A Deterministic Chaos-Model-Based Gaussian Noise Generator

Abstract

The abilities of quantitative description of noise are restricted due to its origin, and only statistical and spectral analysis methods can be applied, while an exact time evolution cannot be defined or predicted. This emphasizes the challenges faced in many applications, including communication systems, where noise can play, on the one hand, a vital role in impacting the signal-to-noise ratio, but possesses, on the other hand, unique properties such as an infinite entropy (infinite information capacity), an exponentially decaying correlation function, and so on. Despite the deterministic nature of chaotic systems, the predictability of chaotic signals is limited for a short time window, putting them close to random noise. In this article, we propose and experimentally verify an approach to achieve Gaussian-distributed chaotic signals by processing the outputs of chaotic systems. The mathematical criterion on which the main idea of this study is based on is the central limit theorem, which states that the sum of a large number of independent random variables with similar variances approaches a Gaussian distribution. This study involves more than 40 mostly three-dimensional continuous-time chaotic systems (Chua’s, Lorenz’s, Sprott’s, memristor-based, etc.), whose output signals are analyzed according to criteria that encompass the probability density functions of the chaotic signal itself, its envelope, and its phase and statistical and entropy-based metrics such as skewness, kurtosis, and entropy power. We found that two chaotic signals of Chua’s and Lorenz’s systems exhibited superior performance across the chosen metrics. Furthermore, our focus extended to determining the minimum number of independent chaotic signals necessary to yield a Gaussian-distributed combined signal. Thus, a statistical-characteristic-based algorithm, which includes a series of tests, was developed for a Gaussian-like signal assessment. Following the algorithm, the analytic and experimental results indicate that the sum of at least three non-Gaussian chaotic signals closely approximates a Gaussian distribution. This allows for the generation of reproducible Gaussian-distributed deterministic chaos by modeling simple chaotic systems.

1. Introduction

Chaos theory has radically revolutionized our understanding of the potential for complex, unpredictable behavior within seemingly simple deterministic systems. This phenomenon, known as chaos, arises in nonlinear systems and is characterized by extreme sensitivity to initial conditions, often leading to the iconic “butterfly effect”. Chaotic dynamics are remarkably pervasive, manifesting in diverse domains such as social, biological, and technical systems [1,2,3,4,5,6].

Security threads and challenges prompt scientists to research how chaotic signals are suitable to resolve these issues. Numerous applications have been proposed, including secure and covert communication, pseudo-random number generators (PRNGs) and cryptography [7,8,9,10,11,12,13,14]. Despite the diverse applications and varying advantages and drawbacks in each of these scopes, chaotic systems are used as sources of entropy, a fundamental property crucial for security applications.

Chaos generators, present in both hardware and software implementation, produce time series that either mask information-bearing signals or undergo postprocessing for various cryptographic purposes. Chaotic trajectories form strange attractors in phase space, which, in contradiction to the limit cycle that represents regular behavior, have fractional or fractal dimensions [1]. There are several variants of fractal dimensions, generally measuring the average closeness, density, and distribution of points on chaotic attractors in phase space. Chaos sources are categorized into two groups, including continuous time and discrete map systems. Due to differences in attractor properties, the trajectories of a chaotic system exhibit different statistics reflected in their probability density functions (PDFs). Chaos applications necessitate a high-quality entropy source, emphasizing the necessity to maximize the entropy of chaotic signals. Statistical analyses have been proposed by Yong Wang et al. to increase the quality of chaos for the same purpose as in this paper [15]. Concerning PDFs, the most important are a number with a uniform PDF and Gaussian-distributed signals with a bell-shaped PDF. The first requirement is common in cryptography; the second is useful in communication. Both of them maximize the uncertainty of range-limited and variance-limited signals, respectively [16].

A two-dimensional discrete-time white Gaussian noise generator was proposed in [17]. To obtain an independent time series with a Gaussian probability density function, a special transformation was used on the tent map. A similar approach to the same tent map demonstrated the efficiency of the transformation method [18].

A Gaussian distribution is not the sole target widely sought in signal processing. Contributions in this direction also focus on achieving a uniform probability density. In [19], it is proven that the folding sums of chaotic trajectories, bounded within a given interval by modulus operation, tend to approach uniformity. It is worth noting that certain discrete chaotic systems yield series with a uniform probability density function without requirement of additional transformations [20,21,22,23,24,25].

Significant achievements have been attained in the realm of optical chaos generators [26,27,28,29,30,31]. Wideband optical chaos exhibits a heightened propensity for Gaussian-distributed probability density functions. This characteristic has prompted suggestions for harnessing optical chaos generators as high-quality, ultra-fast entropy sources [28,29]. A recently reported experimental technique focuses on generating a Gaussian-invariant distribution [32]. Such sources hold potential for diverse applications spanning communication [33,34], random number generation [26,27], and cryptography [30,35].

Covert communication, originally designed to conceal and render the transmission undetectable, is now being harnessed for commercial and private applications to provide physical-level protection. In contrast to conventional communication, where Gaussian white noise diminishes the signal-to-noise ratio, covert communication employs it as a countermeasure against eavesdroppers. Thermal noise is utilized to randomize the transmitted signal in a Gaussian-distributed spread spectrum [36]. Additionally, extending this concept, the utilization of artificially generated Gaussian-distributed noise has been proposed across diverse scenarios [37,38,39,40]. Despite the fact that hundreds of chaotic systems have been designed and studied, few of them can produce Gaussian-distributed signals. Therefore, finding an approach for obtaining the most complex chaotic signal based on simple and well-studied nonlinear systems is important. In this paper, we propose a method to obtain Gaussian-distributed signals. To achieve this, we study the statistical properties of some chaotic signals to find out how much of them are necessary to form a Gaussian-distributed waveform with the following experimental investigation and achievement verification that allow for an implementation of “deterministic noise” generators. With this aim, we have suggested an algorithm that combines the most valued statistical characteristics for the considered signal assessment.

The manuscript is organized as follows. In Section 2, we describe the quality metrics used to evaluate and compare chaotic signals. The properties of time series produced by mathematic models of Chua’s circuit and the Lorenz system are studied in Section 3. Experimental verification of the results is provided in Section 4. The discussion and conclusion are presented in Section 5. Appendix A presents the results of computing the statistical properties of more than 40 continuous-time chaotic systems found in the literature.

2. Materials and Methods

2.1. Central Limit Theorem

The central limit theorem (CLT) relates to probability theory and explains why Gaussian noise is widespread in nature. For identically distributed independent samples, the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed. There are several variants of the CLT. In its common Lindeberg–Lévy form, the CLT is as follows [41,42]:

Theorem 1.

Suppose $\{X_1,\cdots ,X_n\}$ is a sequence of independent and identically distributed random variables with expectation $\mathbb{E}\left[X_i\right]=\mu$ and variance $Var\left[X_i\right]={\sigma }^2<\infty$. Then, as n approaches infinity, the random variables $\sqrt{n}(\frac{1}{n}{\sum }_{i=1}^n{X}_i-\mu )$ converge in distribution to a normal $\mathcal{N}(0,{\sigma }^2)$.

The CLT shows a way to obtain a variable with a Gaussian-like distribution from several others by summation. The key condition for particular components is their limited variance, which chaotic signals satisfy due to the existence of their attractors in bounded phase space. The question is to find how many independent chaotic signals are necessary and which criteria the resulting signal should meet to be considered Gaussian-like. Therefore, the following consideration of the basic distributions will present the possibility to develop an algorithm for testing chaotic signals.

2.2. Basic Distributions

As a reference signal, we chose white Gaussian noise whose statistical properties can be described by three basic distributions which are the PDF of the shape, the envelope and the phase of the signal.

The term “Gaussian noise” or “Gaussian signal” refers to a normal or bell-shaped probability density function of a signal. Such signals can be either random noise or information-bearing. A Gaussian variable has the following PDF

$$p\left(x\right)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{{(x-\mu )}^2}{2{\sigma }^2}},$$

(1)

where $\sigma$—standard deviation; $\mu$—mathematical expectation or mean.

The envelope of white noise has a Rayleigh distribution

$$p(x,\sigma )=\frac{x}{{\sigma }^2}{e}^{-\frac{x^2}{2{\sigma }^2}},$$

(2)

where $\sigma$—standard deviation of the Gaussian distribution (1).

A uniform distribution describes the phase of a signal

$$p\left(\varphi \right)=\frac{1}{2\pi },$$

(3)

where $0\le \varphi \le 2\pi$.

2.3. Measures of Similarity of Probability Density Functions

To evaluate the similarity of any signal to WGN, normality tests that include computing of skewness and excess kurtosis are conventionally utilized [43].

Skewness is a measure of the asymmetry of a PDF around its mean and is defined as the normalized third central moment of a signal

$${\tilde{\mu }}_3=\mathbb{E}\left[{\left(\frac{x-\mu }{\sigma }\right)}^3\right],$$

(4)

where $\mathbb{E}$—expectation operator.

Excess kurtosis is the metric of the height and sharpness of a PDF relative to that of a standard bell curve

$${\tilde{\mu }}_4=\mathbb{E}\left[{\left(\frac{x-\mu }{\sigma }\right)}^4\right]-3.$$

(5)

For WGN, both the skewness and excess kurtosis are equal to zero.

It is worth noting that if a signal exhibits a symmetric distribution with zero excess kurtosis, it by no means guarantees that its probability density function is normal. There are non-Gaussian distributions with ${\tilde{\mu }}_3=0$ and ${\tilde{\mu }}_4=0$ [44,45]. Therefore, at least a graphical analysis is necessary to confidently assert the normality of the distribution, or confirmation should be achieved through more sophisticated tests.

The envelope and phase of a signal can be found by using the Hilbert transform [46]. For a given function $x\left(t\right)$, the Hilbert transform is given as

$$\widehat{x}\left(t\right)=\mathbb{H}\left(x\left(t\right)\right)=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{x\left(t\right)}{t-\tau }d\tau .$$

(6)

Then, the envelope and phase of $x\left(t\right)$ are

$$a\left(t\right)=\sqrt{x^2\left(t\right)+{\widehat{x}}^2\left(t\right)},$$

(7)

$$\varphi \left(t\right)=arctan\frac{\widehat{x}\left(t\right)}{x\left(t\right)}.$$

(8)

If $x\left(t\right)$ is WGN, the envelope $a\left(t\right)$ has a Rayleigh PDF (2), and the phase $\varphi \left(t\right)$ has a uniform PDF (3).

The informational properties of signals can be assessed by their differential entropy. For a variable x with a probability density function $p\left(x\right)$, the differential entropy is given by

$$h=-{\int }_{-\infty }^{\infty }p\left(x\right)logp\left(x\right)dx.$$

(9)

The Gaussian distribution (1) attains the maximum differential entropy, which, for $\mu =0$ and variance ${\sigma }^2$, is equal to

$$h_g=\frac{1}{2}ln2\pi e{\sigma }^2.$$

(10)

For any other signal with the same variance ${\sigma }^2$, the differential entropy $h\left(x\right)$ does not exceed the value given in (10), i.e., $h\left(x\right)\le h_g\left(x\right)$.

Entropy power, a concept pioneered by Shannon [47], allows us to compare the probability density functions of a signal and Gaussian noise:

$$k_1=\frac{e^{2h}}{2\pi e},0\le k_1\le 1,$$

(11)

where h—entropy of signal.

The entropy power of a distribution coincides with the variance of a Gaussian distribution possessing an identical entropy.

To compare two arbitrary PDFs, we used the entropy power ratio, mathematically defined as follows for distributions $p_1\left(x\right)$ and $p_2\left(x\right)$ bearing entropies $h_1$ and $h_2$, respectively:

$$\frac{k_1\left(h_2\right)}{k_1\left(h_1\right)}=e^{2\Delta h},$$

(12)

where $\Delta h=h_1-h_2$—the difference between differential entropies of two distributions.

To restrict the value of ratio (12) into interval $0\le k_1\le 1$, the following expression can be applied:

$$k=e^{-2|\Delta h|}.$$

(13)

The entropy power (13) increases when the difference between entropies of distributions becomes smaller.

To evaluate the similarity of probability density functions with an amplitude and a phase with the reference distributions, we utilize the entropy power relative to these baseline distributions. Thus, the entropy power of the envelope of the signal is determined as:

$$k_2=e^{-2|h_e-h_R|},$$

(14)

where $h_e$—entropy of the envelope of signal; $h_R=1+ln\frac{\sigma }{\sqrt{2}}+\frac{\gamma }{2}$—entropy of a Rayleigh PDF (2) [44]; and $\gamma \approx 0.577$—Euler–Mascheroni constant.

The entropy power for evaluating the similarity of the phase distribution of the signal and a uniform one would be determined as:

$$k_3=e^{-2|h_{ph}-h_u|},$$

(15)

where $h_{ph}$—entropy of the phase of the signal; $h_u=ln2\pi$—entropy of a uniform PDF (3) [44].

Furthermore, the cross-entropy and Kullback–Leibler divergence can serve as additional metrics within the considered algorithm [48]. While not explicitly highlighted in the text, their analysis has been performed preliminarily, revealing clear results that they do not hold a decisive role in characteristic assessments for this type of study.

3. Properties of Sums of Chaotic Signals

The approach for generating a Gaussian-distributed chaotic signal is schematically illustrated in Figure 1. Selected chaotic outputs undergo normalization, followed by summation, to achieve a transformation from their original probability density function to a Gaussian PDF. The normalization step may be omitted depending on the properties of the original chaotic signals and the desired variance of the resulting sum.

There are lots of different nonlinear systems exhibiting chaotic behavior. We investigated more than 40 mostly three-dimensional chaotic systems through computing the PDFs, skewness, excess kurtosis, and entropy power of their output signals. Before estimation of the characteristic mentioned in the algorithm, each chaotic signal was centered around its zero mean and normalized to have unit variance. The results show that the majority of the studied chaotic systems exhibit characteristics that are far from being considered Gaussian-like signals.

The obtained results for a variety of chaotic systems appear in Appendix A, and a detailed description of the algorithm is analytically and experimentally demonstrated below with examples from two widely used and extensively studied systems: Chua and Lorenz. They were specifically chosen since their signals, among all examined, possess the most favorable parameters to serve as a platform for generating deterministic chaotic Gaussian signals.

3.1. Chua’s Circuit

Chua’s circuit is one of the most well-known systems that generate deterministic chaos and can be easily implemented as an electrical circuit [49]. Moreover, various types of nonlinearities can be utilized in this circuit [50,51].

The mathematical model of Chua’s circuit in dimensionless form is a three-differential equation system

$$\begin{array}{c}\dot{x}=\alpha (y-x-f\left(x\right))\\ \dot{y}=x-y+z\\ \dot{z}=-\beta y\end{array}$$

(16)

where x, y, z—output variables; $\alpha$, $\beta$—parameters of the model; and $f\left(x\right)=m_{2}x+0.5((m_0-m_1)\left(\right|x+a_1|-|x-a_1\left|\right)+(m_1-m_2)\left(\right|x+a_2|-|x-a_2\left|\right))$—nonlinear function, where $a_1=1$, $a_2=6.88$, $m_0=-1.238$, $m_1=-0.6665$, $m_2=500$. The connection between the circuit’s and the model’s parameters can be found in [49,52].

A key requirement to obtain independent chaotic systems is the absence of any connection between them. In analytical analysis, the independence of chaotic systems is achieved through different initial conditions of the systems; in experiments, it occurs due to the difference between the system parameters and the initial conditions, which appears naturally.

We studied Chua’s circuit with a piecewise linear characteristic in the nonlinear part, which provides various dynamical modes, including a single- or double-scroll chaotic attractor. As shown in Figure 2, the circuit consists of inductance, two capacitors, and nonlinearity and is implemented on two operational amplifiers [52].

When $\alpha =10$ and $\beta =14.6$, the circuit operates in double-scroll mode and all the outputs possess a symmetric PDF with skewness close to zero. The y output produces a low excess kurtosis ${\tilde{\mu }}_4=0.1421$, which also has the highest entropy power of $k_1=0.9039$, $k_2=0.9756$, and $k_3=0.998$. Nevertheless, there are distinct visual differences between the PDFs of chaotic and basic distributions, which can be seen in Figure 3a,e,i and numerically from

Table A1. Statistical properties of signals of Chua’s circuit.

Chaotic System	Output Variable	${\tilde{\mathit{\mu }}}_3$	${\tilde{\mathit{\mu }}}_4$	${\mathit{k}}_1$	${\mathit{k}}_2$	${\mathit{k}}_3$
Chua’s circuit [49]
$\dot{x}=10(y-x-f\left(x\right))$	x	−0.0116	−1.6609	0.3626	0.4754	0.8435
$\dot{y}=x-y+z$	y	−0.0028	−0.1421	0.9039	0.9756	0.9980
$\dot{z}=-14.6y$	z	0.0084	−1.1152	0.7541	0.8128	0.9056
where $f\left(x\right)=m_{2}x+0.5((m_0-m_1)\left(\right|x+a_1|-|x-a_1\left|\right)+(m_1-m_2)\left(\right|x+a_2|-|x-a_2\left|\right))$; $a_1=1$; $a_2=6.88$; $m_0=-1.238$; $m_1=-0.6665$; $m_2=500$.
Pairwise sum of two signals
$x_1+x_2$		0.0033	−0.8287	0.7040	0.9349	0.9696
$y_1+y_2$		0.0029	−0.0717	0.9961	0.9604	0.9989
$z_1+z_2$		−0.0007	−0.5581	0.9648	0.9976	0.9965
Pairwise sum of three signals
$x_1+x_2+x_3$		−0.0059	−0.5519	0.8741	0.8897	0.9873
$y_1+y_2+y_3$		0.0015	−0.0367	0.9996	0.9893	0.9986
$z_1+z_2+z_3$		0.0089	−0.3715	0.9901	0.9798	0.9987
Pairwise sum of four signals
$x_1+x_2+x_3+x_4$		−0.0090	−0.4088	0.9467	0.9783	0.9982
$y_1+y_2+y_3+y_4$		0.0009	−0.0034	0.9998	0.9908	0.9985
$z_1+z_2+z_3+z_4$		0.0105	−0.2700	0.9955	0.9907	0.9994
Experiment with Chua’s circuit
$y_1$	y	−0.0518	−0.4249	0.9437	0.9381	0.9950
$y_1+y_2$		0.0339	−0.3837	0.9897	0.9192	0.9986
$y_1+y_2+y_3$		0.0511	−0.2393	0.9962	0.9629	0.9986
$y_1+y_2+y_3+y_4$		0.0420	−0.1846	0.9979	0.9720	0.9985
Chua’s circuit [49]
$\dot{x}=9.273(y-x-f\left(x\right))$	x	−0.0763	−1.4807	0.5463	0.6532	0.9087
$\dot{y}=x-y+z$	y	−0.0070	−0.9334	0.8739	0.5006	0.9989
$\dot{z}=-16.8y-0.0042z$	z	0.0465	−0.8164	0.8676	0.8860	0.9808
where $f\left(x\right)=m_{2}x+0.5((m_0-m_1)\left(\right|x+a_1|-|x-a_1\left|\right)+(m_1-m_2)\left(\right|x+a_2|-|x-a_2\left|\right))$; $a_1=1$; $a_2=6.88$; $m_0=-1.238$; $m_1=-0.6665$; $m_2=500$.

.

The central limit theorem states that the summation of two or more chaotic signals yields a PDF of the sum that exhibits a greater resemblance to a Gaussian PDF compared to that of a single signal. The evolution of the PDFs when the number n of independent chaotic signals increases is shown in Figure 3. To minimize the number of chaotic signal in the total signal, it is efficient to use the variable y. When only three independent outputs of y are added, the PDF of the sum, along with its envelope and phase, closely resembles the basic PDFs, i.e., normal, Rayleigh, and uniform PDFs. The entropy powers of the sum $k_1,k_2,k_3$ exceed $0.99$.

The quality metrics of the sum of signals are contingent upon the number of signals incorporated, as illustrated in Figure 4. The excess kurtosis decreases to zero, and all entropy powers converge to 1 with an increasing value of n. This suggests that achieving chaos resembling Gaussian noise necessitates the addition of a minimum of three original chaotic signals from Chua’s circuit.

A change in circuit parameters with preservation of the chaotic double-scroll mode causes significant changes in the quality metrics, particularly the excess kurtosis and entropy power. For example, when $\alpha =9.273$ and $\beta =16.8$, Chua’s circuit has the following metrics, which are quite different from ones mentioned above: $E=-0.9334$, $k_1=0.8739$, $k_2=0.5006$ and $k_3=0.999$. This result suggests that the PDFs of signals are sensitive to the parameters of the chaotic system.

3.2. Lorenz System

While the previously considered Chua’s circuit is one of the simplest both as an analytical model and an experimental implementation, the Lorenz system is a well-known and extensively investigated source of deterministic chaos. The system was originally developed to model atmospheric convection in meteorology and allows for the description of different atmospheric phenomena [53]. The Lorenz system is usually applicable in its mathematical representation (a system of three differential equations with nonlinearity (17)) and has found widespread use in diverse fields such as physics, engineering, biology, economics, and even art. However, its hardware implementation is inherently more complex than Chua’s circuit due to the involvement of analog multipliers (Figure 5) [54].

$$\begin{array}{c}\dot{x}=\sigma (y-x)\hfill \\ \dot{y}=(r-z)x-y\hfill \\ \dot{z}=xy-bz\hfill \end{array}$$

(17)

where x, y, z—system variables; $\sigma =10$, $r=28$, and $b=\frac{8}{3}$—parameters.

We simulated system (17) with two different values of the parameter r, $r=28$ and $r=60$, keeping other parameters constant. The statistic parameters of the Lorenz system are shown in

Table A2. Statistical properties of signals of Lorenz systems.

Chaotic System	Output Variable	${\tilde{\mathit{\mu }}}_3$	${\tilde{\mathit{\mu }}}_4$	${\mathit{k}}_1$	${\mathit{k}}_2$	${\mathit{k}}_3$
Lorenz [53]
$\dot{x}=10(y-x)$	x	0.0003	−0.7093	0.8989	0.8535	0.9830
$\dot{y}=(28-z)x-y$	y	0.0005	−0.1573	0.9013	0.9368	0.8877
$\dot{z}=xy-\frac{8}{3}$	z	0.2023	−0.8499	0.8974	0.5618	0.9731
Pairwise sum of two signals
$x_1+x_2$		−0.0102	−0.3529	0.9893	0.9587	0.9995
$y_1+y_2$		−0.0103	−0.0801	0.9938	0.9580	0.9865
$z_1+z_2$		0.0014	−0.4291	0.9852	0.9921	0.9983
Pairwise sum of three signals
$x_1+x_2+x_3$		−0.0053	−0.2393	0.9966	0.9677	0.9989
$y_1+y_2+y_3$		−0.0059	−0.0570	0.9993	0.9857	0.9978
$z_1+z_2+z_3$		0.0397	−0.2867	0.9946	0.9631	0.9994
Pairwise sum of four signals
$x_1+x_2+x_3+x_4$		−0.0031	−0.1840	0.9981	0.9759	0.9987
$y_1+y_2+y_3+y_4$		−0.0040	−0.0468	0.9997	0.9937	0.9999
$x_1+x_2+x_3+x_4$		0.0002	−0.2122	0.9975	0.9852	0.9987
Experiment with Lorenz system
$x_1$	x	0.0131	−0.5906	0.9292	0.8454	0.9606
$x_1+x_2$		0.0150	−0.1978	0.9937	0.9995	0.9976
$x_1+x_2+x_3$		0.0204	−0.0523	0.9989	0.9996	0.9993
$x_1+x_2+x_3+x_4$		0.0246	0.0499	0.9995	0.9886	0.9996
Lorenz [53]
$\dot{x}=10(y-x)$	x	−0.0034	−0.9328	0.8961	0.7523	0.9649
$\dot{y}=(60-z)x-y$	y	−0.0038	−0.3519	0.9801	0.9247	0.8355
$\dot{z}=xy-\frac{8}{3}z$	z	0.0821	−0.4826	0.9736	0.7515	0.9786

. Two of the three outputs in the Lorenz system, namely x and y, exhibit symmetric PDFs. Similarly to Chua’s circuit, the results indicate a significant impact of the model parameters on the appearance and characteristics of the PDFs of the output signals. For the investigated parameter values, the signal most resembling Gaussian noise is y, with $\sigma =10$, $r=28$, and $b=\frac{8}{3}$, featuring coefficients ${\tilde{\mu }}_4=-0.1573$, $k_1=0.9013$, $k_2=0.9368$, and $k_3=0.8877$.

The original histograms of normalized output signals from the Lorenz system are shown in Figure 6. Despite the asymmetry in the output signal z, combining all three outputs can yield a normal PDF when summing only three–four signals. The relationship between the excess kurtosis coefficient and entropy powers for different values of n is presented in Figure 7 and confirms this conclusion.

The entropy power values ($k_1,k_2,k_3$) for the sum of three and four y variables exceed 0.98 and 0.99, respectively. Furthermore, a comparison of the PDFs in Figure when $n=4$ suggests there are no significant discrepancies between the PDFs of the sum and the basic PDFs.

Analyses of Chua’s circuit and the Lorenz system reveal that the summation of chaotic non-Gaussian signals proves to be an effective method for obtaining deterministic chaotic Gaussian signals.

4. Experimental Verification

4.1. Experiment with Chua’s Circuit

The following results were obtained for the Chua circuit with a piecewise linear–nonlinear element implemented using TL082CP operational amplifiers, as depicted in Figure 2. Four circuits were assembled with slightly different parameters. The parameter difference was determined by the used components’ features—for the used capacitors, $C1=100$ µF and $C2=10$ µF; for the inductor, $L=18$ mH; and they both have a tolerance of $\pm 5\%$. The parameters of the nonlinear element were as $R2,R3=220$ Ohm, $R4=2.2$ kOhm, $R5,R6=22$ kOhm, and $R7=3.3$ kOhm.

Additionally, a potentiometer was utilized to provide a resistance $R1=1580$ Ohm, the value of which could be adjusted in increments of $\pm 100$ Ohm to change the chaotic circuit behavior. The voltages across the capacitors $C_2$, corresponding to the variable y in the model of the circuit, were recorded and processed for further analysis. A set of signal records (50 samples of voltages across the capacitors $C_2$) was obtained, where each record contained $2\times {10}^6$ sweep points.

The experimental results confirmed the theoretical assumptions and analytical outcomes. As depicted in Figure 8, increasing the number of chaotic systems rapidly drives the distribution of their summed signals towards a normal distribution. This observation extends to the envelope and phase of the signal, where Rayleigh and uniform distributions were observed, respectively.

4.2. Experiment with a Lorenz System

An experimental investigation was performed using a simplified electronic realization of a Lorenz oscillator, constructed in accordance with the specifications outlined in [54,55] and the circuit in Figure 5. We used an AD633 analog multiplier and a TL084CN operational amplifier. The other circuit elements were as follows: $R1,R2=100$ kOhm, $R3,R6=10$ kOhm, and $R7=1$ MOhm, with a tolerance of 5%, and $R4=370$ Ohm, with a tolerance of 0.5%. The variable resistor $R5$ allows for tuning of the parameter r and was set at $R5=35.72$ kOhm.

Four circuits were assembled on a breadboard, and the appropriate oscillograms were recorded. Subsequently, they were subjected to the suggested set of tests. The probability density functions of the normalized output x from the electronic circuit of the Lorenz system are illustrated in Figure 9. A comparison between Figure 9 and Figure 6 reveals a good agreement between the analytic and experimental results.

5. Discussion and Conclusions

The results underscore a promising approach for generating deterministic Gaussian-like signals using an optimal number of chaotic systems. The PDFs of the original chaotic signals exhibit essential deviations from a normal distribution, indicating their inherently lower information capacity compared to fully random Gaussian noise. This observation prompts the need to establish criteria for selecting chaotic systems to achieve Gaussian signal formation through the summation of chaotic signals. This requires an optimization process to minimize the number of independent simple chaotic signals. The performed analysis demonstrates that adherence to the following guidelines significantly enhances the likelihood of a successful chaotic system and signal selection:

i.

Prioritize signals possessing symmetric probability density functions.

ii.

Minimize the excess kurtosis of the selected signals, ideally aiming for ${\tilde{\mu }}_4<1$.

iii.

Ensure that all three entropy powers ($k_1,k_2,k_3$) of the original chaotic signal surpass a value of $0.85$.

While strict adherence to these guidelines does not ensure Gaussian signal achievement, it substantially increases the probability of generating a Gaussian chaotic signal with the desired entropy coefficients $k_1,k_2,k_3>0.95$. Although, theoretically, there are no limitations to the number of chaotic signals that can be summed to achieve a Gaussian distribution, practical implementation has restrictions due to the lack and value of resources and the size limitation of devices. The lowest limit is defined by using the algorithm considered in this paper to merely sum three or four independent chaotic signals.

It is crucial to underscore that chaotic signals are generated by deterministic systems, rendering their summation equally deterministic. This underscores the capability of simple summation to yield intricate, noise-mimicking chaotic signals that retain their deterministic nature. The most innovative aspect of this method is the possibility of attributing the output voltage and the current of random chaotic circuits of a well-known signal as Gaussian noise, the properties of which are well known and exploitable in many applications. The possibility of obtaining a Gaussian-like signal description presents the opportunity to design and model chaotic circuits’ output voltages, preserving all the general properties of the Gaussian distribution. Moreover, since many chaotic circuits do not have output signals linked to Gaussian noise, this method allows for widening the field of investigation, even to those circuits that apparently do not have these properties. It is also considerable that more than 40 chaotic systems have been evaluated, leading to the confirmation of this theory and confirming the validity of this method.

The output signals obtained via this process hold considerable value across a variety of applications, including generating deterministic Gaussian noise, electronic warfare systems, coherent and/or covert communication systems, radars, statistical modeling, machine learning, cryptography, and more.

The possibility of obtaining a pseudo-random number generator through the simple sum of chaotic signals predisposes this method to a wide range of applications, bypassing any possible complications of chaotic signal processing. All the possible applications mentioned can support both hardware and software implementation of chaotic circuits, with the addition in this case of greater properties and guidelines that can be exploited during the implementation guaranteed by the traceability to Gaussian noise. Since the main aim is to present more known laws of circuits that in certain conditions are dominated by chaos, other aspects and elaborations of the output signals of the proposed chaotic circuits are also being investigated, which could lead back to the properties of pseudo-Gaussian noise. Future investigations will aim at trying to reconstruct more well-known properties of apparently chaotic signals, trying to make it easier to implement the chosen application.

The results of the study can find applications in the enhancement of security in communication systems. These include:

i.

Covert communication—there are approaches based on thermal noise or AI noise; therefore, our recommendation is to use artificial Gaussian-distributed chaotic signals for hidden communication;

ii.

Radio countermeasure purposes—the deterministic nature of chaotic systems can be used to reproduce and compensate for the influence of chaos in “friendly devices” and remains incomprehensible for “enemies”. This means that the same signal can be at least neutral for one device and harmful to others;

iii.

PRNG for cryptography purposes—a huge number of scientific studies are concerned with this question. By increasing the number of chaotic signals, we increase the keyspace of the encrypted information.

The Gaussian noise model serves as a foundational and widely applicable framework for analyzing signal randomness. However, it is important to recognize that while Gaussian noise is a common and versatile model, quite a few diverse noise models exist that can be useful. Examples include pink noise, brown noise, and various others, which are rather a special case of the Gaussian noise model, but require additional consideration to use them as a reference for the development of a random signal testing algorithm.