Cepstrum Analysis

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The cepstrum (pronounced KEP-strum or SEP-strum) is a signal processing technique performed on a time-domain signal to look at periodic content in data, such as harmonics, sidebands, and orders.

The cepstrum is sometimes referred to as a “spectrum of a spectrum” because it involves calculating the spectrum of the time signal and then performing another spectral operation.  It is calculated by first taking the Fourier transform of a time signal to create a frequency spectrum, converting it to a log scale, and then performing an inverse Fourier transform on the spectrum (Figure 1).
 
Because the cepstrum comes from an inversion of spectrum, the name “cepstrum” is derived by reversing the first few letters of “spectrum”. 

Cepstrum analysis is widely used in a number of applications including diagnostic of gear and bearing noise, voice recognition software, and seismic analysis.

In gear analysis, they are useful for 

• Fault Detection: Gears output a specific vibration signature that directly relates to the speed of rotation, the gear ratio, and the number of teeth or elements a gear can have. Gears can create what are called “Side Bands” that are unique spectral signatures that can indicate if a gear is defective. In complex rotating systems that exhibit a lot of noise and vibration, cepstral processing can help identify the noise issues. 
• Condition monitoring: Cepstral analysis can be used for preventative maintenance, allowing parts to be replaced or repaired before they fail. This is done by tracking cepstral changes over time for gears and rotating components as they wear. 

 
This article will provide a brief history, some background, examples, and applications of the cepstrum:
1. Background
   1.1 Common Terms
   1.2  History
   1.3 Harmonics
   1.4 The Fourier Transform
2. Cepstral Algorithm
3. Signals and Their Cepstrum Transforms
4. Application Examples
   4.1 Gear Pair Defect
   4.2 Sideband Analysis  
   4.3 Order Analysis and Cepstrum


1.    Background 

Background, history, and terminology for cepstrum analysis.

1.1 Terms

Since the name cepstrum comes from reordering the first few letters of “spectrum”, many of the terms related to the cepstral domain have also been altered to indicate that cepstral processing has occurred. Some common ones include:

• Quefrency from frequency
• Saphe from phase
• Liftering from filtering
• Rahmonics from harmonics

Because only the log of the amplitude is kept when the signal is in the frequency domain, it doesn’t translate back to the same time signal when a inverse Fourier transform is performed. Instead, the signal is now in the “quefrency” domain.

1.2 History

Cepstrum analysis has been described in several published papers for voice recognition and defect detection:

• 1963 - Mathematicians Bogert, Healy, and Tukey. Its original application was in voice analysis. Human voices often consist of multiple harmonics, making cepstral analysis particularly useful for analyzing voice pitch, and even separating multiple voices present on a recording.
• 1980s and onward - Professor Robert Randall published multiple papers on use of cepstrum to identify faults in gears.

1.3 Harmonics

To understand exactly how the cepstrum works, it is helpful to understand the concept of harmonics. Harmonics are integer multiples of a fundamental frequency. Imagine striking a 440 Hz tuning fork against an object and listening to the sounds it makes. The spectrum of the tuning fork sound will not only contain the fundamental 440 Hz frequency but harmonics of 440 Hz as well (Figure 2),
 

• What is Total Harmonic Distortion (THD)?
• Harmonics + Harmonic Cursor in Simcenter Testlab
• Harmonic Removal

• STEP 1: First take the time signal and apply a Fast Fourier Transform (FFT) on the data to translate the time data into frequency data. 
• STEP 2: Next, in the frequency domain, discard the phase portion of the FFT by converting the amplitude to a Log¬10¬ of the data. This step reduces the dynamic range of the data by compressing it to a logarithm and makes it easier to find harmonic components that may otherwise not appear due to the amplitude differences between noise and the signal. 
• STEP 3: Convert the frequency signal to the quefrency domain using an Inverse Fourier Transform. This is the same process as a Fourier Transform, it just works in reverse. The key difference is that the phasing portion of an Inverse Fourier Transform rotates in the opposite direction of a Fourier Transform. In a regular application, applying an Inverse Fast Fourier Transform (IFFT) to the frequency domain would give the original time signal. In this instance, since only the amplitude was kept and the log of the amplitude was applied, it converts to the quefrency domain. An inverse Fourier transform is being performed on a Fourier transform, which is why the cepstrum is often described as a “spectrum of a spectrum”.

The formula for a cepstrum is shown in Equation 1 below:
 
The color coding in Equation 1 corresponds to the basic steps shown in Figure 4.

Note that the inverse Fourier transform works exactly the same as a Fourier transform, except that the phasor (j versus the -j in the equation) is rotating in the opposite direction. The equations are nearly identical, and for the purposes of cepstrum, function the exactly same way, which is matching frequency content with its corresponding sinewave. 

Performing cepstral analysis involves moving between time, frequency, and the cepstral domain as shown in Figure 5 below. 
  The convolution of two signals (Gx and Fx) in the time domain, is equal to multiplying the signals in the frequency domain. Due to the logarithmic nature of the cepstrum, this is equal to the addition of two signals (Gx + Fx) in the cepstral domain. This can make cepstrum useful for separating signals in a mathematically more efficient way. This is commonly used in voice analysis for determining a voice against a background. Though more complicated, this principle is how a voice track can be separated from a music track in karaoke for example. 

Important note: While the units for the cepstral domain technically appear in seconds, it is a mistake to interpret them as time. This is because the phase portion was lost when converting back from the frequency domain, since the log magnitude of the signal is used. Instead, the units are called “quefrency”. In this domain, all periodic frequencies are located at a quefrency of 1/f (where f is the distance in frequency between harmonics). 

3. Signals and Their Cepstrum Transforms

In Figure 6, both the Fourier transform and cepstrum of different signals were analyzed.
 
Sinewaves, square waves, sawtooth waves, and random signals were analyzed:

• Sine: For a sinewave, it appears periodic in the time domain (it oscillates up and down at a constant frequency). In the frequency domain, this is a single peak, since a sinewave is made up of one frequency. Because there are no harmonics in a single sinewave, the cepstrum will appear a noisy flat spectrum. 
• Sawtooth: A sawtooth wave is sum of multiple harmonics. If the fundamental frequency of a sawtooth wave is 100 Hz, the waveform is made up a combination of sinewave harmonics of 100 Hz. So in this case, it would be the same as adding 100 Hz, 200 Hz, 300 Hz, 400 Hz, etc. In the time domain, these appear as a jagged sawtooth wave. In the frequency domain, these appear as infinite evenly spaced peaks that are 100 Hz apart. Because the frequency domain contains evenly spaced periodic components, the cepstrum has peaks that correspond to the frequency domain spacings between harmonics.
• Square: A square wave is similar to the sawtooth wave, except that instead of all harmonics, square waves are a combination of oddly spaced harmonics. So if the square wave had a fundamental frequency of 100 Hz, it would be a combination of 100 Hz, 300 Hz, 500 Hz, etc. In the time domain, it produces a square wave. In the frequency domain, it is a series of uniformly spaced peaks that are spaced apart by twice the frequency of the original signal (since even harmonics are skipped).  For a 100 Hz square wave, the harmonics would be spaced 200 Hz apart. In the cepstral domain, there will be a dominant peak corresponding to the spacing between harmonics. All other peaks shown are mathematical artifacts of the Fast Fourier transform algorithm and can ultimately be ignored. The same is true for a sawtooth wave. 
• Random: A random wave, depending on frequency content, typically has a flat and noisy frequency spectrum, especially in the case of white noise. Since there are no periodic components in this wave’s frequency domain, a flat and noisy cepstrum can be expected here as well. 

4. Application Examples

This section contain several examples of how the cepstral analysis can be utilized to diagnosis noise and vibration issues in rotating machinery.

4.1 Gear Pair Defect

Consider a simple two gear system. In this example, time and FFT analysis will not identify a defect in the gear pair as readily as a cepstral analysis.

The driving or pinion gear, Gear A, has 42 teeth and spins at 68.57 RPM, while Gear B has 72 teeth and rotates at 40 RPM (Figure 7). 
 

• f_{Gear B} is the rotational speed of gear B 
• f_{Gear A} is the rotational speed of gear A 
• N_A is number of teeth on gear A
• N_B is number of teeth on gear B

The gear mesh frequency is calculated using Equation 3:
 
The same holds for Gear B as shown in Equation 5:
 

• Other Components: It is likely that the equipment that contains the gear pair has other rotating components (valves, motors, etc) that also produce additional frequency content. 
• Other Noise Sources: The microphone recording might be performed in a factory with other noise producing equipment nearby. The additional noise can disguise the harmonics of the gear pair in the object being tested.

As a result of these additional noise sources, it is difficult to identify the frequency content related to the gear pair by simply looking at the frequency spectrum. The presence or lack of harmonics is disguised by the additional frequency content from other sources in the microphone recordings.

• The blue traces are production measurements with no defects and thus no sidebands.
• The red trace shows sidebands around the fundamental frequency of the gear pair.

The blue "normal" production vibration spectrums do contain harmonics, but the defective gear measurements have additional harmonic content from the sidebands.

In this case, tracking the sideband energy may be enough to identify a defective gear pair from a normal production gear pair. In other cases, the sidebands may not distinctly stand out in the spectrum in which case a cepstrum analysis would be helpful as will be shown in the next section.

• Simcenter Testlab Neo: Modulation Metrics
• Interpreting Colormaps

4.3 Order Analysis and Cepstrum

If the speed of the machine is changing, even small amounts over time, the frequency will change as well. This will cause the frequency domain to smear and prevent the cepstrum from functioning properly. The results are shown in Figure 16.
 

• Blue: Normal production data without defects.
• Orange: The orange traces shows the cepstral order plot of a defect from gear pair 1.
• Pink: The pink traces show shows the cepstral order plot of a defect from gear pair 2.

The peaks in the cepstral domain are specific to the gear ratios of the gear pairs.  This makes it possible to put limits on the lines in the cepstrum domain, to ensure that gear function in the machine is healthy.