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Abstract
We propose a time delay signature extraction method for optical chaos systems based on a convolutional neural network. Through transforming the time delay signature of a one-dimensional time series into two-dimensional image features, the excellent ability of convolutional neural networks for image feature recognition is fully utilized. The effectiveness of the method is verified on chaos systems with opto-electronic feedback and all optical feedback. The recognition accuracy of the method is 100% under normal conditions. For the system with extremely strong nonlinearity, the accuracy can be 93.25%, and the amount of data required is less than traditional methods. Moreover, it is verified that the proposed method possesses a strong ability to withstand the effects of noise.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Chaos systems have been extensively researched in the past two decades because of its widespread applications including image encryption [1], secure communications [2–4], chaotic radar [5], random number generation [6,7], and so on.
Optical time delayed chaos system has following advantages: broad bandwidth, unpredictability, high complexity, and easy to implement [8]. There are two types of delayed feedback chaotic systems in optical domain. One is laser with delayed feedback, which use the intrinsic nonlinearity of a semiconductor laser. If we introduce optical feedback or optoelectronic feedback to a semiconductor laser, the output of the laser will exhibit complex nonlinear dynamics. By changing the parameters such as feedback strength and carrier concentration, we can obtain optical chaos [9]. The other is opto-electronic oscillator (OEO). In this type of delayed feedback system, the laser output is stable, but the intensity of the light that is transmitted through the external nonlinear device (such as a Mach-Zehnder modulator) is chaotic [10,11].
For secure communication systems, the useful information is loaded on the chaotic carrier, and then chaotic synchronization is realized. That is, the same or similar chaotic carrier is generated at both the transmitting end and the receiving end if their parameters are well matched, and the chaotic communication can be completed [12]. The feedback time delay is a particularly important parameter in these systems. It plays a pivotal role in generating rich dynamical behaviors. The time delay is also considered as an encryption key. Once the delay is revealed by an unauthorized attacker, the key space of the system will be significantly reduced. Meanwhile, other parameters of the system can be easily estimated using low-complexity computational reconstruction methods [13,14].
Up to now, many mathematical statistics analysis methods have been proposed to unveil the time delay signature (TDS) directly through a segment of time series, such as extreme value statistics (EVS) [13], fill factor [15], autocorrelation function (ACF) [16], delay mutual information (DMI) [17], and permutation information analysis (PIA) [18,19]. EVS is based on the characteristic location of quadratic extrema in the time series and the statistical analysis of time intervals between them. If the system has inertial properties, the quadratic shape of extrema is typical and in practice one can analyze all extrema in the time series [13]. As for PIA, it extracts TDS of the related physical system based on the embedding delay of symbol reconstruction. By estimating the fastest time of the minimum required sampling time as the embedded delay value, the permutation entropy has a monotonically increasing behavior around the domain [18]. Among these methods, ACF and DMI are the most popular ways due to their effectiveness and robustness. They are also widely used in time domain signal processing and analyzing. ACF shows the correlation relationship of a signal with different time delay. In another word, it can be seen as a function of the similarity between two observations over different time points between them. DMI is a useful way to measure association. In this method, the relationship between two signals is quantified by the information and entropy [20]. Generally, through these measures, it is easy to find the linear relationship between two signals, but it may be invalid for a system with strong nonlinearity [21].
On the other hand, machine learning (ML) techniques such as neural networks (NNs) have been proved to be an effective method to explain the nonlinear transformation of uncertain system [21]. In practical applications, NNs are successfully used as the equalizer [22] and monitor [23] in communication system, the aider to improve chaotic synchronization communication performance [24,25], a tool to attack chaos-based image encryption [26] and the methods to reconstruct nonlinear dynamics [27]. Whether for linear models, non-linear models, or problems that are difficult to be modeled, machine learning methods have been proven effective in feature extraction. Accordingly, delay can be seen as a feature of these chaotic systems. Therefore, using ML to identify the delay feature of a time-delay chaotic system is an attractive idea.
In this work, an effective delay-analysis method for the time-delay chaotic system is proposed. We succeed to use a convolutional neural network (CNN) to extract the accurate TDS of an intensity chaotic OEO system, a phase chaotic OEO system, and a semiconductor laser chaos system with optical feedback.
2. Principle and methods
Under the chaotic secure transmission scenario, the principle of the TDS extraction strategy based on CNN is shown in Fig. 1. Chaotic signal emitted from a time-delayed system is transmitted from the transmitter to receiver. Suppose the type of chaotic system is known, which is a reasonable assumption according to Kerckhoffs’s principle [28]. The chaotic time series signal can be captured from the transmission link. According to some information such as the amplitude and spectrum of the captured waveform, some basic parameters can be obtained at an approximate range. With the known system model and the estimated parameters, a similar system can be built with an arbitrarily chosen time delay. A large amount of training data will be generated in this system to train a CNN. Then the time series captured from the link is put into the trained CNN, and the TDS can be recognized.
The principle of the method is based on the following facts. A chaotic signal ${x_\tau }(t )$ is generated from a time-delayed mathematical model. There is a functional relationship between the time delayed signal ${x_\tau }({t - T} )$ and the original one if the time delay T is equal to the loop delay $\tau $, otherwise the relationship is weak. For each set of one-dimension time series signals, it is difficult to find the relevance among them. However, due to the definite functional relationship between the time delayed signal and the original one, common features exist in two-dimension (2D) images which are formed by the original signals and their delayed copies with correct time delay, even if these images are generated from the systems with different loop delays. Thus, these common features can be regarded as the TDS. The features may be obvious for naked eye if the functional relationship is relatively simple. For some complex models, complex functional relationship makes it difficult to describe these features at the semantic level. Owing to the excellent image recognition capabilities, we use CNN to identify such features that reflect in the image as contours, shapes or some indescribable characteristics.
In our scheme, the detailed TDS extraction process is divided in two stages. Stage 1 is the training process of CNN, which is completed offline ahead of schedule. Firstly, we set an exact loop delay τ in a similar system to generate a chaotic signal ${x_\tau }(t )$, where this delay τ is arbitrarily chosen. Secondly, we use one-dimension chaotic time series to build 2D image training samples through reconstruction method based on delay. ${x_\tau }(t )$ is preprocessed to generate copies for different signal delays $T = {t_i}\; ({i = 1,2,3, \cdots } )$. Here these copies are denoted as ${x_\tau }({t - {t_i}} )$, as shown in Fig. 1(a). Without losing generality, we assume ${t_1} = \tau $ is the correct time delay (${t_i} \ne \tau $ for $i \ne 1$). Then 2D images ${M_i}$ are drawn by ${x_\tau }({t - {t_i}} )$ and ${x_\tau }(t )$ (horizontal axis: ${x_\tau }({t - {t_i}} ),\; \textrm{vertical}\; {\textrm{axis}}:{\; }{x_\tau }(t )$), as shown in Fig. 1(b). The size of each image is $256 \times 256$. These training sample images are divided into two types, the ones with correct time delay is labeled by Y and the others are labeled by N, as shown in Fig. 1(c). A two-layer CNN is built, including an image input layer, two convolution layers with 8 filters of 3×3, two 2×2 max pooling layers and an output layer with two outputs ($Y$ and $N$), as shown in Fig. 1(d).
Stage 2 is the recognition process. The signal to be measured is ${x_{\tau ^{\prime}}}(t )$ with unknown loop delay$\; \tau ^{\prime}$. Once ${x_{\tau ^{\prime}}}(t )$ is received, its time-delayed copies ${x_{\tau ^{\prime}}}({t - {t_i}} )$ are generated by traversing different time delays $T = {t_i}$ ($i = 1,2,3, \cdots )$ in a large range. Then, preprocessing is conducted to generate images $M{^{\prime}_i}$, which is similar to that in the training process. These images are then sent into the trained CNN model. According to the outputs of the CNN, if an image $M{^{\prime}_n}$ is recognized as type Y, the corresponding time delay ${t_n}$ is regarded as the correct time delay $\tau ^{\prime}$.
3. Results and discussions
In general, there are two types of optical chaos system with delayed feedback. One is chaotic OEO. In this type of systems, chaotic behavior can be induced when the continuous-wave (CW) light emitted by a laser goes through an opto-electronic feedback loop containing a nonlinear optical device, such as a broadband Mach-Zehnder electro-optic intensity modulator (IM) or a phase modulator (PM). The other one is the laser system with internal nonlinearity. In this type of systems, the light emitted from a semiconductor laser is directly fed back in all optical way by using a mirror or in optoelectronic way by using a photodiode. To verify the effectiveness of our TDS extraction method, some typical optical chaos systems have been considered in our experiment and simulation.
3.1 Opto-electronic oscillator chaos system
Figure 2(a) shows the structure of a widely researched OEO chaotic system [3,10,24,29–30]. We have implemented such a system experimentally. The CW light of a laser diode (LD, TLG-200) is modulated by the Mach-Zehnder intensity modulator (IM, EOSPACE AX-OMSS-20-PFA-PFA-LV) with a sinusoidal nonlinear transmission function after a polarization controller (PC). The laser light is transmitted through a segment of optical fiber to introduce the time delay. A photodiode (PD, PD-18G-V) is used to detect the signal after the injection power is adjusted with a variable optical attenuator (VOA). The electric signal is divided into two portions by an electrical splitter (ES). One signal is amplified by an amplifier (AMP, GT-PA-1M1G, 1MHz∼1 GHz). Then, it is fed back to the electrode of the IM. Another signal copy is received by the oscilloscope (Tektronix, DSA 72504D Digital Serial Analyzer, 25 GHz, 100 GS/s). The experimental waveform (50 GS/s) and spectrogram of intensity chaos in opto-electronic system are shown in Fig. 2(b) and Fig. 2(c) separately.
In the experimental setup, the nonlinear strength of the IM is restricted due to the limit of our devices. Therefore, we have also made a simulation of the intensity chaotic system with increasing nonlinearity. Such system can be described by the following equation [10],
In our simulation, the parameter values are set as γ=5.3 ps, θ=5.3 µs, T0=30 ns, Φ0=0. Equation (1) is solved by fourth-order Runge-Kutta algorithm in MATLAB. The step size is set as 0.625 ps. Then, a 160Ghz sampling sequence y(t) is obtained by down sampling the obtained x(t). As the control group, Figs. 3(a) and 3(b) show the TDS extraction results of ACF and DMI on different nonlinear coefficient β. It can be seen that the peak value of ACF and DMI gradually decreases as β increases. In addition, ACF is failed when β increases to 15, while DMI begins to be ineffective when β increases to 60. When the peak at the delay cannot be distinguished from the background in ACF and DMI curves, it means ineffective [31].
Fig. 3. The simulation results of ACF (a), DMI (b) and the proposed CNN method (c) on different nonlinear coefficient β when the delay is 55 ns and SNR = 10 dB. T: true, F: false.
In the proposed CNN method, a data y(t) of 60,000,000 points (375 µs) with T0=30 ns is generated through simulation. Every 60,000 points (375 ns) are used to reconstruct a sample (2D image). Through the way, 2800 samples with the correct delay are obtained (here, t=30 ns, as the true delay). Next, by setting the 0.05ns delay interval, that is to say, t=29.85 ns, t=29.90 ns, t=30.05 ns and so on are set as the false delay separately, a large number of false delay image samples are obtained. Then, 2800 samples are evenly extracted as false delay image sample set. What needs to be emphasized is that these 5600 samples are used for training. Then, another set of y(t) with delay time T0=55 ns is generated. Similarly, true delay and false delay image samples are obtained. Next, 2400 samples are used for testing.
The results on different nonlinear coefficient β is showed in Fig. 3(c). Here, the red graphs are those with correct delay, and the blue ones are those with incorrect delay. The difference between the red graph and the blue graph is distinguishable by naked eye when β is smaller than 15. While CNN has a high recognition accuracy for TDS on a much larger range of β. Even when β increases to 100, CNN still has a correct recognition rate up to 93.25%. Note that it is hard to reach such a large β using current devices. However, it can be used as an indicator of nonlinearity to verify the effectiveness of the proposed method. If the amount of CNN training data increases, the result can be further improved. Here, the data size used in testing is the same for ACF, DMI, and CNN.
Next, we focus on the influence of noise on the recognition performance. Figure 4 shows the TDS extraction results of CNN method, ACF and DMI under different SNR. The coefficient β is set as 10, which is large enough if we consider the implementation capability of actual devices. It can be seen that the peak value of ACF and DMI is gradually decreasing as the SNR decreases. Similarly, the difference between the red and blue graph is gradually decreasing in the outline. Even when SNR is low to 6, the recognition rate of proposed CNN method can still be 100%. It shows that our proposed method possesses a strong ability of withstanding the effects of noise.
Fig. 4. The simulation results of the proposed CNN method (a), ACF (b), and DMI (c) on different SNR when the delay is 55 ns and nonlinear coefficient β = 10. T: true, F: false.
Last, we focus on the minimal data size needed in the extraction process. Table 1 shows the comparison results when time delay is set as 30 ns, which corresponds to 4800 data points in the generated time series. These results indicate that the required data size of CNN is less than ACF and DMI.
Table 1. The minimal data size to extract the TDS effectively.
Figure 5 shows analysis of the experimental results of intensity chaos in OEO system. Figure 5(a) shows recognition results of traditional method ACF where the delay is 19.2 ns. In our method, a data of 60,000,000 points (1.2 ms) is generated experimentally. With setting the 0.1 ns delay interval, Fig. 5(b) shows part of the sample set where the red graph is true delay image sample. The difference between the image with the correct delay and false delay can be clearly seen on constructing the sample set. Similar to simulation, in CNN training process, 5600 samples are used for training and 2400 samples are used for testing. In this case, the accuracy of TDS recognition for intensity chaos is 100%.
Fig. 5. Experimental analysis of intensity chaos in OEO system: (a) ACF, (b) delayed reconstruction image sample set of the intensity chaos. T: true; F: false.
Our proposed method has also been verified on phase chaos experimentally. Figure 6(a) shows the structure of a phase chaos in OEO system with PM. Here, the CW light of LD (TSL-C89-P-FA-M) is modulated by a PM (KG-PM-15-10G). The fiber Bragg grating (FBG, TDCMB-C000-NC-BF01) is aimed at performing a phase-to-intensity conversion. The signal is split into two portions after passing through an optical coupler (OC). One part of the light is detected by PD1 after passing through a VOA, and then amplified by an AMP (Centellax, OA4MVM3). Another signal is captured by the oscilloscope after PD2. Figures 6(b) and 6(c) show the waveform and spectrogram of implemented system (100 GS/s) separately.
Figure 7 shows the TDS extraction results for phase chaos in OEO system. As can be seen, the ACF of the time series displays an obvious peak at t=73.64 ns. Here, the setup of CNN method is similar to that in the aforementioned intensity chaos. The difference is that we set the delay interval as 0.04 ns. In this case, the recognition accuracy for the phase chaos experimental data is 100%.
Fig. 7. Experimental analysis of phase chaos in OEO system: (a) ACF, (b) delayed reconstruction image sample set of the phase chaos. T: true; F: false.
3.2 Laser chaos system with optical feedback
Figure 8(a) shows the structure of a widely researched laser chaotic system with optical feedback [32–37]. The laser is connected to an OC and a variable fiber reflector (VFR). The output of the laser is detected by a PD through an optical fiber isolator (FISO). And the effectiveness of the proposed method for that system is also studied through simulation. The temporal dynamics of intensity fluctuation in this system can be modeled by the Lang-Kobayashi equation as follows [32]:
In simulation setup, τ0=100.2 ns, GN=8.40×10−13 m3s-1, N0=1.40×1024 m-3, τp=1.927×10−12 s, τs=2.04×10−9 s, κ=2.75 ns-1, α=3.0, J=1.098×1033 m-3s-1 and ɛ=2.5×10−23. Figures 9(b) and 9(c) show the simulation waveform and spectrogram of Eq. (2).
Fig. 9. (a) Constructed image set, (b) ACF, and (c) DMI of the laser chaotic system with optical feedback.
Figure 9 shows the TDS extraction results. Here, the delay τ0=100.2 ns is used for training, and τ0=135 ns is used for testing. As for ACF and DMI, both of them are effective and the delay is 135 ns. Correspondingly, it can be seen from Fig. 9(a) that the correct delay image set (red graph) is different from the false delay image set (blue graph). The recognition accuracy of CNN method is 99.92%.
We also investigate the influence of noise on the recognition performance for the laser chaos. Figure 10 shows the simulation results, where it can be seen that even when SNR is 6, the recognition accuracy is up to 95%.
Fig. 10. The simulation results of the proposed CNN method on different SNR when the delay is 135 ns. T: true, F: false.
4. Conclusion
In this paper, we propose a CNN-based deep learning method to extract the TDS in a time-delayed chaotic system, and verify the effectiveness of the method by applying it to several common chaotic systems. By turning traditional TDS into image features, the advantages of CNN are fully utilized in time-delayed chaotic systems. As a result, the recognition accuracy of our proposed method is up to 100% for chaos OEO systems with external nonlinear feedback that we have implemented experimentally. At the same time, under the condition of strong nonlinearity, the traditional ACF and DMI may fail, however our method is still applicable. And the proposed method possessing a strong ability of withstanding the effects of noise. In the future work, we will study the effectiveness of the proposed method on existing delay concealment schemes.
Funding
National Key Research and Development Program of China (2018YFB1801304); Key Project of R&D Program of Hubei Province (2017AAA046).
Acknowledgments
The authors are grateful to those who are fighting against COVID-19 on the frontlines, especially in Wuhan. It is because of their selflessness, bravery and perseverance that the revision process could be finally finished. We firmly believe that the epidemic will come to an end soon.
Disclosures
The authors declare no conflicts of interest.
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