Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 150–153___________________________________________________________________________________________________________Stochastic Resonance in an Array of Dynamical Saturating Nonlinearity with Second-OrderYumei Ma

College of Computer Science & Technology, Qingdao University, 308 Ningxia Road, Shinan District
Qingdao, 266071, China
Lin ZhaoCollege of Automation and Electrical Engineering, Qingdao University, 308 Ningxia Road, Shinan District
Qingdao, 266071, China
Zhenkuan Pan

College of Computer Science & Technology, Qingdao University, 308 Ningxia Road, Shinan District
Qingdao, 266071,China
Jinpeng Yu

College of Automation and Electrical Engineering, Qingdao University, 308 Ningxia Road, Shinan District
Qingdao,266071, China
E-mail: mayumei_qdu@163.com; zhaolin1585@163.com;zkpan@qdu.edu.cn; yjp1109@hotmail.com
www.qdu.edu.cn

Abstract
The transmission of weak noisy signal by parallel array of dynamical saturating nonlinearities with secondorder is studied. Firstly, the numerical results demonstrate that the output SNR can be enhanced by parallel array ofdynamical saturating nonlinearities with second-order by tuning the internal noise. Secondly, the SR effects can beoptimized by the self-coupling coefficient of the dynamical nonlinearity. Then, the SR effects when the nonGaussian noise acts as the external noise are superior to that with external Gaussian noise.
Keywords: stochastic resonance, dynamical nonlinearity, second-order, signal-to-noise ratio

Collins designed the neuron network model, summing
nonlinear units into a parallel array, the system
performance can be enhanced by adjusting the couplingstrength and the array noise intensity [3]. Subsequently,
various SR effects as types of array SR are investigated,
for instance, the Suprathreshold SR [4].The positive roleof the array noise has been found in some complex
networks, e.g. stochastic pooling networks [5], scalefree networks [6] and small-world networks

1. Introduction
Stochastic resonance (SR) establishes a phenomenon
where the additive noise can enhance the performanceof some certain nonlinear systems [1-16]. Benzi initiallyobserved the SR effect in climate model several decades
ago [1]. Then, the existence of SR is proved withexperiments by McNamara [2].Later, some new types ofSR models are applied in many research fields. WhenCopyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 150–153___________________________________________________________________________________________________________[7].Recently, the SR in some physical system focuses on
the noisy bistable system, such as the bistablefractrional-order system, asymmetric bistable systemand fractional harmonic oscillator and so on [8-10]. The
SR effects can be measured by signal-to-noise ratio, thefisher information [11], etc.
In this paper, the SR effect in parallel array ofdynamical saturating nonlinearities with second-order isfirstly studied. It is demonstrated that, the SR effectoccurs in arrays of dynamical saturating nonlinearities
with second-order as increasing the array size and thearray noise intensity. And diverse forms of the output
SNR appear which against the array noise intensity andthe array scale. When the Gaussian noise acts as theexternal noise, the self-coupling coefficient has a greater
impact on the output SNR. While the self-coupling
coefficient takes smaller value, the SR effect isobviously visible. With larger value of the self-coupling
coefficient, the bell-shape behavior of the output is notclearly. As the array size N → ∞ , the output of thenonlinear systems with the external Laplacian noise
precedes that with the external Gaussian noise.

100

output SNR

80

0Rout =

| 〈 E[Z(t)]exp(−i 2p t / T )〉 |〈 var[Z(t)]〉 H (1 / Ts )∆B

2(3)Similarly, the input SNR is given by2| 〈 s (t) exp(−i 2p t / T )〉 |(4)Rin =s θ2 ∆t ∆B
When a sinusoidal signal buried in white noise as
the input signal, the input SNR is written as
A2(5)Rin =24σ θ ∆t ∆B
When the array size N → ∞ , the array output SNR canbe defined as| 〈 E[yi (t)]exp(−i 2p t / T )〉 |2∞Rout=(6)〈 E[yi (t) y j (t)] − E 2 [yi (t)]〉 H (1/ Ts ) ∆B

to a white noise θ (t) , which is independent of s (t) witha probability density function (PDF) fθ and variance∫ x fθ (x) dx . The maximal amplitude
2−∞of s (t) is A ( | s (t) |≤ A ) and the period is T . Next, theinput mixture γ =(t) s (t) + θ (t) is applied to eachidentical subsystem of an uncoupled parallel array.

α i (t) plays a role of array noise, independent of γ (t) ,with the same PDF and variance σ α2 .Then the output ofa subsystem as=yi (t) k[γ (t) + α i (t)]54321
noise RMS amplitude saFig. 1. Output SNR as a function of the RMS amplitude σ αof the array noise α i (t) .

We consider a weak periodic sinusoid signal s (t) added

∞

4020

2. Theoretical Model

2=σ θ2 E=θ [x ]

60

3. Experiment Results(1)

In this section, we consider a dynamical saturating
nonlinearity with second-order as Eq.(7)2dxdx=− − xi (t) + C tanh[ω xi (t)] + s (t) + θ (t) + a i (t)
2dtdt
(7)
where C is the self-coupling coefficient and ω as aslope parameter. Here, s (t) = 0.2sin(2π t/ T) is adeterministic sinusoid with period T. The component
θ (t) is the external noise and it is the zero-mean
generalized Gaussian noise, and some general cases

Then the system output Z (t) is written as
1 N(2)Z (t) = ∑ yi (t)N i =1The output SNR can be a measure of the system
performance, which is defined as the power contained in
the output spectral line at fundamental frequency
1 / T divided by the power contained in the noisebackground in a small frequency bin ∆B around 1 / T ,that is151

Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 150–153___________________________________________________________________________________________________________120350(a)100300output SNR

80250output SNR

604020012345noise RMS amplitude sa

2001501005012000(b)100output SNR

80C23Fig. 3.Output SNR as a function of the self-coupling
coefficient C of Eq.(7).

6040of Eq.(7) takes C=0.5 in Fig.1. While C=1 in Fig.2(a)and C=2 in Fig.2(b).In Fig.1 and Fig.2, the parameterµ of Eq.(8) is set µ = 2 , i.e., the Gaussian noise acts
as the external noise. The external noise level σ θ = 0.3 .
The output SNR Rout is revealed for N=1, 5, 10, 50
and ∞ from the bottom up in Fig.1 and Fig.2.The SR
effect occurs upon increasing the array noise level σ α
and the array size N . It is shown that the bell-shape
behavior is obviously as the array size N ≥ 1 in Fig.1.
But the SR effect appears lightly in Fig.2 with a certain
small range of the array noise intensity when N > 1 inFig.2. The SR effect is more unstable while increasing
the self-coupling coefficient C in Fig.2. By comparison,
it is shown that the SR effect can be obtained by tuning
the self-coupling coefficient C of the dynamical
saturating nonlinearity with second-order.
Figure 3 illustrates the output SNR as a function of
the self-coupling coefficient as the array size
N → ∞ with different external noises. The output SNR
is represented by triangles when Gaussian noise acts as
the external noise, while that is represented bypentagrams with external Laplacian noise. The RMS
amplitude of the external noise σ θ = 0.3 and the arraynoise intensity is set σ α = 1/ 3 . The system output
with external Laplacian noise is obviously superior tothat of the nonlinear array when Gaussian noise is theexternal noise.

200112345noise RMS amplitude saFig. 2. Output SNR as a function of the RMS amplitude σ αof the array noise α i (t) .The self-coupling coefficient C =1 in(a) and C =2 in (b).

such as Gaussian noise are contained in the class. ThePDF of the generalized Gaussian noise θ (t) is defined
asc1x µ(8)=fθ (x)exp(− c 2 || )σθσθwhere c1 and c 2 is defined as Eq.(9) and Eq.(10)µ13Γ 2 (3µ −1 ) / Γ 2 ( µ −1 )c1 =2(9)c2 =[Γ(3µ −1 ) / Γ( µ −1 )] 2(10)µThe array noise terms α i (t) are zero-mean uniformly
distributed over [− 3σ α , 3σ α ] with RMS amplitude
σα .
Figure 1 and Figure 2 demonstrate the output SNR ofEq.(3)as a function of the root-mean-square (RMS)
amplitude σ α .The slope parameter ω in Eq.(7) is set
ω = 5 . The self-coupling coefficient of the subsystem

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 150–153___________________________________________________________________________________________________________4. Discussion

5.

In this paper, SR effect is investigated firstly in an
uncoupled array of dynamical saturating nonlinearities
with second-order for transmitting weak noisy signal.
Firstly, the SR effect occurs in parallel array ofdynamical saturating nonlinearities with second-order as
increasing the array size and the array noise intensity.
Then, the numerical results demonstrate that higher
output SNR can be obtained with smaller selfcoefficient. The output SNR with the Laplace noise as
the array noise is superior to that with the Gaussian
noise as the internal noise.The dynamical saturating
nonlinearity with second-order can be applied in thefield of physical systems actual signal processing.The
results from the dynamical saturating nonlinearity withsecond-order may be beneficial to theoretical researchand signal processing.

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Acknowledgements
This work is partially supported by the National Natural
Science Foundation of China (61501276, 61573204),
theChina Postdoctoral
ScienceFoundation(2016M592139), the Qingdao Postdoctoral Application
Research Project(2015120).

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