Glossary

is the name given to the physical variable represented by the output values of the Fourier transformation if the input signal is an acceleration signal such as is measured, for example, by a piezoelectric vibration pick-up. If integrated over a frequency interval, the acceleration density produces a frequency-specific acceleration in much the same way as the power density. The usual unit is 1 millimetre per second squared per Hertz = 1 mm ⁄ s2 ⁄ Hz.

is an error that occurs if frequencies occur in a signal that are higher than half the sampling rate. In this case the signal from the sampling can no longer be clearly reconstructed (Nyquist theorem). These frequencies are reflected in the spectrum as so-called image frequencies.

is the angle between the line along which the balls of a ball bearing touch the running surface and the plane that is perpendicular to the axis of the bearing. While the angle of contact is always close to zero in the case of bearings designed exclusively for radial loads, it can be significantly larger with bearings that also bear axial loads. It therefore depends both on the geometry and on the current load on the bearing and has an effect on the observable damage frequencies due to the pitch diameter. These are therefore not constant in the case of bearings for axial loads.

unwanted changes in the signal that result from errors in the processing, for example due to aliasing.

correction that takes into account the number of the degrees of freedom when estimating statistical moment coefficients from a series of data. Specifically, for example, the standard deviation is corrected by multiplying it by the factor sqrt(n/(n-1)), the skew by sqrt(n*(n-1)/(n-2)) and so on. The factor is generally negligible if n is a larger number.

designates one channel of a multi-channel signal output. The designation is used in particular with transformations that convert signals, such as the FFT or the formation of the histogram.

is a transformation based on frequency analysis that emphasises periodic elements in the spectrum due to harmonics or amplitude modulations. Distinction is made between the power cepstrum and the complex cepstrum.

is an artefact that can occur when signals are modified in the frequency domain and then transformed back into the time domain by means of an inverse FFT (Overlap-Add method). The modification can be described as a multiplication in the frequency domain, which generally corresponds to filtering in the time domain. This is equivalent to a cyclically defined folding with the pulse response of the filtering. If the pulse response is too long, then signal portions belonging to the beginning of the time period appear at the end of the section and vice versa. The reason for this is the cyclic definition of the discrete Fourier transformation. Extensive modifications in the frequency domain can thus lead to artefacts. As countermeasure the time signal can be supplemented by zeros before processing (zero padding), so that a reserve is created for the extension of the signal.

in this case: specification of the required resources of an algorithm (computing time and, if necessary, memory space). Condition Monitoring functions are called with vastly different data quantities; while a short-term FFT may be called with only 32 values, it may be useful, for example, to calculate a cepstrum for 16000 values. Therefore, in the case of a variable number of input data n, the algorithm is observed to see how it behaves with an increasing amount of data; in computer science this is normally described by the notation O(f(n)) (also called ‘Landau Notation’). This notation states that the complexity does not grow significantly faster than a function f(n) as n increases. An algorithm with the computing time complexity O(n) thus requires, for example, eight times the computing time for eight times the data amount n, while an algorithm with the complexity O(n2) already requires sixty-four times the computing time. An FFT of the complexity O(n * log2 n) conversely requires 112 times the computing time for n=16384 compared to n=256. With small amounts of data the computing time is usually dominated by a portion that is independent of the number of input data.

relationship between the peak value and the RMS value of a signal, normally expressed in decibels.

are characteristic frequencies that occur when certain machine elements are damaged. For example, certain frequencies are assigned to damage to the rolling elements, inner race, outer race and cage in roller bearings and these frequencies are proportional to the speed of rotation of the axis, depending on the angle of contact.

logarithmic scale for evaluating the intensity of oscillations or of intensity ratios. A decibel (symbol dB) is defined as one tenth of the auxiliary unit of measurement Bel. If x is a power value, then the value y in decibels = 20 * log10(x/x0). The value 1 or a defined reference value is used for x0.

or Fast Fourier Transformation: Fast Fourier Transformation, a calculation method for calculating the discrete Fourier transformation. Strictly speaking several such calculation methods exist, wherein the common implementations permit only power-of-two numbers as the input length (Cooley-Tukey algorithm). The common feature is a complexity of the order O(n * nlog(n)), i.e. the calculation of an FFT with 2048 points is a little more than four times as complex as for 512 points.

is a transformation that enables a time signal to be decomposed into different frequency portions, thus forming the basis for many frequency analysis methods. Instead of the continuous Fourier transformation, which represents a continuous function of an infinite signal, the discrete Fourier transformation (DFT) is normally used in practice as it is defined for a discrete, periodic signal. An efficient implementation of the discrete Fourier transformation, which is of great practical importance, is the Fast Fourier Transformation (FFT).

or frequency space is the name given to the representation of a signal on the basis of the values of the FFT. Since the complex Fourier spectrum of every signal can be clearly represented and can be transformed back into an equivalent time signal without losses, frequency domain and time domain (as so-called ‘orthonormal bases’ in the function space) represent equivalent representations of the same signal. Many operations for the analysis of signals can be performed more simply and efficiently in the frequency domain than in the time domain.

are oscillations that occur as integer multiples of a basic frequency. They are characteristic of pulse-type excitations and non-linear effects at the origin of the oscillation and in this case can typically be recognised by groups of lines in the spectrum with a constant distance between one another.

transformation that efficiently determines the ninety-degree phase-shifted signal from an oscillation signal. The Hilbert Transformation is used, for example, for the calculation of the analytical signal.

(sometimes also curtosis or curvature): indicator of the ‘impulsiveness’ or ‘peakness’ of a statistical distribution of values, determined from the fourth central statistical moment. For better evaluation of distributions, often the distance between curtosis of the measured distribution and curtosis of the normal distribution (value is 3) is used. This is then called excess curtosis. A Gaussian distribution accordingly has the excess curtosis zero, a distribution with many outliers achieves a value much greater than zero.

is the name given to methods that aim to automatically switch a plant off as quickly as possible if monitoring parameters exceed a critical threshold. In this way accidents and damage can be avoided.

is a collective term for statistical values such as mean value, standard deviation, skew and kurtosis of statistical variables. They are called that because they can be calculated from the central statistical moments of the probability distributions or histograms of these variables.

is a symbolic constant that marks invalid or missing values according to the IEEE 754 standard. The following points rank among the main characteristics of NaN values: All arithmetic operations that use NaN as input data return NaN as the result. All relational operators =, !=, > < >= <= always return the value False if at least one of the operands is NaN. The standard function isnan or _isnan returns the value True if the argument has the value NaN. The expression isnan(a) is equivalent to the expression !(a == a) or NOT(a = a). The fact that NaN values reproduce themselves when used in further calculations is advantageous in that invalid values cannot be overlooked

a theorem from communication technology and signal processing that states, slightly simplified, that a continuous signal must be sampled with a frequency greater than double that of the highest frequency contained in the signal so that the original signal can be reconstructed without loss of information or ambiguity from the time-discrete signal obtained in this way. This maximum frequency is called the Nyquist frequency. In practice filters are integrated into most D/A convertors that limit the maximum frequency of the input signal to a value smaller than half the sampling rate.

a method that enables a signal to be decomposed initially into short-term spectra without loss of information, then to process it further (e.g. to filter it) in the frequency domain and then to reconstruct it as a continuous time signal again.

is the designation of a value that is determined from a statistical variable. First of all its empirical frequency distribution (density function) is determined and from this the cumulative frequency distribution (also called cumulative distribution function) is calculated. The value of the percentile q is the maximum value which the random variable reaches in q percent of all cases, but does not exceed. This value is determined by the formation of the inverse function of the cumulative frequency distribution. The only difference between quantiles and percentiles is that quantiles use the decimal fraction instead of the corresponding percentages. The value of the 50-percent percentile is also called the median.

is the name given to the time axis that results from the calculation of the cepstrum. As a ‘scrambled’ reversal of the term ‘Frequency’, the name suggests the operations ‘inversion’ and ‘re-sorting’ which are characteristic of the cepstrum. As a result of two successive Fourier transformations, a transformation into the frequency domain initially results, with the assigned unit 1 Hertz. The second transformation leads in turn to a time domain in which, however, it is no longer the absolute time that lies on the axis, but the periodic durations determined by means of the cepstrum. The unit of quefrency is a second.

name for the analysis for the determination of primary causes of damage. This is of particular importance in the case of roller bearings, since primary damage leads to more complex consequential damage. Determination of the causes allows the emergence of damage to be effectively reduced.

is the frequency with which the analog signal is originally sampled and converted into digital values. This conversion takes place in steps of a constant length of time called the sampling period. The inverse value of the sampling period is called the sampling frequency and is expressed in Hertz. See also ‘Nyquist theorem’.

is the effect that the precise spectral value of narrow-band signals (for instance a sine signal or that of a calibrator) depends on which part of the FFT channel the frequency of the channel lies. The extent of the effect depends on the window function.

measurement of the asymmetry of a statistical distribution, determined from the third central statistical moment. A symmetrical distribution has a skew of zero.

denotes the representation of a signal using the temporally sampled values, as is originally available following a measurement. Since the Fourier spectrum of every signal can also be clearly represented and can be transformed back into an equivalent time signal without losses, the time domain and frequency domain (as so-called ‘orthonormal bases’ in the function space) represent equivalent representations of the same signal.

or meshing frequencies denotes the frequency with which the pairs of teeth in a gearbox touch each other. This contact causes the so-called meshing oscillation.

functions that are used, for example, in conjunction with a frequency analysis ( Welch method) to decompose long input signals without the addition of artificial jumps. As standard the Hann window can be used in almost all cases. The choice of window function affects the frequency and time resolution of the frequency analysis.

is the name given to the calculation step of the multiplication by a window function (see above).

denotes a processing step that is applied when an FFT with a certain length is to be calculated from a smaller number of samples. To do this the values of the time series are filled at the front and rear with zeros until the desired number of the values is attained. This usually requires the windowing of the signal e.g. according to the Welch method, so that no false jumps are created in the time series. Zero padding increases the frequency resolution of an FFT, which is equal to the sampling rate divided by the number of FFT points, but the information content of the original signal is, of course, not increased.