User:Mbasti01/sandbox

Hi, ... this is my current draft section

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LITERATURE

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LINKS

https://www.bksv.com/media/doc/233-80.pdf

https://infosys.beckhoff.com/english.php?content=../content/1033/tf3600_tc3_condition_monitoring/27021598926718347.html&id=

https://www.phmsociety.org/sites/phmsociety.org/files/Tutorial%20Diagnostics%20Randall.pdf

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NEW

Cepstrum (plural cepstra) is the result of a mathematical transformation in the field of Fourier Analysis. The concept was introduced 1963 in an article of Bogert, Healy und Tukey. It serves as a tool to investigate periodic structures within frequency spectra. Such effects are related i.e. to echos/reflections in the signal or to the occurence of harmonic frequencies (partials, overtones). Mathematically it deals with the problem of deconvolution of signals in the frequency space.

The name cepstrum was derived by reversing the first four letters of "spectrum". Operations on cepstra are labelled quefrency analysis (aka quefrency alanysis ), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with "kepstrum", which also exists (see below).

The cepstrum is the result of following sequence of mathematical operations:


 * 1) transformation of a signal from time domain to freqency domain
 * 2) log of the spectral amplitudes
 * 3) transformation to quefrency domain, where the final independent variable, the quefrency, has actually a time scale.

The concept of the "Cepstrum" lead to a large number of applications :
 * dealing with reflection inference (radar, sonar applications, earth seismology)
 * estimation of speaker fundamental frequency (pitch)
 * speech analysis and recognition
 * medical applications in analysis of electroencephalogram (EEG) and brain waves
 * machine vibration analysis based on harmonic patterns (gearbox faults, turbine blade failures, ...)

Abbreviations
Following abbreviations are used in the formulas to explain the cepstrum in more detail:

Power Cepstrum
The "Cepstrum" was originally defined as "Power Cepstrum" by following relationship:


 * $$C_{p}=\left|\mathcal{F}^{-1}\left\{\log\left(\left|\mathcal{F}\{f(t)\}\right|^2\right)\right\}\right|^2$$

The power spectrum has main applications in analysis of sound and vibration signals. It is a complementary tool to spectral analysis.

Sometimes it is also defined as:


 * $$C_{p}=\left|\mathcal{F}\left\{\log\left(\left|\mathcal{F}\{f(t)\}\right|^2\right)\right\}\right|^2$$

Due to this formula, the spectrum is also sometimes called the spectrum of a spectrum. It can be shown that both formulas are consistent with each other as the the frequency spectral distribution remains the same, the only difference being a scaling factor which can be applied afterwards. Some articles prefer the second formula ref name="Randall_2002" />.

Real Cepstrum
The real cepstrum is defined as the complex spectrum, where the phase has been set to zero.


 * $$C_{r}=\mathcal{F}^{-1}\left\{\log_e(\mathcal{|F|})\right\}$$

Related definitions
A short-time cepstrum analysis was proposed by Schroeder and Noll for application to pitch determination of human speech.

The real cepstrum is based on the spectrum instead of the power spectrum. The "inner" squaring to achieve the power spectrum converts due to the log-operation just to factor of 2. And it can also be debated, whether the final squaring of the power spectrum is really needed [3].


 * (log x²)² = (2 log x)² = 4 (log x) ², if x is real, thus:


 * 4(real cepstrum)2 = power cepstrum

Thus: Many texts define the process as FT → abs → log → IFT, i.e., that the power cepstrum is the "inverse Fourier transform of the log-magnitude Fourier spectrum".

The phase cepstrum is given by the "inverse transform of the phase of the complex logarithm" [2] and may be used to verify the shape of the power spectrum in case of uncertainities due to phase unwrapping problems [2].

The autocepstrum is defined as the cepstrum of the autocorrelation. The autocepstrum is more accurate than the cepstrum in the analysis of data with echoes.

Applications
The cepstrum can be seen as information about the rate of change in the different spectrum bands. It was originally invented for characterizing the seismic echoes resulting from earthquakes and bomb explosions. It has also been used to determine the fundamental frequency of human speech and to analyze radar signal returns. Cepstrum pitch determination is particularly effective because the effects of the vocal excitation (pitch) and vocal tract (formants) are additive in the logarithm of the power spectrum and thus clearly separate.

The cepstrum is a representation used in homomorphic signal processing, to convert signals combined by convolution (such as a source and filter) into sums of their cepstra, for linear separation. In particular, the power cepstrum is often used as a feature vector for representing the human voice and musical signals. For these applications, the spectrum is usually first transformed using the mel scale. The result is called the mel-frequency cepstrum or MFC (its coefficients are called mel-frequency cepstral coefficients, or MFCCs). It is used for voice identification, pitch detection and much more. The cepstrum is useful in these applications because the low-frequency periodic excitation from the vocal cords and the formant filtering of the vocal tract, which convolve in the time domain and multiply in the frequency domain, are additive and in different regions in the quefrency domain.

Recently cepstrum based deconvolution was used to remove the effect of the stochastic impulse trains, which originates an sEMG signal, from the power spectrum of sEMG signal itself. In this way, only information on motor unit action potential (MUAP) shape and amplitude were maintained, and then, used to estimate the parameters of a time-domain model of the MUAP itself.

Cepstral concepts
The independent variable of a cepstral graph is called the quefrency. The quefrency is a measure of time, though not in the sense of a signal in the time domain. For example, if the sampling rate of an audio signal is 44100 Hz and there is a large peak in the cepstrum whose quefrency is 100 samples, the peak indicates the presence of a fundamental frequency that is 44100/100 = 441 Hz. This peak occurs in the cepstrum because the harmonics in the spectrum are periodic and the period corresponds to the fundamental frequency, since harmonics are integer multiples of said fundamental frequency.

Note that a pure sine wave can not be used to test the cepstrum for its pitch determination from quefrency as a pure sine wave does not contain any harmonics and does not lead to quefrency peaks. Rather, a test signal containing harmonics should be used (such as the sum of at least two sines where the second sine is some harmonic (multiple) of the first sine, or better, a signal with a square or triangle waveform, as such signals provide many overtones in the spectrum.).

Filtering
Playing further on the anagram theme, a filter that operates on a cepstrum might be called a lifter. A low-pass lifter is similar to a low-pass filter in the frequency domain. It can be implemented by multiplying by a window in the quefrency domain and then converting back to the frequency domain, resulting in a modified signal.

Convolution
A very important property of the cepstral domain is that the convolution of two signals can be expressed as the addition of their complex cepstra:


 * $$x_1 * x_2 \mapsto x'_1 + x'_2.$$

Further reading: - Brühler und Kjaer - Beckhoff

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OLD

A cepstrum (plural cepstra) is the result of taking the inverse Fourier transform (IFT) of the logarithm of the estimated signal spectrum. There is a complex cepstrum, a real cepstrum, a power cepstrum, and a phase cepstrum. The power cepstrum in particular has applications in the analysis of human speechand vibration diagnosis at machines, especially gearbox fault diagnosis REF1 REF2.

The name "cepstrum" was derived by reversing the first four letters of "spectrum". Operations on cepstra are labelled quefrency analysis (aka quefrency alanysis ), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with "kepstrum", which also exists (see below).

Origin and definition
The complex cepstrum was defined by Oppenheim in his development of homomorphic system theory and is defined as the inverse Fourier transform of the logarithm of the complex-valued Fourier transform of the signal (with unwrapped phase):


 * complex cepstrum of signal $= IFT(log(FT(the signal)) + j2πm$),

where $m$ is the integer required to properly unwrap the angle or imaginary part of the complex log function. It is also sometimes called the spectrum of a spectrum.

The real cepstrum uses the logarithm function defined for real values. The real cepstrum is related to the power by the relationship
 * 4(real cepstrum)2 = power cepstrum

and to the complex cepstrum as
 * real cepstrum = 0.5 × (complex cepstrum + time reversal of complex cepstrum).





The kepstrum, which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has the same relation to it as expected value has to statistical average, i.e. cepstrum is the empirically measured quantity, while kepstrum is the theoretical quantity. It was in use before the cepstrum.