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We propose the piecewise re-scaled vibrational resonance (VR) method and the piecewise twice sampling VR method to amplify the weak linear frequency-modulated (LFM) signal. The system used to amplify the weak LFM signal is a typical bistable system with fractional-order deflection nonlinearity. The concrete procedures of both the piecewise re-scaled VR method and the piecewise twice sampling VR method are explained in detail. Through studying the effect of the factional-order exponent on VR, we find that the traditional bistable system is not the optimal model to improve the weak LFM signal. By investigating different parameters on the VR phenomenon, we verify the effectiveness of the two proposed methods.

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The authors construct the piecewise vibrational resonance (VR) method by introducing the piecewise idea and combine the re-scaled and twice sampling methods for the linear frequency modulated (LFM) signal. Importantly, they put forward a new spectral amplification factor to quantify the VR. The new spectral amplification factor has some advantages as compared with the cross-correlated coefficient, which can also be used to quantify the aperiodic VR. The top value of the spectrum amplification factor is the necessary and sufficient condition for the occurrence of the optimal VR phenomenon. Here, they discuss the effects of different parameters on VR by using some numerical examples. The method described in this study provides an effective way to improve the LFM signal, and even other kinds of frequency modulated signals.