault

Fast kurtogram algorithm was developed as an extension of the kurtogram, especially considering that it was costly and inefficient to analyze all possible combinations of frequency and windows length. Fast kurtogram computes spectral kurtosis using digital filters, instead of short-time Fourier transform, following a dyad-decomposition so-called 1/3-binary tree. This decomposition is similar to discrete wavelet packet transform, where frequency bands are divided into bands with one half of their previous width, but here, divisions by 1/3 are

As an alternative for fast kurtogram, the wavelet kurtogram algorithm was developed. In this case, nonorthogonal complex Morlet wavelets are used for signal decomposition and it is considered that the optimum combination center frequency and bandwidth for envelope analysis could be found based on a 1/nth-octave wavelet analysis. In general, the sequence 1/1, 1/2, 1/3, 1/4, 1/8, 1/12, …, 1/*n*th-octave is used, although, any sequence could be applied. Besides, before wavelet decomposition, the original signal power spectral density is prewhitened by an autoregressive model in order to enhance the fault detection into the envelope spectrum. Additionally, applied complex Morlet wavelet was optimized, since several filter banks are tested and the selected for envelope analysis is the one that maximizes the SK. The scheme of signal decomposition by means of filter bank for SK optimization is similar to that one used

Wavelets are used because they present an impulse response with a constant damping ratio, which is more suitable for impulsive signals analysis in comparison with STFT. Besides, complex Morlet wavelet is analytic; therefore, its Fourier transform presents only positive frequencies. Thus, SK for each wavelet filter can be calculated considering that the product of the Morlet wavelet coefficients and their complex conjugate corresponds to the squared envelope of the filtered signal [17]. Here, it is also important to notice that using the quadratic envelope has been more advantageous for bearing signal analysis [28], which will be

detection, when compared with Wiener and matched filters [28].

stationary signals [29].

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also included [30].

in kurtogram [17].

discussed in the next section.

During the envelope analysis, existing random or discrete noise components can make it difficult to identify components related to bearing failure. That is why a major constraint of envelope analysis is related to signal-to-noise ratio. A way to overcome this limitation is by using squared envelope. In this case, envelope spectrum presents a higher harmonic reduction, which cannot be obtained by a common filtering operation [28].

A method for computing squared envelope from an analytic signal was depicted in Ref. [28]. There, squaring envelope process is defined as a convolution of an analytic signal and its complex conjugate. Thus, squared envelope spectrum can be calculated by the convolution of the analytic signal and its complex conjugate corresponding spectra. In this case, spectrum of squared envelope does not present a sum of frequency components, since the analytic signals have only positive frequency components. Besides, the squared envelope spectrum has the same frequency range as if it was calculated using Hilbert transform and zero padding [8].

It is also important to highlight that the integral of spectral correlation of all considered frequencies is equivalent to spectrum of the squared envelope, where the spectral correlation is a twodimensional Fourier transform calculated on the two-dimensional autocorrelation function [33].
