A Novel Detection Method for Weak Harmonic Signal with Chaotic Noise

To detect weak signal, it is necessary to examine the law of noise and the features of signal, and then extract and measure the weak signal features from the chaotic noise background, using a series of signal processing methods [3]. Before signal detection, the chaotic signal, network signal, and noise signal should be separated from the collected equal interval signal. Then, the noise data should be removed from the signal to facilitate the detection of the target signals. Finally, the weak harmonic signal can be detected accurately based on the chaotic oscillator and periodic features of the chaotic signal.

2.1 Real-time signal acquisition and processing

To detect weak harmonic signal in the wireless network, the first step is to acquire the signal transmitted in the network at different intervals. The input signal of the wireless network can be described as:

$s(k)=A \sin \left(2 \pi f k+\varphi_{0}\right)$  (1)

where, A, f, and φ₀ are the amplitude, frequency, and original phase of the input signal, respectively [4]. Then, the sampling interval of wireless network signal can be calculated by:

$T_{N}=\frac{1+N}{N f}$  (2)

where, N is the conversion coefficient of wireless network signal.

Let 1/T_s be the acquisition rate of wireless network signal. Then, the collected weak signal can be expressed as a time sequence:

$s(y)=A \sin \left(2 \pi f y\left(\frac{1}{N f}+\frac{1}{f}\right)+\varphi_{0}\right)$ (3)

where, y is the number of signal samples. Combining formulas (1) and (3), the collected weak signal can be depicted as:

$\left\{\begin{array}{l}T=\frac{N+1}{f} \\ f_{\varphi}=\frac{f}{N+1} \\ s_{\varphi}=A \sin \left(2 \pi f_{\varphi} y+\varphi_{0}\right)\end{array}\right.$  (4)

where, f_φ is the frequency of the collected weak signal. Then, the initial modal components of the collected signal can be separated by:

$s_{\rho}=\sum_{j=1}^{n} c_{j}(t)+r_{n}(t)$  (5)

where, c_j(t) is the j-th modal component separated from the collected signal; r_n(t) is the residual signal after the n-th separation [5].

2.2 Separation of chaotic noise signal

The collected weak signal contains some chaotic noise signal, for the wireless network is disturbed by chaotic noise. The chaotic noise signal will dampen the detection accuracy of weak harmonic signal [6]. Therefore, the chaotic noise signal should be separated from the collected signal, and eliminated based on the chaotic features.

2.2.1 Feature analysis of chaotic oscillator

Suppose f(x) features continuous self-mapping in the closed interval [a, b]. Then, a noise is a chaotic noise, if it satisfies the following two conditions: (1) Periodic points exist in f(x); (2) There exists an uncountable subset $S \in I$ that satisfies:

$\left\{\begin{array}{l}\forall x, y \in S \& x \neq y, \lim _{n \rightarrow \infty} \sup \left|f^{n}(x)-f^{n}(y)\right|>0 \\ \forall x, y \in S, \liminf _{n \rightarrow \infty}\left|f^{n}(x)-f^{n}(y)\right|=0 \\ \forall x \in S, \liminf _{n \rightarrow \infty}\left|f^{n}(x)-f^{n}(y)\right|=0\end{array}\right.$  (6)

where, sup(⋅) and inf(⋅) are the upper and lower bounds, respectively; fⁿ(⋅) is the n-th iteration of f. The noise content of the collected signal can be measured by signal-to-noise ratio (SNR) and noise level. The modal components of the collected signal can be expressed as a time sequence:

$s(t)=x(t)+n(t)$  (7)

where, x(t) and n(t) are pure time sequence and additive noise, respectively. Then, the noise level of a modal component can be computed by:

$N_{l}=\sqrt{\frac{\sigma(n(t))}{\sigma(x(t))}}$  (8)

where, σ(⋅) is the variance of the time sequence. The trajectory of chaotic signal varies with the noise disturbances [7]. Taking white noise for example, the features of the wireless network signal with white noise are compared in Figure 1 below.

14.1.png

Figure 1. The features of the wireless network signals with white noise

As shown in Figure 1, the chaotic oscillator is immune to noise: the changing noise intensity merely affected the trajectory in the phase diagram, and had little to do with the state of the chaotic signal.

2.2.2 Feature extraction of chaotic noise

Based on the results of feature analysis and the properties of chaotic oscillator, the features of chaotic noise can be extracted [8], including Lyapunov exponent, correlation dimension, and Kolmogorov entropy. Among them, the correlation dimension can be calculated by:

$\left\{\begin{array}{l}\dot{x}=\eta(y-x) \times N_{l} \\ y=-x+7 x-y \times N_{i} \\ \dot{z}=x y-b z \times N_{l}\end{array}\right.$  (9)

where, η, b and r are three constant terms. Then, x, y and z were initialized as 1, 0, and 0.1. Substituting the chaotic noise points in the time sequence of chaotic signal into formula (9), the correlation dimension of chaotic noise can be obtained. The values of other features can be obtained in a similar manner.

2.2.3 Suppression of noise signal

The extracted features of chaotic noise were utilized to determine the minimum embedding dimension of the time sequence for wireless network signal, and to differentiate between deterministic and stochastic time sequences. In this way, the chaotic noise can be separated from the collected signal. In addition, the noise in the wireless network signal was suppressed by the noise cancellation device in Figure 2, where the inputs are the collected signal and the predicted signal after noise suppression; e_φ(n) is the error sequence in noise cancellation.

16.2.png

Figure 2. Structure of the noise cancellation device

2.3 Calculation of effective values of harmonic components

Let g(t) and g(n) be the weak signal of the wireless network after noise suppression and its corresponding digital signal, respectively. Then, the following can be derived by weighting the wavelet basis function:

$\int x(t)^{2}=\sum_{i=0}^{\eta} \sum_{k=0}^{2^{j}-1}\left(d_{j}^{i}(k)-g(t)\right)^{2} \times(\dot{x}+\dot{y}+\dot{z})$  (10)

where,  dⁱ_j(k) is the coefficient of the scale function. Then, the root mean square (RMS) of the wireless network signal can be expressed as:

$X_{\text {тт5}}=\sqrt{\sum_{i=0}^{2^{j}-1}\left(V_{j}^{i}\right)^{2}}$  (11)

where,  Vⁱ_j is the effective value of harmonic components in different frequency bands:

$V_{j}^{i}=\sqrt{\frac{1}{2^{N}} \sum_{k=0}^{2^{N-j-1}}\left(d_{j}^{i}(k)\right)^{2}}$  (12)

Then, the effective value of harmonic components in the wireless network signal can be obtained by reverse deduction.

2.4 Solution of harmonic frequency

Based on the effective values of harmonic components in the wireless network signal, the harmonic matrix was constructed, and the eigenvalue of covariance matrix R was computed. Then, the likelihood function can be solved by substituting the eigenvalue into the following formula:

$\Lambda(n)=V_{j}^{i} \frac{1}{M-n} \times \frac{\sum_{i=n+1}^{M} \lambda_{i}}{\left(\prod_{i=n+1}^{M} \lambda_{i}\right)^{\frac{1}{M-n}}}$ (13)

where, M is the number of array elements of the signal; n is the number of harmonic components; λ_i is the eigenvalue of covariance matrix R of signal X. Next, the function expressed by formula (13) was represented by a curve. The number of inflection points on the curve is the number of harmonic signals in the wireless network signal [9]. Then, the eigenvalues of covariance matrices were sorted in descending order, creating the eigenvector matrix. On this basis, the frequency of each harmonic component can be estimated by:

$f_{k}=\frac{\operatorname{angle}\left(\lambda_{i}\right)}{2 \pi T_{s}}$  (14)

where, λ_k is the composite eigenvalue of the eigenvector matrix.

2.5 Determination of the amplitude and frequency for weak signal detection

Under the background of chaotic noise, the frequency of weak periodic signal can be expressed as:

$r(t)=\frac{k(t)+m(t)}{f_{k}}$  (15)

where, k(t) is the weak sine harmonic signal of the wireless network; m(t) is the white noise signal with an SNR<-20dB. To determine the detection frequency, the frequency of weak sine harmonic signal should be estimated first [10]. Considering the structural variation in chaotic coefficient, the SNR curve with synchronization error can be obtained (Figure 3).

16.3.png

Figure 3. The SNR-frequency curve

As shown in Figure 3, the SNR exhibited a decline with the growing frequency of the input signal [11-15]. Considering the synchronization error of the chaotic noise background, the detection threshold for weak harmonic signal was set to -15dB. Under the current parameters, the effective interval of detection frequency was obtained as [0Hz, 200Hz].

The critical threshold for chaotic signal and detection amplitude of sine signal were derived from the effective values of harmonic components. To obtain an accurate critical threshold, the position of the threshold was approximated based on the amplitude bifurcation [16-20].

Let P₁ and P₂ correspond to the chaotic state and periodic state of the noise signal, respectively. The threshold could fall on the halfway between P₁ and P₂, i.e. x= $\frac{P_{1}+P_{2}}{2}$ . Since $\frac{P_{1}+P_{2}}{2}$  and $\frac{P_{1}+P_{2}}{2}$ + 0.1 both correspond to the chaotic state and $\frac{P_{1}+P_{2}}{2}$ +0.2 corresponds to the periodic state, the interval of f can be determined as $\left[\frac{P_{1}+P_{2}}{2}+0.1, \frac{P_{1}+P_{2}}{2}+0.2\right]$ . In this interval, the compensation was increased with a step of 0.001, such as to find the transition point from the chaotic state to the periodic state. This point was taken as the critical threshold for signal detection.

At the critical threshold, the target wireless network signal was mixed with noise, and used to perturb the chaotic oscillator. By observing the variation of phase trajectory, all the sine signals of the weak harmonics were identified from the mixed signal. After that, the amplitude position was adjusted again and the noisy signal was controlled at the critical threshold of chaotic state, creating a new threshold. Therefore, the sine signal amplitude of the input mixed signal can be computed by:

$A=r(t) \times\left(\gamma_{1}-\gamma_{2}\right)$  (16)

where, γ₁ and γ₂ are the initial critical threshold and the new threshold, respectively.

2.6 Detection of weak harmonic signal

According to the effective values of harmonic components, detection amplitude and detection frequency, the collected signal that contains chaotic noise and the determined detection signal were inputted at the same time. Then, we have:

$\left\{\begin{array}{l}\ddot{x}+\alpha_{1}\left[\left(x^{2}+y^{2}+z^{2}\right)-1\right] \dot{x}+\beta_{1} x= \\ f_{\mathrm{e}} \cos (\omega t+\theta)+a \cos (\omega t+\theta)+n(t) \\ \dot{y}+\alpha_{2}\left[\left(x^{2}+y^{2}+z^{2}\right)-1\right] \dot{y}+\beta_{2} y= \\ f_{\mathrm{e}} \cos (\omega t+\theta)+a \cos (\omega t+\theta)+n(t) \\ \ddot{z}+\alpha_{3}\left[\left(x^{2}+y^{2}+z^{2}\right)-1\right] \dot{z}+\beta_{3} z= \\ f_{\mathrm{e}} \cos (\omega t+\theta)+a \cos (\omega t+\theta)+n(t)\end{array}\right.$  (17)

where, acos(ωt+θ) is the weak harmonic signal of the wireless network; a is the amplitude of the weak harmonic signal. If the weak sine signal and the chaotic noise signal are of the same period, frequency, and phase, the chaotic signal will be sensitive to the weak sine signal, while immune to the noise. Hence, the phase trajectory of the resulting chaotic signal will quickly change from chaotic state to periodic state. Once the change takes place, the weak harmonic signal can be detected successfully [21, 22].

If the chaotic noise in the background is uniform, the detection amplitude and frequency for nonuniform chaotic noise background can be adjusted before detecting the weak harmonic signal in the wireless network. In addition, the transient signal of the weak harmonic signal can be described by:

$\left\{\begin{array}{l}H_{0}: x(n)=c(n) \\ H_{1}: x(n)=c(n)+s(n)\end{array}\right.$  (18)

where, H₀ and H₁ are two states of wireless network. The two states have different transient signals. Therefore, the harmonic signals detected at the two states must be different. Similarly, the periodic signal of weak harmonic signal can be detected, according to the chaotic moving features of the harmonic signal in the wireless network.

The high BER is a common problem among the existing detection methods for weak harmonic signal. To overcome the problem, this paper proposes a novel method to detect weak harmonic signal in wireless network under chaotic noise. The theoretical basis of our method lies in the discrimination between chaos and noise. Drawing on the features of harmonic signal and chaos under noise, the harmonic signal was detected in the wireless network, and used to adjust the operation parameters of the network in time, making the network operation safe and stable. Experimental results show that our method outputted reliable results at different SNRs, and achieved a much lower BER than the existing methods. Of course, the experimental results have certain limitations, because the input signal was only disturbed by Gaussian white noise. The future research will improve and verify the effect of our method in the presence of colored noise and other noises.