O Cepstro

While FFT analyzes frequencies, O Cepstro analyzes periodicities present in the spectrum itself.

This technique allows you to automatically identify:

spacing between side bands;
gear frequencies;
modulation frequencies;
harmonious families.

It is particularly useful in:

reducers;
helical gears;
multipliers;
planetary boxes.

Comparison between FFT and Cepstro in automatic identification of spacing between sidebands.

When if analyzes the vibration of a gearbox or a damaged bearing, It is common to come across a espectro FFT loaded with spikes: the gear frequency, its harmonics, and — around each of them — entire families of equally spaced lateral bands. Manually interpret this “forest” of spikes, identify how many sideband families there are, what spacing they correspond to and what mechanism each one belongs to, It is a time-consuming and error-prone task, especially when multiple modulation sources overlap in the same spectrum.

This is exactly the problem that cepstro was created to solve. Instead of looking directly at the frequency spectrum, the cepstrum applies a second transformation to it, revealing in an immediate and compact way the periodicity that is “hidden” inside it — that is, the family of harmonics and equally spaced sidebands that, not original spectrum, appears dispersed across dozens of individual peaks.

Origin and definition

the term cepstro was born as a deliberate anagram of “spectrum” — a play on words that signals the central idea of the technique: apply a spectral analysis to the spectrum itself. Formally, the cepstrum is defined as the power spectrum of the logarithm of the power spectrum</cite> of the original signal. In the practice of vibration diagnosis, is typically calculated as the inverse Fourier transform of the logarithm of the signal power spectrum.

This step of taking the logarithm of the spectrum before the second transform is not a minor technical detail — it is what gives the cepstrum its most useful property., explained below.

The domain resulting from the transformation is called quefrency (again a pun, this time with “frequency”) and has units of time. A peak in the cepstrum at a given frequency indicates that there is, not original spectrum, a family of components spaced regularly in frequency, with an interval equal to the inverse of this qufrequency.

Because it is particularly useful in diagnosing gears and bearings

When a gear has a defect — a crack in the root of a tooth, wear, eccentricity — the normal meshing pattern is modulated, in amplitude or in phase, by the rotation frequency of the defective shaft. The result in the FFT spectrum is a family of sidebands, equally spaced, around the gear frequency and its harmonics.

Conventional spectral analysis is not sufficient to accurately identify gear defects, bearings and turbine blades where sidebands and harmonics are present, and this approach also depends on the transmission path between the source and the sensor.</cite> The cepstrum solves this in two complementary ways:

By transforming the vibration spectrum into the qufrequency domain, separates the effects of the vibration source from the effects of the transmission path, allowing to easily identify the families of harmonics and sidebands associated with the defect.</cite>
Represents the overall power content of an entire family of harmonics and sidebands in a single value, even when several sideband families are present simultaneously</cite> — what in the FFT spectrum would appear as dozens of distinct peaks, in the cepstrum typically concentrates on a single dominant line at the frequency corresponding to the frequency of repetition of the defect.

This second property is the one that is most interesting in practice.: instead of the analyst having to manually count and group dozens of sidebands in the spectrum, the cepstrum does it automatically, summarizing the entire family in a single quantitative indicator.

Insensitivity to the transmission path

One of the most cited advantages of the cepstrum is its relative insensitivity to the location of the measurement point and other factors related to the signal transmission path between the vibration source and the sensor.. <cite index=”26-1″>The cepstrum is largely insensitive to the location of the measurement point, focusing only on the periodic content of the signal.

The mathematical reason for this lies precisely in the operation of taking the logarithm: when a signal passes through a transmission path, the effect of this path is combined with the original signal by convolution in the time domain — which corresponds to a multiplication in the frequency domain. When applying the logarithm to this multiplication, it turns into a sum. And since the cepstrum is essentially a second Fourier transform, this sum separates into distinct additive components — one associated with the source (periodic, and therefore concentrated in a clear peak of frequency) and another associated with the transmission path (typically smooth and low frequency). This additive separation is what allows us to distinguish the approximate frequency components that, otherwise, would be mixed into the conventional spectrum.

Practical applications

The most classic and most documented application of the cepstrum is the gearbox diagnostics, theme that Robert B. Randall already consolidated in the 1990s 1980 in Brüel reference publications & Darling, and which continues to be the most cited use case in the technical literature. But the technique extends to other contexts:

bearings, where the harmonic families associated with the characteristic defect frequencies (inner track, outer track, rolling elements) also produce identifiable periodic structures in the cepstrum.
Turbine blades on ships and submarines, where the cepstrum is cited as one of the best techniques available for diagnosing defects, next to gears and bearings.
Detection of multiple defects in wind turbine gearboxes, where defects are often buried in the intense vibratory energy generated by the high number of gears and bearings operating under severe conditions — a scenario in which conventional filtering and demodulation techniques are no longer enough, and cepstral analysis helps distinguish frequency components that would otherwise be confused.
Rocket engine turbomachinery, where it has already been implemented as a single quantitative parameter, automatically calculated from stored timestamps, for equipment acceptance/rejection decisions before flight — showing the robustness of the technique even in machines with very high rotational speeds.

The variant for non-stationary regimes: the ceptrum of order

Just like classic envelope analysis, the traditional cepstrum implicitly assumes that the rotational speed is constant during acquisition. On machines that start, stop or change speed — a situation already discussed with regard to envelope limitations — this premise does not hold, and the classical technique loses precision.

The answer to this limitation is the ceptrum of order (order cepstro): instead of sampling the signal at constant time intervals, the signal is resampled at constant rotation angle intervals — converting, in practice, a stationary signal in the time domain into a stationary signal in the angle domain. Only after this conversion is cepstral analysis applied.. This approach has proven to be effective in diagnosing gear wear during starting processes., avoiding the phenomenon of “blur” in frequency (frequency fuzziness) that traditional spectral analysis cannot solve in these conditions, and has been successfully combined with classifiers based on neural networks for automatic diagnosis under acceleration..

Where do you fit among advanced diagnostic techniques

In a progression of techniques for handling increasingly complex vibration signals — from the classic envelope, passing through spectral kurtosis and fast kurtogram, to cyclostationary analysis and artificial intelligence — the cepstrum occupies a well-defined intermediate place: is the tool of choice specifically for disentangle harmonic families and sidebands, a problem that demodulation techniques alone do not solve so directly. For this reason, it is common to find it combined with envelope analysis — using the envelope to expose the amplitude modulation associated with impacts, and the cepstrum to quantify and separate the resulting different periodic families — in a complementary rather than competitive approach.

Conclusion

The cepstrum is one of the most elegant techniques in vibration analysis precisely because it solves, in a mathematically direct way, two problems that would otherwise require laborious manual interpretation: the counting and grouping of complex sideband families, and the separation between what is characteristic of the vibration source and what is merely the effect of the transmission path to the sensor. More than four decades after Randall's first work on the subject,, continues to be a reference tool in diagnosing gears and bearings — and, with the order variant, also extends to machines operating at variable speed regimes, remaining fully relevant in the modern conditioned maintenance toolbox.