3.4 Cepstrum analysis
Cepstrology was originally proposed by Bogert et al. [12] as a better alternative than autocorrelation function for detecting echo delay time, especially for seismic signals. At that time, it was defined as the power spectrum of the logarithmic power spectrum. The "power cepstaphedr" [13] was later redefined as an inverse Fourier transformation of the logarithmic power spectrum, partly because it is more logical to use an inverse transformation between the frequency function and the time function, and partly because it can then be inverted to the power spectrum (e.g., after editing). At about the same time, the "complex cepstruf" is defined as the inverse Fourier transform of the complex logarithm of the complex spectrum [13], which can be inverted into a time function, e.g. allowing the removal of echoes from the time signal. Because the cepstrum relates to the Fourier transform of the spectrum, it is the "spectrum of the spectrum", which is actually one of the terms coined by Bogert et al. in the original paper, which will be discussed in the terminology of Section 3.4.1 in this section. However, the autocorrelation function is the inverse Fourier transform of the corresponding autospectra (see Section 3.2.6.5) and is therefore also a spectrum of the spectrum. What really sets the cepstrum apart is the logarithmic transformation of the spectrum before the second transformation. In the response spectrum, this forces the multiplication relationship between the action function and the transfer function (from force to response) to an additive relationship that remains in the cepstrum spectrum. This gives rise to one of the main applications of the chamfer. For SIMO (Single Input, Multiple Output) systems, addition in the cepstrum corresponds to the convolution of the action function and the impulse response function in the time domain. Note that this does not apply to MIMO (multiple-input, multiple-output) systems, as each response is then the sum of the convolutions.
Another case of converting convolutions to addition in the time domain is the case of echoes. As shown in Figure 3.31 [4] (see also Figure 3.11), a signal with an echo can be modeled as a convolution of the original signal (called f1(t)) and a signal containing a unit δ function at the origin and a scaling δ function at the delay τ (called f2(t)):
Cases with spectrum
As shown in Figure 3.31, the latter is the sum of a fixed vector and a smaller rotation vector, performing one cycle per 1/τ advance along the frequency axis. Therefore, its amplitude (and logarithmic amplitude) and phase are periodic in frequency, with a period of 1/τ. After taking logarithms, the logarithmic spectrum of the total signal is the sum of the original spectrum and an additional periodic component with a frequency period of f = 1/τ. The inverse Fourier transform gives the cepstrum of the original function f1(t), plus the multiples τ corresponding to the Fourier series component of the periodic component. This has applications in detecting and removing echoes, as well as measuring their delay time.
One of the main applications of cepstrum analysis in machine condition monitoring is the analysis of signals containing harmonic and sidelobe families (evenly spaced), where faults are characterized as a whole rather than a single frequency component. For example, in the case of local faults in gears (see Figure 2.12 in Chapter 2), harmonics and/or side lobes are generated throughout the spectrum and are more pronounced in the logarithmic amplitude spectrum (greatly reducing the effect of the transfer function of the relative arbitrary structure between the source and measurement points). This will be discussed in more detail in Section 5.4 of Chapter 5. Cepstral can also be used for harmonics from bearing failures, but only if they are well separated. This is discussed further in Section 5.5 of Chapter 5, stating that it is generally better to use envelope analysis because it does not have such limitations.
3.4.1 Terms and Definitions
As mentioned above, in the original paper [12], the author coined the word "cepstrum" by inverting the first syllable of "spectrum", on the grounds that it is "spectrum of spectrum". Similarly, the word "quefrency" is taken from "frequency", and the author proposes a number of other words, including:
Among them, quefrency, rahmonic, and lifter are useful in clarifying that the operation or feature refers to the cepstrum rather than the spectrum or time-domain signal, and are still frequently used in the literature, as well as in this book. The usefulness of other terms may be even more questionable.
Therefore, the definition of a cepstrum is:
thereinto
Defined by its amplitude and phase, therefore
In this case, when X(f) is a complex number, the cepstrum of equation (3.49) is called a "complex chambled spectrum", although since ln(A(f)) is an even function and φ(f) is an odd function, the complex chambled spectrum is real.
Note that, in contrast, the autocorrelation function can be derived as an inverse transformation of the power spectrum, or
When the power spectrum is used in place of the spectrum X(f) in equation (3.49), the resulting cepstrum is referred to as the "power cepstrum" or "real cepstran", and the expression is:
Thus, the real cepstrum is a scaled version of the compound calf, where the phase of the spectrum is set to zero. Sometimes the term "real calbede" is used to denote an inverse transformation of logarithmic amplitude, so there is no factor 2 in equation (3.53).
Note that the phase function φ(f) must be expanded into a continuous function of frequency before calculating the compound cepstruf, which is often difficult, so it is easier to use a real cepstaphef.
Another type of cepsep that is useful in some cases is "differential cepstrums", which is defined as the inverse transformation of the derivative of the logarithm of a spectrum. It is easiest to define in terms of the z-transform (a Fourier transform that replaces the sampling function) as follows:
where n is the quefrency indicator, which can be calculated directly from the time signal as follows:
One of the advantages of this is that the phase of the (logarithmic) spectrum does not have to be expanded.
If the frequency spectrum X(f) in Eq. (3.49) is the FRF (where |ai|, |bi|, |ci|, |di|) is represented by the gain factor K and the zero and pole (ai and ci) within the unit circle and the zero and pole (1/bi and 1/di) outside the unit circle < 1), then Oppenheim and Schafer have shown that the calbacial spectrum is given by the following analytic formula:
The cepstruptal coefficient n is used as an exponent.
Since the cepstrum is a real number, the complex exponential terms in (3.56) can be grouped in pairs, e.g., a pair of ci terms can be replaced by (2/n)Cin cos(nαi), where Ci = |ci |, αi = ∠ci. This represents an exponentially decayed sine wave, further attenuated by a hyperbolic function of 1/n. Figure 3.32 compares the cepstrum versus impulse response function of an SDOF system with a pair of poles and no zero point. On the logarithmic amplitude scale, the zero point (inverse resonance) of the FRF is like an inverted pole (resonance), so it is not surprising that the corresponding inverted canopy term has an inverted sign.
Performing a derivative operation on the spectrum in the z-domain to obtain a differential cepstrum results in n-multiplication in the cepstruf, so the typical term becomes 2 Ain cos(nαi), i.e., an exponentially decayed sine wave, without the weighting of hyperbolic functions, and thus resembles an IRF in form. This is useful for the direct application of techniques that have been developed to fit parametric curves to IRF directly to differential cepstral. This is the second advantage of differential cepstral, but it is mainly used for operational modal analysis [16]. The so-called "mean differential cepstral" [17] has additional advantages in this application, but will not be discussed in detail here.
Note that for functions with minimal phase properties, this applies to FRFs of many physical structures, where no pole or zero is located outside the unit circle (bi and di disappear), so there is no negative cepstrum term in Eq. (3.56), so the cepstrum (and differential cepstruf) is causal. According to the normal Hilbert transform relation (see Section 3.3), this means that the real and imaginary parts of the corresponding FT, i.e. the logarithmic amplitude and phase of the spectrum, are correlated by the Hilbert transform, and only one of them needs to be measured. It also means that a compound calf can be obtained from the corresponding real cepstrum (real and even) by doubling the positive and negative cepstrum terms to zero. In this case, the phase of the spectrum also does not need to be measured or expanded.
3.4.2 Typical applications of cepstrums
Figure 3.33 compares the cepstrum spectrum and autocorrelation functions in the case of high-speed bearing failure. This illustrates the difference in taking the logarithm of the power spectrum before performing the inverse cepstrum transformation. In the logarithmic power spectrum in Figure 3.33(a), there is a harmonic family of BPFOs (ball passing frequency, outer ring) with an interval of 206 Hz. Since they are all below the −20 dB line, which corresponds to 1% of the full scale on the linear (amplitude squared) scale, they are barely visible in the power spectrum. Thus, the cepstrum spectrum (Fig. 3.33(c)) is dominated by the rahmonics corresponding to the BPFO (with an interval of 4.84 ms), while the autocorrelation function (Fig. 3.33(b)) simply shows the beat frequency between the two largest components in the spectrum, independent of bearing failure.
Note that the quefrency of the cepstral component provides very good accuracy because it represents the average of the intervals across the spectrum. It should also be noted that cepstral can only be used for bearing fault diagnosis if the fault produces discrete harmonics in the spectrum. This is usually true in the case of high-speed machines, as the resonances excited by faults tend to be the relatively low harmonic order of the frequency at which the ball passes, but this is not usually the case for low-speed machines, where the order can reach hundreds or even thousands, and these high harmonics are often mixed together. It is worth noting that in "Envelope Analysis", the spectrum analysis of the envelope obtained by amplitude demodulation of the bandpass filter signal can be used in any situation.
Figure 3.34 shows EDF inspecting a steam turbine with missing blades [18]. This is work carried out by EDF, a French electric power company that has experienced the loss of one or a handful of small blades on a large turbine with little effect on the overall vibration level (sometimes the loss of one blade causes the blades on the other side to "pass by", thus giving the turbine approximately balance). The malfunctioning blades emit incorrectly oriented steam interacts with a series of stator blades, and an accelerometer mounted on the housing detects a pulsed event at each revolution of the rotor. This significantly increases the harmonics of the shaft speed (50 Hz) in the mid-frequency range used to detect the phenomenon.
An example of the application of compound cepstrum in detecting and removing echoes is shown in Figure 3.35 [4]. The figure shows that the overlapping echoes can be removed because the cepstrum is shorter than the impulse response. Although this is a numerically generated case, echoes can also be removed in the actual signal [4]. With real part cepsant, the effect of echoes can be removed from the logarithmic spectrum by making similar edits to the cepstrand. Note that in a composite cepstrucest, the scale of the logarithmic amplitude spectrum should be nepers (natural logarithm of amplitude ratio) to correspond to the radians in the phase spectrum.
3.4.3 Practical considerations for cepstrum analysis
There may be some human factors when calculating the cepstruf, and care must be taken to ensure that they do not influence the results too much. Some of these effects are illustrated in Figure 3.36.
In Figure 3.36(a), it is shown that the noise level in the spectrum affects the detection of harmonic components. It is clear that if harmonics are completely drowned in noise, they will not be detected in the cepstral. A variety of techniques can be employed to make the discrete frequency components more prominent from the background noise. One way to do this is to use a narrower bandwidth, as described in section 3.2.8.4 above, which can be implemented using a scaling analysis. Another approach is to use synchronous averaging or other means to reduce the effect of noise relative to discrete frequency components. Note that synchronous averaging is only available for harmonic components and is limited to cases with one fundamental period. If the sideband family does not pass zero frequency (i.e., the sideband is not harmonic), it cannot be used to enhance equally spaced sidelobes. Some alternative techniques for separating discrete frequencies and random signals are discussed in Section 3.6. In any case, the scaling of the rahmonic group in the cepstrum depends on the signal-to-noise ratio, so it is usually only valid when comparing the cepstrum using the same analysis parameters, and care should be taken to ensure that the noise level in the measurement has not changed.
The results of the cepstrum depend on the characteristics of the filter used in the original spectrum analysis, and Figure 3.36(b) illustrates a number of situations where this can directly lead to erroneous results. It is generally assumed that if the harmonic or sidelobe groups in the spectrum rise, then this will result in an increase in the corresponding components in the cepstrum spectrum. However, if the spectral components are "bridged" within a fixed number of decibels below the peak, as shown in Figure 3.36(b), then the high cepstrum components will not change. Two potential causes of "bridging" are that the components in the spectrum are not good enough in resolution (separation), or that the filter characteristics are very poor (e.g., FFT using a rectangular window). The filter characteristics of the Hanning window are quite good, and if they are separated by at least eight analysis lines, they are able to separate adjacent components within a range of 50 dB. This is usually enough for the noise to dominate in the "bridging" between adjacent components. When interpreting the values of the different components in the cepstrum spectrum, which are generated by the harmonic series in the spectrum, it should be borne in mind that this is similar to performing a frequency analysis on a series of pulses with different spacing but constant widths. The greater the interval between the components, the more rahmonics will be generated, and the smaller the magnitude of the individual rahmonics will be for the protrusion (in decibels) of the harmonic series from the noise level in a given spectrum. Again, it is only valid when comparing cepstrum values using the same analytical parameters, and even then, it is still difficult to compare between components of different quefrencies.
Another practical issue shown in Figure 3.36(c) is the choice of vibration parameters (velocity or acceleration; displacement is rarely used). Since the difference between them is, in principle, a moderate difference in the slope of the logarithmic spectrum (corresponding to the integral from acceleration to velocity), this is a very "low quefrency" effect that does not affect the high quefrecy values that contain diagnostic information. However, if the spectral value is lower than the dynamic range of the analysis, the latter may be affected considerably. Therefore, in general, it is best to choose the parameter with the most uniform spectral level over the frequency range of interest. This is usually velocity, but occasionally it can be acceleration. When discussing dynamic range, it should be borne in mind that a large negative deviation in the spectrum (measured in decibels) has the same effect on the cepstrum spectrum as a positive deviation, but may have little physical significance. Therefore, sometimes it may be a good idea to limit the negative deviation and apply an artificial "noise level" corresponding to the effective dynamic range of the measurement (80-100 dB). The reference value for the logarithm should also be chosen carefully. In principle, it has no effect on any value other than the zero quefrency value in the cepstral spectrum, so it is better to choose a medium range for the logarithmic value of the spectrum. If you choose it as the average decibel value of the spectrum, the zero quefrency value will be zero and will have minimal effect on the useful portion of the cepstrum spectrum.
Due to the need for good resolution of the harmonic/sidelobe components in the spectrum, it is often advantageous to use scaling to obtain the latter. Unwanted components can also be excluded by selecting a portion of the spectrum. The latter may include low harmonics of shaft speeds that are affected by phenomena such as misalignment in order to separate them from the modulated sidelobes around the gear's meshing frequency. When performing a cepstrum analysis on a scaled spectrum, the left end of the spectrum no longer corresponds to the zero frequency, and the resulting cepstrum can give confusing results. Figure 3.37 shows an example in which two slightly offset scaling spectra are used to obtain the cepstrum of the side lobes surrounding the second harmonic of the meshing frequency of the gear. Since the sidelobe family no longer passes through the effective "zero frequency" at the left end of the spectrum, the rahmonics in the real part cepstaphedrum are no longer the positive peaks corresponding to the sidelobe spacing. The quefrency corresponding to the 25 Hz spacing is close to zero crossover in both cases. The quefry corresponding to the 8.3 Hz spacing has a positive peak in one case and a negative peak in the other. This problem can be solved by utilizing the principle of the Hilbert transform, since the true spacing will be indicated by the peak in the amplitude of the complex (analytical) signal obtained by inversely transforming a complex number with a one-sided logarithmic spectrum (the zero filling of the negative frequency component of the solution zero frequency). For convenience, such a cepstrum can be called an "analytic champocephalus" to distinguish it from a real chamfer, which is a real number.
Whenever the sidelobe family passes through the zero frequency, or even the true zero frequency, the exact same phenomenon will be encountered. For normal gears, the gear meshing frequency is a harmonic of the speed of the two shafts, so the side lobe family modulated by the speed of these shafts is also a harmonic and thus passes through the zero frequency. However, in planetary gears, the modulation frequency is not a sub-doubling of the gear meshing frequency, so the sidelobe family does not necessarily pass through the zero frequency.
The specific application of cepstrum analysis in gear diagnostics is discussed in more detail in Section 5.4 of Chapter 5.