Open access peer-reviewed chapter

Optimization of Double-Well Bistable Stochastic Resonance Systems and Its Applications in Cognitive Radio Networks

By Di He

Submitted: April 5th 2017Reviewed: August 4th 2017Published: November 29th 2017

DOI: 10.5772/intechopen.70517

Downloaded: 544

Abstract

In this chapter, the optimization method of double-well bistable stochastic resonance (SR) system and one of its applications in cognitive radio networks are introduced, especially in the energy detection problem. The chapter will be divided into five sections. Firstly, the conventional double-well bistable stochastic resonance system is introduced with its special properties. Then based on the conventional discrete overdamped double-well bistable SR oscillator, the optimization method and the analyses results are given especially under low signal-to-noise ratio (SNR). In the applications, a novel spectrum sensing approach used in the cognitive radio networks (CRN) based on SR is proposed. The detection probability is also derived theoretically under a constant false-alarm rate (CFAR). Moreover, a cooperative spectrum sensing technique in CRN based on the data fusion of various SR energy detectors is proposed. Finally the whole chapter is summarized.

Keywords

  • stochastic resonance
  • optimization
  • cognitive radio networks
  • spectrum sensing
  • energy detection
  • cooperative spectrum sensing

1. Introduction of conventional double-well bistable stochastic resonance system

In many different dynamic systems, it can be found that the stochastic resonance (SR) is a kind of complex nonlinear phenomenon with many applications [1, 2]. In this kind of dynamic system, it possesses some good performances, while it can help to increase the periodic driving signal power under some special conditions. A lot of researches have demonstrated that the SR system may help to convert some power of the state variable signal in the SR system into the spectral power of the single-frequency driving signal in the SR system [13]. So, the SR system has been widely used in many applications, such as the weak target identification, weak signal detection and estimation, and so on [46].

In the dynamic SR processing, according to the SR noise power influence to the SR system, it can be found that the improvement effects, which include the signal power amplification and the signal-to-noise ratio (SNR) enhancement, have great relationships between the SR driving sinusoidal signal power and the SR noise power [3].

Mathematically, an SR system in a continuous form can be written as [3]

dxt/dt=fxt,rt,E1

while in the above equation, f [⋅] is the dynamic SR system, x(t) is the state vector, and r(t) is the driving signal of the SR system.

In many SR systems, it can be found that the quartic double-well bistable system is a widely used SR system with many researches and discussions, and it has been applied in many applications. It can be expressed as

dxt/dt=axtbx3t+krt.E2

In the expression above, x(t) is the state variable of the SR system, parameters a and b determine the properties of the SR system, and the driving parameter k influences the effect of driving signal r(t) seriously. In many studies, r(t) is set as a single-frequency sinusoidal signal, which is also influenced by some additive noise n(t), which is

rt=εsinωst+nt,E3

while in the above equation, the parameters ε and ωs are the corresponding signal amplitude and signal angular frequency of the driving signal; n(t) is the additive noise. To simplify the analyses, n(t) can always be supposed to obey the Gaussian distribution, which possesses mean 0 and variance σn2. So, the SNR of the driving signal r(t) (or the SNR of the input SR system) can be expressed by

SNRi=ε2/2σn2.E4

According to the linear response theory of SR system [3], the output of the SR system state variable x(t) can be expressed as a sum of two components as

xt=εosinωst+ϕo+not,E5

where εo is amplitude of the output signal, φo is the phase of the output signal at the input frequency point ωs, and no(t) is the additive noise in the output signal.

Based on the above assumptions, when ωs→0 or even ωs = 0, the output SNR of the SR system may be estimated by [2]

SNRo2aε2c2k3σn4e2U0k2σn214a2ε2c2π2k3σn4e4U0k2σn2/2 a2π2e4U0k2σn2 + ωs21,E6

where c=a/band U0 = a2/4b are constants corresponding to the selection of parameters a and b in (2). It can also be found in (6) that in many real applications, the parameter k is the only parameter which can be adjusted, and also it cannot influence the parameter SNRi in (4), so it is a very important factor which can determine the SR phenomenon [2].

2. Optimization of double-well bistable stochastic resonance system

2.1. System optimization and performance analyses

As described in last section, to make the SR system more applicable to the weak target identification or detection problems, we investigate a kind of optimization method to the quartic double-well bistable SR system in (2), and the target is to guarantee the enhancement of the signal SNR and also reach a maximal output SNR at the same time.

Although the result in (6) is based on the assumption ωs → 0, when under some conditions that ωs → 0 cannot be guaranteed, some traditional down-conversion methods can be applied if the frequency of the sinusoidal signal cannot fulfill ωs → 0. Without loss of generality, an additive SR noise n1(t) is also introduced into the SR system, which possesses mean 0 and variance 1; then the quartic double-well bistable system in (2) can be rewritten as

dxt/dt=axtbx3t+k1rt/rt+k2n1t=axtbx3t+k1εsinωst/rt+k1nt/rt+k2n1t,E7

where k1 and k2 are the positive driving parameters corresponding to r(t) and n1(t), respectively. r(t) is normalized by ‖r(t)‖ to simplify the analyses, which is defined by

r(t)=deflimN1Nt=1Nr2(t)=12ε2+σn2,E8

where N is the sampling number. And when SNRi is small enough, we have

σ̂n2rt.E9

Based on the analyses in Ref. [4], if we want to reach an optimal result, it requires that the SR noise should be symmetric, and then n1(t) can also be chosen as a kind of noise signal with Gaussian distribution. So, (7) can be rewritten as

dxt/dt=axtbx3t+k3εsinωst+k4nSRt,E10

where the parameters are defined by

k3=defk1/rt,E11
k4=defk12σ̂n2/rt2+k22,E12

and nSR(t) in (10) is a Gaussian noise with mean 0 and variance 1.

With the assumptions above, the SNRo in (6) can be rewritten as

SNRo2ak32ε2c2k44e 2U0k4212k32ε2c2k441=2ak32ε2c2k442k32ε2c2e2U0k42.E13

Firstly, to ensure the SNR improvement effect of the SR system, it requires SNRo > SNRi, so

2ak32ε2c2k442k32ε2c2e2U0k42>ε22σn2.E14

And when SNRi is low enough, (14) can be simplified to

k32>k44e2U0k42/22U0σn2.E15

When U0 and σn2are fixed, it is obvious that k4=U0will lead to the maximal value of the right side expression of (15). So when we have:

k32>U0e2/22σ̂n2,E16

the SNR enhancement can be achieved.

What is more, to reach the maximum output SNR of the system, we can set up an optimization problem, where we suppose (13) as the corresponding objective function and let k1 be fixed, and then we let

SNRo/k42=0.E17

And the result can be changed to

k46U0k44+2U0k32ε2c2=0,E18

or k4 is the solution of the above equation.

By calculating the discriminant Δ of (18), we have

Δ=U02k34ε4c4227U04k32ε2c2=U02ak32ε2c2216b2216k32ε2ba3.E19

Then the optimization result can be decided by the power or the amplitude of the driving sinusoidal signal. It can be found that k3 should also satisfy (16), so we can choose a reasonable k3 big enough to fulfill Δ > 0 and guarantee that the optimization result of k4 can be achieved. When substituting the optimal values of k3 and k4 into (11) and (12), the optimal driving parameters k1 and k2 can finally be achieved.

2.2. Computer simulations

To testify the effectiveness of the above proposed optimization method, we give out a testifying example and carry out corresponding computer simulation results based on the analyses in the last section.

To simplify the simulations, a single-frequency sinusoidal signal corrupted with additive white Gaussian noise (AWGN) is assumed as the signal r(t), and the amplitude and angular frequency of the signal are chosen as ε = 1, ωs = 0.01, respectively. The sampling number is N = 1 × 105; and the parameters are chosen as a = 1 and b = 1 in the SR system.

In the following computer simulations, the maximum likelihood estimate (MLE) method [7] is applied to estimate the amplitude of the signal as

ε̂=α̂12+α̂22,E20

where we have

α̂1=2Nt=0N1rtcosωst,α̂2=2Nt=0N1rtsinωst.E21

Eq. (7) is changed to the following difference equation for simulations [8]:

xt+1=xt+taxtbx3t+k1rt/rt+k2n1t,t=0,1,,N2,E22

where the parameter Δt is chosen as 0.0195.

Figure 1 gives the SNRo vs. SNRi comparison performance through the proposed method, while the range of SNRi is between −25 dB and 10 dB. From the result, it can be discovered that the SNR of r(t) has been enhanced especially under low SNR, for example, SNRi < 0 dB.

Figure 1.

SNRo vs. SNRi by using the proposed optimal SR approach.

Figure 2 shows a result regarding to the SNR enhancement through the proposed optimal SR method. The SNRi also changes from −25 dB to 10 dB. It can be found that the SNR enhancement can also be reached even under low SNR.

Figure 2.

SNR improvement by using the proposed optimal SR approach.

Figure 3 shows the SNRo performance when the parameters k1 and k2 are adjustable under SNRi = −25 dB. It can be discovered clearly that a maximal SNRo can be reached with some certain optimal k1 and k2 values. Figure 4 shows the performance of SNRo vs. k2 under optimal k1 under the condition SNRi = −25 dB. Figures 5 and 6 give the same computer simulation results under the condition SNRi = −20 dB, and they also verify the reliability of the proposed method.

Figure 3.

Performance of SNRo under SNRi = −25 dB when k1 and k2 are adjustable.

Figure 4.

SNRo vs. k2 under optimal k1 and SNRi = −25 dB.

Figure 5.

Performance of SNRo under SNRi = −20 dB when k1 and k2 are adjustable.

Figure 6.

SNRo vs. k2 under optimal k1 and SNRi = −20 dB.

3. Applications in the energy detection problem in cognitive radio networks

3.1. Energy detection problem in cognitive radio networks

In the past years, the research works in the area of cognitive radio (CR) network have been widely reported with fast progress. A lot of novel research developments make the research topics in the related areas more and more attractive [9]. As is known, in CR, the spectrum sensing approaches play an important role in CR network because it helps the secondary or opportunistic users (SUs) to detect the existence of the primary users (PUs) and define whether they can transmit the information or not [10]. Without loss of generality, we suppose that only the overlay mode in CR networks is considered in this discussion. The main target of spectrum sensing is to define the presence of PUs under some unpredictable noisy wireless communications conditions. So when the PUs are detected to be absent, the SUs are permitted to use the spectrum holes on an opportunistic basis which are occupied by PUs before, so that it can enhance the spectrum utility significantly [11]. In other words, the spectrum sensing can be regarded as a base of CR networks seriously.

In the literatures, many approaches have been proposed to ensure the performance of spectrum sensing and minimize the interference to some other users, including the PUs [9]. With the previous studies, it is found that the energy detection is a very general spectrum sensing method which does not need any prior knowledge of PUs; and based on the traditional Neyman-Pearson criterion [7], the spectrum sensing problem can be converted to a detection problem as the following two hypotheses:

H0:rt=nt,t=0,1,,N1H1:rt=hst+nt,t=0,1,,N1,E23

where r(t) is the signal at the receiver, s(t) is the PU signal, and it is assumed that s(t) obeys the distribution with mean 0 and variance σs2and h is the channel fading factor between the transmitter (PU) and the receiver (SU). In the wireless communications applications, it can always be assumed that it has Rayleigh distribution with the second-order moment Eh2=mh2independent to PU, and n(t) is the additive noise independent to s(t) and h. Simultaneously, sometimes the co-channel interference or multiuser interference of the PU signal can also be regarded as another additive part of n(t). So to simplify the analyses, we suppose that h is predictable or can be estimated properly at the receiver and n(t) also obeys the additive white Gaussian noise (AWGN) distribution with mean 0 and variance σn2.

For the traditional Neyman-Pearson detection, the assumption or decision H1 can be made when the likelihood ratio exceeds a certain threshold γ, as follows:

Lr=prH1/prH0>γ,E24

where r = [r(0), r(1),  … , r(N − 1)]T is the receiving signal vector and p(r; H0) and p(r; H1) represent the probability density functions (PDFs) of the receiving signal vector r under H0 and H1, respectively, while L(r) is the likelihood ratio to be calculated.

Based on the analyses above, under two different hypotheses, the receiving signal r obeys Gaussian distribution with different variances, which can be expressed by

r~N0σn2IunderH0,E25
r~N0mh2σs2+σn2IunderH1.E26

Thus, H1 is decided when

Tr=t=0N1r2t>2lnγNlnσn2mh2σs2+σn2σn2mh2σs2+σn2mh2σs2=γED,E27

where T(r) is the statistic of the traditional energy detector and γED is the threshold to satisfy Pfa = α for a given CFAR α. Because the Neyman-Pearson detector calculates the energy of the receiving signal r(t), it is also called an energy detector.

In the following, the corresponding false alarm rate Pfa(ED) and the detection probability Pd(ED) of the above energy detector can be given as

PfaED=PrTr>γEDH0=PrTrσn2>γEDσn2H0=QχN2γEDσn2,E28
PdED=PrTr>γEDH1=PrTrmh2σs2+σn2>γEDmh2σs2+σn2H1=QχN2γEDmh2σs2+σn2,E29

while QχN2is the right-tail probability function with N degrees of freedom.

It can be found in the researches that this kind of energy detection method could perform well under high SNR. But its performance degrades seriously when SNR is reduced, especially when SNR < −10 dB. For example, the value of detection probability Pd under N = 103 and Pfa(ED) = 0.1 will decrease from 0.795 to 0.283 when the SNR changes from −10 dB to −15 dB, which may be a very general case in CR networks [12].

3.2. SR-based spectrum sensing approach

In this subsection, we propose a novel spectrum sensing method with the combination of traditional energy detector and the SR processing. First, let the receiving signal pass the SR system, and the amplified signal can be observed at the output of the SR system. Then the amplified signal goes through the conventional energy detector to get the final spectrum sensing decision.

In this proposed scheme based on SR, first, we set the normalized signal of r(t) in (23), say r0(t), as the input of an SR system f [⋅]; then we have

ẋt=fxt,r0t+n0t,E30
r0t=rt/varrt,t=0,1,,N1E31

where x(t) is still the SR system status vector and n0(t) is the SR noise with mean 0 and variance σn02, so r0(t) + n0(t) can be taken as the drive signal of the SR system.

Based on the SR linear response theory [3], the status vector of SR system can be divided into two independent additive parts, say

xt=sSRt+nSRt,E32

where sSR(t) is the system response signal corresponding to the normalized PU signal hst/var[r(t)]and nSR(t) is the system response signal corresponding to the noise signal nt/var[r(t)]+n0t. It can be found that the additive channel noise n(t) also plays a part role of SR noise.

From the above analyses, to reach a maximal SNRo, the optimal variance of the introduced SR noise σn0opt2can be calculated according to the derivations in the last section.

3.3. Experimental and comparison results

In the following, we present some experimental and comparison outcomes. In the computer simulations, the discrete overdamped bistable oscillator in (22) is used as the dynamic SR system model.

As is known QPSK and QAM are the mostly used modulation methods [13, 14] in the broadcasting systems. So in the computer simulations thereafter, a QPSK signal as the PU signal together with a co-channel interference QPSK signal with AWGN through the Rayleigh fading channel is utilized as the driving signal of the SR system, which can be expressed by

rt=hAPsinωPt+φP+AMsinωMt+φM+nt,E33

where h is the Rayleigh channel gain with mean 1; AP, ωP, and φP are the amplitude, angular frequency, and phase of the PU sinusoidal carrier signal; AM, ωM, and φM are the amplitude, angular frequency, and phase of the multiuser interference sinusoidal carrier signal, respectively. Here, φP , φM ∈ {π/4, 3π/4, 5π/4, 7π/4} in QPSK. In this case, the input SNR can be calculated by [15]

SNRi=12mh2AP2/12mh2AM2+σn2.E34

In the following simulations, we choose ωP = 0.04π, ωM = 0.2π, and σn2=1. So the optimal variance of the introduced white Gaussian SR noise with mean 0 can be calculated through the analyses in the last section, and we can get σn0opt2=1k2which requires k ≤ 1.

Figures 7 and 8 give the performance comparison results of the receiver operating characteristic (ROC) plots between the traditional energy detector and the proposed SR-based energy detector under the conditions SNR = −15 dB and SNR = −20 dB. The total sampling number is N = 103. In the figures, both the theoretical results and the computer simulation results of the above two methods are given, and the theoretical results of detection probability of the proposed method are calculated based on (29). It can be discovered that the detection probabilities of the proposed approach are higher than the energy detector, especially under low SNR as SNR < −10 dB, which is a good performance to the real applications; and it can also be discovered that even under SNR = −20 dB which is also very common in CR networks, the proposed detection method can still perform better than the energy detection method with a significant detection probability enhancement.

Figure 7.

ROC curves of different spectrum sensing approaches under SNR = −15 dB.

Figure 8.

ROC curves of different spectrum sensing approaches under SNR = −20 dB.

Besides the ROC curve performance comparison, the results of the detection probability versus SNR under CFAR are also presented. Figures 9 and 10 give the performance comparison results between the proposed detection method and the conventional energy detection method under Pfa = 0.05 and Pfa = 0.1, respectively. The total sampling number is still selected as N = 103. In the following simulations, the input SNR changes from −20 dB to 0 dB. And both the theoretical analyses results and the computer simulation results are given in Figures 9 and 10. It is obvious that the detection probability of the proposed SR-based method can be improved, especially under low SNR of SNR < −10 dB, and also it can be discovered that a 5 dB SNR enhancement can be achieved. Based on the simulation results, the main problems of the conventional energy detection method can be solved.

Figure 9.

Detection probability versus SNR under Pfa = 0.05.

Figure 10.

Detection probability versus SNR under Pfa = 0.1.

4. Application of cooperative stochastic resonance in the energy detection problem in cognitive radio networks

4.1. Cooperative SR-based spectrum sensing approach

In these years’ studies, the spectrum sensing techniques in physical layer can be divided into two classes: noncooperative sensing techniques and cooperative sensing techniques. Recently it has become a new direction by introducing some cooperation methods into the spectrum sensing or PU signal detection procedure with the cooperation of different secondary user (SU) sensing results [16, 17]. So based on the results in the last two sections, here we introduce the chaotic stochastic resonance (CSR) system to improve the spectrum sensing performance especially under low SNR circumstances.

At the first step, we could randomly select K SUs to carry out the spectrum sensing process independently to the communication channel. To realize the data fusion, we carry out two kinds of sensing methods at each SU at the same time. One is the traditional energy detection method with the statistics A(x), and another is the CSR energy detector with the statistics B(x). While in the real applications, it is also very difficult to determine what kind of SR noise will be best or optimal, so we choose different types of CSR noise as the CSR noise signal candidate β0(t) in each of the CSR systems, but the CSR systems in each SUs are all the same. To get some certain fusion result, some commonly used noise types in the wireless communication systems can be used, for example, AWGN signal, lognormal distribution noise, Weibull distribution noise, etc. Here we list the noise-type candidates as {β1(t), β2(t),…, βK(t)}. When some certain CSR system f [.] and the noise types are fixed, the corresponding optimal parameters with these noise-type candidates can also be calculated.

Theoretically and without loss of generality, it can be assumed that all the receiving signal at different SU obey the same distribution, so (23) is suitable for each SU. To simplify the analyses thereafter, we suppose that h = 1. So we have

H0:EA1x=EA2x==EAKx=EAx=σn2,H1:EA1x=EA2x==EAKx=EAx=σs2+σn2,E35

while A1(x), A2(x), …, and AK(x) are the statistics of SUs 1, 2, …, K, respectively.

In the data fusion processing, we introduce the traditional Bayesian fusion method to realize the cooperative spectrum sensing. Simultaneously, if the same traditional energy detection method and the same threshold γED are used at each SU detector, the expectation result E[A1 , 2 ,  ⋯  , K(x)] of the Bayesian fusion can be written as

EA1,2,,Kx=k=1KEAkxAkx=1Nt=1Nxk2tk=1KEAkxAkx=1N1t=1N1xk2tEA1,2,,KxAkx=1N1t=1N1xk2t=EAx,E36

where xk(t) is the output of the kth SU’s CSR system.

Let the receiving signal r(t) goes through the dynamic CSR system with different CSR noise β1(t), β2(t),…, βK(t), and we can denote the output of each SU’s CSR energy detector to be B1(x), B2(x), …, BK(x), respectively. Introducing the conventional Bayesian fusion method to fuse all K SUs’ statistical results {A1(x), A2(x), …, AK(x)}, {B1(x), B2(x), …, BK(x)}, and A1,2…,K(x), then the following Theorem 1 [18] could verify the effectiveness of the proposed cooperative spectrum sensing method.

The cooperative spectrum sensing approach proposed by using the Bayesian fusion to all K SUs’ statistics {A1(x), A2(x), …, AK(x)}, {B1(x), B2(x), …, BK(x)}, and A1,2,…,K(x) shown in Figure 1 can improve the sensing performance of conventional energy detection method.

Proof: Please refer Theorem 1 in Ref. [18] for details.

4.2. Computer simulation results

In the following, some computer simulations are carried out to certify the correctness of the proposed method. Here, a QPSK signal is selected as the PU signal, that is

st=APsinωPt+φP,E37

where AP, ωP, and φP are the amplitude, angular frequency, and phase of the PU signal and φP ∈ {±π/4, ±3π/4} in QPSK. In the following simulations, we set AP = 5 and ωP = 0.02π.

Also in the simulations, a kind of conventional discrete overdamped bistable oscillator is utilized as the CSR system, that is [19]

xit+1=gxitxi3texi2t/h+drt+βit.E38

In the equation above, xi(t) is the state variable and g and h are the corresponding parameters which determine the performance of the system seriously. In the simulations, we choose g = 2.85 and h = 10. d is the driving parameter of the CSR system.

The additive channel noise n(t) is supposed to be composed by a sinusoidal interference signal and an AWGN signal in the computer simulations as

nt=n0t+εsinωεt,E39

while n0(t) is the AWGN signal, and the amplitude and angular frequency of the sinusoidal signal are set as ε = 0.1 and ωε = 0.8π.

Simultaneously, we choose the following types of CSR noise: uniform distribution noise, Weibull distribution noise, and lognormal distribution noise. While the uniform distribution noise is evenly distributed within the range [−1,+1].

The pdf of the Weibull distribution noise is

gxuv=uvuxu1ex/vu,E40

where u = 2 and v = 1. The pdf of the Lognormal distribution noise is

gx;µ,σ=elnxµ2/2σ2/xσ2π,E41

where the parameters of are fixed as μ = 1 and σ = 1.

In the computer simulations, the total sampling number is N = 106, and the Bayesian fusion process is performed under the CSR energy detection spectrum sensing driven by these three various kinds of noises, respectively.

Both Figures 11 and 12 give the ROC curves of different spectrum sensing results under SNR = −20 dB and −15 dB, respectively. It can be found obviously that the proposed cooperative approach can achieve some better performance than the conventional noncooperative spectrum sensing methods.

Figure 11.

ROC curves of different spectrum sensing approach under SNR = −20 dB.

Figure 12.

ROC curves of different spectrum sensing approach under SNR = −15 dB.

5. Summary

In this chapter, some conventional double-well bistable SR systems are introduced first. Then based on the conventional discrete overdamped double-well bistable SR oscillator, the optimization method and the corresponding analyses results are given especially under low SNR circumstances. Besides, a novel spectrum sensing approach used in CRN based on SR is proposed. And a cooperative spectrum sensing technique in CRN based on the data fusion technique is also proposed. The last section summarizes the whole chapter.

The optimization approach introduced is especially applicable under low SNR, which are familiar in the wireless communications. In the applications, the performance analyses and computer simulations show that the effectiveness of the proposed spectrum sensing approach is better than the traditional energy detection methods, and this methodology can be extended to some other problems with the same two-hypothesis decisions.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Di He (November 29th 2017). Optimization of Double-Well Bistable Stochastic Resonance Systems and Its Applications in Cognitive Radio Networks, Resonance, Jan Awrejcewicz, IntechOpen, DOI: 10.5772/intechopen.70517. Available from:

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