A Method of 2FSK Signal Detection using Duffing Oscillator

Huang Yiran, Yin chengqun, Liu Li, Bi hongbo School of Electrical and Electronic Engineer,

North China Electric Power University, Baoding, Hebei, 071003, China hyr_2000@sina.com, adam_157@163.com

Abstract

Based on the principle of the chaos theory of weak signal detection, a new method is presented for Duffing oscillator used to demodulate 2FSK signal in this paper. The method can demodulate the 2FSK signal under strong noise background in the signal-to-noise rate at -25dB. Finally, the feasibility and the validity of the method were testified by the simulation. 1. Introduction

Technology of Modulation of 2FSK, which have many merits such has high power efficiency, convenient realization, reliable control, low cost and so on [1], is widely applied in digital communication fields. The traditional measurement technologies of 2FSK signal carrier frequency which include the zero crossing periodic detection method, the frequency mixing filter method, the coherent detection method and so on [2] are not satisfactory with the increasing noise. When the noise and other disturbances are serious under the power line environment, it is difficult to measure the carrier frequencies of 2FSK signal accurately with the traditional methods. So the new theory and technology are needed to guarantee reliable communication.

With the development of the detection technology, the weak signal detection has become more and more attractive in this field. The chaos theory, successfully introduced to the signal detection field, makes the detection technology of weak signal get a new breakthrough and inaugurates a new way in weak signal detection [3,4]. In this paper, a new method of detecting signal is brought forward based on the principle of the chaos theory. The method is used to demodulate 2FSK signal using Duffing chaotic oscillator. Simulation shows that it has a good anti-noise performance and is effective to demodulate the 2FSK weak signal. 2. Principle of detection using chaotic oscillator

Duffing oscillator is one of the chaotic systems that have been extensively studied, the normal form of the Duffing equation is:

3( cos( ))

x y

y ky x x f

ω

ω ω τ

•

•

⎧ =⎪⎨⎪ = − + − +⎩

(1)

Where ω is the frequency of to-be-detected periodic signal, fcos(ωτ) is the main sinusoidal driving force, k is damp ratio, -x+x3 is nonlinear restoring force.

In the Duffing equation system, when the k is fixed and system in the critical state (chaos, but on the verge of changing to great periodic), the condition of the oscillator will change with the f of the driving power, which is sensitive to certain signal and immune to noise at the same time [5]. Suppose the form of the to-be-detected signal is: acos(ωt+φ)+zs(t), where a is the amplitude of the useful signal, zs is the average amplitude of noise, and add to Duffing oscillator, the expression of the system is :

3( cos( ) cos( ))

x y

y ky x x f a t

ω

ω ω τ ω

•

•

⎧ =⎪⎨⎪ = − + − + +⎩

(2)

The detection of signals in noise based on chaotic oscillator is to set f=fd , (fd refers to the bifurcation value of the system).The noise has no effect on the system and acos(ωt+φ) plays a key role in changing the system state[6]. If the to-be-detected signal buried in the noise environment whose frequency is the same to or have little difference from the driving power, the consequence of f will be greater than fd .The system state will vary from chaotic motion to great periodic state and the trajectory of the orbits in the phase traces out a closed curve as in Figure 1 (a).Otherwise, the system continues to be at the chaotic state and phase plane orbit will tend to fill up a section of the phase space is shown in Figure 1 (b). Finally, through the method of x-y phase plane based on image recognition [7] or envelope extraction [5] to distinguish between chaotic state and periodic state, the weak signal was detected.

2008 ISECS International Colloquium on Computing, Communication, Control, and Management

978-0-7695-3290-5/08 $25.00 © 2008 IEEE

DOI 10.1109/CCCM.2008.277

510

2008 ISECS International Colloquium on Computing, Communication, Control, and Management

978-0-7695-3290-5/08 $25.00 © 2008 IEEE

DOI 10.1109/CCCM.2008.277

510

Figure 1 phase plane orbits of Duffing oscillator

3. The method of 2FSK signal detection by chaotic oscillator 3.1. Principle of 2FSK signal detection using chaotic oscillator

The carrier frequency of the 2FSK changes along with

the modulated signal (“1” or “0”), “1” corresponds to the carrier frequency f1 and“0” corresponds to the carrier frequency f2. The expression of the modulated 2FSK signal is:

2 1 2( ) [ ( )]cos [ ( )]cosFSK n n n nn n

S t a g t nT t a g t nT tω ω= − + −∑ ∑ (3)

Where,ω1=2πf1, ω2=2πf2, _

na is the reverse code of na

, generally, g(t) is single rectangular pulse. In fact, the 2FSK signal demodulation is to detect the

carrier frequency under each code, or to distinguish between the carrier frequency f1 and f2 under each code with known the carrier frequency. According to the principles above, a new method of 2FSK signal demodulation using the Duffing oscillator is brought forward. The principle of the detection is as follows: First, we put the chaos detection system into the critical state (chaos, but on the verge of changing to great period), then the 2FSK signal as to-be-detected signal is added into formula (4):

3( ) ( ) ( ) ( ) ( )x t kx t x t x t A t+ − + = (4) (1) When the transmission code is “0”, A(t)=

fd(ω2t)+acos(ω2t+φ).If the phase difference of 2FSK signal and forced power signal do not meet: π-arccos(a/2fd)≤φ≤π+arccos(a/2fd),it may cause transition of the system state from chaotic state to period state.

(2) When the transmission code is “1”, A(t)= fd(ω2t)+acos(ω1t+φ) .Because ω2≠ω1, the system state will be in the chaotic state or show intermittent chaotic state ((ω2-ω1)/ ω2≤0.03) [9].

At last, according to the change of state in the chaos system, the carried information of 2FSK signal can be detected.

3.2. The method of using Duffing oscillator to demodulate 2FSK signal

The method to demodulate 2FSK signal using Duffing

oscillator is described as follows:

Figure 2 flow chart of 2FSK Signal Detection According to the Figure 2, to judge the system state

under each code-width is necessary. The method of observing the phase plane can’t meet the practical engineering appliance, so we need to use the method of identify each code automatically. In this paper, the envelope of phase shift x-wave [5] is extracted to judge the chaos system state, to identify the system state automatically by program, so it can be used in practical engineering. The concrete thought is: due to the Duffing oscillator signal detection method has a better SNR, when the system is in the great periodic state, the perturbation of x-wave caused by noise is extremely small; but the chaotic state corresponds to the complex phase contrail switching between three singularity, and phase shift x has a great change between 0 and peak value. Waves of phase shift x in the periodic state with strong noises and the chaotic state are shown in Figure3 (a) and Figure 3 (b). Observing the Figure3, we can find that the characteristic of the two waveform envelope have obvious difference: The envelope fluctuant in waves of phase shift x in the periodic state with strong noises is far less than the chaotic state. In this paper, the judge threshold of the envelope is 0.3 times to the minimum of envelope wave. By the output of each judgment, the demodulation of 2FSK signal is realized.

Figure 3 x-waves of period state and chaos state 4. Simulation 4.1. Simulation model

(a) phase plane orbit in chaos state (b) phase plane orbit in period state

Set the driving force frequency of Duffing oscillator to be one ofthe two carrier frequencies and adjust the driving force into thecritical state.

Output and judge the system state.

Realize the detection of 2FSK signal by chaotic system state.

Take the 2FSK signal under strong noise background as to-be-detected signal to add to the chaos system.

(a) x-waves of period state

2

1

0

-1

-20 10000 20000 30000 40000 n

x2

1

0

-1

-2

x

(b) x-waves of chaos state

0 10000 20000 30000 40000 n

511511

Take the power line carrier communication as an

example, the two carrier frequencies of 2FSK signal are f1=59000Hz (corresponding to”1”) and f2=60000Hz (corresponding to”0”), Sampling frequency is fs=24000000Hz, the code rate is 600bit. Based on equation (1), the simulation model under the Matlab Simulink environment is shown in Figure 4, where “Driver” is the driving force of the chaos system, “Signal” is the to-be-detected signal, and adopt four-order Runge-Kutta to solve the Duffing equation, the simulation step is h=1/ fs /N (namely, each cycle N=400 points).In this paper, ω1=120000π, the critical state(chaos, but on the verge of changing to great periodic) fd is 0.82583, k is 0.5, k1=ω2, k2=0.5×ω, k3=1/ω .

Figure 4 Duffing oscillator 4.2. Simulation results

Figure 5 waves of 2FSK signal and chaos envelope

By Figure 5 we can intuitionally see, in each code-width, if the envelope waves of phase shift x is smooth, we can say the system is in the periodic state; and if the envelope waves of phase shift x has a great fluctuation, we can say the system is in the chaotic state. By the judging

program can output the demodulate results. So using the method proposed to demodulate 2FSK signal is feasible.

4.3. Anti-noise performances

In order to validate the anti-noise performance of the

method proposed in this paper, gauss white noise is added to the collection data. Simulation shows that the proposed method has strong noise-suppression in noise background, especially in the condition of weak signal detection in the strong noise environment. It is still effective in the signal-to-noise rate at -25dB.

Figure 6 2FSK error rat curves

The comparison between chaotic oscillator method and

the traditional methods of 2FSK is given in Figure6. The results indicate that when the SNR is equal, the bit

error rate of the proposed method is significantly lower than that of the traditional methods. So 2FSK signal can be directly and accurately detected by this method that isn’t necessary to take measures of filter and amplification and so on.

5. Conclusions

In this paper, according to the characteristics of 2FSK

signal, a new method to detect weak 2FSK signal by Duffing chaotic oscillator is presented. 2FSK weak signal detection using chaotic oscillator is realized by simulation. At last, the comparison between chaotic oscillator method and traditional demodulation method of 2FSK is given. The results indicate that the proposed method has strong noise-suppression especially in the condition of low SNR. It provides reference and guidance for the advanced research and implementation.

Acknowledgment

(a) Waves of to-be-detected 2FSK signal

1

0

-1 0 0 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0

0 0 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 (b) Envelope waves of x under the demodulation code

2.5

0

-2.5

SNR (dB) -30 -26 -22 -18 - 14 - 10 -6 -2 2 6 10 14

1

10-1

10-2

10-3

10-4

10-5

10-6

Erro

r rat

e (B

ER)

Chaotic demodulation Incoherent demodulation Coherent demodulation

k1

k3k2

1

gain_f

1

gain_a

x

y

sigdata

Signal

1

s

1

s

u-u^3

Driver

512512

This work was supported by the Young Teacher Foundation of North China Electric Power University through the grant number 93201710.

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