Energy Analysis of a Class of Nonlinear Detection

Systems with Chaotic Oscillators

Sun Wen-jun, Rui Guo-sheng

(Graduate Students’ Brigade,Naval Aeronautical and Astronautical University,

Yantai Shandong, 264001)Abstract: In areas of weak signal detection,

considering the algorithm complexity andhuge computation in phase transitionidentification of Duffing oscillator, a novelmethod based on energy distribution wasproposed in this paper. Firstly, chaoticdetection theory was analysised, then TeagerEnerg Operator was associated with nonlineardynamic theory, which makes symbolidentification achievable. Finally, therobustness and real-time property of thealgorithm were analyzed. Simulation resultsshow that, compared with the traditionalmethods, the computation and real-timeproperty of this method has significantlyimproved to meet the needs of the weak signaldetection under the strong noise background.

Index Terms: chaotic detection; energyoperator, algorithm complexity; real-timeI Introduction

With facing more complex spaceelectromagnetic environment, the transmissionchannel susceptible to environmental factors,the input signal to noise ratio is too low fordemodulation at the receiving end. And hencethe information transfering is blocked,thedifficulty of demodulation becomes larger,the error rate increase consumedly. In fact,enhancing the ability to detect weak digitalsignals effectively, improving the reliability ofdigital demodulation in strong noisebackground has been a hot issue of theory andengineering research. However, the ‘linear’demodulation technology based on matchedfiltering,and the best correlation receiving hasget to the extreme performance, closing to thetheoretical limit. If wanting a further "lowpoint " walk, there must have a newtheoretical breakthrough, just as the researchof nonlinear demodulation techniques.

In recent years, chaos theory has beenwidely applied in the field of signal detection,using the chaotic oscillator to detect weaksignals is a complex mathematical process,whose implementation core is chaotic fractalidentification method. At present, domestic

and foreign scholars have made a lot of work,but there are still some problems: the phaseplane trajectory method proposed in theliterature [1,2] was viewed as the criterionphase transition, but its vulnerability is insimulation time and other factors, the error islarge and inefficient, the dual oscillatorsdifference method in literature [3] improvedthe trajectory method, but still not trip out ofimage observation scope, Lyapunov exponentmethod in literature [4] can accurately identifythe system in phase space trajectories andstates, but Jacobian matrix of systemequations is very difficult to resolve. TheMelnikov method in literature [5] is relativelysimple, but the identification accuracy isworse than others.

Considering these problems, a novelmethod based on chaotic energy was proposedin this paper. Firstly, chaotic detection theorybased on Duffing oscillator was analysised in-depth. Then Teager Energ Operator wasassociated with nonlinear dynamic theory,which makes symbol identificationachievable. Finally, the robustness and real-time property of the algorithm were analyzed.Simulation results show that the non-lineardemodulation algorithm based on chaos theoryhas a good anti-noise performance fordemodulating digital signals at low SNRbackground. II Chaotic energy detection principle

2.1 Duffing chaotic oscillators

The main chaotic dynamic modes are:Duffing, Lorenz, Vanderpol and etc, thereinto,research and application of Duffing equationhas been in emphases, and was widely used inweak signal detection. Considering thedetection model based on Duffing equation:

( ) ( ) ( )x t x t u a tδ γ η′′ ′+ + = + +

(1)

Where, we defined the system drivingforce: ( )cosa tω= , damping ratio: δ ,restoring force: 3u x x= − + , circumstance

978-1-4673-5225-3/14/$31.00 ©2014 IEEE

noise: ( )tη , signal to-be-measured:( )cosd tγ ω . Besides, the frequencies of

system and signal-to-be-measured are equal.The chaotic system based on Duffing

equation have a very high sensibility toperiodic driving force, when the damping ratiois fixed, the system will represent amplenonlinear fractal states orderly: chummage,bifurcation, chaos, critical and periodic trace[6]. However, the system trace in chaos issome anomalistic approaches, andcorrespondingly one simple closed curve inperiodic state. By the differences between thetwo states, the weak signal could be detected.

2.2 Energy tracking model

In the 1990s, Kaiser firstly preferred theconception of the energy operator[7], andapproved it could track the nonlinear signalaccurately, particularly envelope of amplitudemodulation signals and instantaneous featureof frequency modulation signals[8].

To the continue signal ( )x t , we definedits operator cψ :

( ) ( ) ( ) ( )2

c x t x t x t x tψ ′ ′′= − (2)

Where, ( )x t′ , ( )x t′′ are individuallythe first derivative and second derivative ofthe signal ( )x t , the physics sense representsenergy is proportional to square of amplitudeand frequency.

Energy detection algorithm is defined onsmall sample signals, which makes theapproach of nonlinear system’s instantaneousfrequency, instantaneous amplitude, envelopebe one real-time processing.

2.3 Fractal states identification

Chaotic system is a high-order nonlinearstochastic differential equation, which doesn’thave accurate analytic solution. And hence wecould only resolve the numerical solution bydiscretize- tion approach. By the approach ofEuler-Maruyama[9] method，recurrence formis as follows:

( 1) ( )

( 1) ( ( ) ) 2 k

x k hy k

y k y k h D hδ µ γ ε

+ =

+ = − + + +(3)

Where, h is integration step, kε is standardGauss random series, D is the intension ofwhite noise.

For the system energy in stable state,continuously sample n times as the collectedassemble, each integration step is considered

as one point ( )x k , and continuous-time t isdisplaced with discrete-time variable nT :

( ) ( ) ( )1 1 2x t x k x k T′ ↔ + − − (4)

So we can get the discrete signal energyoperator:

( ) ( ) ( ) ( )2 1 1d x k x k x k x kψ = − − + g (5)

In different background noise circumst-ances, the system energy operator calculatedby the algorithm varies correspondingly. Inorder to better distinguish chaotic phasechange, we introduce statistics knowledge inquantitative analysis of system energy.III Simulation experiment

3.1 Feature analysis and Detection method

Build chaotic system model consisting ofHolmes type Duffing equation, resolve systemby Teager energy algorithm to get the energydistribution characteristics, the results areshown in Figure 1, Figure 2. As we can seefrom the figures, the energy distributioncharacteristics of large-scale periodic state,after the initial fluctuations , the envelopesoon goes into convergence and becomesstability. By contrary, in chaotic state , thereare a lot of negative pulses present in energyfluctuations . From the above analysis, thedifference between the the energy distributioncharacteristics under two states is conspicuous, which will be used as one fractal statesidentification method.

Fig.1 Chaotic energy distribution characteristic

Fig.2 Periodic energy distribution characteristic

Based on the above analysis, the chaoticenergy algorithm could be described asfollows:

(1) Adjust the intensity of the cycle drivingforce to make the system be in a critical state,

(2) Input the to-be-measured signal withnoise, and record the system output,

(3) Use the energy operator to track thesystem output, obtain the energy distributionof the sustem characteristics, and calculatephase change threshold in accordance withEquation (5),

(4) Comparing the phase transitionthreshold average and the minimum energyvalue, if 0E > , the system has a phasetransition; Instead, the system phase transitiondoesn’t occur.

3.2 Real-time analysis

In this chapter, weak electrical carriersignals are used in simulation. Simulationparameters: the electrical carrier signal’sintensity 0.02V, frequency 1 rad/s, calculationstep 0.01h = s. When the phase of to-be-measured signal transfers from “0” and “π ”,the Duffing system switches between fromchaos and large-scale periodic state. Thedetection equation is derived to obtain thephase detecting range[11]:

πγτπγ +≤≤− )2arccos()2arccos( aa

(6)When the phase τ is located in phase

detection range, the Duffing system .can beused for signal detection.

Currently, Lyapunov exponent method isone of the most widely used classical phasetransition algorithms.Therefore compared with

the chaotic energy methom, the real-timefeature will be analysised as follows.

The simulation time is 0 ~ 200s, within 100Monte Carlo experiments, the singlesimulation computation time results withdifferent data sizes are shown in Figure 3.

Fig.3 Fractal states identification methods’

simulation-time with different data sizes

In the simulation results, the averagesimulation computation time of single energydetection method is 0.0146s, while Lyapunovexponent calculation method is 6.1621s, themethod proposed in paper reduced than thelatter by approximately two orders ofmagnitude. Compared with the method inliterature [1,2], the method proposed in thispaper has a higher detection efficiency.Further more, it’s more conducive to thecomputer automatically detects and quickjudgment to avoid the influence of humanfactors.

3.3 Reliability analysis

In order to quantitatively analyze thealgorithm 's anti-noise performance, one groupof weak electric-power line carrier signal(1000 random symbols) under strongbackground noise is detecd in experiments.

The detection model is Duffing system, themian parameters of simulation is the same tothe former chapter. By the approach of Energyalgorithm, we obtain the energy distributioncharacteristics of system for each symbol,which is shown in Figure 4. To reduce thealgorithm complexity furtherly, the sampleratio k is introduced.

Fig.4 Different ratios of sampled data for detection

However, as shown in Figure 5, with thenoise intensity increasing, the accurateidentification probability is decreasing. Inaddition, the identification probability is indirect proportion to the sample ratio k. Thesimulation data are 10 groups of weak powerline carrier signals, traditional detectionmethods are indaptable, iteration method isEuler-Maruyama approach and the miansimulation parameters is the same to theformer.

Fig.5 Chaotic Energy idetification probability

with different Noise-intensity

IV Conclusion

For detecting weak signals in strong noiseaccurately, chaotic method is combined withTeager energy operator in paper, named aschaotic energy algorithm . With extremely lowcomputational complexity and excellent noiseimmunity, this method can achieve the reliabledetection requiements of 0dB~-30dB signals.Results show that the alrorithm is a very

effective detection method for signalsidentification in strong noise background.

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