A quadratic polynomial receiving scheme for sine signals enhanced by stochastic resonance

Liu Guang-Kai; Quan Hou-De; Kang Yan-Mei; Sun Hui-Xian; Cui Pei-Zhang; Han Yue-Ming

doi:10.7498/aps.68.20190952

Abstract

Aiming at the reception of the intermediate frequency signal of sine wave of radio and communication system at extremely low signal-to-noise ratio (SNR), a quadratic polynomial receiving scheme for sine signals enhanced by stochastic resonance (SR) is proposed. Through analyzing the mechanism of sine signals enhanced by SR and introducing the decision time, the analytic periodic stable solution with time parameters of the Fokker-Planck Equation (FPE) is obtained through converting the non-autonomous FPE into an autonomous equation. Based on the probability density function of the particle of SR output, a quadratic polynomial receiving scheme is proposed by analyzing the feature of energy detector and matching filter receiver. By maximizng the deflection coefficient, the binomial coefficients and the test statistic are obtained. For further reducing the bit error, by combining the thought of " the average of N samples”, a quadratic polynomial receiving scheme for sine signals enhanced by SR is proposed through the hypothesis under Gaussian distribution approximation of the law of large N. And the conclusion is obtained as follows. When N is 500 and the SNR is greater than –17 dB, the bit error rate is less than 2.2 × 10^–2, under the constraint of the parameters of the optimally matched SR.

Keywords:

PACS:

Authors and contacts

1.
Department of Electronics and Optical Engineering, Army Engineering University, Shijiazhuang 050003, China
2.
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
3.
The Troop of 66389, Shijiazhang 050000, China

Corresponding author: Liu Guang-Kai, dreamer_gk@163.com

Funds: Project supported by the Natural Science Foundation of Hebei Province, China (Grant No. F2017506006)

FullText HTML

References

[1]	Quan H D, Zhao H, Cui P Z 2015 Wirel. Pers. Commun. 81 1159 Google Scholar
[2]	赵寰, 全厚德, 崔佩璋 2015 系统工程与电子技术 37 671 Zhao H, Quan H D, Cui P Z 2015 Syst. Eng. Electron. 37 671
[3]	Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A 14 453 Google Scholar
[4]	Gammaitoni L, Hänggi P, Jung P 1998 Rev. Mod. Phys. 70 223 Google Scholar
[5]	Chapeau B F, Godivier X 1997 Phys. Rev. E 55 1478 Google Scholar
[6]	Tougaard J 2002 Biol. Cybern. 87 79 Google Scholar
[7]	Kang Y M 2016 Phys. Lett. A 380 3160
[8]	王珊, 王辅忠 2018 67 160502 Google Scholar Wang S, Wang F Z 2018 Acta Phys. Sin. 67 160502 Google Scholar
[9]	Krauss P, Metzner C, Schilling A 2017 Sci. Rep. 7 2450 Google Scholar
[10]	Galdi V, Pierro V, Pinto I M 1998 Phys. Rev. E 57 6470 Google Scholar
[11]	Zozor S, Amblard P O 2002 Signal Process. 82 353 Google Scholar
[12]	Zozor S, Amblard P O 2003 IEEE Trans. Signal Process. 51 3177 Google Scholar
[13]	Chen H, Varshney P K, Kay S M 2007 IEEE Trans. Signal Process. 55 3172 Google Scholar
[14]	Chen H, Varshney P K, Kay S M 2008 IEEE Trans. Signal Process. 56 5031
[15]	Wang J, Ren X, Zhang S 2014 IEEE Trans. Wireless Commun. 13 4014 Google Scholar
[16]	Zhang Z, Kang Y, Xie Y 2014 Commun. Theor. Phys. 61 578 Google Scholar
[17]	Zhang L, Song A 2018 Phys. A 503 958 Google Scholar
[18]	胡岗 1994 随机力与非线性系统 (上海: 上海科技教育出版社) 第220−225页 Hu G 1994 Stochastic Force and Nonlinear Systems (Shanghai: Shanghai Scientific and Technological Education) pp220−225 (in Chinese)
[19]	卢志恒, 林建恒, 胡岗 1993 42 1556 Google Scholar Lu Z H, Lin J H, Hu G 1993 Acta Phys. Sin. 42 1556 Google Scholar
[20]	Socha L 2005 Appl. Mech. Rev. 58 178
[21]	Askari M, Adibi H 2015 Ain Shams Eng. J. 6 1211 Google Scholar
[22]	胡茑庆 2012 随机共振微弱特征信号检测理论与方法 (北京: 国防工业出版社) 第85, 86页 Hu N Q 2012 The Theory of Detection and Estimation Using Stochastic Resonance (Beijing: National Defence Publishing) pp85, 86 (in Chinese)
[23]	康艳梅, 徐健学, 谢勇 2003 52 2712 Kang Y M, Xu J X, Xie Y 2003 Acta Phys. Sin. 52 2712
[24]	史蒂文 M L 著 (罗鹏飞译) 2014 统计信号处理基础-估计与检测理论 (北京: 电子工业出版社) 第397−400页 Steven M K (translated by Luo P F) 2014 Fundamentals of Statistical Signal Processing (Beijing: Publishing House of Electronics Industry) pp397−400 (in Chinese)
[25]	Picinbono B 1995 IEEE Trans. Aerosp. Electron. Syst. 31 1072 Google Scholar
[26]	盛骤, 谢式千, 潘承毅 2008 概率论与数理统计 (北京: 高等教育出版社) 第147−153页 Sheng Z, Xie S Q, Pan C Y 2008 Probability Theory and Mathematical Statistics (Beijing: Higher Education Press) pp147−153 (in Chinese)

图 1 SR系统增强正弦信号的二次多项式接收结构

Figure 1. Quadratic polynomial receiving structure for sine signals enhanced by SR.

DownLoad: Full-Size Img PowerPoint

图 2 正弦信号经SR系统增强前后的时频域波形(输入SNR = –18 dB, 噪声功率 ${\sigma ^2} = 4$ , 信号幅度 $A = 0.25$ , SR系统参数 $a = 1 \times {10^4}$ , $b = 3.3856 \times {10^{12}}$ )　(a) 输入信号时域波形; (b)输入信号频域幅值谱; (c)输出信号时域波形; (d)输出信号频域幅值谱

Figure 2. Waveform of time and frequency zone of sine wave enhanced by SR (input SNR = –18 dB, the noise intensity ${\sigma ^2} = 4$ , signal amplitude $A = 0.25$ , parameters of system $a = 1 \times {10^4}$ , $b = 3.3856 \times {10^{12}}$ ): (a) The waveform of input signal in time zone; (b) the amplitude of input signal in frequency zone; (c) the waveform of output signal in time zone; (d) the amplitude of output signal in frequency zone

DownLoad: Full-Size Img PowerPoint

图 3 粒子处于不同位置时的概率密度(输入SNR = –14 dB dB, 噪声功率 ${\sigma ^2} = 4$ , 信号幅度 $A = 0.4$ , SR系统参数 $a = 1 \times {10^4}$ , $b = 2.6406 \times {10^{12}}$ )　(a)未经SR处理的粒子的分布概率; (b)经SR处理后粒子的分布概率; (c)经SR处理后粒子的分布概率局部图

Figure 3. Probability density function of particles of SR (input SNR = –14 dB, the noise intensity ${\sigma ^2} = 4$ , signal amplitude $A = 0.4$ , parameters of system $a = 1 \times {10^4}$ , $b = 2.6406 \times {10^{12}}$ ): (a) The probability density of particles before SR processed; (b) the probability density of particles after SR processed; (c) the partial of probability density of particles after SR processed

DownLoad: Full-Size Img PowerPoint

图 4 不同N时 $g\left( {{x_0}} \right)$ 的输出值(输入SNR = –18 dB, 噪声功率 ${\sigma ^2} = 4$ , 信号幅度 $A = 0.25$ , SR系统参数 $a = 1 \times {10^4}$ , $b = 3.3856 \times {10^{12}}$ )　(a) N = 1时检验统计量的时域波形; (b) N = 10时检验统计量的时域波形

Figure 4. Output of $g\left( {{x_0}} \right)$ at different N (input SNR = –18 dB, the noise intensity ${\sigma ^2} = 4$ , signal amplitude $A = 0.25$ , parameters of system $a = 1 \times {10^4}$ , $b = 3.3856 \times {10^{12}}$ ): (a) The waveform of test statistics when N = 1; (b) the waveform of test statistics when N = 10

DownLoad: Full-Size Img PowerPoint

图 5 不同N时g(x₀)的输出概率密度(输入SNR = –14 dB, 噪声功率 ${\sigma ^2} = 4$ , 信号幅度A = 0.4, SR系统参数 $a = 1 \;\times$ 10⁴, $b = 2.6406 \times {10^{12}}$ , $g\left( x \right) = {x^2} + 0.0701x$ )　(a) N = 1时粒子的分布概率; (b) N = 10时粒子的分布概率; (c) N = 50时粒子的分布概率; (d) N = 100时粒子的分布概率

Figure 5. Output probability density function of $g\left( {{x_0}} \right)$ at different N (input SNR = –14 dB, the noise intensity ${\sigma ^2} = 4$ , signal amplitude $A = 0.4$ , parameters of system $a = 1 \times$ 10⁴, $b = 2.6406 \times {10^{12}}$ , $g\left( x \right) = {x^2} + 0.0701x$ ): (a) The output probability density when N = 1; (b) the output probability density when N = 10; (c) the output probability density when N = 50; (d) the output probability density when N = 100

DownLoad: Full-Size Img PowerPoint

图 6 不同接收结构的系统输出误码率

Figure 6. Output bit error ratio of different receiving structure

DownLoad: Full-Size Img PowerPoint

图 7 不同N时的二次多项式接收结构的误码率

Figure 7. Output bit error ratio of quadratic polynomial receiving structure at different N

DownLoad: Full-Size Img PowerPoint

map

[1]	Quan H D, Zhao H, Cui P Z 2015 Wirel. Pers. Commun. 81 1159 Google Scholar
[2]	赵寰, 全厚德, 崔佩璋 2015 系统工程与电子技术 37 671 Zhao H, Quan H D, Cui P Z 2015 Syst. Eng. Electron. 37 671
[3]	Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A 14 453 Google Scholar
[4]	Gammaitoni L, Hänggi P, Jung P 1998 Rev. Mod. Phys. 70 223 Google Scholar
[5]	Chapeau B F, Godivier X 1997 Phys. Rev. E 55 1478 Google Scholar
[6]	Tougaard J 2002 Biol. Cybern. 87 79 Google Scholar
[7]	Kang Y M 2016 Phys. Lett. A 380 3160
[8]	王珊, 王辅忠 2018 67 160502 Google Scholar Wang S, Wang F Z 2018 Acta Phys. Sin. 67 160502 Google Scholar
[9]	Krauss P, Metzner C, Schilling A 2017 Sci. Rep. 7 2450 Google Scholar
[10]	Galdi V, Pierro V, Pinto I M 1998 Phys. Rev. E 57 6470 Google Scholar
[11]	Zozor S, Amblard P O 2002 Signal Process. 82 353 Google Scholar
[12]	Zozor S, Amblard P O 2003 IEEE Trans. Signal Process. 51 3177 Google Scholar
[13]	Chen H, Varshney P K, Kay S M 2007 IEEE Trans. Signal Process. 55 3172 Google Scholar
[14]	Chen H, Varshney P K, Kay S M 2008 IEEE Trans. Signal Process. 56 5031
[15]	Wang J, Ren X, Zhang S 2014 IEEE Trans. Wireless Commun. 13 4014 Google Scholar
[16]	Zhang Z, Kang Y, Xie Y 2014 Commun. Theor. Phys. 61 578 Google Scholar
[17]	Zhang L, Song A 2018 Phys. A 503 958 Google Scholar
[18]	胡岗 1994 随机力与非线性系统 (上海: 上海科技教育出版社) 第220−225页 Hu G 1994 Stochastic Force and Nonlinear Systems (Shanghai: Shanghai Scientific and Technological Education) pp220−225 (in Chinese)
[19]	卢志恒, 林建恒, 胡岗 1993 42 1556 Google Scholar Lu Z H, Lin J H, Hu G 1993 Acta Phys. Sin. 42 1556 Google Scholar
[20]	Socha L 2005 Appl. Mech. Rev. 58 178
[21]	Askari M, Adibi H 2015 Ain Shams Eng. J. 6 1211 Google Scholar
[22]	胡茑庆 2012 随机共振微弱特征信号检测理论与方法 (北京: 国防工业出版社) 第85, 86页 Hu N Q 2012 The Theory of Detection and Estimation Using Stochastic Resonance (Beijing: National Defence Publishing) pp85, 86 (in Chinese)
[23]	康艳梅, 徐健学, 谢勇 2003 52 2712 Kang Y M, Xu J X, Xie Y 2003 Acta Phys. Sin. 52 2712
[24]	史蒂文 M L 著 (罗鹏飞译) 2014 统计信号处理基础-估计与检测理论 (北京: 电子工业出版社) 第397−400页 Steven M K (translated by Luo P F) 2014 Fundamentals of Statistical Signal Processing (Beijing: Publishing House of Electronics Industry) pp397−400 (in Chinese)
[25]	Picinbono B 1995 IEEE Trans. Aerosp. Electron. Syst. 31 1072 Google Scholar
[26]	盛骤, 谢式千, 潘承毅 2008 概率论与数理统计 (北京: 高等教育出版社) 第147−153页 Sheng Z, Xie S Q, Pan C Y 2008 Probability Theory and Mathematical Statistics (Beijing: Higher Education Press) pp147−153 (in Chinese)

[1]	Jiao Shang-Bin, Ren Chao, Huang Wei-Chao, Liang Yan-Ming. Parameter-induced stochastic resonance in multi-frequency weak signal detection with stable noise. Acta Physica Sinica, 2013, 62(21): 210501. doi: 10.7498/aps.62.210501
[2]	Zhang Lu, Zhong Su-Chuan, Peng Hao, Luo Mao-Kang. Stochastic resonance in an over-damped linear oscillator driven by multiplicative quadratic noise. Acta Physica Sinica, 2012, 61(13): 130503. doi: 10.7498/aps.61.130503
[3]	Gao Shi-Long, Zhong Su-Chuan, Wei Kun, Ma Hong. Overdamped fractional Langevin equation and its stochastic resonance. Acta Physica Sinica, 2012, 61(10): 100502. doi: 10.7498/aps.61.100502
[4]	Gao Shi-Long, Zhong Su-Chuan, Wei Kun, Ma Hong. Weak signal detection based on chaos and stochastic resonance. Acta Physica Sinica, 2012, 61(18): 180501. doi: 10.7498/aps.61.180501
[5]	Zhang Li, Yuan Xiu-Hua, Wu Li. Stochastic resonance for pulse signal modulated by noise in a single-mode laser system. Acta Physica Sinica, 2012, 61(11): 110501. doi: 10.7498/aps.61.110501
[6]	Leng Yong-Gang, Lai Zhi-Hui, Fan Sheng-Bo, Gao Yu-Ji. Large parameter stochastic resonance of two-dimensional Duffing oscillator and its application on weak signal detection. Acta Physica Sinica, 2012, 61(23): 230502. doi: 10.7498/aps.61.230502
[7]	Zhu Guang-Qi, Ding Ke, Zhang Yu, Zhao Yuan. Experimental research of weak signal detection based on the stochastic resonance of nonlinear system. Acta Physica Sinica, 2010, 59(5): 3001-3006. doi: 10.7498/aps.59.3001
[8]	Wang Mao-Sheng. Frequency-dependent stochastic resonance in a two-dimensional neural map. Acta Physica Sinica, 2009, 58(10): 6833-6837. doi: 10.7498/aps.58.6833
[9]	Ning Li-Juan, Xu Wei. Stochastic resonance under modulated noise in linear systems driven by dichotomous noise. Acta Physica Sinica, 2009, 58(5): 2889-2894. doi: 10.7498/aps.58.2889
[10]	Guo Li-Min, Xu Wei, Ruan Chun-Lei, Zhao Yan. Stochastic resonance for dichotomous noise in a second derivative linear system. Acta Physica Sinica, 2008, 57(12): 7482-7486. doi: 10.7498/aps.57.7482
[11]	Zhang Liang-Ying, Jin Guo-Xiang, Cao Li. Stochastic resonance of frequency modulated signals in a linear model of single-mode laser. Acta Physica Sinica, 2008, 57(8): 4706-4711. doi: 10.7498/aps.57.4706
[12]	Zhou Bing-Chang, Xu Wei. Stochastic resonance in an asymmetric bistable system driven by mixed periodic force and noises. Acta Physica Sinica, 2007, 56(10): 5623-5628. doi: 10.7498/aps.56.5623
[13]	Zhou Xiao-Rong, Luo Xiao-Shu, Jiang Pin-Qun, Yuan Wu-Jie. The second super-harmonic stochastic resonance in the neural networks with small-world character. Acta Physica Sinica, 2007, 56(10): 5679-5683. doi: 10.7498/aps.56.5679
[14]	Lin Min, Huang Yong-Mei. Modulation and demodulation for detecting weak periodic signal of stochastic resonance. Acta Physica Sinica, 2006, 55(7): 3277-3282. doi: 10.7498/aps.55.3277
[15]	Xu Wei, Jin Yan-Fei, Xu Meng, Li Wei. Stochastic resonance for bias-signal-modulated noise in a linear system. Acta Physica Sinica, 2005, 54(11): 5027-5033. doi: 10.7498/aps.54.5027
[16]	Jin Yan-Fei, Xu Wei, Li Wei, Xu Meng. Stochastic resonance for periodically modulated noise in a linear system. Acta Physica Sinica, 2005, 54(6): 2562-2567. doi: 10.7498/aps.54.2562
[17]	Han Li-Bo, Cao Li, Wu Da-Jin, Wang Jun. Stochastic resonance in a single-mode laser driven by the direct signal-modulated correlated colored noise. Acta Physica Sinica, 2004, 53(7): 2127-2132. doi: 10.7498/aps.53.2127
[18]	Leng Yong-Gang, Wang Tai-Yong, Qin Xu-Da, Li Rui-Xin, Guo Yan. Power spectrum research of twice sampling stochastic resonance response in a bistable system. Acta Physica Sinica, 2004, 53(3): 717-723. doi: 10.7498/aps.53.717
[19]	Zhu Heng-Jiang, Li Rong, Wen Xiao-Dong. Extracting information signal under noise by stochastic resonance. Acta Physica Sinica, 2003, 52(10): 2404-2408. doi: 10.7498/aps.52.2404
[20]	Leng Yong-Gang, Wang Tai-Yong. Numerical research of twice sampling stochastic resonance for the detection of a weak signal submerged in a heavy Noise. Acta Physica Sinica, 2003, 52(10): 2432-2437. doi: 10.7498/aps.52.2432

Catalog

Vol. 68, Issue 21 - 2019-11-05

Metrics

Abstract views: 8573
PDF Downloads: 59

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板