# A Novel Piecewise Tri-Stable Stochastic Resonance System Driven by Dichotomous Noise

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## Abstract

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## 1. Introduction

## 2. The PTSR System and Parameters

#### 2.1. Numerical Simulation

#### 2.2. Verification by Simulation

#### 2.3. Dichotomous Noise Drives Different Sr Models

## 3. SNR-GM of PTSR System Driven by Dichotomous Noise

#### 3.1. Impact of System Parameters $m$, $p$, and $q$

#### 3.2. Impact of Amplitude $A$ and Characteristic Frequency ${f}_{d}$ on SNR-GM

#### 3.3. Impact of Dichotomous Noise Parameters a and b on SNR-GM

## 4. Performance of the PTSR System Driven by Dichotomous Noise Compared with Driven by AWGN

## 5. Discussion

## 6. Conclusions

- Dichotomous noise as a driving source can still cause SR phenomena in the PTSR system, and dichotomous noise can transfer energy to periodic signals for signal enhancement.
- Compared with classical bistable SR and standard tri-stable SR, PTSR has higher signal enhancement when dichotomous noise is the driving source.
- PTSR system parameters $m,p,q$, periodic signal parameters $A,{f}_{d}$, and dichotomous noise parameters $a,b$ have an obvious effect on the SNR-GM of the system. The increase in $m$, $q,$ and ${f}_{d}$ cause the SNR-GM to show a downward trend, but the effect of $p$, $a,$ and $b$ is the opposite; amplitude $A$ has little influence on the size of SNR-GM but affects the size of optimal $D$.
- PTSR systems have better signal enhancement when they are driven by AWGN, and the range of $D$ adapted is relatively small when driven by dichotomous noise.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Dichotomous noise (a = 2, b = −1) (

**a**) D = 0.01 (

**b**) D = 0.02 (

**c**) D = 0.03 (

**d**) D = 0.04 (

**e**) D = 0.05 (

**f**) D = 0.06.

**Figure 3.**Original signal and output signal: (

**a**) input time domain signal, (

**b**) input frequency domain signal, (

**c**) output time domain signal, (

**d**) output frequency domain signal (m = 3.0, p = 0.5, q = 1.2).

**Figure 4.**SNR-GM of system: (

**a**) Just change m (p = 0.5, q = 1.2), (

**b**) Just change p (m = 2.0, q = 1.2), (

**c**) Just change q (m = 2.0, p = 0.5).

**Figure 11.**Input signal with dichotomous noise of different D (

**a**) time domain (D = 0.001), (

**b**) frequency domain (D = 0.001), (

**c**) time domain (D = 0.003), (

**d**) frequency domain (D = 0.003), (

**e**) time domain (D = 0.005), (

**f**) frequency domain (D = 0.005), (

**g**) time domain (D = 0.007), (

**h**) frequency domain (D = 0.007).

**Figure 12.**Input signal with AWGN of different D (

**a**) time domain (D = 0.5), (

**b**) frequency domain (D = 0.5), (

**c**) time domain (D = 0.7), (

**d**) frequency domain (D = 0.7), (

**e**) time domain (D = 0.9), (

**f**) frequency domain (D = 0.9), (

**g**) time domain (D = 1.1), (

**h**) frequency domain (D = 1.1).

**Figure 13.**Output signal with dichotomous noise of different D (

**a**) time domain (D = 0.001), (

**b**) frequency domain (D = 0.001), (

**c**) time domain (D = 0.003), (

**d**) frequency domain (D = 0.003), (

**e**) time domain (D = 0.005), (

**f**) frequency domain (D = 0.005), (

**g**) time domain (D = 0.007), (

**h**) frequency domain (D = 0.007).

**Figure 14.**Output signal with AWGN of different D (

**a**) time domain (D = 0.5), (

**b**) frequency domain (D = 0.5), (

**c**) time domain (D = 0.7), (

**d**) frequency domain (D = 0.7), (

**e**) time domain (D = 0.9), (

**f**) frequency domain (D = 0.9), (

**g**) time domain (D = 1.1), (

**h**) frequency domain (D = 1.1).

**Figure 15.**Comparison of high-value $D$ (

**a**) input time spectrum with dichotomous noise, (

**b**) input frequency spectrum with dichotomous noise, (

**c**) output time spectrum with dichotomous noise, (

**d**) output frequency spectrum with dichotomous noise, (

**e**) input time spectrum with AWGN, (

**f**) input frequency spectrum with AWGN, (

**g**) output time spectrum with AWGN, (

**h**) output frequency spectrum with AWGN.

System | Classical Bistable SR | Standard Tri-Stable SR | PTSR |

The Amplitude of the Enhanced Signal | 0.9658 | 1.41 | 1.449 |

Parameters | $m$ | $p$ | $q$ | $A$ | ${f}_{d}$ | $a$ and $b$ |

The change in SNR-GM | reduce | increase | reduce | The increase or decrease is not obvious, which affects the best matching $D$ | reduce | increase |

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**MDPI and ACS Style**

Zhao, S.; Shi, P.
A Novel Piecewise Tri-Stable Stochastic Resonance System Driven by Dichotomous Noise. *Sensors* **2023**, *23*, 1022.
https://doi.org/10.3390/s23021022

**AMA Style**

Zhao S, Shi P.
A Novel Piecewise Tri-Stable Stochastic Resonance System Driven by Dichotomous Noise. *Sensors*. 2023; 23(2):1022.
https://doi.org/10.3390/s23021022

**Chicago/Turabian Style**

Zhao, Shuai, and Peiming Shi.
2023. "A Novel Piecewise Tri-Stable Stochastic Resonance System Driven by Dichotomous Noise" *Sensors* 23, no. 2: 1022.
https://doi.org/10.3390/s23021022