Bistable Stochastic Resonance Enhanced 4-ary PAM Signal Detection under Low SNR

Linlin Liang＊,Nina Zhang,Haiyan Huang,Zan Li

1 School of Telecommunications Engineering,Xidian University,Xi'an,Shaanxi 710071,China

2 Xianyang Detachment of Chinese Armed Police Force,Xianyang,Shaanxi 712000,China

3 School of Electronic and Information Engineering,Lanzhou Jiaotong University,Lanzhou,Gansu 730070,China

4 The Collaborative Innovation Center of Information Sensing and Understanding at Xidian University,Xi'an,Shaanxi 710071,China

Abstract: To boost the performance of 4-ary pulse amplitude modulated (PAM) at low signal-to-noise ratio (SNR),bistable stochastic resonance (BSR) system is introduced into digital communications system and get a reliable signal detection scheme.In this paper,we first analyse BSR system for different amplitudes of 4-ary PAM signals.The steadystate of the bistable system will be statistically distinct,and the feasibility of the proposed detection scheme is confirmed.On this basis,we present a detailed study on steady-state transitions of the BSR system,and an explicit expression of the bistable system parameters is derived.By setting the bistable system parameters,bistable system,4-ary PAM signal,and noise reach the resonance state,and the BSR-based detection scheme is implemented.Moreover,we derive an analytical expression to calculate the symbol error rate (SER) of 4-ary PAM signals with the BSR-based detection under additive white Gaussian noise (AWGN).Finally,the simulation results validate that BSR-based detection scheme can improve the detection performance while efficiently reducing the symbol error rate.

Keywords: 4-ary pulse amplitude modulated signal detection;Low SNR;Symbol error rate (SER);Bistable stochastic resonance (BSR)

I.INTRODUCTION

As one of the commonly used digital modulation modes,pulse amplitude modulation (PAM) has attracted extensive attentions in the field of information theory and communications systems [1]-[2].It is well known that,each binary PAM (BPAM) symbol can transmit one bit amount of information,and eachM-ary PAM (MPAM) symbol can transmitkbits (M=2k) amount of information.Therefore,MPAM signal has the advantage of higher efficiency of information transmission compared with BPAM signal under the same symbol transmission rate conditions.Though the reliability and the validity of information transmission is a pair of contradictions,the anti-noise performance of MPAM signal is much less than that of BPAM signal [3].In recent years,the existed researches on PAM signal detection mainly focus on the condition of high signal-to-noise ratio (SNR) [4]-[6].However,with the increasing noise and interference sources,useful signals are often submerged in strong background noise,and SNR of the received signals is usually very low,which will lead to the PAM signal detection performance degradation by exploiting the traditional detection methods.To guarantee higher efficiency of information transmission and improve the detection performance of the PAM signal under low SNR conditions,a scheme of signal detection technique based on bistable stochastic resonance (BSR) will be investigated in this paper.

Some researchers have put forward the concept of BSR to resolve the problem that noise is how to play the role of help in the process of restore signal [7]-[8].From then on,this theory is of great conceptual significance and attracts increasing attentions of researchers.Focusing on this phenomenon,the nonlinear dynamics method is used on the theoretical study of communications system in this paper.The paper attempts to combine these two theories (the BSR theory and the information theory) together,and focuses on analysing the bistable system response to informative aperiodic signals [15]-[19].It shows that the“non-coherent”noise can be used to enhance the restoration of a“coherent”signal of a known form,the implementation of turning the noise from a nuisance into a benefit [12]-[14].To date,some researchers are giving a profound study of aperiodic binary signal processing [15]-[19].But what I want to emphasize is the two articles [20] and [22].In [20],an analytical expression of the parameters of parameter-tuned stochastic resonator (PSR) was derived.For BPAM noisy signal detection,the results are showed that the BER performance of the nonlinear PSR-based receiver was superior to the linear optimal receiver at low SNR.In [22],the mechanism of the BSR system response to 4-ary PAM signal inputs is elucidated,and the feasibility of this method was demonstrated.In [23],the passage between the stable points of the system was strengthen by the increasing of the noise strength or the forcing frequency,and the system responses also displayed the asymmetry for the asymmetric dichotomous noise.Further studies conclusively showed that the switching from logic gate AND to OR can be performed by changing the system parameter in a single gene network model [24].It is observed that the increase of the signal amplitude or Lévy noise intensity in a certain range could lead to the optimal occurrence of the SR phenomenon [25].The followed study manifests that the occurrence of the SR phenomena in the FHN system [26].

It is still unknown whether 4-ary PAM signal can be detection by the BSR-based detection scheme.Moreover,the symbol error rate formula has not yet been researched.According to the characteristics of the 4-ary PAM signal,this article focuses on analysing and discussing the BSR signal detection technique under AWGN.For the transformation between steady states of the bistable system corresponding to the jump between the symbols of different polarity (for example,the two symbols +1 and -1),a method named as the BSRbased detection scheme is proposed for different 4-ary PAM signals.Furthermore,through a further research on steady-state transitions of the BSR system,an explicit expression of the bistable system parameters is derived.By configuring the bistable system parameters,bistable system,4-ary PAM signal,and noise reach the resonance state,which in order to restore the input source signals.

The structure of this paper is organized as follows.Section II introduces the system model and the feasibility of BSR-based detection scheme for different 4-ary PAM signals.Performance analysis of the BSR-based detection scheme are presented in Section III.In Section IV,the calculation of the symbol error rate of the BSR-based detection scheme is given.Finally,conclusions are drawn in Section V.

II.SYSTEM MODEL AND FEASIBILITY ANALYSIS

It is assumed that {I} is a binary message sequence,and {I} is mapped onto theM-ary PAM signal (M= 2k).M-ary PAM signals(t) is written asfordenotes the amplitudes ofM-ary PAM signal,which corresponds tok-bit symbols.g(t) is a rectangular pulse,and each pulse lasts for T.Therefore,the duration of each symbol interval isT.In this paper,the signal of background noise detection is additive white Gaussian noiseξ(t).ξ(t) is a zero-mean with the noise intensityD,and autocorrelationwhereδ(τ) is the dirac function.

The main research objects of this paper is the 4-ary PAM signal.Refer to the book [Digital Communication 3rd Edition by John R.Barry] for more information about the basic theory of 4-ary PAM signal and its uses as BSR system detect object.For a transmitted binary message sequence {111001010100},mapping onto the 4-ary (M= 4) PAM signal,the amplitudes {A1,A2,A3,A4} of transmitted signal represent respectively the transmitted symbolsSi∈{11,10,01,00}.The amplitudes of 4-ary PAM signal are usually selected as {± ±1,3}.Hence,Amare set to be

The 4-ary PAM signals(t) and the background noiseξ(t) are applied to a nonlinear dynamic bistable system,and the output signalx(t) evolves according to Langevin equation [19].

Fig.1.The quartic potential of BSR system (a=1,b=1,s(t)=Am,ξ(t)=0).

Whereaandbare the parameters of the bistable system.For the bistable potential,dx/dt=-dU/dxis governed by the quartic potential,whereU(x) denotes the symmetric bistable potential field.The mathematical expression ofU(x) is expressed asU(x)=-ax2/2 +bx4/4 -Amx.It is important to point out that,without the signals(t) and the noiseξ(t) motivation (s(t)= 0,ξ(t) =0),ΔU=a2/4bis the height of the quartic potential,andare two potential wells of bistable system.

Taking no account of the noise,the quartic potential is shown in figure 1.In view of a static threshold for the bistable system[16],[19].Under the condition of the signal amplitude to be small enough andit is impossible to trigger transitions between two steady states without noise.However,there is a strange phenomenon taking place,which a moderate amount of noise can help to increase the transition probability.The input signal accepts assistance from the noise.In terms of energy,part of the energy of incoherent noise is converting into the coherent signal energy,which is called sub-threshold stochastic resonance phenomenon.Under the condition ofit is sufficient to trigger transitions without the aid of noise.We called this phenomenon as super-threshold stochastic resonance.It is demonstrated that the amplitudes (A1,A2,A3,A4) of different 4-ary PAM signal corresponds to the four steady states (c1,c2,c3,c4) of the BSR (dotted lines in f igure 1),respectively.Next,the feasibility of BSRbased detection scheme can be further determined for different 4-ary PAM signals.

III.IMPLEMENTATION AND PERFORMANCE ANALYSIS OF THE BSRBASED DETECTION SCHEME

3.1 Related theory analysis of steady-state transitions

The 4-ary PAM symbols are transmitted with a rate of one symbol everyT.We propose the BSR-based detection scheme in which all symbols are decoded through the bistable system.It is assumed that the transmitted symbol durationTis known at the receiving terminal.The bistable system signalx(t) is in the form ofXi=x(ti+T).In order to restore the symbols exactly,a requirement for respectively resting on the stable statesc1,c2,c3,c4at the decoding timet=ti+T.The transmitted 4-ary PAM symbolsSiare 11,10,01,00,which respectively corresponding to the steady statesc1,c2,c3,c4.

The bistable system response timeTris the time that is taken by the particle transfer from one potential well to the other.Based on SR system,signal detection algorithms are shown in Eq.(1).To implement our proposed decoding scheme,the relationship betweenTrand symbol intervalTneeds to be further studied.There exists two different kinds of system response time.One is to describe the time taken by the particle transition without the aid of noise under the condition ofIn these cases,the response time meets theTr=Tsuper.The other one is to describe the time taken by the particle cannot transfer from one potential well to another with the aid of noise under the condition ofIn these cases,the response time also meets theTr=Tsub.

From Eq.(1) with the condition ofwe have

According to Ref [16],we have the following corollary about the response timeTsuper.

Wherex-is the unique real root of equationax-bx3+A4=0.x+,x1,x2are respectively the real root and two conjugate complex roots of equationax-bx3+Am=0,andC1,C2,C3are the integration constants.

Wherex-is the unique real root of equationax-bx3+A3=0,x+,x1,x2are respectively the real root and two conjugate complex roots of equationax-bx3+Am=0,andare the integration constants.

Under the condition of noise andthe method proposed in Ref [19] is adopted to analyse the system response timeTr=Tsub.The parameters of adiabatic approximation meetT≫ 1,D≪1.The probability density function (PDF) of the internal statex(t) of the PSRρ(x,t) is governed by Fokker-Planck (FP) equation [19],[21].

Whereρ±(xt,) obeys natural boundary conditions such that it vanishes at largexfor anyt.

We can get the steady state solution of the actual situation FPK equation.

WhereCis normalized constants,and meet the conditions

We can get the total probability in the two potential well of the system.The specific calculation formula is as follows.

And satisfy the condition thatp+(t)+p-(t)=1.

We have integrate the function Eq.(5) on that region (-∞,0),and the rate of change ofp-(t) is as follows.

The rate of change ofp-(t) is only with regard to the nature ofρ(x,t) near the pointx=0.When only consider the situation that probability f low fromp+(t) top-(t),the initial conditions of probability distributions at the pointtare as shown below.

Because of the probability flow occurs at unstable pointx=0,we have make a linear approximation near the pointx=0.We can get the probability distribution of the momentt+Δt.

Fig.2.The process of input-output signal states hopping under the condition of the same transmitted symbol.(a = 1,b = 1,ξ( t ) =0) (a) the amplitudes of adjacent signals are the same and both of them are equal to A1;(b) the amplitudes of adjacent signals are the same and both of them are equal to A4;(c) the amplitudes of adjacent signals are the same and both of them are equal to A2;(d) the amplitudes of adjacent signals are the same and both of them are equal to A3.

We substitute Eq.(12) into Eq.(10),and obtain formula as follows.

Based on the above analysis,we get the following conclusion that the key to implement our proposed decoding scheme is the following relationship hold.

According to the amplitude of 4-ary PAM signal changing,steady state of the BSR system should be jumped so that it can detect the 4-ary PAM signal [22].

Remark1: In f igure 2,when the amplitudes of the input 4-ary signal areA1,A2,A3,A4,the bistable system output signalx(t) will rest on corresponding steady statesc1,c2,c3,c4.When the amplitudes of adjacent signals are the same,the output signalx(t) will oscillate in the corresponding steady states vicinity of the system.Therefore,the system response timeTris equal to zero,and Eq.(14) is constant hold in this cases.

Remark2: In f igure 3(a),horizontal coordinates of the red particle isx(t) =-1 .5.It is obvious that the red particle is in a potential well of bistable system.In other word,it rests on the steady statexc=4.Besides,the amplitude of the current transmitted symbol isA4=-1.8.When the amplitude of the next symbol isA1=1.8,amplitude change fromA4toA1.As the change of the input signal amplitude,the output signalx(t) is changed.The red particle fi rstly moves to the position of the blue particle and crosses the quartic potential to reach stable statexc=1represented by the black particle.Ultimately,the output signalx(t) will oscillate in the near area ofx(t) 1.5= .The steady state transitions in fi gure 3(b) is similar to that in fi gure 3(a).In fi gure 3(c),horizontal coordinates of the red particle isx(t) 1.2=- .It is obvious that the red particle is in a potential well of bistable system.In other word,it rests on the steady statexc=3.Besides,the amplitude of the current transmitted symbol isA3=-0.3.When the amplitude of the next symbol isA2=0.3,amplitude change fromA3toA2.As the change of the input signal amplitude,the output signalx(t) is changed.The red particle fi rstly moves to the position of the blue particle and crosses the quartic potential to reach steady statexc=2represented by the black particle.Ultimately,the output signalx(t) will oscillate in the near area ofx(t) 1.2= .The steady state transitions in figure 3(d) is similar to that in fi gure 3(c).

The system response timeis equal to 0.8474s [see f igure 3(a)] when the amplitudes of the transmitted symbols change fromA4toA1(Am=A1).Because of the symmetry of the steady-state transitions process,the system response time[see figure 3(b)] is the same asby numerical comparison.According to theory of sub-threshold stochastic resonance (SR) [19],we getTsub＜Tk=12.07s[see f igure 3(c),(d)].

Remark3: In f igure 4(a),horizontal coordinates of the red particle isx(t) = 1.5.It is obvious that the red particle is in a potential well of bistable system.In other word,it rests on the steady statex=c1.Besides,the amplitude of the current transmitted symbol isA1=1.8.

Fig.3.The process of input-output signal states hopping under the condition of different polarity transmitted symbol with the same amplitude.(a = 1,b = 1,ξ(t ) =0) (a) the amplitude of the transmitted symbol is changed from A4 to A1.) (b) the amplitude of the transmitted symbol is changed from A1 to A4;(c) the amplitude of the transmitted symbol is changed from A3 to A2;(d) the amplitude of the transmitted symbol is changed from A2 to A3.

Fig.4.The process of input-output signal states hopping under the condition of different transmitted symbol.(a = 1,b = 1,ξ( t ) =0) (a) the amplitude of the adjacent transmitted symbols switches from A1 to A3;(b) the amplitude of the adjacent transmitted symbols switches from A3 to A1;(c) the amplitude of the adjacent transmitted symbols switches from A2 to A4;(d) the amplitude of the adjacent transmitted symbols switches from A4 to A2.

When the amplitude of the next symbol isA3=0.3,amplitude change fromA1toA3.As the change of the input signal amplitude,the output signalx(t) is changed.The red particle fi rstly moves to the position of the blue particle and crosses the quartic potential to reach steady statexc=3represented by the black particle.Ultimately,the output signalx(t) will oscillate in the near area ofx(t) 1.2=- .The steady state transitions in fi gure 4(d) is similar to that in fi gure 4(a).In fi gure 4(b),horizontal coordinates of the red particle isx(t) 1.2=- .It is obvious that the red particle is in a potential well of bistable system.In other words,it rests on the steady statexc=3.Besides,the amplitude of the current transmitted symbol isA3=0.3.When the amplitude of the next symbol isA1=1.8,amplitude change fromA3toA1.As the change of the input signal amplitude,the output signalx(t) is changed.The red particle fi rstly moves to the position of the blue particle and crosses the quartic potential to reach steady statexc=1represented by the black particle.Ultimately,the output signalx(t) will oscillate in the near area ofx(t) = 1.5.The steady state transitions in figure 4(c) is similar to that in f giure 4(b).

Fig.5.The process of input-output signal states hopping under the condition of different polarity transmitted symbol with the different amplitude.(a = 1,b = 1,ξ(t ) =0) (a) the amplitude of the transmitted symbol is changed from A2 to A1;(b) the amplitude of the transmitted symbol is changed from A1 to A2;(c) the amplitude of the transmitted symbol is changed from A3 to A4;(d) the amplitude of the transmitted symbol is changed from A4 to A3.

The system response timeis equal to 0.7134s [see f igure 4(b)] when the amplitude of the transmitted symbols switches fromA3toA1.Because of the symmetry of the steadystate transitions process,the system response timethat the amplitude of the transmitted symbols switches fromA2toA4[see f igure 4(c)] is the same asby numerical comparison.According to theory of sub-threshold stochastic resonance (SR) [19],we also get＜Tk[see f igure 4(a),(d)].

Remark4: In fi gure 5(a),horizontal coordinates of the red particle isx(t) 1.2= .It is obvious that the red particle is in a potential well of bistable system.In other words,it rests on the steady statexc=2.Besides,the amplitude of the current transmitted symbol isA2=0.3.When the amplitude of the next symbol isA1=1.8,amplitude change fromA2toA1.As the change of the input signal amplitude,the output signalx(t) is changed.The red particle firstly move to the position of the black particle and rests on the steady statexc=1.Ultimately,the output signalx(t) will oscillate in the near area ofx(t) 1.5= .The steady state transitions in Fig .5 (b),(c) and (d) is similar to that in figure 5(a).Therefore,the system response timeTris so small that approximates to zero,and in these cases Eq.(14) is constant hold.

But above all,Tsub＜Tk＜Tandwhich can assure that the signalx(t) respectively resting on the steady statesc1,c2,c3,c4at the decoding timet=ti+T.The bistable system,4-ary PAM signal,and noise reach the resonance state.

3.2 The feasibility analysis of the BSR-based detection scheme

With the parameters of the bistable systema=1 andb=1,the intervalTand the noise intensityDsatisfying Adiabatic approximation condition,which can be calculated asT=T0=100 andD=D0=ΔU=a2/4b=0.25.We focus on determining the parameters of the bistable system for noisy 4-ary PAM signal detection on the basis of the referenced stochastic resonance model (RSRM).

Considering the fact that the 4-ary PAM signal intervalT≪1 and the AWGN intensityD≫1 in practical application scenario,we do linear transformation of Eq.(1) by the following formτ=at,Thus,we get

Combining Eq.(15) with Eq.(16),an analytical expressions of parameters for the bistable system is obtained as

The implementation steps of the BSRbased detection scheme can be summarized as follows:

1) Under the condition of adiabatic approximation (TD≫ ≪1,1),a referenced stochastic resonance model (RSRM) is given in the form of Eq.(17).The model can assure that the bistable system,4-ary PAM signal,and noise reach the resonance state.

2) By doing linear transformation,the dynamical equation Eq.(1) can be transformed into Eq.(16).By comparing of Eq.(15) and Eq.(16),we obtain the expression of the bistable system parameters,which is in the form of Eq.(17).

3) We set parameters (aandb) of the BSR system according to Eq.(17),which assure that the BSR system,the 4-ary PAM signal,and noise reach the resonance state in practical application scenario (TD≪ ≫1,1).

4) When the 4-ary PAM noise signal (r(t) =s(t) +ξ(t)) is entered into the BSR system and decoding the resonated signalx(t),the framework of the BSR-based detection scheme is implemented.

3.3 Symbol error rate performance analysis

We build 4-ary PAM signal detection scheme based on BSR system,which compares with the traditional 4-ary PAM signal detection scheme (the received signal unprocessed by the BSR system).To objectively evaluate the proposed detection scheme,the symbol error rate (SER) is the most important performance indicators.

We began with an introduction to the SER of detection scheme without BSR system.The Gaussian noiseξ(t) obeys the zero-mean variance 2Dnormal distribution.So,we obtain the conclusion that the received signal (r(t) =s(t) +ξ(t)) obeys the normal distribution.

The probability density function (PDF) of the received signalr(t) has the following expression.

It is quite clear that the transmitted signals have four amplitudes {1,3}±±.In the light of distinguishing transmitted signals with different polarity,the optimal threshold can be easily set asr0=0.In order to distinguish the same polarity signals,by means of the maximum-likelihood (ML) criterion,the optimal decision threshold can be derived as

From Eq.(20),we can get the optimal threshold of signal detection.r+= (A1+A2)/2 is to distinguish the transmitted signal amplitudesA1andA2,andr-= (A3+A4)/2 is to distinguish the transmitted signal amplitudesA3andA4.

Assume the transmitted 4-ary PAM signals are identically distributed,i.e.,P(s(t) =A1) =P(s(t) =A2) =1/4,P(s(t) =A3) =P(s(t) =A4) =1/4.The SER of the transmitted 4-ary PAM signal can be calculated byPe1,Pe2,Pe3,Pe4respectively.

Due to the symmetry of the PDF of the transmitted 4-ary PAM signals,A1=-A4andA2=-A3,we getPe1=Pe4,Pe2=Pe3.

From Eq.(20),and Eq.(21),the total SER of detection scheme without BSR system can be calculated as follows.

Concrete analysis is as follows: firstly,we mainly expounds the probability density function (PDF) of the system outputpxt(,).Second,we derive formula that the SER expression based onpxt(,).Considering the amplitude of the transmitted 4-ary PAM signal is actually a constant,we use the following formula to express the quartic potential of PSR.

As stated previously,the mechanism of the BSR system are described in the following sections.At the decoding timet=ti+T,the internal statex(t) of PSR has rested on the four steady statesc1,c2,c3,c4.When the input 4-ary PAM signal amplitudes to beAm(m=1,2,3,4),p(x,t) is equal to the steady state solution of FP equation.

WhereCm,m= 1,2,3,4 are normalized constants.

In the light of the problem of decision threshold,the analysis method could be reference for the method used in the traditional 4-ary PAM signal detection scheme.The optimal threshold can be easily set asr0=0,which to distinguishes transmitted signals with different polarity.In order to distinguish the same polarity signals,by means of the maximum-likelihood (ML) criterion,the optimal decision threshold can be derived as

From Eq.(25),x+=Dln(C1/C2) /(A1-A2) is the optimal threshold and in order to distinguish transmitted signal amplitudesA1andA2,andx-=Dln(C3/C4) (A3-A4) is the optimal threshold and in order to distinguish transmitted signal amplitudesA3andA4.

Based on the analyses above,the SER formula of BSR-based detection scheme is derived as follows.

The SER of the BSR-based 4-ary PAM signal detection can be calculated byrespectively.

For the symmetry of the PDF of the BSR processed transmitted 4-ary PAM signals(t)=A1ands(t)=A4,s(t)=A2ands(t)=A3,we get

From Eq.(25) and Eq.(27),the total SER of the BSR-based 4-ary PAM signal dete0ction scheme can be calculated as follows

Where the expression of the bistable system parametersaandbhas been given in Eq.(17).For Eq.(28) containing non-integrable terms ∫exp(x2 )dxand ∫exp(-x4 )dx,it cannot be further simplif ied.

IV.SIMULATION RESULTS

In the following,some simulations are given to validate the effectiveness of BSR-based 4-ary PAM signal detection scheme.

In figure 6,the parameters of the bistable system are seta= 1,b=1,and the AWGN intensityD=0.25.We choose the duration to beT=100s,the amplitude of 4-ary BPAM signal to beA1= 3,A2= 1,A3=-1 ,A4=-3,and the number of BPAM symbol isN=100.Bringing the above parameters into Eq.(24).The optimal decision threshold can be calculated asx-=-1 .5,x+=1.5.

Fig.6.PDF curves of the BSR system output signal x(t) with different inputs.

Fig.7.Symbol error rate of the BSR-based 4-ary PAM signal detection scheme.

Fig .7 shows that in the condition of the low SNR varying from -10dB to 5dB,the SER performance comparison between the traditional 4-ary PAM signal detection and the BSR-based 4-ary PAM signal detection,in which the SNR is defined asM=4.is the average power of the 4-ary PAM signal,andPξ=2Dis the average power of the AWGN.The transmitted 4-ary PAM signal symbol interval is set to beT=0.0001s,i.e.,the symbol rate isRBM= 1 /T=10000 symbols per second.According to the expression in Eq.(17),the bistable system parameterais set to bea=106,and the parameterbis changing according to different SNR,i.e.,according to different AWGN intensityD.The simulation results show that the SER performance of the BSR-based 4-ary PAM signal detection is superior to the traditional 4-ary PAM signal detection under low SNR conditions.For example,under the condition SNR=-5dB,about 5dB SER performance can be improved through the BSR-based 4-ary PAM signal detection scheme,when compared with the traditional 4-ary PAM signal detection method.

V.CONCLUSIONS

In this paper,to simplify theoretical analyses,a 4-ary PAM signal detection scheme using BSR technique is proposed.Based on a detailed theoretical study on steady-state transitions of the BSR system,an explicit expression of the bistable system parameters is derived,which can assure the bistable system,4-ary PAM signal,and noise reach the resonance state,and the BSR-based detection scheme is easy to be implemented.The simulation results show that the SER performance of the BSR-based detection scheme is superior to the traditional detection scheme under low SNR conditions.In addition,the proposed 4-ary PAM signal detection scheme can be extended to M-ary PAM (M＞4) signal detection easily.

ACKNOWLEDGEMENT

This work is supported by the National Natural Science Foundation of China (61631015,61501354,61501356,and 61573202),the Fundamental Research Funds of the Ministry of Education (7215433803),the Foundation of State Key Laboratory of Integrated Services Networks (ISN1101002),Higher School Subject Innovation Engineering Plan (B08038),Science and Technology Innovation Team Key Plan of Shaanxi Province (2016KCT-01).The Fundamental Research Funds of the Ministry of Education,China (Grant No.JB160101).The Key Laboratory Foundation of Ministry of Industry and Information Technology (KF20181912).China Postdoctoral Science Foundation (2018M631122).