Uses and mis-uses of energy operators for machine diagnostics

• Published in: Mechanical systems and signal processing 2019-11, Vol.133, p.106199, Article 106199
• Main Authors: Randall, R.B., Smith, W.A.
• Format: Article
• Language: English
• Subjects: Advantages; Algorithms; Amplitude demodulation; Amplitudes; Bandwidths; Bearing; Bearing diagnostics; Carrier frequencies; Demodulation; Differentiation; Estimation; Filtration; Fourier transforms; Frequency demodulation; Frequency domain analysis; Frequency domain energy operator; Frequency ranges; Frequency weighted energy operator; Gear diagnostics; Hilbert transformation; Monitoring; Operators (mathematics); Phase distortion; Post-production processing; Potential energy; Real time; Signal processing; Speed determination; Teager Kaiser Energy Operator; Time domain analysis; Waveforms
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2019.06.017

•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. 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 t