Uses and mis-uses of energy operators for machine diagnostics. (1st November 2019) - Randall, R.B.; Smith, W.A.
Highlights: TKEO approximately equal to squared envelope of the derivative of a signal. Useful for real-time applications but of limited value for machine diagnostics. More accurate/efficient energy operator can be calculated in the frequency domain. Non-causal processing allows use of ideal filters and exact differentiation. Mono-component requirement can be relaxed for gear and bearing diagnostics. Abstract: The Teager Kaiser Energy Operator (TKEO) was originally proposed for use in speech analysis as representing the total energy (i.e. kinetic plus potential energy) in a signal. It was shown that for a mono-component carrier, with slowly changing amplitude and frequency, the TKEO is approximately equal to the product of the squares of the instantaneous amplitude and frequency. The TKEO is only strictly defined for mono-components, i.e. signals that can be modelled as a single carrier frequency, modulated in amplitude and frequency in such a way that they can be represented as the real part of an analytic signal, with a one-sided spectrum. The traditional way of estimating the TKEO was by an efficient time domain operation involving only three adjacent samples, and this can be done in real time, but this implies that all filtering and other processing must use causal processing to retain this advantage. However, causal filters give phase distortion and non-ideal filter characteristics. It is easily shown that the TKEO is approximately equal to the squared envelope of the derivative of the signal, which can alternatively be calculated by efficient non-causal Hilbert transform techniques via the frequency domain, incidentally giving a more accurate result, as well as being virtually as efficient. When combined with other non-causal processing, such as ideal filtering by choice of a specified band in the frequency domain, and ideal differentiation/integration by jω operations in the frequency domain, this approach has many advantages in cases where real-time processing is not required, and where the processing can be carried out by post-processing of recorded signals, which can be very long. Machine diagnostics is one area where real-time processing gives no advantage, and even numerous disadvantages, which accompany causal processing, such as mentioned above. Even in the single situation in machine monitoring where a result might be required rapidly, online monitoring of critical equipment, there is little practical difference in the processing time for causal and non-causal techniques (a maximum of a second or so) as this would rarely be sufficient time to make a decision on whether to shut a machine down, or for its speed to reduce significantly even if it were. The disadvantage of non-causal (batch) processing via Fourier transforms comes from the intrinsic circularity of the latter, where all functions in both time and frequency domains are assumed periodic. However, this has been dealt with since the birth of the FFT algorithm in 1965, and usually means that time records (or spectra) just have to be extended a small amount to allow truncation of wraparound effects. There are already a considerable number of papers published recommending the use of the TKEO and its variants for machine diagnostics, many claiming that this gives advantages over traditional approaches, for example of amplitude and frequency demodulation based on Hilbert transforms. However, this paper demonstrates that the claimed advantages are invariably false, for the following reasons: 1) The formulas derived for estimating the instantaneous amplitude and frequency of a mono-component using the TKEO actually give the values for the derivative of the signal, which are not the same. It is true that the time domain TKEO gives better results for a single chirp sweeping over a wide frequency range from zero (because of huge wraparound effects) but this situation does not apply to machine signals because of interference between multiple harmonics of shaft speeds, meaning that the maximum speed range in one record is 2:1. 2) By employing Hilbert transform and non-causal processing techniques, the errors and excessive time of causal time domain processing (for example time domain convolutional filtering, differentiation, etc) are avoided and virtually all other parameters and features of machine faults are estimated more accurately and faster, with considerably more control of frequency bandwidth and waveforms. Multiple differentiations can be achieved with equal accuracy in one operation. 3) Many of the proposed applications of the TKEO do not require the signal to be mono-component, such as the application to bearing diagnostics (since bearing signals do not have continuous phase) and where the only advantage of differentiation (increasing weighting with frequency) can only be realised with a frequency range much greater than the maximum 2:1 limit for a mono-component. This paper demonstrates all the above claims with a range of typical signals and applications to gear and bearing diagnostics, and rebuts many of the false claims previously made.