2017 2nd International Conference on Computer, Network Security and Communication Engineering (CNSCE 2017) ISBN: 978-1-60595-439-4

**The Fractional Order Duffing Oscillator Detection Method for Nanovolt **

**Level Weak Signal **

### Nan LI

*### and Ming-yan LIU

College of Information Engineering, Northeast Electric Power University, Jilin City, Jilin Province, 132012, China

*Corresponding author

**Keywords: **Fractional order Duffing oscillator, Nanovolt (nV) level weak signal, Symmetric Alpha

stable distribution noise, Weak signal detection.

**Abstract. **The traditional weak signal detection makes it difficult to detect nV level signal under
strong noise background, so it proposes a new detection method of improving fractional order
Duffing oscillator (IMFR-Duffing). Using variable substitution method for improving conventional
fractional order Duffing equation obtained a mathematical model of IMFR-Duffing oscillator. It
studied anti-noise performance of applying this mathematical model to low frequency nV level
weak signal detection in the Gaussian and non-Gaussian noise background. The detection results of
contrasting the IMFR-Duffing with the integer order Duffing oscillator verify the effectiveness of
the proposed method, and the minimum signal-to-noise ratio (SNR) threshold is reduced by 10 dB.
The simulation results show that the lowest SNR threshold of the IMFR-Dufffing oscillator reaches
-73 dB in Gaussian white noise, -57 dB in Gaussian colored noise and -153 dB in symmetric Alpha
stable distribution noise, and the amplitude detection has higher precision.

**Introduction**

Weak signal is useful signal submerged in strong noise, the Duffing equation was used for weak signal detection since 1990s, and it has been widely applied for mechanical fault diagnosis[1], line-spectrum detection of underwater radiated noise[2], fault line selection in small current system[3] and other fields now. In engineering application, the nV level signal is extremely weak, it is hard detected from the strong noise with the common time-domain method, but it can better on achieving the nV level signal detection by using Duffing oscillator. Literature[4] a new chaotic system which is composed of autocorrelators and Duffing chaotic oscillator is used to detect 1 nV signal, and the SNR threshold value is up to -10 dB under the background of Gaussian white noise; SNR threshold value can reach -1 dB in Gaussian colored noise. Literature[5] the mixed detection system of Phase-Locked Loops for detecting weak signal frequency and Duffing system for detecting the amplitude, can test the nV level weak signal containing the Gaussian white noise, and the minimum SNR is -22.23 dB. In order to further reduce the SNR threshold value of nV level weak signal detection, the weak signal detection method of IMFR-Duffing oscillator is proposed. The purpose is to research on the detection ability of the nV level weak signal in the Gaussian white noise, Gaussian colored noise and symmetric Alpha stable distribution noise background.

**IMFR-Duffing Oscillator Mathematical Model **

On the basis of the integer order Holmes type Duffing equation, it introduces the fractional order
differential operator, and the equation of the fractional Duffing oscillator is shown in Eq. 1:_{ }

Dαx(t)+kDβx(t)-x(t)+x3(t)=Fcos(t). (1)

Dq1x(ωτ)=ωq1y

Dq2y=ωq2[-ky+x(ωτ)-x3(ωτ)+Fcos(ωτ)]. (2) In the Eq.2, q2=α-β,q1=β. The mathematical model is used to study the dynamic characteristics

and nV level signal detection.

**IMFR-Duffing Oscillator Weak Signal Detection Method **

The weak signal detection principle of IMFR-Duffing oscillator system is that the system is at the critical chaos state of Fd and built-in cycle driven frequency is set to the weak periodic measured signal frequency. If the system phase is from chaos into large-scale periodic state, the measured signal is included in the frequency of weak periodic signal, but if the system still maintains the chaotic state, the detection is failure or weak periodic signal detection does not contain the measured signal. Estimation value of weak signal is the amplitude of the driving force which can make system jump into chaotic state again, and Fc is this moment amplitude, then the weak signal amplitude estimation is Fd-Fc[7,8]. In the end the detection performances are measured by the relative error of amplitude and minimum SNR threshold.

**The Detection Performance Analysis of Low Frequency nV Level Signal Using IMFR-Duffing **
**Oscillator **

**Low Frequency nV Level Signal Detection in Gaussian White Noise Background**

In the simulation, measured signal is s(t)=acos(2πft)+n(t), n(t) is the variance as 1×10-14 of the Gaussian white noise signal, the amplitude is a=1×10-10 V, frequency is set to 150 Hz, and the waveform of measured siganl is shown in Figure 1. The power spectrum analysis of measured signal is shown in Figure 2, and it can not identify the nV level measured signal containing 150 Hz signal.

0 0.2 0.4 0.6 0.8 1 1.2 -0.5

0 0.5 x 10-10

time/s a m p lit u d e /V measured signal

0 0.2 0.4 0.6 0.8 1 1.2 -5

0 5x 10-7

time/s a m p lit u d e /V

Gaussian white noise

0 0.2 0.4 0.6 0.8 1 1.2 -5

0 5x 10

-7 time/s a m p lit u d e /V

measured signal with Gaussian white noise

00 100 200 300 400 500 600 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10

[image:2.612.128.266.419.489.2]-9 frequency p o w e r power spectrum

Figure 1. Time domain waveform of the measured signal. Figure 2. Power spectrum diagram.

The selection of integer order Duffing oscillator is to achieve nV level weak signal detection, the system is set to the critical chaotic state as shown in Figure 3 (a) and (b) and Fd is 0.6719578790. When the measured signal is incorporated into the detection system, the Duffing oscillator system enters into a large-scale periodic state from chaotic state, and the phase trajectories are shown in Figure 4 (a) and (b). According to the observation of phase diagrams, it is shown that the measured signal contains 150 Hz periodic weak signal. Then it can detect that the amplitude of nV level is 8.4×10-11 V and the relative error of the amplitude is 16%.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5 -1 -0.5 0 0.5 1 1.5 x(t) y (t ) phase diagram

-2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x(t) y (t ) Poincare graph

-1.5-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 x(t) y (t ) phase diagram

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x(t) y (t ) Poincare graph

[image:2.612.82.527.511.713.2](a) (b) (a) (b)

Figure 3. The phase locus of critical chaotic state. Figure 4. The phase locus of signal detection.

[image:2.612.89.528.607.704.2]Table 1. The results of integer order Duffing oscillator detection in Gaussian white noise.

a/(V) f / (Hz) σ2 SNR/(dB ) δ/ (%)

1×10-10

1 1×10-16 -43 12 5 1×10-17 -33 23 25 1×10-15 -53 32 75 1×10-17 -33 21 100 1×10-16 -43 29 150 1×10-15 -53 16 200 1×10-14 -63 28

The results from the table are that the maximum relative error of amplitude is 32% and the minimum SNR threshold is -63 dB.

The IMFR-Duffing oscillator detection method for nV level signal aimes to further improve the relative error of amplitude and the minimum SNR threshold. The parameters are as follows: q1=0.7, q2=1.2, k=0.2. The critical chaotic state are shown in Figure 5 (a) and (b) and Fd is 0.5748134327. The phase trajectories are shown in Figure 6 (a) and (b), and the system can determine the 150 Hz weak periodic signal existing the measured signal. When Fc is 0.574813432610, the detecting amplitude of the measured signal is Fd-Fc=9×10-11, and the relative error is 10%.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

x(t)

y

(t

)

phase diagram

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x(t)

y

(t

)

Poincare graph

-2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5

-1 -0.5 0 0.5 1 1.5 2

x(t)

y

(t

)

phase diagram

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

x(t)

y

(t

)

Poincare graph

(a) (b) (a) (b)

Figure 5. The phase locus of critical chaotic state. Figure 6. The phase locus of signal detection.

After a large number of simulations, the detecting results of IMFR-Duffing oscillator detection are shown in Table 2.

Table 2. The results of IMFR-Duffing oscillator detection in Gaussian white noise.

a/(V) f / (Hz) σ2 SNR/(dB ) δ/ (%)

1×10-10

1 1×10-14 -63 17 5 1×10-15 -53 19 25 1×10-13 -73 23 75 1×10-16 -43 16 100 1×10-16 -43 12 150 1×10-14 -63 10 200 1×10-13 -73 31

The Table 2 shows that the detection frequency is between 1 and 200 Hz, the maximum relative error of amplitude is 31%, and the minimum SNR threshold is -73 dB .

**Low Frequency nV Level Signal Detection in Gaussian Colored Noise Background **

In this section, the Gaussian colored noise is attained by Gaussian white noise through the band pass filter. Using the above analysis method, the simulation results of the integer order Duffing and IMFR-Duffing oscillator detection are shown in Table 3 and Table 4.

Table 3. The results of integer order Duffing oscillator detection in Gaussian colored noise.

a/(V) f / (Hz) σ2 SNR/(dB ) δ/ (%)

1×10-10

[image:3.612.173.435.651.749.2]Table 4. The results of IMFR-Duffing oscillator detection in the Gaussian colored noise.

a/(V) f / (Hz) σ2 SNR/(dB ) δ/ (%)

1×10-10

1 2.26×10-15 -57 24 5 2.15×10-16 -46 29 25 2.3×10-16 -47 13 75 2.25×10-16 -47 26 100 5.47×10-16 -50 0 150 2.2×10-15 -56 18 200 5.42×10-16 -50 10

By the integer order Duffing oscillator detection, the maximum relative error of estimated amplitude can reach 39%, and the minimum SNR threshold can reach -50 dB. Using the IMFR-Duffing oscillator detection, the maximum relative error of estimated amplitude reaches 29%, and the minimum SNR threshold can reach -57 dB.

**Low Frequency nV Level Signal Detection in the Symmetric Alpha Stable Distribution Noise **

Alpha stable distribution noise is a kind of non-Gaussian distribution noise, there is no two order statistics, so the mixed signal to noise ratio (MSNR) is introduced to measure the relative intensity of the signal and noise to be measured as shown in Eq.3[9]:

MSNR=10lg(σs2/γv). (3)

In Eq.3 σs2 is the variance of the measured signal and γv is the dispersion coefficient.

In the simulation, the parameters of the symmetric Alpha stable distribution noise are as follows: α=1.8,β=0,a=0 and variable γ. The results of the integer order Duffing oscillator and IMFR-Duffing oscillator detection are shown in Table 5 and Table 6.

Table 5. Detection results of integer order Duffing oscillator in symmetric Alpha stable distribution noise.

a/(V) f / (Hz) γ MSNR/(dB ) δ/ (%)

1×10-10

1 1×10-7 -133 18 5 1×10-7 -133 18 25 1×10-6 -143 24 75 1×10-7 -133 25 100 1×10-7 -133 9 150 1×10-7 -133 26 200 1×10-6 -143 23

Table 6. Detection results of IMFR-Duffing oscillator in symmetric Alpha stable distribution noise.

a/(V) f / (Hz) γ MSNR/(dB ) δ/ (%)

1×10-10

1 1×10-6 -143 18
5 1×10-6 -143 16
25 1×10-5 -153 13
75 1×10-6 -143 20
100 1×10-6 _{-143 } _{19 }

150 1×10-6 -143 24 200 1×10-5 -153 23

It can be seen that the maximum relative error of amplitude is 26%, and the minimum SNR threshold reaches -143 dB by integer order Duffing oscillator detection; the maximum relative error of amplitude using the IMFR-Duffing oscillator detection is 24%, and the minimum SNR threshold is -153.

**Conclusion **

weak signal detection by the IMFR-Duffing oscillator is -73 dB which is reduced by 10 dB in Gaussian white noise. However, in the Gaussian colored noise, the minimum SNR threshold of two Duffing oscillator weak signal detection is not significant difference. In the symmetric Alpha stable distribution noise background, IMFR-Duffing oscillator can achieve the lowest SNR threshold is -153 dB, and it also reduces 10 dB.Therefore, the IMFR-Duffing oscillator is better than the integer order Duffing oscillator detection method, the accuracy of amplitude detection is higher, and the system has better anti-noise performance.

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