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

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## Abstract

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.

## Keywords

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

## Notions in **Duffing oscillator**

*x*,*y*The state variables;

*δ*The damping ratio;

*x*−*x*^{3}The nonlinear restoring force;

*f*,*ω*The amplitude and frequency of the reference driving force;

*s*(*t*)The to-be-detected signal;

*f*_{e},*ω*_{e},*θ*_{e}The amplitude, frequency and phase of the to-be-detected signal;

*N*(*t*)A Gaussian white noise.

## Notions in **Particle Swarm Algorithm**

*n*The number of particles;

*m*The number of generations(iterations);

*d*The number of dimensions;

- \(v^{k}_{ij}\)
The velocity of the

*i*th particle in the*j*th dimension at the*k*th iteration;- \(p^{k}_{ij}\)
The position of the

*i*th particle in the*j*th dimension at the*k*th iteration;*v*_{max}The maximum velocity;

*ξ*The inertia weight factor;

*c*_{1},*c*_{2}The acceleration constant;

*r*_{1},*r*_{2}The random number between 0 and 1;

- \(\mathit{pbest}^{k}_{ij}\)
The

*p*_{best}of the*i*th particle in the*j*th dimension at the*k*th iteration;- \(\mathit{gbest}^{k}_{j}\)
The

*g*_{best}of the swarm of the*j*th dimension.

## Notes

### Acknowledgements

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.

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