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Higher-order spectra

Higher-order spectra

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Over the past decades, higher-order spectra (HOS), also called polyspectra, have established a status as a suitable mathematical and signal processing tool for nonlinear system analysis. However, a major problem with this kind of signal processing tool application is the interpretation of the obtained results, and much uncertainty still exists about the relation between HOS contribution compared with the second-order statistics. This chapter provided an important opportunity to advance the understanding advantages of HOS. The classical power spectrum (PS), which is defined as the Fourier transform (FT) of the autocorrelation sequence (the second-order cumulant), does not give any information about the phase of system frequency response; therefore, it is unable to give any indication about system nonlinearity. However, the HOS [1-11] are defined as the multidimensional FT of higher-order cumulants of a stationary random process and can overcome the inability of PS to detect these nonlinearities.

Chapter Contents:

  • 4.1 Introduction
  • 4.2 Higher-order statistics analysis: definitions and properties
  • 4.2.1 Higher-order moments
  • 4.2.2 Power spectrum
  • 4.2.3 Bispectrum and bicoherence
  • 4.2.4 Estimation
  • 4.3 Bispectrum use for harmonic signals' nonlinearity detection
  • 4.3.1 Case 1: a simple harmonic wave at frequency F0
  • 4.3.2 Case 2: sum of two harmonic waves at independent frequencies F0,F1; and with F1 = 2F0
  • 4.3.3 Case 3: sum of three harmonic waves at coupled frequencies, F2 = F0 + F1
  • 4.3.4 The use of bispectrum to detect and characterize nonlinearity
  • 4.3.4.1 QPC detection
  • 4.3.4.2 Robustness against the presence of additive Gaussian noise
  • 4.4 Practical applications of bispectrum-based fault diagnosis
  • 4.4.1 BRB fault detection
  • 4.4.1.1 Simulation and experimental tests for BRB fault
  • 4.4.1.2 Model of the BRB stator current
  • 4.4.1.3 Numerical simulation
  • 4.4.2 Bearing multi-fault diagnosis based on stator current HOS features and SVMs
  • 4.4.2.1 Bearing defect signatures
  • 4.4.2.2 BDs stator current bispectrum: a theoretical approach
  • 4.4.2.3 Features extraction and reduction
  • 4.4.2.4 Bearings *multi-fault classification proposed method
  • 4.4.2.5 BD classification based on SVM
  • 4.4.2.6 Experimental results
  • 4.4.2.7 Training and test vectors
  • 4.4.3 Bispectrum-based EMD applied to the nonstationary vibration signals for bearing fault diagnosis
  • 4.4.3.1 Nonstationary nature of defective REB vibration response
  • 4.4.3.2 Brief description of EMD
  • 4.4.3.3 Experimental results
  • 4.4.4 The use of SK for bearing fault diagnosis
  • 4.4.4.1 SK and its application for bearing fault diagnosis
  • 4.4.4.2 SESK proposed method
  • 4.4.4.3 Experimental results
  • 4.5 Conclusions and perspectives
  • Appendix A
  • Appendix B
  • References

Inspec keywords: random processes; signal processing; frequency response; correlation methods; Fourier transforms; higher order statistics

Other keywords: signal processing tool; stationary random process; nonlinear system analysis; frequency response; autocorrelation sequence; mathematical tool; higher-order spectra; second-order statistics; HOS; power spectrum; higher-order cumulants; multidimensional Fourier transform; second-order cumulant

Subjects: Signal processing theory; Other topics in statistics; Signal processing and detection; Other topics in statistics; Integral transforms in numerical analysis; Integral transforms in numerical analysis

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