Application of Lyapunov exponent algorithm in balise signal chaotic oscillator detection

ZHANG Hongyan，WANG Ruifeng

(School of Automation and Electrical Engineering，Lanzhou Jiaotong University，Lanzhou 730070,China)

Abstract：To improve the detection accuracy of the balise uplink signal transmitted in a strong noise environment,we use chaotic oscillator to detect the balise uplink signal based on the characteristics of the chaotic system that is sensitive to initial conditions and immune to noise.Combining with the principle of Duffing oscillator system used in weak signal detection and uplink signal feature,the methods and steps of using Duffing oscillator to detect the balise signal are presented.Furthermore,the Lyapunov exponent algorithm is used to calculate the critical threshold of the Duffing oscillator detection system.Thus,the output states of the system can be quantitatively judged to achieve demodulation of the balise signal.The simulation results show that the chaotic oscillator detection method for balise signal based on Lyapunov exponent algorithm not only improves the accuracy and efficiency of threshold setting,but also ensures the reliability of balise signal detection.

Key words：Duffing oscillator；Lyapunov exponent；Jacobian algorithm；small data sets；balise uplink signal

0 Introduction

The signal transmission is an important link to ensure the traffic safety in the train control system.The balise system is the main equipment for the vehicle-ground information transmission.Its on-board subsystem named the balise transmission module (BTM)is responsible for receiving and demodulating the signals sent by the ground balise,decoding them to balise messages,and transmitting them to automatic train protection (ATP)on-board equipment.Whether BTM can demodulate the signals with high accuracy and accurately decode them is the key factor to ensure safe and reliable communication between vehicle and ground.Due to the fast running speed of the train,strong electromagnetic interference[1]and bad information transmission environment,the balise signals received by BTM are relatively weak.Therefore,it is necessary to study a method of balise signal demodulation with high detection accuracy.At present,most BTM devices adopt traditional coherent demodulation method,difference demodulation method,incoherent orthogonal demodulation method,discrete short time Fourier transform (DSTFT),EEMD[2]and so on.These traditional methods follow the idea of filtering first and then demodulation,and the demodulation accuracy is affected by the performance of the filter.

With the development of chaos theory,it has been applied in weak signal detection[3-6].Refs.[7-9] present the chaotic oscillator subsystem for the detection of balise signals,and the feasibility of the chaotic demodulation method is proved.The core of chaotic detection system is to determine the critical threshold of the system,and the accuracy of the critical threshold determines the accuracy of the detection signal[10].In this study,the Lyapunov exponent algorithm is adopted for chaotic oscillator detection of balise signal,which can accurately determine the critical threshold of the system,quantitatively judge the system output results,improve the detection accuracy of the balise signal,and reduce the bit error rate.

1 Basic theory

As a new signal detection technology,chaos theory abandons the traditional method of trying to eliminate or suppress noise by filtering and other technologies,and makes use of the sensitivity of chaotic system to initial conditions and its “immunity”to noise to extract useful signals from strong noise,which has good anti-noise performance.Duffing oscillator is one of the commonly used models for studying chaos,and its dynamic expression is given by[11]

(1)

wherekis the damping ratio,Fis the periodic driving force amplitude,andωis the driving force angular frequency.

The chaotic oscillator is very sensitive to the system periodic driving force amplitudeF,and the phase diagram state of the system output will vary with the change ofF.The basic idea of using Duffing oscillators to detect weak signals with known frequencies is as follows:Firstly,determine the driving force amplitudeFdto make the Duffing system with a specific frequency in a chaotic critical state;Then add the signal to be tested for detection,if the signal to be tested does not produce the signal with the same frequency as the Duffing system,the phase diagram output by the system is still in a chaotic state,and on the contrary,even if a small amplitude signal with the same frequency is detected under a lot of noise,the phase diagram output by the system will turn into a periodic state;Finally,judge the system output phase diagram to realize the detection of weak signals.

In the process of detecting weak signals by the chaotic oscillator,the core step is to accurately determine the critical threshold of the system and accurately determine the state of the output phase diagram of the system.The Lyapunov exponent refers to the degree to which two adjacent trajectories with different initial conditions are attracted or separated at an exponential rate over time in the phase space[12],thus the Lyapunov exponent can be used to judge whether the system is stable.By analysis of the Duffing oscillator system shown in Eq.(1),the output chaotic state phase diagram is shown in Fig.1(a),and the corresponding Lyapunov exponential spectrum is shown in Fig.1(c).It can be seen that in the chaotic state,the system motion trajectory is unstable,and the corresponding Lyapunov exponent is greater than 0;on the contrary,as shown in Fig.1(b),when the system outputs the phase diagram of the periodic state,it indicates that the system motion is in a local stable state,and the corresponding Lyapunov exponent spectrum is shown in Fig.1(b).In Fig.1(d),the Lyapunov exponent is less than 0 in this state.

(a)Chaotic state phase diagram

2 Chaotic demodulation method for balise signal

The ground balise signal is transmitted to the vehicle-mounted equipment in a 2FSK modulation mode,and its mathematical expression is given by

SFSK(t)=Ascos{2π[f+vm(t)Δf]t},

(2)

whereAsrepresents the signal carrier amplitude,and its value is 1 in this study;frepresents the center frequency of the carrier,and according to the technical index requirements of the balise,its value is 4.234 MHz;Δfrepresents the frequency deviation,and its value is 282 kHz;vm(t)represents binary input modulation signal,and the value of this signal is ±1[13].That is,when the symbol “0”is transmitted,the transmission carrier frequency is 3.948 MHz,and when the symbol “1”is transmitted,the transmission carrier frequency is 4.512 MHz.

The chaotic demodulation method of the balise signal is based on the characteristics of the Duffing oscillator system that is sensitive to specific frequency signals and immune to noise.According to the state of the Duffing system output phase diagram (chaotic state or large-scale periodic state),it is determined that within a certain time interval whether the corresponding transmission symbol of the balise signal transmission carrier is “0”or “1”.That is to say,if the internal driving force frequency in the Duffing vibrator detection system is set to the carrier frequency corresponding to the symbol “0”in the chaotic state,for the symbol “0”in the balise signal,the corresponding transmission within a period of time;if the internal and external driving forces of the detection system have the same frequency,the phase diagram output by the system will be transformed from a chaotic state into a large-scale periodic state;if the transmission symbol is “1”,due to the different frequencies of the internal and external driving forces,the phase diagram of the Duffing oscillator system will remain in a state of chaos.

In this study,the Lyapunov exponent method is used to distinguish the critical threshold of the system and the output phase diagram.The specific detection process is shown in Fig.2.To calculate the Lyapunov exponent the Jacobian method is used to determine the critical threshold of the system,and the Lyapunov exponent of the output time series is calculated by a small data amount algorithm,and the output state of the system is judged accordingly.

Fig.2 Flow chart of Duffing oscillator detection for balise signal

2.1 Jacobian method to determine critical threshold

We use the Jacobian method to determine the threshold of the Duffing oscillator detection system at a specific frequency.Firstly,the Duffing system model (Eq.(1))needs to be transformed into a third-order autonomous system,as shown given by

(3)

(4)

PerformQRdecomposition forY,Y=QR,namely

(5)

Substituting Eq.(5)into Eq.(4),we can get

(6)

Multiplying Eq.(6)left byQTand right byR-1,we can get

(7)

(8)

Defined by the Lyapunov exponent[14],we can get

(9)

2.2 Small data amount method to determine system output state

After the critical threshold of the system is determined,the balise signal to be tested is added to the Duffing oscillator detection system.To accurately determine the transmission symbol,it is necessary to accurately determine the output state of the Duffing oscillator detection system within the transmission time period of each symbol.Here,the small data amount method is used to calculate the maximum Lyapunov exponent of the system output sequence within the transmission time of each symbol,and the output state of the system is quantitatively determined according to the magnitude of the Lyapunov exponent,thereby demodulating the signal.

The key of the small data amount method is the phase space reconstruction,that is,the embedding dimensionmand the delay timeτare determined through the time series.For the output time series of the system,the Takens theorem[15]is used to reconstruct the phase space,namely

X={Xi|Xi=[xi,xi+τ,…,xi+(m-1)τ]T}.

(10)

The embedding dimensionmand time delayτare determined by the C-C method[16].

Secondly,we need to find the nearest neighbor pointXj′of each reference pointXjin the reconstructed phase space.To avoid that the reference point and its nearest neighbors are on the same trajectory,the short-term separation is restricted[17],namely

dj(0)=min‖Xj-Xj′‖ and |j-j′|>p,

(11)

wherepis the average period of the time series,and the value ofpcan be estimated by the inverse of the average frequency of the power spectrum.

For each reference point in the phase space,the distance of its adjacent points after discrete time steps is calculated by

dj(i)=min‖Xj+i-Xj′+i‖.

(12)

The maximum Lyapunov exponent can be estimated by the average divergence rate of each point on the basic orbit[18].Assuming that the reference point diverges exponentially at the rate of the largest Lyapunov exponent,namely

dj(i)=dj(0)eλi(i·Δt),

(13)

taking the logarithm of both sides of Eq.(13),we can get

lndj(i)=lndj(0)+λi(iΔt).

(14)

For eachi,we need to find the lndj(i)of allj,then average them and divide them by Δt,thus we can get

(15)

whereqis the number of non-zerodj(i),the regression line is made by the least square method,and the slope of the line obtained is the maximum Lyapunov exponent.

3 Simulations and analysis

3.1 Simulation analysis based on Matlab

According to the Duffing oscillator detection process of the balise signal,we first set the internal driving force frequency of the Duffing oscillator to be consistent with the carrier frequency corresponding to the symbol “0”,namelyω=2πf0=789 600π rad/s,and damping ratiok=0.5.Then the critical threshold of the Duffing oscillator detection system can be determined.In the simulation experiment,the fixed step size is 0.01,and the initial value is (0,0,0).The simulation gives the relationship curve between the Lyapunov characteristic index and the detection system’s driving force amplitude,as shown in Fig.3.

Fig.3 Relationship between driving force F and Lyapunov exponent

The critical chaotic threshold of the system is in the range of [0.82,0.83].To determine the high-precision threshold,Fstarts from 0.82 and increments by 10-6each cycle to calculate the Lyapunov corresponding to eachFIndex.Part of the results are shown in Table 1.It can be seen that whenF<0.825 843,the maximum Lyapunov exponent of the system is greater than 0,and the system is in a chaotic state;whenF>0.825 843,the maximum Lyapunov exponent of the system is less than 0,and the system is in a large-scale periodic state.Therefore the chaos critical threshold is set to beFd=0.825 843.Compared with the critical threshold 0.825 8 in Ref.[7],it can be seen that using the Lyapunov exponent method to determine the critical threshold of the chaotic detection system not only improves the efficiency,but also improves the accuracy of the threshold.

Table 1 Partial numerical table of driving force F and Lyapunov exponent

Then the balise uplink signal with noise (Fig.4)is added for detection.Setting the Duffing system driving force amplitudeFd=0.825 843，internal driving force frequencyω=789 600π rad/s，discretization step sizeh=1/20 320 000,we can get the output time-domain waveform diagram of the Duffing oscillator system,as shown in Fig.6.It can be seen from Fig.4 that the balise signal is completely submerged in noise,and it is difficult to distinguish whether the transmission signal is “1”or “0”in a certain period of time.However,it can be seen from Fig.5 that through the detection of the Duffing oscillator system,different time-domain waveforms are presented in different time periods,and the balise signal is detected from the noise,but it is difficult to distinguish whether the output state of the system is a chaotic state or a periodic state in a certain period of time.Thus the maximum Lyapunov exponent of the system output time series is calculated by the small data amount method,and the result is shown in Fig.6(a).According to the comparison between the Lyapunov exponent and the threshold,the threshold is set to 0 here,and the demodulated signal of the balise uplink is shown in Fig.6(b).

Fig.4 Balise uplink 2FSK signal with noise

Fig.5 Output time domain waveform of Duffing oscillator detection system

(a)Maximum Lyapunov index

3.2 Performance analysis

To verify the superiority of proposed method,it is compared with the method in Ref.[7] by calculating the bit error rate of the system under different signal-to-noises(SNRs).The bit error rate curve obtained through Monte Carlo simulation experiment is shown in Fig.7.Under the same SNR,the bit error rate of the balise uplink signal using the proposed method is lower than that in Ref.[7] because the threshold precision in Ref.[7] is lower than that in this study.With the decrease of SNR,some code elements do not fully enter the periodic state within a period of time,or the periodic state duration is too short.However,the calculation amount of the phase diagram discrimination algorithm based on Lyapunov index method is significantly larger than that of the power spectrum entropy method used in Ref.[7],therefore the Lyapunov index algorithm needs to be further improved.

Fig.7 Bit error rate performance curve

4 Conclusions

The Lyapunov exponent is applied to quantitatively determine the critical threshold and output state of the system for balise signal chaotic oscillator detection.The simulation shows that the proposed method not only overcomes the subjectivity of intuitionistic phase diagram judgment on threshold determination,but also improves the efficiency and accuracy balise signal chaotic oscillator detection.