Research on the Strategy for the Flexible Configuration of Chaotic Signal Probability Distribution and Its Application

Abstract

Given the constraints on the invariant distribution in chaotic systems, flexibly setting the probability distribution of chaotic signals poses a significant challenge. To tackle this issue, this paper proposes a strategy that transforms the task into solving and modifying the probability density function of the chaotic intrinsic signal. Initially, kernel density estimation algorithms are employed to address the issue of obtaining smooth probability density functions for high-dimensional chaotic signals. Any chaotic signal can serve as the intrinsic signal source, with its probability density function and distribution function being solvable using this algorithm. Subsequently, a graph-based transformation algorithm is introduced for the flexible adjustment of chaotic signal probability distribution. This algorithm can convert the intrinsic signal into a chaotic signal with the desired distribution type based on the characteristics of the target distribution, providing an analytical expression for the transformation relationship. Finally, the effectiveness of this strategy is validated by generating uniform distribution chaotic signals using a Chua chaotic signal as the intrinsic source. The outstanding performance of this signal in suppressing common-mode conducted electromagnetic interference in high-frequency converters is highlighted. The experimental results demonstrate this strategy’s ability to flexibly configure probability distribution types of chaotic signals. Additionally, chaotic signals with a uniform distribution can achieve uniform power spectrum shaping, with a suppression effect on maximum common-mode conducted electromagnetic interference reaching 16.56 dB.

1. Introduction

Random signal generators enable the flexible setting of the probability distribution of random signals, greatly promoting the widespread application of random signals across various industries [1,2,3,4,5,6,7,8,9,10,11,12,13]. Chaotic signals exhibit quasi-randomness, are relatively simple to implement, and have lower costs. In many applications, they can serve as alternatives to random signals, thus attracting extensive attention [14,15,16,17,18,19]. However, due to the constraints imposed by the invariant distribution of chaotic systems [16,20], it is not easy to flexibly set the probability distribution of chaotic signals, which to some extent limits the application scope of chaotic signals.

Currently, a substantial portion of the literature on the probability distribution of chaotic signals is predominantly dedicated to achieving a uniform distribution of such signals [21,22,23,24,25,26,27,28,29,30,31].

One method involves seeking chaotic systems with invariant distributions (probability density functions) that demonstrate uniformity, aiming to generate chaotic signals with uniform distributions [22,23,24,25,26]. Research reveals that the logistic map exhibits a uniform invariant distribution when the parameter is set to 4 [22,23,24], as does the Chebyshev map [25]. Similarly, both the tent map and the n-way Bernoulli shift system display uniform distributions in their invariant distributions [26]. Nonetheless, the range of chaotic systems with uniformly distributed invariant distributions is quite limited, especially in higher-dimensional chaotic systems with greater dynamical complexity, where their invariant distributions tend to be non-uniform. Consequently, the array of chaotic signals with uniform distributions obtainable through this approach is quite restricted.

To diversify the range of chaotic signals exhibiting uniform distributions, some researchers have advocated for employing composite systems [27,28]. The research outlined in [27] reveals that one-dimensional composite mappings can generate chaotic signals with uniform distributions, exemplified by the logistic-tent system, logistic-sine system, and tent-sine system. Likewise, [28] indicates that utilizing a three-dimensional logistic-sine cascading mapping can produce even more uniformly distributed chaotic signals. Nevertheless, these composite systems still require the inclusion of chaotic systems with limit distributions that demonstrate uniformity. Hence, the assortment of chaotic signals with uniform distributions derived from this approach remains relatively constrained.

To broaden the spectrum of uniformly distributed chaotic signals, some researchers have proposed an alternative approach involving the manipulation of non-uniformly distributed chaotic systems to obtain such signals [29,30,31]. Reference [29] successfully obtained a type of uniformly distributed chaotic signal by accumulating folded chaotic signals. However, the oscillation frequency of the resulting chaotic signal is closely tied to the sampling frequency and the size of the folding domain, necessitating iterative adjustments of these parameters to achieve a more desirable uniform distribution. Conversely, reference [30] employed a method of output distortion and conjugate mapping to synthesize uniformly distributed chaotic signals using multiple chaotic systems. Yet, this method is operationally complex, and implementing inverse mapping without obtaining the analytical distribution function is challenging. In reference [31], a method of extracting the rightmost bit using the fixed-point arithmetic mode yielded a type of uniformly distributed chaotic signal. Nevertheless, the diminishing usage of fixed-point arithmetic due to its low precision is gradually supplanted by floating-point arithmetic. In summary, although they are effective in obtaining uniformly distributed chaotic signals, these methods are incapable of generating chaotic signals with other probability distribution types.

In the literature, several studies explore Gaussian distributed chaotic signals. Reference [32] suggests that within certain parameter ranges, the limit probability distribution of the logistic mapping resembles a Gaussian distribution. Reference [33] utilizes multi-value quantization to generate Gaussian distributed chaotic signals in binary-phase chaotic spread spectrum sequences. Reference [34], on the other hand, approximates the normal distribution using chaotic expressions with the Weierstrass function, resulting in an approximate Gaussian distributed chaotic signal. Reference [35] employs a conjugate mapping transformation to derive a Gaussian distributed chaotic signal from the tent map. Moreover, the research outlined in reference [36] demonstrates two methods: first, based on the central limit theorem, synthesizing Gaussian distributed chaotic signals within specific ranges using chaotic sequences; and second, utilizing the Von Neumann method to select samples from a uniformly distributed chaotic sequence to fit within a specific Gaussian distribution range. Although there has been some progress in the research on the probability density function of chaotic signals, there is still a lack of reports on how to flexibly configure its probability distribution type. Therefore, further research is needed on both the methods and applications of flexibly configuring the probability distribution types of chaotic signals.

Hence, this paper introduces a graphical language-based strategy enabling the flexible adjustment of probability distribution types for chaotic signals. It converts the task of setting these distribution types into solving and transforming the probability distribution of intrinsic chaotic signals, effectively resolving the challenge of flexible distribution setting for chaotic signals.

In Section 2, we delve into an innovative approach that allows for flexible configuration of the probability distribution types of chaotic signals. Specifically, in Section 2.1, we demonstrate how the kernel density estimation algorithm is applied to overcome the challenge of obtaining smooth probability density functions of high-dimensional chaotic signals. Subsequently, in Section 2.2, we introduce a novel strategy based on graphical language that achieves highly flexible configuration of the probability distribution types of chaotic signals. Furthermore, Section 2.3 showcases, through an example, how our strategy can realize a uniform distribution of chaotic signals. Moving into Section 3, we explore the application of this strategy in suppressing common-mode conducted electromagnetic interference (EMI) in high-frequency power converters. The outstanding performance of this strategy not only validates its effectiveness but also highlights its broad prospects in practical applications.

2. Strategy for Flexible Setting Probability Distribution of Chaotic Signals Based on Graphical Language

Currently, the most common method for solving the probability density function of high-dimensional chaotic signals is the histogram method. However, the shape of the density function obtained using the histogram method is closely related to the number of bins, resulting in a stepped appearance lacking smoothness. This paper employs the kernel density estimation algorithm to address the challenge of obtaining smooth probability density functions for high-dimensional chaotic signals.

2.1. Kernel Density Estimation for Solving the Probability Density Function of Chaotic Signals

Here, we use a sample of size $N$ independently and identically distributed from a population with probability density function $f_x\left(x\right)$ and distribution function $F_x\left(x\right)$. The sample values are $x_1$, $x_2$, …, $x_i$, …, $x_N$, where $i=1,2,\cdots ,N$. Then, the estimation of the probability density function $f_x\left(x\right)$ is

$${\widehat{f}}_x\left(x\right)=\frac{1}{Nh}{\displaystyle \sum _{i=1}^{N}K\left(\frac{x-x_i}{h}\right)}$$

(1)

where $K(\cdot )$ is the kernel function and $h$ is the bandwidth.

The distribution function $F_x\left(x\right)$ is

$$F_x\left(x\right)={\displaystyle \underset{a}{\overset{x}{\int }}{f}_x\left(\tau \right)d\tau },x\in \left(a,b\right)$$

(2)

where the range of values for $x$ is between $\left(a,b\right)$.

The kernel function $K(\cdot )$ must satisfy the normalization condition:

$$\{\begin{array}{l}K\left(x\right)\ge 0,{\displaystyle {\int }_{-\infty }^{+\infty }K\left(x\right)dx=1}\\ \sup K\left(x\right)<+\infty ,{\displaystyle {\int }_{-\infty }^{+\infty }{K}^2\left(x\right)dx<+\infty }\\ \underset{x\to \infty }{\lim }K\left(x\right)·x=0\end{array}$$

(3)

Common types of kernel functions include the Gaussian kernel function, uniform kernel function, triangular kernel function, and Epanechnikov kernel function, etc. The kernel density estimation results are shown in Figure 1.

Figure 1 shows that the probability density functions obtained by four different kernel functions overlap significantly and are almost indistinguishable, indicating that the kernel density estimation algorithm is not sensitive to the choice of kernel functions. Therefore, the Gaussian kernel function is used in the examples of this study.

The smoothness of kernel density estimation is closely related to the bandwidth $h$. Two common methods for choosing the bandwidth $h$ are as follows:

$$\{\begin{array}{l}{h}_1=1.059s{N}^{-0.2}\\ h_2=0.785\left(q_{0.75}-q_{0.25}\right)N^{-0.2}\end{array}$$

(4)

where $s$ is the sample standard deviation and $q_{0.75}$ and $q_{0.25}$ are the 0.75 quantile and 0.25 quantile, respectively. The kernel density estimates for the two commonly used bandwidths are shown in Figure 2.

Figure 2 shows that for irregular distributions with multiple peaks in the probability density function, the approximation degree of the bandwidth $h$ shown in Equation (4) is not ideal.

After balancing between smoothness and error, this paper takes one-thousandth of the range width of the chaotic signal $x$ as the value of the bandwidth $h$, as shown in Figure 3.

Figure 3 shows that the proposed method can provide a good approximation for probability density functions with multiple peaks. In the details, it is shown that the probability density function obtained by the kernel density estimation algorithm is smoother than that obtained by the histogram method.

This paper selects the second component of the Chua’s chaotic system (see Appendix A) as the intrinsic signal $x$, whose probability density function and distribution function are shown in Figure 4.

2.2. Algorithm for Probability Distribution Variation Based on Graphical Language

To tackle the challenge of flexibly setting probability distribution types for chaotic signals, this section introduces a graph-based algorithm for probability distribution transformation. Utilizing this algorithm, chaotic intrinsic signals can seamlessly undergo conversion into chaotic signals exhibiting the desired probability distribution type. Thus, the objective of the flexible adjustment of probability distributions for chaotic signals is effectively attained.

Let the probability density function of the intrinsic chaotic signal $x$ defined on $\left(a,b\right)$ be $f_x\left(x\right)$, and its distribution function be $F_x\left(x\right)$; let $y$ be the chaotic signal with the expected probability distribution defined on $\left[c,d\right]$, with the probability density function $f_y\left(y\right)$ and distribution function $F_y\left(y\right)$.

The graphical transformation from the intrinsic chaotic signal $x$ to the target distribution chaotic signal $y$ is illustrated in Figure 5.

According to Figure 5, the specific steps of the graphical-based chaotic signal probability distribution transformation algorithm are described as follows:

Step 1: Choose an intrinsic chaotic signal $x$ as the signal source, and let its domain be (a, b). Extract a sample of $N$ independent identically distributed values, denoted as $x_1$, …, $x_N$.

Step 2: Use the kernel density estimation algorithm as shown in Equation (1) to calculate its probability density function $f_x\left(x\right)$, and then calculate the distribution function $F_x\left(x\right)$.

Step 3: Based on the probability distribution type of the expected probability distribution chaotic signal $y$ (defined on the interval $\left[c,d\right]$), calculate its probability density function $f_y\left(y\right)$ and distribution function $F_y\left(y\right)$.

Step 4: Select M nodes in the interval $\left(a,b\right)$: $x_1=a$, …, $x_i$, …, $x_M=b$. Using the graphical method shown in Figure 5 and following the principle of equal distribution function values, calculate the corresponding $M$ nodes on the interval $\left[c,d\right]$ as $y_1=c$, …, $y_i$, …, $y_M=d$.

Step 5: By using piecewise linear interpolation, the analytical expression of the transformation relationship $y=g\left(x\right)$ can be obtained

$$g\left(x\right)={\displaystyle \sum _{j=1}^M{y}_j{l}_j\left(x\right)}$$

(5)

where the basis function $l_j\left(x\right)$ is

$$l_j\left(x\right)=\{\begin{array}{ccc}\frac{x-x_{j-1}}{x_j-x_{j-1}}& ,& x_{j-1}\le x\le x_j\left(j\ne 1\right)\\ \frac{x-x_{j+1}}{x_{j+1}-x_j}& ,& x_j\le x\le x_{j+1}\left(j\ne M\right)\\ 0& ,& x\in \left(a,b\right), but x\notin \left[x_{j-1},x_{j+1}\right]\end{array}$$

(6)

Step 6: By using the transformation relationship $y=g\left(x\right)$, the intrinsic signal $x$ can be transformed into chaotic signals $y$ with the desired probability distribution.

In Appendix B, a set of parameter values for the analytical expression of the transformation relationship of transforming the chaotic signal of Chua into a uniform distribution chaotic signal composed of 100 nodes is provided (as shown in

Table A1. Values of parameter $x_i$.

${\mathit{x}}_{\mathit{i}}$	1	2	3	4	5	6	7	8	9	10
$x_1~x_{10}$	−1.0345	−1.0136	−0.9927	−0.9719	−0.9510	−0.9301	−0.9092	−0.8883	−0.8674	−0.8465
$x_{11}~x_{20}$	−0.8257	−0.8048	−0.7839	−0.7630	−0.7421	−0.7212	−0.7003	−0.6794	−0.6586	−0.6377
$x_{31}~x_{40}$	−0.6168	−0.5959	−0.5750	−0.5541	−0.5332	−0.5124	−0.4915	−0.4706	−0.4497	−0.4288
$x_{41}~x_{50}$	−0.4079	−0.3870	−0.3662	−0.3453	−0.3244	−0.3035	−0.2826	−0.2617	−0.2408	−0.2199
$x_{51}~x_{60}$	−0.1991	−0.1782	−0.1573	−0.1364	−0.1155	−0.0946	−0.0737	−0.0529	−0.0320	−0.0111
$x_{61}~x_{70}$	0.0098	0.0307	0.0516	0.0725	0.0933	0.1142	0.1351	0.1560	0.1769	0.1978
$x_{71}~x_{80}$	0.2187	0.2395	0.2604	0.2813	0.3022	0.3231	0.3440	0.3649	0.3858	0.4066
$x_{81}~x_{90}$	0.4275	0.4484	0.4693	0.4902	0.5111	0.5320	0.5528	0.5737	0.5946	0.6155
$x_{91}~x_{100}$	0.6364	0.6573	0.6782	0.6990	0.7199	0.7408	0.7617	0.7826	0.8035	0.8244

,

Table A2. Values of parameter $y_i$.

${\mathit{y}}_{\mathit{i}}$	1	2	3	4	5	6	7	8	9	10
$y_1~y_{10}$	−10.0000	−9.9856	−9.9511	−9.9132	−9.8738	−9.8318	−9.7866	−9.7400	−9.6901	−9.6374
$y_{11}~y_{20}$	−9.5824	−9.5265	−9.4600	−9.3550	−9.2434	−9.1292	−9.0064	−8.8603	−8.7031	−8.5355
$y_{21}~y_{30}$	−8.3647	−8.1874	−8.0048	−7.8156	−7.6175	−7.4130	−7.2036	−6.9892	−6.7697	−6.5397
$y_{31}~y_{40}$	−6.2994	−6.0524	−5.8014	−5.5429	−5.2698	−4.9833	−4.6915	−4.3963	−4.0937	−3.7777
$y_{41}~y_{50}$	−3.4583	−3.1235	−2.7815	−2.4320	−2.0662	−1.6889	−1.3285	−0.9667	−0.5894	−0.2003
$y_{51}~y_{60}$	0.1738	0.5621	0.9396	1.3026	1.6635	2.0382	2.4027	2.7535	3.0977	3.4315
$y_{61}~y_{70}$	3.7520	4.0748	4.3835	4.6801	4.9723	5.2522	5.5229	5.7822	6.0373	6.2875
$y_{71}~y_{80}$	6.5292	6.7612	6.9822	7.1960	7.4033	7.6069	7.8041	7.9925	8.1758	8.3511
$y_{81}~y_{90}$	8.5220	8.6891	8.8495	8.9962	9.1223	9.2421	9.3537	9.4588	9.5288	9.5848
$y_{91}~y_{100}$	9.6394	9.6912	9.7394	9.7877	9.8308	9.8709	9.9096	9.9478	9.9838	10.0000

and

Table A3. Values of parameter $k_i$.

${\mathit{k}}_{\mathit{i}}$	1	2	3	4	5	6	7	8	9	10
$k_1~k_{10}$	0.0031	0.0076	0.0086	0.0091	0.0099	0.0109	0.0115	0.0126	0.0136	0.0146
$k_{11}~k_{20}$	0.0152	0.0185	0.0300	0.0327	0.0344	0.0381	0.0467	0.0518	0.0569	0.0599
$k_{21}~k_{30}$	0.0643	0.0685	0.0735	0.0799	0.0856	0.0913	0.0974	0.1041	0.1141	0.1250
$k_{31}~k_{40}$	0.1349	0.1445	0.1572	0.1760	0.1964	0.2138	0.2319	0.2565	0.2908	0.3214
$k_{41}~k_{50}$	0.3716	0.4233	0.4888	0.5881	0.7130	0.8260	1.0538	1.5070	2.4712	5.7961
$k_{51}~k_{60}$	−13.6799	−3.1122	−1.6946	−1.1749	−0.9366	−0.7394	−0.5988	−0.5069	−0.4325	−0.3707
$k_{61}~k_{70}$	−0.3371	−0.2939	−0.2594	−0.2363	−0.2106	−0.1904	−0.1711	−0.1586	−0.1471	−0.1348
$k_{71}~k_{80}$	−0.1230	−0.1116	−0.1031	−0.0957	−0.0901	−0.0839	−0.0771	−0.0722	−0.0667	−0.0627
$k_{81}~k_{90}$	−0.0593	−0.0551	−0.0489	−0.0407	−0.0376	−0.0340	−0.0312	−0.0202	−0.0157	−0.0149
$k_{91}~k_{100}$	−0.0138	−0.0126	−0.0123	−0.0107	−0.0097	−0.0092	−0.0089	−0.0082	−0.0036	−0.0036

).

In summary, the flexible setting strategy for the probability distribution of chaotic signals based on graphical language can be summarized as follows: firstly, using the principle that the distribution function values of $x$ and $y$ are equal to obtain corresponding pairs of points $\left(x_i,y_i\right)$; then, using the piecewise linear interpolation method to obtain the transformation relationship $y=g\left(x\right)$; and finally, using the transformation relationship to transform the intrinsic signal $x$ into the target chaotic signal $y$.

2.3. Implementation of Uniformly Distributed Chaotic Signals

Uniform distribution chaotic signals have broad applications in many research fields, such as sampling survey research, simulation modeling, information encryption, image processing, and so on.

This section takes the implementation of a uniform distribution chaotic signal as an example to verify the effectiveness of the proposed strategy. Given the boundedness of the chaotic signal $x$, this article sets the domain of the target chaotic signal $y$ to a symmetric interval $\left[c,d\right]=\left[-10,10\right]$. The probability density function of an ideal uniform distribution is represented as

$$f_y\left(y\right)=\frac{1}{20},y\in \left[-10,10\right]$$

(7)

The probability density function and distribution function of the ideal uniform distribution are shown in Figure 6.

The transformation relationship from the intrinsic signal $x$ to the uniform distribution chaotic signal $y$ is shown in Figure 7.

The waveform, probability density function, and distribution function of the chaotic signal $y$ are shown in Figure 8.

By comparing Figure 4a with Figure 8a, it can be seen that the distribution of $y$ is more uniform than the distribution of $x$. By comparing Figure 6 with Figure 8b, it can be seen that the probability density function of chaotic signal $y$ follows a uniform distribution.

Therefore, the proposed strategy can transform the intrinsic signal $x$ into a uniform distribution chaotic signal $y$.

The autocorrelation function and power spectral density of chaotic signals $x$ and $y$ are shown in Figure 9a,b,c,d, respectively.

Figure 9a,c show that the autocorrelation functions of the chaotic signals before and after the distribution transformation are very small, and the difference is not significant. Figure 9b,d show that the power spectral density before and after the distribution change is a continuous distribution, with no obvious discrete spectral lines. Therefore, the proposed strategy does not change the “quasi-random” characteristic of the chaotic signal, and the resulting chaotic signal y remains a chaotic signal.

In conclusion, the proposed strategy can flexibly adjust the probability distribution of chaotic signals based on the characteristics of the target distribution of chaotic signals.

3. Results and Discussion

The extensive adoption of silicon carbide (SiC) devices and other wide-bandgap devices has led to a continual rise in the switching frequency of power converters. This trend towards higher frequencies exacerbates EMI challenges encountered by conventional PWM power converters. Research suggests that employing uniformly distributed chaotic signals for frequency modulation of conventional PWM converters can effectively enhance the electromagnetic compatibility performance of high-frequency converters [21,37,38].

This section emphasizes the outstanding performance achieved by employing uniformly distributed chaotic signals to suppress common-mode conducted EMI in an active clamp flyback converter, thus underscoring the strategy’s significant potential application value.

The utilization of Chua’s chaotic signals for EMI suppression is highly favored [39]. This section conducts comparative experiments using one component of Chua’s chaotic signals as the control group and analyzes the mechanism of employing uniformly distributed chaotic signals to suppress EMI.

3.1. Construction of Experimental Prototype and Implementation Scheme of Chaotic Spread Spectrum Modulation PWM

The flyback converter is a widely used topology. Active clamp technology can enhance its performance. This section uses Texas Instruments’ (TI) active clamp voltage mode PWM controller LM5025A (Texas Instruments, Dallas, TX, USA) as an example to illustrate the implementation of chaotic spread spectrum PWM control technology in an analog control system. Based on the schematic diagram shown in Figure 10, an active clamp flyback converter based on LM5025A was constructed, and the prototype is shown in Figure 11. The main parameters of the prototype are shown in

Table 1. The main parameters of the prototype.

Parameter	Specification or Value
Controller	LM5025A
Optocoupler	TLP250H
Precision Programmable Reference IC	TL431C
$\mathrm{MOSFET}{S}_1$	CS9N90
$\mathrm{MOSFET}{S}_2$	SMF5N60
$\mathrm{Diode}{D}_{\mathrm{o}}$	AIDW10S65C5
$\mathrm{Input}\mathrm{voltage}{u}_{\mathrm{in}}$	180 V~235 V alternating current
$\mathrm{Output}\mathrm{voltage}{U}_{\mathrm{o}}$	15 V direct current
$\mathrm{Rated}\mathrm{power}{P}_{\mathrm{o}}$	45 W
transformer magnetic core	PQ3535
$\mathrm{transformer}\mathrm{excitation}\mathrm{inductance}{L}_{\mathrm{m}}$	509.02 μH
$\mathrm{primary}\mathrm{coil}\mathrm{leakage}\mathrm{inductance}{L}_{\mathrm{s}}$	9.74 μH
$\mathrm{resonant}\mathrm{capacitor}{C}_{\mathrm{r}}$	0.1 μF
voltage sensor	LV25-P
current sensor	FC-SCT4.6-1:100
$\mathrm{output}\mathrm{capacitor}{C}_{\mathrm{o}}$	100 μF

.

The controller LM5025A has two methods for implementing spread spectrum modulation, as shown in Figure 12. Among them, the digital spread spectrum implementation has better anti-interference capability, and this implementation scheme is chosen in this paper.

Using a uniform distribution chaotic signal y to modulate the frequency of the test prototype based on LM5025A, the sync pulse and OUTA waveform are shown in Figure 13, in which, the spreading range is [80 kHz, 120 kHz], Channel 2 is the chaotic spread spectrum modulated synchronization pulse, and Channel 1 is the output waveform of OUTA under the fixed conduction time mode.

3.2. Uniformly Distributed Chaotic Signal for Uniform Shaping of Conventional PWM Power Spectrum

Using Chua’s chaotic signal $x$ and the obtained uniform distribution chaotic signal $y$ to perform chaotic spread spectrum modulation on conventional PWM, the measured power spectra are shown in Figure 14. The maximum value is marked by ‘×’.

Figure 14a shows that the power spectrum of conventional PWM is a linear spectrum with a large peak; the power spectra of Chua’s chaotic spread spectrum modulated PWM and uniformly distributed chaotic spread spectrum modulated PWM are both continuous spectra, with the peak being significantly reduced.

Figure 14b shows that Chua’s chaotic spread spectrum modulated PWM power spectrum is non-uniformly distributed, while the uniformly distributed chaotic spread spectrum modulated PWM power spectrum is uniformly distributed, with a smaller peak.

Therefore, the uniformly distributed chaotic signal can uniformly shape the power spectrum of conventional PWM, obtaining a uniformly distributed power spectrum, thus maximally reducing the peak of the power spectrum and effectively suppressing EMI.

3.3. Uniformly Distributed Chaotic Signal for Uniform Shaping of Power Spectrum of Common-Mode Conducted EMI

The actual measured power spectra of common-mode conducted EMI under the operation of the prototype in conventional PWM control mode, $x$ chaotic spread spectrum modulated PWM control mode, and $y$ chaotic spread spectrum modulated PWM control mode are shown in Figure 15.

The measured data of the maximum value of common-mode conducted EMI are as follows: 81.73 dBμV for the conventional PWM control mode; 68.65 dBμV for the $x$ chaotic spread spectrum modulated PWM control mode; and 65.17 dBμV for the $y$ chaotic spread spectrum modulated PWM control mode, as indicated by the ‘×’ mark in Figure 15a.

Figure 15a shows that under the conventional PWM control mode, the power spectrum of the common-mode conducted EMI appears as a discrete line spectrum with the maximum peak. This is the root cause of severe EMI in conventional PWM controlled converters. Under the $x$ chaotic spread spectrum modulated PWM mode and $y$ chaotic spread spectrum modulated PWM control mode, the power spectrum of common-mode conducted EMI is extended, showing a continuous distribution, and the peak is significantly reduced.

Figure 15b shows that the distribution of the power spectrum of common-mode conducted EMI under the $y$ chaotic spread spectrum modulated PWM control mode is the most uniform, with the lowest peak. Compared to conventional PWM control mode, the maximum peak reduction is 16.56 dB. Compared to the $x$ chaotic spread spectrum modulated PWM control mode, there is a 3.48 dB improvement in the suppression effect on common-mode conducted EMI.

In summary, employing the acquired uniformly distributed chaotic signal to modulate the frequency of conventional PWM converters can uniformize the power spectrum, exhibiting uniform distribution characteristics. This approach markedly reduces the peak value of the power spectrum, consequently lowering EMI from the power converter and enhancing its electromagnetic compatibility performance.

4. Conclusions

Based on the inherent invariant distribution characteristic of chaotic systems, configuring the probability distribution of chaotic signals is often challenging. To overcome this limitation, this paper proposes a strategy using graphical language to flexibly configure the probability distribution of chaotic signals. Using the realization of a uniformly distributed chaotic signal as an example, the effectiveness of this strategy is verified. Furthermore, the remarkable performance of utilizing these uniformly distributed chaotic signals in suppressing high-frequency converter common-mode conducted EMI highlights the potential application value of this strategy. The main conclusions are summarized as follows:

(1)

The issue of obtaining smooth probability density functions for high-dimensional chaotic signals is addressed through the use of kernel density estimation algorithms.

(2)

The 2A transformation algorithm based on graphical language is introduced, allowing the transformation of intrinsic chaotic signals into chaotic signals with arbitrary probability distributions. This enables the flexible configuration of chaotic signal probability distribution types, offering feasible solutions for obtaining chaotic signals with specific probability distributions.

(3)

Applying the obtained uniformly distributed chaotic signals for chaotic spread spectrum modulation achieves uniform shaping of the power spectrum. Experimental results demonstrate that compared to traditional Chua’s chaotic spread spectrum modulation techniques, the proposed strategy exhibits superior performance in suppressing high-frequency converter common-mode conducted EMI.