Circuits, Systems, and Signal Processing

, Volume 33, Issue 1, pp 223–239 | Cite as

Phase Dependent and Independent Frequency Identification of Weak Signals Based on Duffing Oscillator via Particle Swarm Optimization

  • Yuan ChangEmail author
  • Yi Hao
  • Chunwen Li


In detecting weak signals based on the Duffing oscillator, it is usually assumed that the frequency is known, which is not always the case. This paper studies the problem of detecting the frequency of the to-be-detected weak signal based on the Duffing oscillator. For this purpose, the variance of the Duffing oscillator’s output is exploited, which has the property of multi-extremum single-maximum (MESM) distribution with the frequency of the periodic signal. The impact of signal’s phase on the MESM distribution is discussed. When the signal’s phase is known, the frequency of the signal can be directly identified as that with the maximal variance, which leads to a nonlinear optimization problem that can be solved by a particle swarm optimization (PSO) algorithm. When the phase is unknown, the π/2-phase-shift method is to be exploited integrated with a PSO algorithm. It is shown that the frequency can be precisely and efficiently identified by this method, whose effectiveness is verified by simulation results in Matlab.


Weak signal detection Duffing Chaotic Oscillator Frequency detection Variance Particle swarm optimization algorithm 


Notions in Duffing oscillator


The state variables;


The damping ratio;


The nonlinear restoring force;


The amplitude and frequency of the reference driving force;


The to-be-detected signal;


The amplitude, frequency and phase of the to-be-detected signal;


A Gaussian white noise.

Notions in Particle Swarm Algorithm


The number of particles;


The number of generations(iterations);


The number of dimensions;


The velocity of the ith particle in the jth dimension at the kth iteration;


The position of the ith particle in the jth dimension at the kth iteration;


The maximum velocity;


The inertia weight factor;


The acceleration constant;


The random number between 0 and 1;


The p best of the ith particle in the jth dimension at the kth iteration;


The g best of the swarm of the jth dimension.



The authors would like to express sincere thanks to professor Re-Bing Wu for comments and editing suggestions. Also, sincere thanks to the anonymous reviewers for their constructive comments and suggestions.


  1. 1.
    D. Birx, S. Pipenberg, Chaotic oscillators and complex mapping feed forward networks (CMFFNS) for signal detection in noisy environments, in Proc. International Joint Conference on Neural Networks, 7–11 Jun. (1992), pp. 881–888 Google Scholar
  2. 2.
    Y. Chang, C. Li, Y. Hao, Variance based identification of phase transition in Duffing oscillator for weak signal detection. Appl. Mech. Mater. 128–129, 354–358 (2011) CrossRefGoogle Scholar
  3. 3.
    R. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in Proc. Sixth International Symposium on Micro Machine and Human Science, 4–6 Oct. (1995), pp. 39–43 CrossRefGoogle Scholar
  4. 4.
    A. Hashemi, M. Meybodi, A note on the learning automata based algorithms for adaptive parameter selection in PSO. Appl. Soft Comput. 11(1), 689–705 (2011) CrossRefGoogle Scholar
  5. 5.
    N.Q. Hu, X.S. Wen, The application of Duffing oscillator in characteristic signal detection of early fault. J. Sound Vib. 268, 917–931 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    B. Jiao, Z. Lian, X. Gu, A dynamic inertia weight particle swarm optimization algorithm. Chaos Solitons Fractals 37(3), 698–705 (2008) CrossRefzbMATHGoogle Scholar
  7. 7.
    C. Li, L. Qu, Applications of chaotic oscillator in machinery fault diagnosis. Mech. Syst. Signal Process. 21, 257–269 (2007) CrossRefGoogle Scholar
  8. 8.
    C. Nie, Y. Shi, Z. Wang, B. Guo, A detection method of signal frequency based on optimization theory. Proc. SPIE 6357, 635701 (2006) CrossRefGoogle Scholar
  9. 9.
    V.N. Patel, N. Tandon, R.K. Pandey, Defect detection in deep groove ball bearing in presence of external vibration using envelope analysis and Duffing oscillator. Measurement 45, 960–970 (2012) CrossRefGoogle Scholar
  10. 10.
    A. Ratnaweera, S. Halgamuge, H. Watson, Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Trans. Evol. Comput. 8(3), 240–255 (2004) CrossRefGoogle Scholar
  11. 11.
    S.L. Sabata, L. Ali, S.K. Udgata, Integrated learning particle swarm optimizer for global optimization. Appl. Soft Comput. 11(1), 574–584 (2011) CrossRefGoogle Scholar
  12. 12.
    S.H. Shi, Y. Yuan, H.Q. Wang, M.K. Luo, Weak signal frequency detection method based on generalized Duffing. Chin. Phys. Lett. 28(4), 040502 (2011) CrossRefGoogle Scholar
  13. 13.
    G.Y. Wang, D. Chen, J. Lin, X. Chen, The application of chaotic oscillators to weak signal detection. IEEE Trans. Ind. Electron. 46(2), 440–444 (1999) CrossRefGoogle Scholar
  14. 14.
    G. Wang, S. He, A quantitative study on detection and estimation of weak signals by using chaotic Duffing oscillators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50(7), 945–953 (2003) CrossRefGoogle Scholar
  15. 15.
    G. Wang, W. Zheng, S. He, Estimation of amplitude and phase of a weak signal by using the property of sensitive dependence on initial conditions of a nonlinear oscillator. Signal Process. 82(1), 103–115 (2002) CrossRefzbMATHGoogle Scholar
  16. 16.
    L. Wang, Z. Li, Measurement the frequency of weak sinusoidal signal based on the genetic algorithm, in Proc. Sixth International Conference on Natural Computation (ICNC), 10–12 Aug. (2010), pp. 3597–3600 Google Scholar
  17. 17.
    W. Wang, Q. Li, G.J. Zhao, Novel approach based on chaotic oscillator for machinery fault diagnosis. Measurement 41, 904–911 (2008) CrossRefGoogle Scholar
  18. 18.
    Website of Bearing Data Center with the Case Western Reverse University: Seeded Fault Test Data [DB/OL].
  19. 19.
    X. Xiang, B. Shi, Weak signal detection based on the information fusion and chaotic oscillator. Chaos 20(1), 013104 (2010) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of AutomationTsinghua UniversityBeijingChina
  2. 2.Datang Microelectronics Technology Co., Ltd.BeijingChina

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