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Abstract

As a promising technology in signal detection, the chaotic detection system can significantly improve the accuracy of weak signal detection in strong background noise. It benefits from its characteristics of the sensitivity to the initial condition and the immunity to the Additive White Gaussian Noise. However, the fundamental challenges of the existing chaotic detection system are the sensitivity to narrow-band noise and the influences of multi-target detection with adjacent frequency, which bring great difficulties in the real application. To address these problems, in this article, we focus on the weak multi-target detection with adjacent frequency under the narrow-band noise, and a novel chaotic detection system that integrates the detection algorithm based on period-chaos duration ratio is proposed. In order to enhance the robustness to narrow-band noise, the Melnikov method is used to analyze the Duffing difference system. To realize the detection of weak multi-target with adjacent frequency, we proposed the detection system using the rule named general critical state. Furthermore, simulation results corroborate that the proposed system based on period-chaos duration ratio can achieve satisfactory performance in terms of the weak multi-target detection under narrow-band noise, and it is well investigated by extensive simulation for testing its effectiveness.

Keywords

Weak signal detection multi-target adjacent frequency chaotic system general critical state

Introduction

Nowadays, weak signal detection has been widely used in various fields of society, such as mechanical fault detection,¹ biomedical science,² deep space exploration,³ geological disaster prediction,⁴ and so on. Furthermore, with the rapid development of the Internet of Things (IoT), the circumstance of signal detection is becoming more and more challenging.⁵ As the main feature of IoT, radio-frequency identification (RFID) is mostly used for passive functions such as tracking and identification, which promote the further growth of weak signal detection technology. As a dominant way in signal detection, the linear detection method mainly includes correlation detection, time-domain averaging, time-frequency domain transform, wavelet transform, and so on.⁶ The objective of these traditional linear methods is to improve the signal–noise ratio (SNR) at the receiver and then recover the target signal by filtering or other technologies. However, they are sensitive to the noise changes and also consume much time, which all limits their widespread application.

During recent years, nonlinear detection technology is becoming a hot research issue in weak signal detection. It provides a new thought of signal detection under ultra-low SNR conditions, which is that the presence or absence of a target signal can be discriminated according to the state change of the nonlinear system. As an essential branch of the nonlinear detection, the chaotic detection has obtained many remarkable achievements,⁷ which is original from “Butterfly Effect” and described by Lorenz⁸ through the mathematical equation, and by now has been widely used in signal detection and chaotic communication.⁹ The significant properties of the chaotic detection system are the sensitivity to the initial condition and the immunity to Additive White Gaussian Noise (AWGN). Comparing with traditional methods mentioned above, it is these characteristics that make the chaotic system suitable for signal detection under ultra-low SNR conditions.^10,11 Moreover, the recent work discussed by scholars has indicated that not only the Duffing oscillator array in chaotic system is an effective scheme to detect target signal fast, but also the Duffing difference system can attain a more obvious distinction in the time-domain waveform of chaotic system.¹² In particular, a lot of research shows that the Duffing difference system with two coupling oscillators can detect the target signal accurately,¹³ such as maximum-variance detection,¹⁴ zero-cross detection, and fractional order Duffing oscillator detection.¹⁵

Researchers exploit many techniques to realize weak signal detection by the nonlinear methods. Liu et al.¹⁶ showed that a distorted weak acoustic signal can be detected in a wide area and the narrow-band noise interference can be overcome. Zhao and Yang¹⁷ proposed Van-Der-Pol Duffing oscillator to detect weak signal under low SNR with other different far-away frequency signals. For further development, Lu et al.¹⁸ combined the extended Kalman filter with a chaotic system to get better performance with lower computational complexity, but it does not extend the detection method to multi-signal detection. The traditional chaotic system can attain accurate detection result under many blind area environment.¹⁹ Although these recent papers have derived different schemes for weak signal detection by the chaotic system, these methods require the detection signal is single due to the property of Duffing oscillator and fail to attain satisfying performance under multi-target signal.

To sum up, these previous proposals focus on the single periodic signal detection, which is a common situation in mechanical fault detection, while it is not applicable in communication areas. On the one hand, although chaotic detection system is immunity to AWGN, it can be affected by the narrow-band noise, which widely exists in communication field.²⁰ Due to the influence of narrow-band noise, there is no apparent distinction in different output state in the time-domain waveform of a chaotic detection system, which made it not applicable at all. On the other hand, the situation of weak multi-target detection with adjacent frequency exists widely in communication, such as RFID in IoT, biomedical signal processing, and target tracking. But the previous chaotic detection techniques are not efficient and reliable enough to achieve detection of multi-target with adjacent frequency. In consideration of these two focuses, based on the analysis of narrow-band noise and multi-target with adjacent frequency, the fundamental challenge of the existing chaotic system is how to make it applicable under this situation. Moreover, the critical state is not apparent because of the influences of multi-target with adjacent frequency, and a new concept has to be introduced to distinguish the chaotic state and the large-scale periodic state.

Different from detecting a single target signal in AWGN hypothesis, the chaotic system suffers a significant performance loss in the detection of multi-target with adjacent frequency in narrow-band noise. Therefore, the previous analysis needs to be reconsidered. In this article, we investigate the obvious distinction in the time-domain waveform of Duffing difference system; we propose the general critical state to discriminate the output state of multi-target effectively; and we obtain an accurate detection in narrow-band noise. Major contributions of this article are threefold.

To the best of our knowledge, we first propose a rule of the probability of a period state that discriminates the output state of the Duffing difference system in narrow-band noise. Here, Melnikov function is adopted and this method is different from existing chaotic detection methods under AWGN.

In our work, a novel chaotic system is proposed to realize the detection of weak multi-target with adjacent frequency. To be specific, the general critical state is introduced to analyze the output state of the chaotic system from a new perspective.

We provide performance analysis of the proposed period-chaos duration ratio (PCDR)-based algorithm. Specifically, we simulate PCDR of the time-domain waveform for assessing the accuracy of multi-target detection with adjacent frequency in narrow-band noise. Also, simulation results provide evidence to support such predominance.

The following parts of this article are organized as follows. In “Basic model and Melnikov method” section, we analyzed the basic Duffing model and its corresponding Melnikov analysis method. “Problem formulation and analysis” section formulated and analyzed the proposed problem. “Detection scheme based on the PCDR” section presents the proposed chaotic detection system and a detection algorithm based on PCDR. Numerical results for evaluating the performance of the proposed system are provided in “Simulation and results” section, which is followed by conclusions in “Conclusion” section.

Basic model and Melnikov method

Basic model of Duffing oscillator

As the most classical model of chaotic system, Holmes-Duffing oscillator has been extensively studied and evaluated by Holmes,²¹ which is expressed as

\overset{\cdot\cdot}{x} (t) + k \overset{\cdot}{x} (t) - x (t) + x^{3} (t) = γ \cos (ω t)

(1)

where $\overset{\cdot\cdot}{x} (t)$ and $\overset{\cdot}{x} (t)$ are the second-order and first-order derivatives of $x (t)$ , $- x (t) + x^{3} (t)$ is nonlinear restoring force, $k$ is damping ratio, $γ \cos (ω t)$ is the driving force of system, and $γ$ represents the amplitude. With $γ$ increasing, the system output state will go into three apparent states: the chaotic state, the critical state, and the large-scale periodic state.

Define $t = ω τ$ , then

x (t) = x (ω τ)

(2)

\overset{\cdot}{x} (t) = \frac{dx (t)}{dt} = \frac{dx (ω τ)}{d (ω τ)} = \frac{1}{ω} \frac{dx (ω τ)}{d τ}

(3)

\overset{\cdot\cdot}{x} (t) = \frac{d \overset{\cdot}{x} (t)}{dt} = \frac{d \overset{\cdot}{x} (ω τ)}{d (ω τ)} = \frac{1}{ω^{2}} \frac{dx (ω τ)}{d τ}

(4)

Substituting equations (2)–(4) into equation (1), we have

\frac{1}{ω^{2}} \overset{\cdot\cdot}{x} (ω τ) + \frac{k}{ω} \overset{\cdot}{x} (ω τ) - x (ω τ) + x^{3} (ω τ) = γ \cos (ω τ)

(5)

The dynamic equation of the Duffing oscillator can be expressed as

{\begin{matrix} {\overset{\cdot}{x}}_{τ} = ω y \\ {\overset{\cdot}{y}}_{τ} = ω (- ky + x - x^{3} + γ \cos (ω τ)) \end{matrix}

(6)

Based on equation (6), the Duffing oscillator system is established by Simulink as shown in Figure 1.

Figure 1.

Duffing oscillator system.

In Figure 1, Sine Wave is the sinusoidal driving force; Fcn is the nonlinear restoring force; Integrator1 and Integrator2 are the first-order and second-order derivatives; Gain0 is the coefficient of the integrator; Gain1 is the coefficient used to adjust the integrator when drawing the phase diagram; and Gain2 is the damping ratio. After simulating, the phase diagram and time-domain waveform of the Duffing oscillator will be displayed on the XY Graph and Scope, respectively.

When $k = 0.5$ , with $γ$ increasing, the phase diagram will go through three states: the chaotic state, the critical state, and the large-scale periodic state. These phase diagrams and the time-domain waveforms corresponding are shown in Figure 2.

Figure 2.

The phase diagrams and time-domain waveforms corresponding of the Duffing oscillator: (a, b) the chaotic state, (c, d) the critical state, (e, f) the large-scale periodic state.

From Figure 2, we can see that the orbits of phase diagram are disordered in the chaotic state, and the time-domain waveform corresponding is non-periodic. Otherwise, the orbits of phase diagram are ordered in the large-scale periodic state and the time-domain waveform corresponding is periodic. Then, the critical state contains the chaotic state and the large-scale periodic state. We can discriminate the time-domain waveform is the chaotic state or the large-scale periodic state intuitively and straightforward, but it is also inefficient, error-prone, and affected by strong background noise easily. The high accuracy of discriminating the output state of the chaotic system is considered a key issue in weak signal detection.

The Melnikov analysis method

Melnikov method^22,23 is a quantitative analysis method to discriminate the output states of the chaotic detection system based on Melnikov function. And it is a criterion to distinguish large-scale periodic state from chaotic state. Melnikov method describes the splitting of the stable and unstable manifolds of the hyperbolic fixed point defined on the cross section.²⁴ For small perturbations, if the stable manifold and the unstable manifold intersect transversely at a certain point, then a horseshoe map may exist and the Melnikov function will oscillate around zero with a relatively large amplitude, which means the system is in the chaotic state. Otherwise, if the system orbit is in the large-scale periodic state, the stable manifold will coincide with the unstable manifold and the Melnikov function will stay zero.

In equation (6), when $k = 0$ , $ω = 1$ , $γ = 0$ , a pair of homoclinic orbits are given by

{\begin{matrix} x_{0} (t) = \pm \sqrt{2} sech t \\ y_{0} (t) = \mp \sqrt{2} sech t \times \tanh t \end{matrix}

(7)

Otherwise, when $k \neq 0$ , $γ \neq 0$ , there exist extra excitation and disturbance, then the dynamic equation is

{\begin{matrix} {\overset{\cdot}{x}}_{0} = y \\ {\overset{\cdot}{y}}_{0} = x - x^{3} + - ky + γ \cos (ω τ) \end{matrix}

(8)

Based on the general expression of nonlinear system, we set

F (X) = (\begin{matrix} f_{1} (X) \\ f_{2} (X) \end{matrix}) = (\begin{matrix} y \\ x^{3} - x^{5} \end{matrix})

(9)

G (X, t) = (\begin{matrix} g_{1} (X, t) \\ g_{2} (X, t) \end{matrix}) = (\begin{matrix} 0 \\ - ky + γ \cos ω t \end{matrix})

(10)

The definition of Melnikov function is as follows

M (t_{0}) = \int_{- \infty}^{\infty} F (q^{0} (t)) \land G (q^{0} (t), t + t_{0}) dt

(11)

where $q^{0} (t)$ is the point on the orbit, and $F \land G = f_{1} g_{2} - f_{2} g_{1}$ . So, the corresponding Melnikov function in Duffing system is

\begin{matrix} M (t_{0}) = \int_{- \infty}^{+ \infty} \overset{\cdot}{x} [- k \overset{\cdot}{x} + γ \cos (t + t_{0})] dt \\ = \int_{- \infty}^{+ \infty} - {ky}_{0}^{2} (t) - y_{0} (t) γ \cos (t + t_{0}) dt \\ = - \frac{4}{3} k - \sqrt{2} π γ sech (\frac{π}{2}) \sin (t_{0}) \\ = M_{1} (t_{0}) + M_{2} (t_{0}) \end{matrix}

(12)

where

M_{1} (t_{0}) = \frac{- 4 k}{3}

(13)

M_{2} (t_{0}) = - \sqrt{2} π γ sech (\frac{π}{2}) \sin (t_{0})

(14)

As for $k < 0$ , then $M_{1} (t_{0}) > 0$ . To achieve $M (t_{0}) = 0$ , from equation (12), we can get that $M_{2} (t_{0}) < 0$ .

It can be found that when the value of $γ$ in $M_{2}$ is small, the Duffing oscillator will be in chaotic state all the time because the value of $t_{0}$ , let $M (t_{0}) = 0$ , does not exist. Therefore, only $γ$ is large enough to get $M (t_{0}) = 0$ , then Duffing oscillator will be in large-scale periodic state. Setting $γ_{d}$ is the critical value when $M (t_{0}) = 0$ and the value of $γ_{d}$ is

γ_{d} = \frac{- 2 \sqrt{2} k}{3 π sech (\frac{π}{2})}

(15)

When $γ > γ_{d}$ , the Duffing oscillator will stay in the large-scale periodic state. Otherwise, the Duffing oscillator will be in the chaotic state. The relationship between Melnikov function and $γ$ when $γ > γ_{d}$ at $t = 50 s$ is shown in Figure 3.

Figure 3.

The relationship between Melnikov function and $γ$ when $γ > γ_{d}$ at $t = 50 s$ .

Problem formulation and analysis

In this section, based on the detection method of the critical state in Duffing system, the property of narrow-band noise is formulated by Melnikov, and the effect of multi-target detection with adjacent frequency on the chaotic system is analyzed.

The critical state of Duffing system

The critical state means chaotic state and large-scale periodic state occur alternately. When a target signal inputs into the Duffing system, the system equation is changed from equation (1) to

\begin{matrix} \overset{\cdot\cdot}{x} (t) + k \overset{\cdot}{x} (t) - x (t) + x^{3} (t) = γ \cos (ω t) \\ + A \cos ((ω + Δ ω) t + φ) \end{matrix}

(16)

where $A \cos ((ω + Δ ω) t + φ)$ is the weak target signal; $A$ and $(ω + Δ ω)$ are the amplitude and angular frequency, respectively; $φ$ is the phase difference between the target signal and the system driving force.

Then, the total periodic driving force of the system is

\begin{matrix} γ_{d} \cos (ω t) + A \cos ((ω + Δ ω) t + φ) \\ = (γ_{d} + A \cos (Δ ω t + φ)) \cos (ω t) - A \sin (ω t) \sin \\ (Δ ω t + φ) \\ = γ_{0} \cos (ω t + θ_{0}) \end{matrix}

(17)

where the amplitude $γ_{0}$ and phase $θ_{0}$ are

γ_{0} = \sqrt{γ_{d}^{2} + 2 γ_{d} A \cos (Δ ω t + φ) + A^{2}}

(18)

θ_{0} = \arctan \frac{A \sin (Δ ω t + φ)}{γ_{d} + A \cos (Δ ω t + φ)}

(19)

The input of target signal causes $γ_{0} < γ_{d}$ or $γ_{0} > γ_{d}$ periodically, and the output state enters the chaotic state and the large-scale periodic state correspondingly. By measuring the period of the critical state $T$ , we can calculate $Δ ω$ through $Δ ω = 2 π / T$ , and then the frequency of the target signal is obtained.

Comparing to the chaotic detection system consisting of a single Duffing oscillator, the Duffing difference system has better performance to discriminate the chaotic state and the large-scale periodic state. The model of Duffing difference system established by MATLAB Simulink is shown in Figure 4. And the time-domain waveform of the critical state in Duffing difference system is shown in Figure 5.

Figure 4.

The model of Duffing difference system.

Figure 5.

Time-domain waveform of the critical state in Duffing difference system.

Comparing to the waveform of critical state in Figure 2(d), Figure 5 has more apparent discrimination between the chaotic state and the large-scale periodic state in a critical state. So, in this article, we use the Duffing difference system to detect the target signal.

The analysis of narrow-band noise

When the background noise $n (t)$ is input into the chaotic system, there exist excitation and disturbance, and the equation (6) can be expressed as

{\begin{matrix} \overset{\cdot}{x} = y \\ \overset{\cdot}{y} = - ky + x - x^{3} + γ \cos (t) + n (t) \end{matrix}

(20)

According to the analysis in “Basic model and Melnikov method” section, we can get the corresponding Melnikov function as shown

\begin{matrix} M (t_{0}) = \int_{- \infty}^{+ \infty} \overset{\cdot}{x} [- k \overset{\cdot}{x} + γ \cos (t + t_{0}) + n (t + t_{0})] dt \\ = - \frac{4}{3} k - \sqrt{2} π γ \underset{(\frac{π}{2})}{sech} \sin (t_{0}) \\ - \int_{- \infty}^{+ \infty} y_{0} (t) n (t + t_{0}) dt \\ = - \frac{4}{3} k - \sqrt{2} π γ sech \frac{π}{2} \sin (t_{0}) \\ - \int_{- \infty}^{+ \infty} \frac{e^{t} - e^{- t}}{{(e^{t} + e^{- t})}^{2}} n (t + t_{0}) dt \\ = M_{1} + M_{2} (t_{0}) + M_{3} (t_{0}) \end{matrix}

(21)

where

M_{1} = - \frac{4}{3} k > 0

(22)

M_{2} (t_{0}) = - \sqrt{2} π γ sech \frac{π}{2} \sin (t_{0})

(23)

M_{3} (t_{0}) = - \int_{- \infty}^{+ \infty} \frac{e^{t} - e^{- t}}{{(e^{t} + e^{- t})}^{2}} n (t + t_{0}) dt

(24)

Obviously, $M_{1}$ is a positive constant. $M_{2} (t_{0})$ is a sine function with the amplitude $| M_{2} (t_{0}) | = \sqrt{2} π γ sech (π / 2)$ , and $| M_{2} (t_{0}) |$ also is an increasing function about $γ$ . However, $M_{3} (t_{0})$ is a random variable due to the excitation and disturbance. When the noise $n (t)$ exists, if $t > 0$ , $M_{3} (t_{0})$ can be written as

M_{3} (t_{0}) = \int_{- t_{0}}^{+ \infty} f (t) n (t + t_{0}) dt

(25)

where $f (t) = (e^{t} - e^{- t}) / (e^{t} + e^{- t})^{2}$ is an odd function and its function graph is shown in Figure 6.

Figure 6.

$f (t)$ function graph varies with time.

The limit of $f (t)$ is zero as shown in equation (26)

lim_{t \to \infty} f (t) = lim_{t \to \infty} \frac{e^{t} - e^{- t}}{{(e^{t} + e^{- t})}^{2}} = 0

(26)

Furthermore, the mean value of $M_{3} (t_{0})$ is

E [M_{3} (t_{0})] = - \int_{- t_{0}}^{+ \infty} f (t) E [n (t + t_{0})] dt = 0

(27)

According to equation (27), in theory, the noise just makes the orbits rough in the phase diagram and it cannot change the output state of Duffing system. But, in real application, $M_{3} (t_{0})$ is a random process because of the noise, so $M_{3} (t_{0}) = M_{3} (t_{0}, n (t))$ , where $t_{0}$ is element of parameter set and $n (t)$ is random variable. With unknown $n (t)$ , it will cause the uncertainty of the critical value $γ_{d}$ to achieve $M (t_{0}) = 0$ , so $γ_{d}$ is a random variable rather than a constant that is used in the previous chaotic detection system. Otherwise, when the noise is known, $M_{3} (t_{0})$ is a function about $t_{0}$ , and $M (t_{0})$ is a function about $t_{0}$ and $γ$ . Therefore, to any certain $γ$ , $M (t_{0})$ is a function about $t_{0}$ , and then it should be judged whether there exists $t_{0}$ making $M (t_{0}) = 0$ .

Due to the influence of noise on $M_{3} (t_{0})$ and $M (t_{0})$ , the output state of Duffing system is also random, which cannot be described with the traditional chaotic method. So, in this article, we proposed a definition named the probability of a period state to describe the large-scale periodic state

P_{ps} = lim_{N \to \infty} [\oint_{σ} f_{n (t)} (n) dn]

(28)

where $f_{n (t)} (n)$ is joint probability density of N-dimension noise, and $σ$ is the solution set of $M (t_{0}) = 0$ constitutes of $n (t)$ , Obviously, equation (28) is a definition that cannot be calculated or analyzed.

Based on the another expression of equation (21)

\begin{matrix} M (t_{0}) = M_{1} + M_{2} (t_{0}) + M_{3} (t_{0}) = 0 \Leftrightarrow M_{3} (t_{0}) \\ = - M_{1} - M_{2} (t_{0}) \end{matrix}

(29)

when $t$ tends to infinity, $lim_{t \to \infty} f (t) = 0$ , and $lim_{t \to \infty} M_{3} (t_{0}) = 0$ , so $M_{3} (t_{0}) < max [- M_{1} - M_{2} (t_{0})]$ . Then, the meaning of a solution of equation (29) is that there exists $t_{0}$ making $M_{3} (t_{0}) > min [- M_{1} - M_{2} (t_{0})] = - M_{1} - | M_{2} (t_{0}) | = M (γ)$ . Therefore, the $P_{ps} (t_{0})$ of $t_{0}$ corresponding is the probability of $M_{3} [t_{0}, n (t)] > M (γ)$ , and then it can expressed as

P_{ps} (t_{0}) = lim_{N \to \infty} \int_{M (γ)}^{\infty} p (x) dx

(30)

where $p (x)$ is the probability density function of $M_{3} (n (t))$ in $N$ -dimension noise. Then, $P_{ps}$ can be calculated as

\begin{matrix} P_{ps} = lim_{M \to \infty} \frac{\int_{- M}^{+ M} P_{ps} (t_{0}) d t_{0}}{2 M} \\ = lim_{M \to \infty} lim_{N \to \infty} \frac{\int_{M (γ)}^{\infty} \int_{- M}^{+ M} p (x) dxd t_{0}}{2 M} \end{matrix}

(31)

Then, the derivative of $P_{ps}$ about $γ$ is

\begin{matrix} \frac{d P_{ps}}{d γ} = lim_{M \to \infty} lim_{N \to \infty} \frac{d \int_{M (γ)}^{\infty} \int_{- M}^{+ M} p (x) dxd t_{0}}{2 Md γ} \\ = lim_{M \to \infty} \frac{1}{2 M} lim_{N \to \infty} \int_{- M}^{+ M} \frac{d \int_{M (γ)}^{\infty} p (x) dx}{d γ} d t_{0} \\ = lim_{M \to \infty} \frac{1}{2 M} lim_{N \to \infty} [- \int_{- M}^{+ M} p (M (γ)) \frac{dM (γ)}{d γ} d t_{0}] \\ = \frac{\sqrt{2}}{2} π \sec h (\frac{π}{2}) lim_{M \to \infty} \frac{1}{M} lim_{N \to \infty} \int_{- M}^{+ M} p (M (γ)) d t_{0} \end{matrix}

(32)

From equation (32), the $P_{ps}$ is a monotone increasing function of $γ$ because of $d P_{ps} / d γ > 0$ . That is to say, the higher the $γ$ value is, the larger the probability of periodic state will be, conversely, the larger the probability of chaotic state will be. Time sequences of $P_{ps}$ with different $γ$ values are shown in Figure 7.

Figure 7.

Time sequences of $P_{ps}$ with different $γ$ values.

From Figure 7, we can see that $P_{ps}$ is close to $1$ when $γ$ is 0.9, which causes the output system state to stay in large-scale periodic state. And when $γ$ is 0.83 and 0.826, which are close to the critical value $γ_{d}$ , $P_{ps}$ fluctuates around $0.5$ , which indicates the system is in critical state. For the other two $γ$ lower than $γ_{d}$ , $P_{ps}$ is nearly $0.1$ with the output state is in chaotic state. In this method, we can discriminate the large-scale periodic state and chaotic state by $P_{ps}$ clearly.

Moreover, the effect of $γ$ on $M_{2} (t_{0})$ is independent of the effect of noise on $M_{3} (t_{0})$ . Through the analysis above, the previous chaotic detection system is not applicable at all, which is based on the constant critical value setting and stable state transition. Therefore, to address this problem, the definition of $P_{ps}$ is proposed to provide more accurate and reasonable description of the output state of the chaotic system.

The detection of multi-target with adjacent frequency

Multi-target detection is a common phenomenon in the field of communication since the fast development of IoT and RFID. Nevertheless, the principle of chaotic detection is that it changes the presence or absence of a target signal into the apparent differences in system output states. So, the Duffing system can only detect the signal that has the same frequency to the frequency of the oscillator. In view of the multi-target detection, it still works when the frequencies are far away from each other because there exists apparent critical state. However, the multi-target with adjacent frequency cannot be detected due to the disorder of the critical state and its form will be decided by the combination of input multi-target.

When the multi-target $\sum_{i = 1}^{K} [A_{i} \cos ((ω + Δ ω_{i}) t + φ_{i})]$ with number of $K$ is input into the Duffing system, the total driving force is changed from $γ \cos (t)$ in equation (1) to

\begin{matrix} γ_{d} \cos (ω t) + \sum_{i = 1}^{K} [A_{i} \cos ((ω + Δ ω_{i}) t + φ_{i})] \\ = γ_{d} \cos (ω t) + \sum_{i = 1}^{K} [ξ \cdot \cos (ω t) - ζ \cdot \sin (ω t)] \\ = γ_{1} \cos (ω t + θ_{1}) \end{matrix}

(33)

where

ξ = \sum_{i = 1}^{K} [A_{i} \cos (Δ ω_{i} t + φ_{i})]

(34)

ζ = \sum_{i = 1}^{K} [A_{i} \sin (Δ ω_{i} t + φ_{i})]

(35)

γ_{1} = \sqrt{γ_{d}^{2} + 2 γ_{d} ξ + ξ^{2} + ζ^{2}}

(36)

θ_{1} = \arctan \frac{ζ}{γ_{d} + ξ}

(37)

According to equation (33), we can conclude that the value of $Δ ω_{i}$ among the multi-target to be detected has an influence on the form of the critical state apparently, that is, the smaller the $Δ ω_{i}$ is, the slower the $\cos (Δ ω_{i} t + φ_{i})$ changes, and has more obvious influence on total amplitude $γ_{1}$ . When $Δ ω_{i}$ , $φ_{i} (i = 1, 2, \dots, K)$ is known, $γ_{1}$ is a function about $t$ , and $θ_{1}$ tends to be zero as the value of $A_{i} (i = 1, 2, \dots, K)$ is very small. So, we set the function $f_{K} (t) - γ_{1} = 0$ and its solution set is the time point of output state changed, which is defined as $t_{1}, t_{2}, \dots$

When $K = 1$ , it means the input signal is single. Then

γ_{1} = f_{K} (t) = \sqrt{γ_{d}^{2} + 2 γ_{d} A \cos (Δ ω t + φ) + A^{2}}

(38)

and the solution set is

t_{1} = \frac{[\arccos (\frac{A}{2 γ_{d}}) - φ - π]}{Δ ω}

(39)

t_{2} = \frac{[\arccos (\frac{A}{2 γ_{d}}) - φ + π]}{Δ ω}

(40)

where $t_{1}$ and $t_{2}$ are the time points of state changed from large-scale periodic state to chaotic state and vice versa, which is the common critical state. During the periodic state [ $0 ~ t_{2}$ ], there is only one point changed at $t_{1}$ , and we can calculate the critical value between the two states easily. But, in the situation of multi-target input into the chaotic system, there are more than two elements in the solution set, more states changed, and more time points of state change exist in a period. In this case, it cannot be detected in the previous chaotic system.

The schematic diagram of time points of state changed in the critical state is shown in Figure 8.

Figure 8.

The schematic diagram of time points of state changed in the critical state.

In Figure 8, when $K = 1$ , there is a stable cycle of chaotic state and large-scale periodic state in the critical state, and the same to the cycle of the whole critical state. While $K = 3$ , the duration of chaotic state and large-scale periodic state is no longer constant, and the cycle of whole critical state cannot be calculated. For example, when we input three signals into the Duffing difference system, the critical state of multi-target with equidifferent frequency and random frequency is shown in Figure 9.

Figure 9.

The critical state of three signals with adjacent frequency: (a) equidifferent frequency, (b) random frequency.

From Figure 9(a), the critical state of multi-target with equidifferent frequency is the same to that of a single signal, and it is impossible to distinguish them from the previous chaotic detection system. In Figure 9(b), the critical state is composed by different periods, which is due to the solutions in the function $f_{K} (t) - γ_{d} = 0$ .

Based on the analysis above, when the multi-target with adjacent frequency is input into the detection system, as for the different combination of multi-target $Γ_{K} : Δ ω_{i}, φ_{i} (i = 1, 2, \dots, K)$ , it will cause the different forms of critical state, and then the critical value of chaotic detection system is also variable. In addition, the signal combination in multi-target is also unknown. As a result, the frequencies of each signal in the multi-target cannot be detected through the critical state changed in the previous chaotic detection system.

Detection scheme based on the PCDR

The problems of narrow-band noise and multi-target with adjacent frequency are analyzed in “Problem formulation and analysis” section. Hence, in this section, we proposed the new concept of general critical state to discriminate the output state of the chaotic system and then detect the multi-target by the detection algorithm based on PCDR.

General critical state and PCDR

The critical state is composed of the chaotic state and the large-scale periodic state. Although, during the period of the critical state, there are infinite combinations of chaotic state and large-scale periodic state, the ratio of these two states’ duration is constant. In the previous chaotic detection system, the period $T$ of the critical state can be obtained from the output time-domain waveform, and

Δ ω = \frac{2 π}{T}

(41)

we can get the frequency difference $Δ ω$ between the frequency of driving force and the frequency of the input signal, and then the frequency of the input signal can be obtained further. However, in the situation of multi-target detection under narrow-band noise base on “Problem formulation and analysis” section, the critical value in the critical state and the ratio of these two states duration are all variables.

Considering the limitation of the critical state, we proposed a new concept named the general critical state and the details of the definition are as follows:

The general critical state is a common state in Duffing oscillator, which is independent of the target signal, the influence of other signal, and the noise. The large-scale periodic state, critical state, and chaotic state are all a special state of the general critical state.

The general critical state is divided into two states, general periodic state and general chaotic state, which is based on the different probability of periodic state duration.

In general critical state, the output state can only be discriminated by the probability of the large-scale periodic state and the chaotic state. As $P_{ps}$ is a function about $γ$ , which is influenced by the input signal and noise, the probability of these two states is variable as for the different cases, such as the number of the input signal and the type of noise.

We further proposed the algorithm based on PCDR of the time-domain waveform to discriminate the large-scale periodic state and the chaotic state. The definition of PCDR is as follows

PCDR = \frac{T_{period}}{T_{total}} = \frac{n_{period}}{n_{total}}

(42)

where $T_{period}$ and $T_{total}$ are the duration of periodic state and total state, $n_{period}$ and $n_{total}$ are the number of sampling points in periodic state and total state.

During the process of detection, we will make real-time monitoring of the change of PCDR until it is stable. The variation range of PCDR is $0 - 1$ , and it approaches 1 when there exists a target signal in multi-target which has the same frequency to the oscillator’s; then, the frequency of the target signal is detected.

Detection algorithm based on PCDR

According to the analysis in “Problem formulation and analysis” section and the definition of PCDR, we proposed a scheme based on the general critical state and PCDR to achieve the detection of the target signal in multi-target under the narrow-band noise, as outlined in the steps explained below.

In situations of narrow-band noise where the multi-target exists, we prefer to arrange the chaotic detection system with Duffing oscillator. Step 1 carries out the Duffing difference system for weak multi-target detection. In this mode, we can get the critical amplitude of driving force $γ_{d}$ , the frequency of the oscillator $f_{d}$ , and the PCDR $R_{0}$ corresponding, which is the basic detection unit. Then, in the simulation, under the circumstance of weak multi-target with adjacent frequency, the detection system will be assigned to step 2, which completes the process of detection in such a situation. In step 2, with multi-target sets input into the system, we will calculate the PCDR of each signal set base on the Melnikov function analysis and then compare it to the threshold value $R_{0}$ . The large-scale periodic state and the chaotic state will be discriminated by the comparison to $R_{0}$ , and then the weak target signal will be detected; furthermore, the multi-target set corresponding will be detected. This multi-target set with a target signal will be designated as the input to step 3.

Step 3 is designed for the frequency detection of each signal in the multi-target set. In this step, the traditional detection system with Duffing oscillators array will be built. As there are many signals in each multi-target set, the Duffing array system with the advantage of less time-consuming can detect the target signals in a time unit. The PCDR values of every oscillator in the Duffing array system can also be obtained, which is the criterion of detection of the weak target signal in the multi-target set by the comparison to $R_{0}$ . Algorithm 1 presents the operational procedure of the chaotic detection.

Algorithm 1. Target signal detection in multi-target with adjacent frequency under the narrow-band noise
1. Step1: Duffing Difference System in Narrow-band Noise
2. Set Duffing difference system in the critical state under narrow-band noise, then we can get the amplitude of driving force $γ_{d}$ , the frequency of the oscillator $f_{d}$ , and the PCDR $R_{0}$ .
3. Step2: Multi-target Detection with Adjacent Frequency
4. Input multi-target set into the system, and the number of the set is $m$ and the number of the multi-target in each set is $n$ .
5. for $i = 1, 2, 3, \dots, m; j = 1, 2, 3, \dots, n$ do
6. Calculate the PCDR of each signal set, set $R_{i}$ ;
7. if $f_{ij} = f_{0}$ then
8. $R_{i} > R_{0}$ , there exists a signal with frequency $f_{d}$ in the signal set $i$ ;
9. else
10. $R_{i} \leq R_{0}$ ;
11. end if
12. end for
13. Discriminate if there is one signal in the multi-target set with the same frequency to that of Duffing oscillator.
14. Step3: Frequency Detection on Each Signal in Multi-target Set
15. Input multi-target set $i$ into Duffing array system with $k$ oscillators, $k \geq n$ .
16. for $j = 1, 2, 3, \dots, n; v = 1, 2, 3, \dots, k$ do
17. Calculate the PCDR of oscillator array, set $R_{v}$ ;
18. if $f_{j} = f_{v}$ then
19. The PCDR corresponding is $R_{v}$ , $R_{v} > R_{0}$ , the signal with frequency $f_{v}$ is detected;
20. else
21. $R_{v} \leq R_{0}$ ;
22. end if
23. The frequencies of multi-target set $i$ are detected.
24. end for

Simulation and results

In this section, the performance of the proposed approach in Algorithm 1 is well investigated, and numerical results are presented to demonstrate the proposed method. The simulation is implemented on MATLAB. All the simulation parameters of the Duffing difference system and the frequency value of multi-target sets are shown in Table 1.

Table 1.

Simulation parameters and values.

Description	Parameters	Values
Amplitude of signal	$a$	$0.1 V$
Symbol rate	$h$	$100 Kbps$
Sampling frequency	$f_{s}$	$5 GHz$
Band of noise	$B_{n}$	$5 GHz$
Frequency of oscillator	$f_{0}$	$50 MHz$
Frequency of signal set1	$f_{set 1}$	$50, 50.2, 53.7 MHz$
Frequency of signal set2	$f_{set 2}$	$50.1, 50.2, 53.7 MHz$
Frequency of signal set3	$f_{set 3}$	$49.9, 50.2, 53.7 MHz$
Frequency of signal set4	$f_{set 4}$	$50.5, 51.2, 51.5 MHz$
Frequency of signal set5	$f_{set 5}$	$50, 50.6, 51.2 MHz$

In this article, we focus on the weak multi-target detection with adjacent frequency under the narrow-band noise; in this case, we choose the different signal sets with different frequencies that are all close to the oscillator frequency $f_{0}$ . In Table 1, as for $f_{0} = 50 MHz$ , we set some of the signal sets to contain the same frequency as the oscillator frequency, which are the signal set1 (50, 50.2, 53.7 MHz) and signal set5 (50, 50.6, 51.2 MHz) with the target signal frequency of 50 MHz. The other three signal sets with other adjacent frequencies to the oscillator frequency are all used in the simulation for comparison. These frequencies are chosen randomly as long as they have adjacent frequency to the oscillator frequency.

In summary, there are two steps in the detection of multi-target with adjacent frequency. First, using the proposed detection system and algorithm above, we can achieve the detection of the multi-target set with the target signal. Second, in this multi-target set, a new Duffing array system is built to detect the frequencies of target signals in the set.

The detection of multi-target sets with target signal

In Table 1, there are five signal sets of multi-target and 15 signals in total. For the simulations, $E_{b} / n_{0} = 10 dB$ , while $SNR = - 30 dB$ in real communication. As the frequency of Duffing oscillator is $50 MHz$ , our goal is to detect the target signal with frequency of $50 MHz$ among these multi-signal sets. When we input these signal sets into Duffing detection system based on step 2 in Algorithm 1, the performance of PCDR of these five multi-target sets in Duffing detection system is shown in Figure 10. We can see that the PCDR of the proposed system reduces with the increasing time, and it becomes stable gradually until the time is large enough. Also, we can see from the simulation result that the different signal sets attain different PCDR value when it is stable. The PCDR of multi-target set1 and set5 is larger than that of the other three multi-target sets—the value of PCDR is above 0.89. However, the PCDR value of the other three sets is only nearly 0.75. This result is dedicated by the fact that both multi-target set1 and set5 contain the same frequency of $50 MHz$ to the frequency of the Duffing oscillator. Using the proposed detection system and algorithm, no matter how close the frequency of multi-signal sets to the oscillator’s, the detection of the multi-target set with the target signal is achieved.

Figure 10.

The performance of PCDR of five multi-target sets in Duffing detection system.

Figure 11 provides the precision probability comparison of target signal detection in multi-signal sets against different amplitude threshold $γ_{d}$ based on the proposed approach, where “ $γ_{d} = 0.7$ ,”“ $γ_{d} = 0.75$ ,”“ $γ_{d} = 0.8$ ,”“ $γ_{d} = 0.85$ ,”“ $γ_{d} = 0.9$ ,” and “ $γ_{d} = 0.95$ ” are investigated. The results show that the precision probability is very low between 0.4 and 0.5 when “ $γ_{d} = 0.7$ ”; as the $γ_{d}$ increases, it gets better varying over time with the best result around 0.7. Afterward, the precision probability improved greatly when “ $γ_{d} = 0.8$ ” and “ $γ_{d} = 0.85$ ,” which achieve perfect detection result. However, in the case of “ $γ_{d} = 0.9$ ” and “ $γ_{d} = 0.95$ ,” the precision probability does not increase anymore; in contrast, it gets worse and worse as the value $γ_{d}$ is far from its threshold.

Figure 11.

Precision probability comparison with different amplitude threshold $γ_{d}$ .

The detection of target signal in a multi-target set

Based on the detection result in the previous subsection, we can know that the target signal is included in the signals set1 and set5. In this subsection, all signal frequency in the signal set1 and signal set5 are obtained by a new Duffing array system. Taking the multi-target set1 as an example, we built a new Duffing array system composed of seven Duffing oscillators, whose frequencies contain not only the same frequency but also very adjacent frequency to the signal sets. By the proposed detection system, no matter how close the signal frequency is to the oscillator’s, the system can only detect the same frequency as the oscillator’s, which is better showed in the PCDR value that varies over time. In Table 1, we can see that the frequencies in signal set1 are 50, 50.2, and 53.7 MHz, so we set the frequencies of the oscillators in Duffing array system as follows: 50, 50.1, 50.2, 50.3, 53.6, 53.7, and 53.8 MHz, which contain two adjacent frequencies (50.1 and 50.3 MHz) to the 50 and 50.2 MHz and two adjacent frequencies (53.6 and 53.8 MHz) to 53.7 MHz, respectively.

According to the step 3 in Algorithm 1, then $k = 7$ , $n = 3$ , when we input signal set1 into Duffing array system, the oscillator with the same frequency to any one of the signal set1 will be with larger PCDR obviously, which, due to its corresponding output state, stays in the large-period state. The performance of PCDR of signal set1 in the Duffing arrays system is shown in Figure 12.

Figure 12.

The performance of PCDR of signal set1 in the Duffing arrays system.

Different from that of all signal sets input into the same oscillator to detect whether the target signal is in these sets or not in the previous subsection, we input the signal set1 containing three frequencies into Duffing array constituted of seven oscillators with different frequency to get the accurate detection on each signal in signal set1. From Figure 12, we can see that the PCDR becomes stable gradually over time; obviously, PCDR value differs according to the different oscillators. The oscillators with frequencies 50, 50.2, and 53.7 MHz have a larger value of PCDR of 0.85 and above; in contrast, the other four oscillators with adjacent frequency (50.1, 50.3, 53.6, and 53.8 MHz) all have PCDR value of 0.78 and below. As a result, all frequencies of signal set1 are detected. Moreover, the simulation on precision probability comparison of target signal detection against different amplitude threshold $γ_{d}$ is shown in Figure 13. The analysis is similar to that of Figure 11, which will not be discussed here repeatedly.

Figure 13.

Precision probability comparison with different amplitude threshold $γ_{d}$ .

Conclusion

In this article, we have proposed chaotic technical-based schemes for achieving super-resolution detection on multi-signal detection with adjacent frequency in narrow-band noise. At first, we construct a typical chaotic detection system by Duffing difference oscillator and analyze its characteristics by Melnikov method. Afterward, as the previous methods are not suitable anymore in discriminating the large-scale periodic state and chaotic state accurately, a novel rule named general critical state is proposed, which is derived from a new point of view. Then, by comprehensive theoretical analysis and derivation, we derive that the large $γ$ can reinforce the immunity to the narrow-band noise. In order to realize super-resolution detection on the weak multi-target with adjacent frequency, we proposed an algorithm based on PCDR to achieve accurate detection on both target signal in multi-signal sets and each signal frequency of a signal set. Simulation results have demonstrated that our proposed system can achieve accurate detection on multi-target with adjacent frequency in narrow-band noise, which also proved the proposed novel rule of general critical state and PCDR are reasonable and effective.

Footnotes

Handling Editor: Hongying Meng

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported in part by the MSIT, Korea, under the ITRC program (IITP-2019-2017-0-01637).

ORCID iD

Dawei Chen

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Detection of weak multi-target with adjacent frequency based on chaotic system

Abstract

Keywords

Introduction

Basic model and Melnikov method

Basic model of Duffing oscillator

The Melnikov analysis method

Problem formulation and analysis

The critical state of Duffing system

The analysis of narrow-band noise

The detection of multi-target with adjacent frequency

Detection scheme based on the PCDR

General critical state and PCDR

Detection algorithm based on PCDR

Simulation and results

The detection of multi-target sets with target signal

The detection of target signal in a multi-target set

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References