Chaotic Waveforms Research Articles

Systematic realisation of inductorless and resistorless Chua’s chaotic oscillator using VDGA

ABSTRACT This paper presents an implementation of inductorless and resistorless Chua’s chaotic oscillator using voltage-differencing gain amplifier (VDGA). The chaotic oscillator circuit is implemented by designing a resistorless Chua’s diode using a single VDGA and a VDGA-based tank oscillator with coupled network circuit. The proposed chaotic oscillator uses two VDGAs and three grounded capacitors only. Although resistors and inductors may also be fabricated, the use of no resistor and inductor makes the circuit suitable for fabrication in terms of cost and area. It offers the advantages of an increased frequency spectrum having a dominant operating frequency of 21.1 MHz. The Chua’s circuit is simulated in Pspice using both macro models of IC MAX435 and 0.18 µm CMOS TSMC Technology parameters. As a result, chaotic waveforms (V1 and V2) and the various attractors (V1 versus V2) are found in the simulation. These results confirm the theoretical predictions of Chua’s chaotic oscillator. Various aspects of the proposed implementation of Chua’s oscillator are compared with the earlier implementations. Finally, the paper concludes with simple remarks on the proposed circuit.

Optimizations of optical chaos in semiconductor lasers based on multiobjective genetic algorithms.

We investigated the frequency bandwidth, autocorrelation function, and complexity of chaotic temporal waveforms in unidirectionally coupled semiconductor lasers with time-delayed optical feedback. The effective bandwidth, peak value of autocorrelation function, and maximum Lyapunov exponent were simultaneously optimized by searching several control parameters of the laser systems based on multiobjective genetic algorithms. We found a conflicting relation between the effective bandwidth enhancement and the time-delay signature suppression, and a detailed relationship between the maximum Lyapunov exponent and the peak value of autocorrelation function.

Challenge-Response Authentication Scheme With Chaotic Lasers

In this paper, we numerically evaluate a hardware-based method for authentication using a pair of twin chaotic lasers. This method belongs to the PUF (Physically Unclonable Function) class; however, it does not require storing the response of the hardware in a database, because both users own the same hardware (one of the twin chaotic lasers), producing, on the fly, the same response, i.e., the same chaotic waveform, when subject to the same stimulus (injection from a third laser) in the same working conditions (local injection, pump current). A bit sequence can be easily obtained from the chaotic waveforms by electronic processing, and authentication consists in comparing the sequences produced by the twin lasers. This scheme is proposed above all for authentication of a client in an unsecure environment to a server in a secure environment, but it can be used also for peer-to-peer authentication. Both the case of open- and close-loop chaotic lasers are considered. Simulations are based on the Lang-Kobayashi model.

Numerical Study of Statistical Properties for Semiconductor Laser Chaos by Exploring the Injection Parameter Space

The statistical properties of semiconductor laser chaos are of great importance for practical applications. By exploring the injection parameter space of an optically injected semiconductor laser, the associated probability density functions (PDFs) are numerically investigated for generation of chaotic waveforms with rogue waves (RWs) or Gaussian distributions. First, generation of RWs are demonstrated by considering chaotic waveforms with long-tailed PDFs. The dynamical behaviors for RW generations are systematically investigated by exploring the entire injection parameter space, where a pattern is identified for parameter regions of nonzero RWs. The occurrence of RWs is also investigated for different injection parameters by evaluating the effects of laser bias current, spontaneous emission noise, and detection bandwidth. Alternatively, Gaussian PDFs are generated by bandwidth-limited detection of chaotic waveforms. The bandwidth-dependent statistical properties are investigated through tuning the injection parameters for different detection bandwidths. Generation of Gaussian PDFs are demonstrated for a large region of injection parameters and for a detection bandwidth as comparable to the laser relaxation resonance frequency. This numerical study reveals the importance of injection parameters for tuning the statistics associated with chaotic lasers, and confirms the feasibility of generating RWs and Gaussian PDFs on demand for potential applications in various fields.

Denoising Chaotic Signals Using Ensemble Intrinsic Time-Scale Decomposition

Processing chaotic signals is a complicated task due to their nonlinear and non-periodical properties. Conventional linear filters do not allow to properly denoise signals generated by chaotic systems, distorting the carrier while removing the noise, which is critical for such applications as coherent chaotic communications. In this paper, we propose a novel denoising algorithm, called Ensemble Intrinsic Time-Scale Decomposition (EITD) using specific chaotic noise generators. We may use specific chaotic noise generators in the known Ensemble Empirical Mode Decomposition (EEMD), as we also show. Considering the examples of Rössler and Lorenz systems as chaotic waveforms generators, we compare the developed algorithm modifications with other filtration algorithms using ITD and EMD. We use the root-mean-square error (RMSE) as a metric to estimate the denoising quality. Signal-to-noise ratio (SNR) range -10...20 dB is examined, and white, pink and chaotic noise generators are utilized to disturb signals under study. As a result, we explicitly show that the developed approach provides the error 2-10 times less in the case of white and pink noise, and is capable of denoising chaotic signals in case of all the considered types of noises, in contrast to Conventional and Iterative ITD and EMD algorithms.

Multi-Dimensional Lorenz-Based Chaotic Waveforms for Wireless Power Transfer

In this paper, we investigate multi-dimensional chaotic signals with respect to wireless power transfer (WPT). Specifically, we analyze a multi-dimensional Lorenz-based chaotic signal under a WPT framework. By taking into account the nonlinearities of the energy harvesting process, closed-form analytical expressions for the average harvested energy are derived. Moreover, the practical limitations of the high power amplifier (HPA) at the transmitter are also taken into consideration. We interestingly observe that for these types of signals, high peak-to-average-power-ratio (PAPR) is not the only criterion for obtaining enhanced WPT. We demonstrate that while the HPA imperfections do not significantly affect the signal PAPR, it notably degrades the energy transfer performance. As the proposed framework is general, we also demonstrate its application with respect to a Henon signal based WPT. Finally we compare Lorenz and Henon signals with the conventional multisine waveforms in terms of WPT performance.

Multi-dimensional Lorenz-Based Chaotic Waveforms for Wireless Power Transfer

In this paper, we investigate multi-dimensional chaotic signals with respect to wireless power transfer (WPT). Specifically, we analyze a multi-dimensional Lorenz-based chaotic signal under a WPT framework. By taking into account the nonlinearities of the energy harvesting process, closed-form analytical expressions for the average harvested energy are derived. Moreover, the practical limitations of the high power amplifier (HPA) at the transmitter are also taken into consideration. We interestingly observe that for these types of signals, high peak-to-average-power-ratio (PAPR) is not the only criterion for obtaining enhanced WPT. We demonstrate that while the HPA imperfections do not significantly affect the signal PAPR, it notably degrades the energy transfer performance. As the proposed framework is general, we also demonstrate its application with respect to a Henon signal based WPT. Finally we compare Lorenz and Henon signals with the conventional multisine waveforms in terms of WPT performance.

STM32F429 Discovery Board-Based Emulator for Lotka-Volterra Equations

Lotka-Volterra equations are commonly used in prey-predator population studies. Simulation programs are commonly used to produce solutions of Lotka-Volterra equations and to examine their initial value dependendence. In literature, chaotic waveform generators, ECG and EEG generators have been made and used for research and education. To the best of our knowledge, such an electrical circuit to produce the Lotka-Volterra waveforms does not exist. Such a circuit can be made using either analog or digital circuit components. However, such a device may be used for education in classroom and also to prove concepts by population researchers. In this study, implementation and experimental verification of the microcontroller-based circuit which solves LotkaVolterra equations in real time and produces its waveforms are presented. Euler method is used to solve the equation system in discrete time. Presented design has been implemented using an STM32F429 Discovery Board, two DACs and four opamps. The microcontroller sends the signals to the outputs of the circuit using digital-to-analog converters and opamps. The waveforms acquired experimentally from the implemented circuit outputs matches well with those obtained from numerical simulations.

0.75\u2009Gbit/s high-speed classical key distribution with mode-shift keying chaos synchronization of Fabry\u2013Perot lasers

High-speed physical key distribution is diligently pursued for secure communication. In this paper, we propose and experimentally demonstrate a scheme of high-speed key distribution using mode-shift keying chaos synchronization between two multi-longitudinal-mode Fabry–Perot lasers commonly driven by a super-luminescent diode. Legitimate users dynamically select one of the longitudinal modes according to private control codes to achieve mode-shift keying chaos synchronization. The two remote chaotic light waveforms are quantized to generate two raw random bit streams, and then those bits corresponding to chaos synchronization are sifted as shared keys by comparing the control codes. In this method, the transition time, i.e., the chaos synchronization recovery time is determined by the rising time of the control codes rather than the laser transition response time, so the key distribution rate is improved greatly. Our experiment achieved a 0.75-Gbit/s key distribution rate with a bit error rate of 3.8 × 10−3 over 160-km fiber transmission with dispersion compensation. The entropy rate of the laser chaos is evaluated as 16 Gbit/s, which determines the ultimate final key rate together with the key generation ratio. It is therefore believed that the method pays a way for Gbit/s physical key distribution.

Differential Chaos Shift Keying-Based Wireless Power Transfer With Nonlinearities

In this paper, we investigate conventional communication-based chaotic waveforms in the context of wireless power transfer (WPT). Particularly, we present a differential chaos shift keying (DCSK)-based WPT architecture, that employs an analog correlator at the receiver, in order to boost the energy harvesting (EH) performance. We take into account the nonlinearities of the EH process and derive closed-form analytical expressions for the harvested direct current (DC) under a generalized Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> block fading model. We show that, in this framework, both the peak-to-average-power-ratio of the received signal and the harvested DC, depend on the parameters of the transmitted waveform. Furthermore, we investigate the case of deterministic unmodulated chaotic waveforms and demonstrate that, in the absence of a correlator, modulation does not affect the achieved harvested DC. On the other hand, it is shown that for scenarios with a correlator-aided receiver, DCSK significantly outperforms the unmodulated case. Based on this observation, we propose a novel DCSK-based signal design, which further enhances the WPT capability of the proposed architecture; corresponding analytical expressions for the harvested DC are also derived. Our results demonstrate that the proposed architecture and the associated signal design, can achieve significant EH gains in DCSK-based WPT systems. Furthermore, we also show that, even by taking into account the nonlinearities at the transmitter amplifier, the proposed chaotic waveform performs significantly better in terms of EH, when compared with the existing multisine signals.

Differential Chaos Shift Keying-based Wireless Power Transfer with Nonlinearities

In this paper, we investigate conventional communication-based chaotic waveforms in the context of wireless power transfer (WPT). Particularly, we present a differential chaos shift keying (DCSK)-based WPT architecture, that employs an analog correlator at the receiver, in order to boost the energy harvesting (EH) performance. We take into account the nonlinearities of the EH process and derive closed-form analytical expressions for the harvested direct current (DC) under a generalized Nakagami-m block fading model. We show that, in this framework, both the peak-to-average-power-ratio of the received signal and the harvested DC, depend on the parameters of the transmitted waveform. Furthermore, we investigate the case of deterministic unmodulated chaotic waveforms and demonstrate that, in the absence of a correlator, modulation does not affect the achieved harvested DC. On the other hand, it is shown that for scenarios with a correlator-aided receiver, DCSK significantly outperforms the unmodulated case. Based on this observation, we propose a novel DCSK-based signal design, which further enhances the WPT capability of the proposed architecture; corresponding analytical expressions for the harvested DC are also derived. Our results demonstrate that the proposed architecture and the associated signal design, can achieve significant EH gains in DCSK-based WPT systems. Furthermore, we also show that, even by taking into account the nonlinearities at the transmitter amplifier, the proposed chaotic waveform performs significantly better in terms of EH, when compared with the existing multisine signals.

Dual feedback IRC ring for chaotic waveform generation

The authors have proposed an inverter resistor capacitor (IRC)-based simple chaotic ring oscillator with dual feedback for the generation of the chaotic waveform. The proposed chaotic system uses three coupled autonomous first-order differential equations and it can be electrically demonstrated using the complementary metal oxide semiconductor (CMOS) inverters with few passive RC elements for the generation of the high-frequency strange attractor. The basic dynamical characteristics and multistability property with complex bifurcation pattern in a wide range of parameters are well observed through theoretical as well as numerical simulation analysis using scaled inverse tangent function. In addition, a simple IRC chaotic model design is investigated using 0.18μm MOS transistor parameters with grounded capacitors and an equivalent CMOS-based scaled inverse tangent function. Moreover, the Cadence post layout simulation gives useful information about the circuit sustainability for IC design with high frequency (MHz), low power dissipation (940 μW) and small chip area (826.3 μm2).

Unstable basis function for joint radar-communication systems

• A combined radar and communications system using basis functions derived from an unstable second-order system is described. • Using unstable systems with different frequencies allows for the creation of a set of orthogonal basis functions. • Both communications and radar signals may be decoded with linear analog circuits, leading to simple inexpensive receivers. • The radar properties are not sensitive to the encoded communications method. • Because of simplicity and low cost, the system could be used in applications where it was necessary to piggyback a radar signal on a communications signal. In this paper, we develop a joint radar-communication waveform based on the unstable basis function. Basis functions of a chaotic waveform are uncertain, vary continuously and hence the use of a traditional matched filter is not practical. Corron et al. [1,2] demonstrated a chaotic circuit for which an analytic matched filter could be constructed, solving the signal detection problem. The Corron circuit produced a narrow band signal, which is not optimal for spread spectrum communications and radar applications. In this paper, we adapt Corron’s concept of a chaotic signal composed of a linear combination of basis functions and show how the basis functions are used to design joint radar-communications systems that still have simple analytic matched filters. The form of the matched filter makes it useful for both the radar and communications systems where it is necessary to minimize weight or power consumption.

Wireless Chaos-Based Communication System: Literature Review

Since the early 1990s, a slew of chaotic-based communication systems have been proposed, all of which take advantage of chaotic waveform properties. The inspiration stems from the substantial benefits that this form of nonlinear signal offers. Many communication schemes and applications have been specifically designed for chaos-based communication systems to achieve this goal, with energy, data rate, and synchronization awareness being taken into account in most designs. However, non-coherent chaos-based systems have recently received a lot of attention in order to take advantage of the benefits of chaotic signals and non-coherent detection while avoiding the use of chaotic synchronization, which has poor performance in the presence of additive noise. This paper provides a thorough examination of all wireless radio frequency chaos-based communication systems. It begins by describing the difficulties of chaos implementations and synchronization processes, then moves on to a thorough literature review and study of chaos-based coherent techniques and their applications.

Entropy rate of chaos in an optically injected semiconductor laser for physical random number generation.

We evaluate the (ɛ, τ) entropy of chaotic laser outputs generated by an optically injected semiconductor laser for physical random number generation. The vertical resolution ɛ and sampling time τ are numerically optimized by comparing the (ɛ, τ) entropy with the Kolmogorov-Sinai entropy, which is estimated from the Lyapunov exponents using linearized model equations. We then investigate the dependence of the (ɛ, τ) entropy on the optical injection strength of the laser system. In addition, we evaluate the (ɛ, τ) entropy from the experimentally obtained chaotic temporal waveforms in an optically injected semiconductor laser. Random bits with an entropy close to one bit per sampling point are extracted to satisfy the conditions of physical random number generation. We find that the extraction of the third-most significant bit from eight-bit experimental chaotic data results in an entropy of one bit per sample for certified physical random number generation.

Design and Analysis of a Novel Six-Dimensional Hyper Chaotic System

The chaotic system has been widely studied. A new six-dimension hyper chaotic system is introduced in this paper. We used a new chaotic system based on a six-dimension for the purpose of increasing chaos in the system, where the new system has eleven positive parameters, complicated chaotic dynamics behaviors and gives an analysis of the new systems. The basic characteristics and dynamic behavior of this system are investigated with a presence of chaotic attractor, Dissipativity, symmetry, equilibrium points, Lyapunov Exponents, Kaplan-Yorke dimension, waveform analysis and sensitivity toward initial conditions. The results of the analysis exhibit that the new system contains three unstable equilibrium points and the six Lyapunov exponents. Maxim non-negative Lyapunov Exponent (MLE) is obtained as 4.72625, and Kaplan-Yorke are obtained as 3.92566, and the new system characteristics with, unstable, high complexity, and unpredictability, the new system dynamics is simulated utilizing MATHEMATICA program. The phase portraits and the qualitative properties of the new hyper chaotic system have been described at the detail.

Precise ranging for the multi regions of two complex-shape targets by using two chaotic polarization components in the optically pumped spin vertical cavity surface emitting laser with optical injection

&lt;sec&gt; The ranging based on the chaotic lidar (CLR) generated by using the nonlinear dynamic of semiconductor with optical feedback or optical injection exhibits many advantages over the ranging using pulse lasers and CW lasers, such as low probability of intercept, strong anti-interference ability and low cost. Moreover, it has high resolution, benefiting from the broad bandwidth of the optical chaos. Finally, it is easily be generated and controlled due to the sensitivity of chaotic radar to laser parameters. &lt;/sec&gt;&lt;sec&gt; The resolution of the correlated chaotic lidar (CLR) ranging which has been reported in many literatures is largely limited by the bandwidth of the chaotic laser. An ultra-fast chaotic laser with large modulation bandwidth is required to further improve the ranging resolution. The recently proposed optically pumped spin-VCSEL has attractive features such as flexible spin control of lasing output, fast dynamics with femtosecond magnitude and large modulation bandwidth. The ultra-fast chaos radar wave emitted from the optically pumped spin-VCSEL with optical injection or optical feedback is expected to be used for improving the resolution and accuracy of target ranging. In addition, since the multi beams of CLRs were utilized in the previous works, the number of ranging targets is limited to a small number of targets. The reported CLR ranging technology cannot completely detect the distance of different regions in the target, and it is not suitable for the accurate ranging of the whole area in the complex shape target. The detection waveform based on the correlation CLR has not been designed before the target ranging, which affects the further improvement of the resolution and accuracy of the target ranging. To overcome these problems, it is necessary to further explore the theoretical and physical mechanism of the CLR ranging for the multi-region in complex shape target, and explore the new scheme and method for its realization. Motivated by these, in this paper, based on the optically pumped spin vertical cavity surface emitting laser with optical injection, we present a novel scheme for the accurate ranging of the multi regions in two complex shape targets, using two chaotic polarization components modulated by the bipolar sinc waveform. Here, these two modulated chaotic polarization probe waveforms possess the attractive features of the uncorrelation in time and space, fast dynamic with femtosecond magnitude. Utilizing these features, the accurate ranging to the position vectors of the multi regions of two complex-shape targets can be achieved by correlating the multi beams of the time-delay reflected chaotic polarization probe waveforms with their corresponding reference waveforms. The further investigations show that the ranging to the multi-region small targets possesses the very low relative error that is less than 0.94%. If the bandwidths of the photodetectors are large enough, their range resolutions are achieved as high as 0.4 mm, and exhibit excellent strong anti-noise performance and strong stability. The multi area target ranging proposed in our scheme has the following attractive advantages: stable and high range resolution, strong anti-noise ability and very low relative error. These characteristics can meet the needs of the position vector ranging of the multi regions in complex shape targets.&lt;/sec&gt;

Polysomnographic characteristics and acoustic analysis of catathrenia (nocturnal groaning)

Objective: Catathrenia is a sleep disorder characterized by nocturnal groaning sounds emitted during prolonged expiration. As a rare condition, its polysomnographic findings were inconsistent. We aimed to present polysomnographic characteristics of catathrenia patients and perform acoustic analysis of groaning sounds. Approach: Twenty-three patients (eight males and 15 females) diagnosed with catathrenia by video-polysomnography were included. They underwent clinical evaluation and physical examination, and answered a questionnaire. Acoustic analyses (oscillograms and spectrograms) of catathrenia and snoring signals were performed by Praat 6.1.09. Sounds were classified according to Yanagihara criteria. Main results: The average age of catathrenia patients was 29.6 ± 10.0 years, with a body mass index of 22.3 ± 5.1 kg m−2. A total of 3728 groaning episodes were documented. Catathrenia events of 16 patients (70%) were rapid eye movement (REM)-predominant. The average duration of groaning was 11.4 ± 4.6 s, ranging from 1.3 to 74.9 s. All signals of groaning were rhythmic or semi-rhythmic, classified as type I and type II, respectively, with formants and harmonics. Snoring events were observed in nine patients. Snoring mainly occurred in the non-REM stage, with a duration of less than 1.5 s. Signals of snoring were chaotic, classified as type III, without harmonics. Significance: Catathrenia occurred in all sleep stages but mainly in REM. Durations of groaning varied greatly across patients. Acoustic characteristics of catathrenia were typical. Groaning had rhythmic or semi-rhythmic waveform, formants and harmonics, indicating vocal origin, while snoring had chaotic waveform.

Implementation of a Microcontroller-Based Chaotic Circuit of Lorenz Equations

Lorenz equations are commonly used in chaos education and studies. Simulation programs can be used to produce solutions of Lorenz equations and to examine its chaotic waveforms. However, sometimes a chaotic signal source can be needed. Such a circuit can be made using either analog or digital circuit components. Recently, a microcontroller-based circuit is suggested to obtain chaotic waveforms of Lorenz equations however only simulations are used to show proof of concept. Such circuit needs experimental verification. In this paper, implementation and experimental verification of the microcontroller-based circuit which solves Lorenz equations in real time and produces its chaotic waveforms are presented. Runge-Kutta method is used to solve the equation system. By using Proteus, microcontroller-based chaotic circuit is simulated and designed. Presented design has been implemented using an Arduino Mega 2560 R3 microcontroller. The microcontroller sends the chaotic signals to the outputs of the circuit using digital-to-analog converters. The waveforms acquired experimentally from the implemented circuit matches well with those obtained from Proteus simulations.

A SPICE-type nonlinear operational floating current conveyor macromodel

This paper address the derivation of a SPICE-type macromodel for imitating the nonlinear behavior of operational floating current conveyors (OFCC) at low-frequency. By using 0.35μm AMS CMOS technology with ±1.5V bias voltage the OFCC is designed and its most influential performance parameters are acquired, like the bandwidth, DC gain, dynamic range, slew-rate together with parasitic components connected to output-input ports. Afterwards, the behavioral model of the OFCC is derived and the SPICE-type macromodel is coded. A saturated nonlinear function series (SNFS) based on OFCC is designed for certifying the SPICE-type macromodel. To do it, chaotic waveforms were used as input signal to SNFS based on OFCC with CMOS transistors and to the deduced macromodel. Time-domain numerical simulations demonstrate that the deduced macromodel is able to predict the nonlinear behavior of analog circuits with little error and reducing the CPU-time in comparison with the macromodel at the transistor level.