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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 137-158. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221215
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 94A40, 94A14, 94A11, 78M31, 70K99, 78A55
Analysis and Simulation of BER Performance of Chaotic Underwater Wireless Optical Communication Systems
I . V . Semernik, O . V . Bender, A . A . Tarasenko, Ch . V . Samonova
In this article, a method for increasing the noise immunity of an underwater wireless optical communication system by applying chaotic oscillations is considered. To solve this problem, it is proposed to use modulation methods based on dynamical chaos at the physical level of the communication channel.
Communication channel modeling is implemented by calculating the impulse response using a numerical solution of the radiation transfer equation by the Monte Carlo method. The following modulation methods based on the correlation processing of the received signal are considered: chaotic mode switching, chaotic on-off keying (COOK). On-off keying (OOK) modulation was chosen as a test modulation method to assess the degree of noise immunity of the modulation methods under study.
An analysis of the noise immunity of an underwater optical communication channel with a change in its length and parameters of the aquatic environment, which affect the absorption and scattering of optical radiation in the communication channel, is carried out.
It is shown that modulation methods based on the phenomenon of dynamic chaos and correlation processing can improve the noise immunity of underwater wireless communication systems. This provides the possibility of signal recovery at negative values of the signal-to-noise
Received August 10, 2022 Accepted November 15, 2022
This work was supported by the Grant Council of the President of the Russian Federation (Project No. MK-2283.2022.4).
Ivan V. Semernik semernikiv@rusgeology.ru Oleg V. Bender benderov@rusgeology.ru Andrey A. Tarasenko tarasenkoaa@rusgeology.ru Christina V. Samonova kristi-rosa@yandex.ru JSC Yuzhmorgeologia
ul. Krymskaya 20, Gelendzhik, 353461 Russia
ratio. It is shown that the considered modulation methods (COOK and switching of chaotic modes) in combination with the correlation processing of the signal at the physical level of the communication channel provide an advantage of about 15 dB compared to OOK modulation.
Keywords: underwater communication, optical communication, wireless communication, dynamical chaos, noise immunity, wideband signals, communication channel modeling, modulation, Monte Carlo method
1. Introduction
Submersible vehicles are currently used for a wide range of different tasks, from games using compact remotely operated vehicle (ROV) to tasks performed by heavy working-class ROV equipped with a large number of tools and actuators, as well as autonomous missions and manned underwater vehicles.
Ensuring stable communication between the ship and the submersible is one of the necessary criteria for the success of the mission. At the same time, the specifics of the task do not always allow connecting an underwater vehicle to a vessel using a communication cable. In the latter case, it is necessary to use wireless technologies to transmit information: radio frequency communication, acoustic communication or optical communication [1-4].
Due to the high attenuation of electromagnetic waves in the marine environment, radio frequency communication methods are not widely used due to the extremely short range (no more than 10 m) [1-5].
High-frequency electromagnetic waves cannot be used for communication over long distances underwater due to high attenuation. In the frequency range from 1 to 3.5 GHz, the power of electromagnetic radiation is attenuated by 50 dB at a distance of about 10 cm [6]. At a frequency of 800 MHz, the attenuation of the RF signal is about 70 dB at a distance of 1 m [7].
So in [8] the results of the implementation of underwater communication by means of electromagnetic waves with a frequency of 433 MHz are presented. As a result of the tests, the authors achieved a connection range of up to 14 m at the sensitivity limit of the receiving path.
In [9], the results of an experimental study on the implementation of magneto-inductive (MI) communication using a compact 3-dimensional coil are presented. The possibility of communication with a data transfer rate of 1 kbps at a distance of up to 10 m at an operating frequency of 3 kHz was demonstrated.
The paper [10] presents the results of the development of a power amplifier to provide a near-field magnetic communication system. It is shown that the theoretically achievable communication range at the sensor sensitivity limit can reach 150 m at a data transfer rate of 2.4 kbps, a carrier frequency of 20 kHz, and binary phase-shift keying (BPSK) modulation method.
In [11], the results of the development of a compact radio frequency modem for underwater communication at a carrier frequency of 433 MHz are presented. The maximum communication range achieved during the experiment is 350 cm at a data transfer rate of up to 100 bps. In handover mode, a connection range of 700 cm was achieved at a data transfer rate of 100 bps.
Thus, electromagnetic waves can only be used to transmit data underwater over very short distances, for example, to provide communication between a submersible and a bottom station or a docking station with a small distance between them.
Acoustic communication systems [12] make it possible to communicate with an underwater vehicle several kilometers away, however, the data transfer rate is no more than 500 kbps at short distances and decreases as the communication range increases to about 10 kbps. Such a low data
transfer rate does not allow real-time transmission of video information and telemetry from an underwater vehicle. This fact greatly limits the functionality of the underwater vehicle.
Attempts are being made to use chaotic signals in hydroacoustics to improve equipment characteristics [13, 14]. At the same time, special attention is paid to the issues of ensuring the secrecy and security of the communication channel based on chaotic dynamics [15].
In [15], the possibility of using chaotic signals for the implementation of a sonar was considered. The results of studying the influence of the aquatic environment, noise, crosstalk on chaotic and chirp signals are presented. It has been established that chaotic signals have better resistance to factors existing in the hydroacoustic path compared to chirp signals. Chaotic signals are less absorbed by the aquatic environment, and their individual shape allows them to remain distinguishable against equally powerful interference signals existing in the channel. The results obtained by the authors can be extended to systems of underwater acoustic communication.
The paper [16] presents the results of an experimental study of an underwater acoustic communication system based on the frequency hopping-multiple frequency shift keying (FH-MFSK) system for signal modulation. The m-sequence, which has high correlation properties, was chosen as a scheme for frequency jumps. The possibility of communication in the frequency range of 35-45 kHz at a distance of more than 500 m at a speed of 200-300 bps is demonstrated.
The work [17] presents the results of modeling a hydroacoustic modem for underwater communication, which is based on the use of a minimum shift keying (MSK) system for signal modulation. The maximum connection range achieved is 250 m at an average signal frequency of 55 kHz and a rate of 4000 baud per second. At the same time, the scheme based on the correlation detector demonstrated the best noise immunity in comparison with the filter-type quadrature demodulator and mathematical discriminator.
The paper [18] considers the possibility of detecting weak acoustic signals both with a known frequency and with an unknown frequency using a differential double coupling Duffing oscillator and Van der Pol - Duffing oscillator based on intermittent chaos theory. The possibility of detecting acoustic signals with low signal-to-noise ratio (SNR) under Rayleigh distribution noise with an average error rate of no more than 3.265 percent has been demonstrated.
The paper [19] explores the possibility of using 5G multicarrier chaotic sequence spread spectrum technology for underwater acoustic communications. The generalized frequency division multiplexing-chaotic sequence spread spectrum (GFDM-CSSS) technology was used as a modulation method. It is shown that the use of chaotic sequence spread spectrum provides an advantage of about 10 dB compared to the traditional GFDM modulation method.
In [20], a variant of implementation of covert acoustic underwater communication based on chaotic signals mimicking ship-radiated noise is proposed. The signal which is generated by the Chebyshev sequence is used as a chaotic signal. The experiment demonstrated the possibility of data transmission at a distance of 10 km at a speed of 40 bps. When transmitting information, the correlation of the chaos signal to embed the secret message is used. The demodulation scheme demodulates the information according to the correlation coefficient of the received signal. At the same time, the simulation results demonstrate the operability of the system even at negative values of the signal-to-noise ratio.
In [21], a demodulation scheme of chaos phase modulation spread spectrum signals using machine learning methods was proposed to provide underwater acoustic communication. For the purpose of high accuracy performance, the authors used a partial-least square (PLS) regression-based classifier as the detector array. During the modeling process, the authors used the multisegment piecewise linear mapping to generate the desired chaos sequence. The simulation results demonstrate the possibility of data transmission over long distances up to 16.5 km at a relatively
high speed of about 4000 baud per second in the presence of external noise sources with a power comparable to the power level of the useful signal, and other distortions that occur during the propagation of an acoustic signal in an underwater communication channel.
In addition, attempts are being made to apply the features of nonlinear dynamics and chaotic systems in the design of networks for the joint operation of a large number of underwater vehicles and various underwater equipment [22]. Thus, in [23], the authors proposed a scheme for using the Chebyshev chaotic maps to create a secure network authentication scheme for subscribers. The proposed scheme turns out to be robust and secure based on the Diffie Hellman Problem (DHP) and the discrete logarithmic problem (DLP). By applying the Burrows-Abadi-Needham (BAN) logic and random oracle model, the authors have certified that the proposed scheme is able to accomplish mutual authentication and negotiate session key among the user, the gateway and the sensor node.
In [24], a variant of the implementation of an underwater acoustic communication system based on the Wi-Fi communication system protocol, adapted to create the differential chaos shift keying (DCSK) method based on a hybrid dynamical system, is proposed. A feature of the approach proposed by the authors is the use of a hybrid system as a chaotic signal generator, which has a simple matched filter to maximize the signal to noise ratio at the receiver end, and a corresponding matched filter to relieve the noise effect on the decoding of information. The proposed modulation scheme achieves better performance than methods such as differential chaos shift keying (DCSK), code shifted differential chaos shift keying (CS-DCSK), high-efficiency differential chaos shift keying (HE-DCSK), phase-separated differential chaos shift keying (PS-DCSK), improved differential chaos-shift keying (I-DCSK), binary phase-shift keying (BPSK) and frequency-modulated differential chaos shift keying (FM-DCSK) with significantly low BER under both the same spreading gain and the same bit transmission rate.
In [25], the possibility of combining the advantages of the code-shifted differential chaos shift keying (CS-DCSK) with the orthogonal frequency division multiplexing (OFDM) method is considered. The proposed scheme has increased resistance to time/frequency-selective fading. Simulation results over underwater wireless acoustic (UWA) channels show that the proposed system has better performance than MultiCode Direct Sequence Spread Spectrum (MC-DSSS), multicarrier M-ary differential chaos shift keying (MM-DCSK), and OFDM-based code shifted chaos shift keying (OFDM-CS-DCSK). The theoretical possibility of providing communication at a distance of up to 1000 m is demonstrated.
Attempts are being made to combine the advantages provided by various methods of data transmission under water [26, 27]. The scope of such hybrid schemes is significantly limited, however, under certain scenarios, the hybrid communication method can provide the required functionality and performance of the equipment.
Thus, acoustic communication systems provide data transmission over long distances, but the data transfer rate does not allow the transmission of a large amount of data in real time. At the same time, the use of chaotic dynamics in the composition of acoustic communication systems provides an increase in the noise immunity of communication systems and, as a result, the possibility of increasing the data transfer rate with a long connection range.
In turn, underwater wireless optical communication [4, 28, 29] systems make it possible to provide high-speed data transmission at a moderate communication range (1-5 Gbps at a distance of no more than 10 m and 1-10 Mbps at a distance of about 100-200 m) [30, 31].
In [32], a platform with a graphical user interface for simulation and the results of modeling an underwater optical communication system are discussed. The proposed system is based on a light-emitting diode (LED) with a wavelength of 532 nm and a power of 1000 mW, an avalanche
photodiode as a receiver, and the OOK modulation method. The possibility of communication at a distance of up to 30 m through a communication channel with a coastal water at a data transfer rate of 10 Mbps was shown.
In [33], the possibility of providing data transmission at a rate of 500 Mbps over a distance of up to 150 m using a digital signal processing (DSP) scheme including partial response shaping and trellis coded modulation (TCM) technology was experimentally demonstrated.
The paper [34] presents the results of an experimental study illustrating an underwater wireless optical communication (UWOC) system using a 520-nm laser diode (LD) and 32-quadrature amplitude modulation (32-QAM) single carrier signals. With a communication channel bandwidth of 200 MHz, the proposed communication system revealed a maximum data transfer rate of 3.48 Gbps at a distance of 56 m.
At the same time, commercial manufacturers offer commercially available models of underwater equipment to provide underwater optical wireless communication.
Thus, the MC500 underwater optical wireless modem manufactured by Shimadzu [35] provides communication at a distance of up to 80 m at a data transfer rate of up to 20 Mbps, depending on the conditions of the aquatic environment. The MC100 modem provides communication over a distance of more than 10 m at a data transfer rate of up to 95 Mbps.
The BlueComm 200 modem manufactured by Sonardyne provides wireless data transmission using an optical signal over a distance of up to 150 m at a data transfer rate of up to 10 Mbps; in the absence of external illumination sources, the communication range, depending on the conditions of the aquatic environment, can exceed 200 m [36].
At the same time, manufacturers of underwater vehicles are actively cooperating with the developers of modems for optical underwater communication due to the fact that the presence of optical communication in addition to traditional communication systems provides wide opportunities, including high-quality video transmission in real time from autonomous and manned vehicles not connected by a communication cable with the operator [37-39].
However, the communication range significantly depends on the parameters of the aquatic environment and in reality is less than theoretically achievable values. In connection with the foregoing, the tasks of increase in the communication range, researching new methods of modulation and signal detection not only do not lose their relevance, but every day they attract more and more attention due to the growing performance requirements for underwater wireless communication technologies [40].
The possibility of implementing a wireless underwater communication system based on Bessel-like beams with enlarged depth of focus based on fiber microaxicon was studied in [41]. The simulation results demonstrate the theoretical possibility of providing underwater optical communication over long distances up to 4000 m.
When performing optical communication under water, it is possible to build an underwater communication channel that is different from the line-of-sight channel, for example, for communication using reflection from ice in the case of underwater equipment operating under ice [42].
In this regard, methods of information transmission based on the dynamical chaos are prospective. Features inherent in dynamic chaos, including increased resistance to noise and interference, the ability to reliably operate at low signal-to-noise ratios, resistance to multipath signal propagation and secrecy of information transmission, are very promising for use in underwater wireless optical communication systems.
This article is organized as follows. A description of the method for modeling an underwater wireless optical communication channel, based on the numerical solution of the radiation transfer equation by the Monte Carlo method, is given in the section "Communication channel modeling".
A description of the parameters of the communication channel and the aquatic environment are given in the section "Communication channel model". A description of the studied modulation methods is given in the section "Modulation Methods". The obtained results of numerical modeling of the noise immunity of an underwater wireless communication system for various signal modulation methods are given in the section "Numerical results".
2. Communication channel modeling
The signal at the output of the communication channel can be calculated by convolving the input signal and the impulse response of the communication channel as follows:
y(t) = x(t) ■ h(t), (2.1)
where y(t) is the output signal, x(t) is the input signal, and h(t) is the impulse response of the communication channel. Thus, the task of modeling the communication channel is reduced to determining the impulse response, which will allow one to calculate the signal at the output of the communication channel for any signal at its input.
The impulse response of a communication channel can be determined in various ways [41, 44, 45], but one of the most common approaches is the numerical solution of the radiative transfer equation using the Monte Carlo method.
In addition to ease of implementation, the Monte Carlo method has significant advantages, such as high solution accuracy and flexibility in setting the model input parameters, allowing photon tracing and dispersion and scattering in the communication channel [46]. These advantages make it possible to apply the Monte Carlo method to simulate the propagation of broadband signals through an underwater optical communication channel [47]. The reliability, accuracy and efficiency of the Monte Carlo method have been confirmed by a large number of theoretical and experimental studies [48].
The impulse response of a communication channel can be approximated by the following function:
h(t) = C1Ate-G2At + C3Ate-G4At, (2.2)
where At = t —10 is the time interval and t0 is the optical signal propagation time.
The coefficients of the approximating function C1 — C4 are determined based on the following condition:
(Ci, C2, C3, C4) = argmin (f [h(t) — hmc(t)f dtj , (2.3)
where hmc(t) is the result of modeling the impulse response of a communication channel by the Monte Carlo method and argmin is a function that returns the values of the parameters that provide the minimum value of the function given in the brackets.
Modeling of an underwater wireless optical communication channel is implemented in MAT-LAB based on the numerical solution model of the radiative transfer equation using a modified Monte Carlo method [47].
While using the Monte Carlo method, the optical beams whose power has become less than a given threshold during propagation are deflected. In contrast, in the modified Monte Carlo method, rays that have gone beyond the boundaries of a given three-dimensional space are deflected, regardless of their power level. This makes it possible to more accurately analyze the effects of dispersion of the optical signal by processing all the rays that came to the location
of the receiver. This approach allows one to estimate the distortions of broadband and chaotic signals that occur during the propagation of optical signals through a communication channel and their effect on the detection process in the receiver.
3. Communication channel model
In this paper, signal transmission through a diffused line-of-sight (Diffused LOS) channel is considered (Fig. 1).
Optical radiation propagation
Fig. 1. Configuration of the diffused LOS UWOC channel
To accurately determine the model parameters, at the initialization stage, the wavelength of the optical signal, the depth of the communication channel, and the dimensions of the analyzed area, with the transmitter and receiver inside it, are set.
Based on the entered value of the depth of the communication channel, the value of the refractive index of water, the temperature and salinity of the water are determined.
The refractive index is determined according to the McNeil approximation based on the data of the water salinity and temperature.
To determine the data of salinity and water temperature, the average values of the corresponding parameters for the Black Sea in the summer period were used [49] depending on the depth.
To take scattering into account, a model based on the Henyey-Greenstein function is used, which has a variable parameter "g" — an anisotropy factor the zero value of which determines the isotropic nature of the scattering of optical radiation, and "1" determines scattering only in the forward direction. The value of 0.9185 allows one to most accurately approximate the experimental data [50], so this parameter value was used in the calculations.
The value of chlorophyll concentration in water is set equal to 0.006 g/m3, which corresponds to the average value for most of the Black Sea, with the exception of the shelf of the western coast [51].
Modeling is carried out for the following types of water: clear ocean water, coastal water, harbor water. The values of the absorption and scattering coefficients used in the calculations for each type of water are given in [47].
An LED with a Lambert directivity pattern was used as a source of optical radiation.
The list and values of the parameters of the model of the underwater wireless optical communication system are given in Table 1.
To check the influence of the dimensions of the analyzed area on the simulation results, test simulation was carried out with identical input data, but for several different values of the analyzed area width from 5 m to 50 m. It was determined that with an increase in the width of the communication channel, the number of scattered beams that have reached the location of
Table 1. Parameters of the UWOC system model
Parameter name Parameter value
Water area Black Sea
Season Summer
Communication channel depth 30 m
Optical signal wavelength 518 nni
Scattering model Henyey - Greenstein function, anisotropy factor g = = 0.9185
Chlorophyll concentration 0.006 g/m3
Dimensions of the area of space that forms the communication channel Channel size 10 x 10 m. The channel length depends on the distance between the source of optical radiation and receiver
Type of optical radiation source Lambertian LED (m = 1)
Maximum number of scattering phenomena on the path from source to receiver 10
Source power 1 W
The number of rays coming from the source 2000
The number of rays produced by each scattering phenomenon
Receiver field of view 40°
Photodetector lens diameter 10 mm
the receiver with a large delay increases, but their power is very small and they have almost no effect on the total received power and the impulse response of the communication channel. At the same time, with the expansion of the communication channel, the simulation time increases significantly. Therefore, in the calculations, the value of the width of the analyzed area of space was taken to be 10 m.
The results of simulation using the Monte Carlo method for the propagation of optical rays through a UWOC channel 50 m long and with the parameters given in Table 1 are shown in Fig. 2.
The results of approximating the impulse response of a UWOC channel with a length of 50 m using Eq. (2.2) are shown in Fig. 3.
4. Modulation methods
Analysis of the simulation results and experimental studies carried out by various authors [47, 52] showed that the greatest connection range can be achieved using on-off-keying (OOK) modulation [53] in conjunction with correction codes that improve the noise immunity of the communication system. Other modulation methods, including orthogonal frequency division multiplexing (OFDM) [53], quadrature amplitude modulation (QAM) [53], pulse-amplitude modulation (PAM) [53], pulse-position modulation (PPM) [53], discrete multitone modulation (DMT) [53], due to multiposition, allow for high data transfer rates, but the connection range does not exceed several tens of meters at best, even when using a collimated light source in the form of laser diodes [47, 52].
50
-250-1-1-1-1-1-1-1-■»-
200 300 400 500 600 700 800 900 1000 1100
t, ns
Fig. 2. The results of simulation using the Monte Carlo method for the propagation of optical rays through a UWOC channel 50 m long
Fig. 3. The results of approximating the impulse response of a UWOC channel with a length of 50 m using Eq. (2.2)
In [47], the authors show that the most promising methods for transmitting information through an underwater optical communication channel using chaotic oscillations are methods
based on matched filtering (correlation reception) on the receiving side. Among these methods, chaotic on-off keying (COOK) [54, 55] and the method of chaotic modes switching [55-58] should be noted. Due to the fact that in the time and frequency domains, as well as in the phase plane, a broadband chaotic signal undergoes strong distortions, data transmission methods through chaotic oscillations based on the analysis of the received signal in these areas are less efficient.
In accordance with the above, for further consideration we will choose the following methods based on the correlation reception of a chaotic carrier oscillation: chaotic OOK modulation (COOK) [54, 55] and switching of chaotic modes [55-58]. Let's consider each of these modulation methods in more detail.
COOK modulation means that with each transmission of a logical "1" the same fragment of a chaotic signal is emitted. In the case of a logical "0" transmission, there is no radiation. On the receiving side, correlation processing of the received signal is carried out to restore the transmitted information.
The block diagram of the COOK modulation method is shown in Fig. 4.
Underwater optical channel
Fig. 4. The block diagram of the COOK modulation method
In the investigation modulation method, the chaotic pulse source, together with the switch, generates a chaotic pulse of a given duration when a command to transmit a logical "1" is received from the information signal source. In the case of a command to transmit a logical zero from the source of the information signal, the formation and emission of a chaotic pulse is not carried out and nothing is emitted into the communication channel.
The underwater optical communication channel is a combination of a module that determines the output signal of the communication channel based on the known impulse response of the channel in the absence of external noise sources, and a module that adds the noise signal to the output signal of the communication channel. Both the noise signal added to the signal for testing the noise immunity of the system and the background noise consisting of black body radiation and underwater ambient light are taken into account [32, 63].
The source of the chaotic pulse on the receiving side of the communication system generates a chaotic pulse corresponding to a logical "1" to provide correlation processing of the received signal. The duration of the chaotic pulse is assumed to be known and corresponds to the duration of the pulse generated on the transmitting side.
The threshold unit compares the result of correlation processing with the set threshold and, if exceeded, generates a logical signal "1" at the output; if the set threshold is not exceeded, the threshold unit output signal corresponds to logical "0".
The method of switching chaotic modes assumes that in the case of transmitting a logical "1", a fragment of one chaotic signal is emitted, and when transmitting a logical "0", a fragment of another chaotic signal orthogonal to the first one is emitted. This method has a high protection against unauthorized access, since an outside observer will only see a continuous noise-like signal.
Switch
Underwater optical channel
Chaotic impulse 1 source
Chaotic impulse 2 source
Information signal source
y(t) = x(t)h(t)
Noise Source
Chaotic
impulse 1
source
Correlator Threshold
unit
Correlator Threshold "0"
unit
"0"
Chaotic impulse 2 source
Fig. 5. The block diagram of the method of switching chaotic modulation
The block diagram of the studied method of switching chaotic modulation is shown in Fig. 5.
In the scheme under consideration, the source of the chaotic pulse 1 and the source of the chaotic pulse 2 form chaotic pulses corresponding to the logical "1" and the logical "0". Depending on the command coming from the information signal source, either one or the second pulse is emitted into the channel.
The underwater optical communication channel is identical to that described above.
Sources 1 and 2 of chaotic pulses on the receiving side of the communication system form similar chaotic pulses corresponding to logical "1" and logical "0" to provide correlation processing of the received signal in correlators. The duration of the chaotic pulses is also assumed to be known and corresponds to the duration of the pulses generated on the transmitting side.
Threshold units compare the results of correlation processing with the set thresholds and, if exceeded, generate a logical signal at their output. In this case, the logical signal at the output of one threshold unit corresponds to "1", and the logical signal at the output of the second one corresponds to logical "0".
At the same time, both methods make it possible to provide full-duplex communication by using various chaotic signals to transmit information in one direction and in the other. This approach also allows multiuser communication and implement multiposition modulation methods by simultaneously emitting various chaotic signals, the separation of which is carried out using correlation processing.
Of note is a high degree of information security since, without information about the chaotic signals corresponding to each symbol, their selection from a mixture of signals and noise is impossible.
OOK modulation was chosen as a test modulation method to assess the degree of noise immunity of the studied methods based on dynamical chaos. The theoretical dependence of the bit error rate on the signal-to-noise ratio for OOK modulation is given by the following expression [59, 60]:
1 . {y/SNJJ^j
BER = - ■ erfc ,
2 I
(4.1)
where BER is the bit error rate and SNR is the signal-to-noise ratio.
To form a broadband chaotic signal, a model of a microwave generator of chaotic oscillations based on an avalanche-transit time (IMPATT) diode [61-63] was used in the form of a system of differential equations with a delayed argument, the solution of which was carried out using the Runge-Kutta method of 2-3 orders.
The system of differential equations describing the dynamics of an IMPATT oscillator in different dynamical modes including chaotic mode has the form
/
dil
~dt = ?'2;
di2 f Rp-n (I, —n (I,
lb" I L ) %2 ~ L Jh+
+ kw2il (t — t ) exp(—jwT),
where I0 is the IMPATT diode supply current, I is the amplitude of the microwave current, il is the microwave current, i2 is the time derivative of the microwave current, k is the module of the inertial feedback transmission coefficient, t is the signal delay in the inertial feedback, Rl = 4.3 Ohm is the load resistance, L = 0.895 nH is the cavity inductance, Rp-n(I, I0), Xp-n(I, I0) is the active and reactive components of the IMPATT diode impedance, XCd = = 0.4 uF is the capacity of the IMPATT diode drift region, w0 is the resonant frequency of the oscillator cavity, and w is the angular frequency.
The third term on the right-hand side of the second equation in Eq. (4.2) takes into account the presence of inertial feedback in the system to ensure the transfer of the deterministic microwave diode oscillator into the dynamical chaos mode and provides the possibility of controlling the basic parameters of the chaotic oscillation.
The oscillogram, phase portrait, amplitude spectrum and autocorrelation function of the chaotic pulse corresponding to the logical "1" are shown in Fig. 6.
The oscillogram, phase portrait, amplitude spectrum and autocorrelation function of the chaotic pulse corresponding to the logical "0" for the method of switching chaotic modes are shown in Fig. 7.
The chaotic impulses corresponding to the logical "1" and "0" shown in Fig. 6 and Fig. 7 are two sections of the same chaotic signal generated by the dynamic system described by Eq. (4.2), corresponding to different time periods. In this regard, the differences between chaotic pulses are minimal, which can provide additional secrecy of the fact of communication.
The duration of one symbol and the time of its transmission through the communication channel were set to 1200 ns, which corresponds to a rate of 0.83 Mbaud per second.
The cross-correlation function of two chaotic pulses shown in Fig. 8 demonstrates the orthogonality of chaotic pulses despite their similarity in time, frequency and phase domains.
5. Numerical results
Let us analyze the noise immunity of an underwater wireless optical communication system using the described modulation methods. Correction codes are not used in this work, since they can be applied to any of the described modulation methods, and therefore, in this work, the main attention is paid to the noise immunity of signal transmission at the physical level of the communication channel.
Figures 9-12 show the dependence of the Bit error rate (BER) on the SNR during signal transmission using the three described modulation methods through an underwater optical com-
/, Mhz r, ns
Fig. 6. The oscillogram (a), phase portrait (b), amplitude spectrum (c) and autocorrelation function (d) of the chaotic pulse corresponding to the logical "1"
munication channel formed by clear ocean water [47, 64], having a length of 5, 20, 100 and 200 m, respectively.
Figure 9 shows that the results of the noise immunity estimation obtained using the developed communication channel and system models for the case of OOK modulation correspond to the theoretical values, which confirms the reliability of the results obtained in the simulation process.
A slight increase in the noise immunity of the communication system for the OOK modulation method with increasing connection range, which can be seen in Figs. 10 and 11, may be a consequence of the expansion of the transmitted impulse due to signal dispersion during propagation through the communication channel, which increases mainly with increasing communication distance from 5 to 100 m. An increase in the communication range from 100 to 200 m leads to an insignificant increase in the pulse duration at the output of the communication channel due to signal dispersion, but a significant weakening of the signal in power, which is reflected in a decrease in the noise immunity of the communication channel with OOK modulation, which can be seen in Fig. 12. A partial contribution can also be made by statistical errors inherent in the Monte Carlo method since, with an increase in the communication distance, the number
0 200 400 600 800 1000 1200 -1
t, ns
50 100
/, Mhz
150 -1500 -1000 -500 0 500 1000 1500
r, ns
Fig. 7. The oscillogram (a), phase portrait (b), amplitude spectrum (c) and autocorrelation function (d) of the chaotic pulse corresponding to the logical "0"
of photons reaching the receiver decreases and the calculation of the impulse response of the communication channel is based on a smaller amount of data.
The obtained results demonstrate that the use of modulation methods based on dynamic chaos makes it possible to provide an increase in noise immunity by 15 dB or more at the physical level. This allows us to consider COOK and chaotic mode switching as promising modulation methods for the implementation of noise-immune systems for underwater wireless optical communication.
Figure 13 shows the results of calculating the dependence of the BER on the SNR for a communication channel with 5 m length in changing the type of water forming the communication channel: clear ocean water, coastal water, and harbor water.
It can be seen from the simulation results that COOK modulation has greater noise immunity compared to the method of switching chaotic modes due to the fact that in the first case, when a logical "0" is transmitted, only a noise signal is received at the receiver input, while in the second case there is an orthogonal chaotic signal having a small correlation with the signal corresponding to the logical "1". The choice of chaotic signals with a lower degree of correlation will increase the noise immunity. In addition to chaotic signals, other orthogonal signals can also be used, for example, pseudo-random sequences.
250
-1500 -1000 -500 0 500 1000 1500
r, ns
Fig. 8. The cross-correlation function of two chaotic pulses
SNR, dB
Fig. 9. Dependence of BER on the SNR at a communication channel length of 5 m with clear ocean water (the solid line is the theoretical dependence of BER for OOK modulation, calculated according to Fig. 4)
Increasing the range of the communication channel from 5 to 200 m and changing the parameters of absorption and scattering of optical radiation over a wide range (changing the
Î ^ ^ ^ * ^ ^ ^ ^ ^ ^ ^ ^
^ + \
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+
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+
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OOK
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\
\
\
*
I
10
-4
COOK 6
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1 I
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-10 -5
0 5 10
SNR, dB
15 20
Fig. 10. Dependence of BER on the SNR at a communication channel length of 20 m with clear ocean water
-10 -5
0 5 10
SNR, dB
Fig. 11. Dependence of BER on the SNR at a communication channel length of 100 m with clear ocean water
type of water from clear to harbor) slightly affects the noise immunity of all modulation methods under study.
i I™ ^ H® ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
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-1
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-3
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\ \
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cook 6
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-10
0 5 10
SNR, dB
15
20
Fig. 12. Dependence of BER on the SNR at a communication channel length of 200 m with clear ocean water
SNR, dB
Fig. 13. Dependence of the BER on the SNR for a communication channel with 5 m length in changing the type of water forming the communication channel: clear ocean water ("o"), coastal water (" *"), harbor water (" +")
The duration of one chaotic pulse and the time of transmission of one symbol through the communication channel during the simulation varied from 1200 ns to 60 ns by compressing the
transmitted chaotic pulse in time and, accordingly, expanding the bandwidth of the transmitted signal.
In the results presented in this paper, the duration of one symbol and the time of its transmission through the communication channel were set to 1200 ns, which corresponds to a rate of 0.83 Mbaud per second.
At the same time, when the chaotic pulse was compressed to 60 ns and the signal spectrum was expanded accordingly, it was restored regardless of the length of the communication channel and the parameters of the aquatic environment, despite the distortions in the time and frequency domains that occur when the signal propagates through the communication channel. Further compression of the chaotic pulse in time made it impossible to detect it on the receiving side.
Thus, the theoretically achievable rate was 16.7 Mbaud per second. The use of error-correcting coding methods and multiposition modulation methods based on the use of chaotic signals can improve the result obtained.
6. Conclusion
Thus, in this paper it is shown that modulation methods based on the phenomenon of dynamic chaos and correlation processing can improve the noise immunity of underwater wireless communication systems, providing the possibility of signal recovery, for example, at negative values of the signal-to-noise ratio.
It is shown that the considered methods of COOK modulation and switching of chaotic modes in combination with correlation signal processing in the receiver provide an advantage in terms of noise immunity of the order of 15 dB compared to OOK modulation. The use of the OOK modulation method allows communication at a noise level exceeding the level of the information signal, with negative values of the signal-to-noise ratio up to —10 dB. The use of the chaotic mode switching method provides less noise immunity and allows communication at a noise level comparable to the useful signal level at signal-to-noise ratios up to —5 dB. The simulation results confirm the theoretical possibility of providing communication through an underwater wireless line-of-sight optical channel up to 200 m long at a rate of up to 16.7 Mbaud per second with a symbol duration of 60 ns without degrading noise immunity.
The use of error-correcting coding at the software level in conjunction with the use of modulation methods based on chaotic signals at the physical level may further increase the noise immunity of UWOC systems, providing protection against background illumination of the receiving device.
In addition, the considered methods at the physical level provide a high degree of protection of information from unauthorized access due to the fact that, without knowledge of the used chaotic signals, it is almost impossible to restore information from a mixture of useful signals and noise.
Further research is aimed at modeling underwater wireless optical communication systems based on various methods of chaotic modulation, investigation of the possibility of using the phenomenon of chaotic synchronization to provide communication through a wireless underwater optical channel, modeling a multisubscriber UWOC communication system, as well as creation, manufacturing and experimental studies of individual nodes and the system as a whole.
Conflict of interest
The authors declare that they have no conflict of interest.
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