# The relationship between spectral correlation and envelope analysis

Jul 27, Li and Qu [9] deduced the cyclic correlation and cyclic spectrum of amplitude However, the explicit relationship between the cyclostationary features of envelope and instantaneous frequency of gear meshing vibration. Abstract In recent years there has been an increasing interest in applying cyclostationary analysis to the diagnostics of machine vibration signals. This is. Antoni J () Cyclic spectral analysis of rolling-element bearing signals: facts () The relationship between spectral correlation and envelope analysis in .

Li and Qu [ 9 ] deduced the cyclic correlation and cyclic spectrum of amplitude modulation signals and applied them to rolling element bearing fault diagnosis. Recently, Antoni et al. To reduce the computational complexity of cyclic energy indicator based on cyclic spectral density, Wang and Shen [ 16 ] proposed an equivalent cyclic energy indicator for rolling element bearing degradation evaluation.

These researches illustrate the effectiveness of cyclostationary analysis in gearbox and bearing fault diagnosis. However, the explicit relationship between the cyclostationary features of vibration signals and the gearbox and bearing dynamic nature still needs further investigation, in order for thoroughly understanding the vibration signal characteristics and thereby effectively diagnosing fault.

Gearbox and rolling element bearing vibration signals usually feature amplitude modulation and frequency modulation AM-FMand the modulation characteristics contain their health status information [ 17 — 21 ]. Cyclic correlation and cyclic spectrum are effective in extracting modulation features from amplitude modulation AMfrequency modulation FMand AM-FM signals.

Feng and his collaborators [ 22 — 24 ] derived the expressions of cyclic correlation and cyclic spectrum for gear AM-FM vibration signals and proposed indicators based on cyclic correlation and cyclic spectrum for detection and assessment of gearbox fault.

Nevertheless, the carrier frequency of rolling element bearing vibration signals resonance frequency is completely different from that of gear vibration signals gear meshing frequency and its harmonics.

Therefore, it is important to investigate the cyclic correlation and cyclic spectrum of bearing vibration signals in depth, considering both the AM and the FM effects due to bearing fault. Meanwhile, how to explain the cyclostationary features displayed by the cyclic correlation and cyclic spectrum and to map the modulation characteristics to gear and bearing fault are still important issues for application of cyclostationary analysis in gearbox and bearing fault diagnosis.

In this paper, we derive the explicit expressions of cyclic correlation and cyclic spectrum for general AM-FM signals, summarize their properties, and further extend the theoretical derivations to modulation analysis of both gear and rolling element bearing vibration signals, thus enabling cyclostationary analysis to detect and locate both gearbox and bearing fault.

Definition The statistics of cyclostationary signals have periodicity or multiperiodicity with respect to time evolution. Cyclic statistics are suitable to process such signals. Among those, second order cyclic statistics, that is, cyclic correlation and cyclic spectrum, are effective in extracting the modulation features of cyclostationary signals. For a signalthe cyclic autocorrelation function is defined as [ 25 ] where is cyclic frequency. Cyclic Correlation of AM-FM Signal During the constant speed running of gearboxes and rolling element bearings, the existence of fault, machining defect, and assembling error often leads to periodical changes in vibration signals.

For gearboxes, such periodical changes modulate both the amplitude envelope and instantaneous frequency of gear meshing vibration.

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For bearings, such repeated changes excite resonance periodically. The excited resonance vanishes rapidly due to damping before next resonance comes, resulting in AM feature. On the other hand, it gives to envelope analysis a new conceptual framework, which may help to better understand its optimal use.

**Lecture 50A: Estimation of Power Spectral Density -1**

This latter aspect is analysed in the next section. ALIASING The conventional way of performing an envelope analysis was to band-pass "lter the bearing signal and then to rectify it to form the envelope. However, when performed digitally, this technique was shown to yield spurious harmonics in the spectrum of the envelope, due to inevitable aliasing [5]. Moreover, the squared magnitude of the signal was shown to produce an envelope without sharp cusps in contrast to the recti"ed signal, thus leading to a spectrum with a "nite number of harmonics and the possibility of avoiding aliasing [5].

The superiority of the envelope obtained from the squared magnitude as opposed to the recti"ed signal now "nds a new basis in the light of the analysis of quasi-cylostationary processes. Interpreting the spectrum of the squared signal as the integrated CSD clearly provides a simple but powerful means to reveal hidden periodicities of a purely stochastic process.

This interpretation also gives an insight on how to pre-"lter the vibration signal in order to avoid aliasing. Figure 7 shows the CSD of a digitised signal sampled at the rate f. Due to sampling, the s CSD is now a periodic function in both a and f and integrating it along the f-axis would make the redundant quadrants 3 and 2 fold with quadrants 1 and 4.

Figure 8 shows three ways to counter this problem: This is typically the s s processing required when performing the squared envelope analysis of a signal. Due to implicit squaring of the signal associated with calculating the CSD, it is now apparent that the sample rate should always be doubled before calculation, regardless of the application.

Cyclic spectral density domains for a digitised signal. The useful domain is distinguished from the redundant one by a di! Three solutions to avoid aliasing of digitised signals.

As can be seen from Fig. As opposed to the previous techniques, this solution does not require the integrated CSD to be post-"ltered in order to cancel the interferences.

Indeed, conventional envelope analysis of rolling element bearings is usually done by "rst band-pass "ltering the vibration signal in a part of the spectrum where the change due to the fault has been greatest, thus ensuring maximum signal-to-noise ratio.

### Cyclostationary Analysis for Gearbox and Bearing Fault Diagnosis

Then, any of the three aforementioned solutions, or a combination of them, may be applied to compute the integrated CSD or equivalently the spectrum of the squared magnitude of the signal. For example, one may "rst band-pass "lter the signal, then frequency shift the bands toward zero and then zero pad on the left and the right side as suggested in reference [5]. Indeed, the CSD really provides a new insight into the techniques which were previously discovered for squared envelope analysis [5].

As an illustration, take a synthesised cyclostationary signal made of two carriers modulated by the same random process, say, x[k]"a [k] cos [2nf k] a[k] cos [2nf k], k3N 25 1 2 where a[k] is a digitised random process, band limited in the range [!

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Note that this signal is not produced 1 2 by model 1its only purpose being to illustrate the above discussion. However, one can easily infer from equation 25 that neither the Fourier transform of the deterministic part which is identically zero for this example nor the PSD of the stochastic part can reveal the hidden periodicities associated with frequencies f and f.

Therefore, the CSD of the signal 1 2 should be analysed. Figure 9 displays the CSD with aliased terms in the upper and lower right corners, while Fig. As can be seen from these "gures, the result to be expected depends closely on which of the aforementioned solutions one decides to use to avoid aliasing.

For the application considered in this paper, determination of bearing modulation frequencies, which are always given by di! Alternatively, the frequency-shifted band-pass signals of Fig. On the other hand, the issue of estimating the spectrum of the squared magnitude of the signal should be investigated in more detail. Looking back at equation 22it can be seen that the exact formula involves the mathematical expectation operator or ensemble average operator. Yet, in most applica- tions, only one realisation of the vibration signal is available and therefore the mathematical expectation cannot be computed.

A natural question is to enquire to what extent this will interfere with the true formula In fact, it has been shown in reference [7] that a consistent estimator, i. The real bene"t of this is that the cyclic frequency resolution also improves as the empirical estimator M K a approaches M a.

Here again, this fact stems from a more general result proved for quasi-cyclostationary processes, which states that the estimated CSD keeps its full resolution with respect to the cyclic frequency a [1]. Therefore, the integrated CSD provides a convenient tool for enhanced spectral analysis. Reference [2] gives an example where the e! The vibration signals were obtained from two rail vehicle bearings, which had been overloaded on a test rig so that faults had developed.

Figure 12 shows the three di! Details of the data are given in Table 1. Table 2 gives the expected frequencies of typical faults, computed from the usual formulae [4].

As a matter of fact, computation of the Fourier transform and the PSD of the signal as given by equations 6 and 15 could not accurately reveal the frequency of the fault.

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That is in agreement with the theoretical results of Section 2. Therefore, both the CSD and the envelope spectrum were employed, in order to yield higher resolution spectral estimates.

The bearing signals under study: The purpose of that operation was twofold, as it "rstly increased the signal-to-noise ratio and secondly enabled a reduction of the number of time samples to be processed for estimating the CSD.

Note that this corresponds to the analysis of an analytic signal. Figure 13 a shows the estimated CSD for a demodulation in the spectral R. Spectrum of the squared magnitude signal of the inner race fault demodulated in the spectral band [ Hz; Hz]. The resolution is 1 Hz on the cyclic frequency a and 10 Hz on the frequency f. One can easily appreciate the pattern made up of parallel lines, which indicates the presence of cyclostationarity. Careful inspection of these reveals that they are sidebands of three carriers located around 0 Hz, a " Hz and 2a " Hz.

The spacing between 2 2 the sidebands is about a "9. Some overlapping of the sidebands occurs in the 1 r bands [50 Hz; 70 Hz] and [ Hz; Hz] due to the fact that the process under study is quasi-cyclostationary rather then purely cyclostationary.

Comparison of these frequencies with Table 2 obviously indicates an inner race fault. Figure 13 b shows the integrated CSD, which retains all the useful information needed for the diagnosis. Neither the Fourier transform nor the PSD of the signal could be used to identify any harmonics related to the frequency of a defect.

As before, a spectral band was selected to perform a CSD analysis and an envelope analysis. The resolution is the same as in the previous case. Figure 15 b displays the integrated CSD computed directly from the squared magnitude signal. Its inspection reveals a peak at a "97 Hz and its second harmonic at 1 2a " Hz, with no sidebands. Looking back at Table 2, these frequencies match with an 1 outer race fault. Here again, the spectrum of the squared magnitude signal maintains all the useful information for diagnostics.

Indeed, for this case, it proved to be more readable than the 2-D-CSD, thus providing a more reliable tool for diagnosis. The authors have been able to repeat this practical observation on a number of other real cases. The examples in the last section supported this idea and even showed that the interpretation of the integrated CSD can be easier than that of the 2-D-CSD.

Eventually, there seems to be no point in analysing the CSD instead of simply the spectrum of the squared magnitude signal, that is, performing a classical envelope analysis. Although this proposition may be true for most vibration analyses of rolling element bearings, it needs to be moderated. Firstly, even though the spectrum of the squared magnitude signal indicates the presence of quasi- cyclostationarity, it cannot be used as R.

Actually, measuring the degree of quasi- cyclostationar- ity would require integration of the squared magnitude of the CSD instead of the raw CSD [8]. However, for narrowband signals such as those used in envelope analysis, the two procedures would give very similar results. Secondly, the 2-D feature of the CSD may prove useful when analysing complex systems, for instance where several bearings and possibly gears contribute to the vibration signal. Then, inspection of the CSD could help to select the best spectral bands for isolating and demodulating the contribution of each sub-system.

In this context, CSD analysis would be used as a precursor to envelope analysis. Possibly, the CSD could also be useful to discriminate between bearing and gear fault signals where both contain modulations at shaft speedas they should yield di! According to the present authors' knowledge, very few examples of such analyses have been reported in the literature, so that these assertions still need to be veri"ed on a practical basis.

The classical way of overcoming this problem was to analyse the envelope and even squared envelope of the bearing signal. It has now been shown that the latter gives the same result as the integration of the cyclic spectral density function over all frequencies, thus establishing the squared envelope analysis as a valuable tool for the analysis of quasi- cyclostationary signals more generally.