Multi-User Detection Based on Improved Cheetah Optimization Algorithm

Abstract

Targeting the issues of slow speed and inadequate precision of optimal solution calculation for multi-user detection in complex noise environments, this paper proposes a multi-user detection algorithm based on a Hybrid Cheetah Optimizer (HCO). The algorithm first optimizes the control parameters and individual update mechanism of the Cheetah Optimizer (CO) algorithm using a nonlinear strategy to improve the uniformity and discretization of the individual search range, and then dynamically introduces a differential evolutionary algorithm into the improved selection mechanism of the CO algorithm, which is utilized to fine-tune the solution space and maintain the local diversity during the fast search process. Simulation results demonstrate that this detection algorithm not only realizes fast convergence with a very low bit error rate (BER) at eight iterations but also has obvious advantages in terms of noise immunity, resistance to far and near effects, communication capacity, etc., which greatly improves the speed and accuracy of optimal position solving for multi-user detection and can achieve the purpose of accurate solving in complex environments.

1. Introduction

In the fields of wireless communications, network security, and electronic warfare, multi-user detection techniques are essential for efficiently processing the acquisition of complex signal information and improving system performance and communication security. With the continuous development of communication technology, especially driven by 5G and 6G communication standards, the demand for research in areas such as multi-user detection technology has further increased. The demand in the field of information countermeasures, among others, has also driven continuous innovation and research on signal processing techniques.

In recent years, significant progress has been made in multi-user detection techniques covering a wide range of areas, including deep learning-based methods that utilize deep neural network structures to learn complex signal features and interference patterns [1,2,3], multitask learning techniques that model the multi-user detection problem as a unified multitask learning problem [4], information sharing between multiple receiver nodes for collaborative signal processing techniques [5,6,7], nonconvex optimization methods that improve the local search capability and convergence speed of the system by introducing nonconvex constraints and optimization methods [8], biomimetic optimization algorithms that solve the optimization problem by simulating the behavior of groups of living organisms in nature [9,10,11,12,13], and modeling the multi-user detection using methods such as graph-based models that model signal transmission relationships and interference relationships for analysis and optimization [14]. These techniques play an important role in improving system performance, reducing complexity, and enhancing real-time performance.

Most of the current research on multi-user detection algorithms is based on the ideal Gaussian channel environment, where the power spectral density of Gaussian noise follows a uniform distribution. However, due to the complexity of the signal interference in the actual transmission process, especially the impact noise interference with shock characteristics, this interference can cause random changes in the signal power, posing significant challenges to the multi-user detection solution process. In this context, studying multi-user detection algorithms for impact noise environments becomes crucial. For example, the non-coherent solution proposed in the literature [15] to address the multi-user detection problem in impact noise environments proves to be an effective measure. Non-coherent detection demonstrates greater robustness in handling complex interference scenarios and can better adapt to actual transmission processes. However, coherent detection can offer higher detection accuracy and performance. Depending on practical application scenarios and requirements, various choices exist, leading to diverse outcomes. Therefore, conducting an in-depth study on multi-user detection algorithms under impact noise environments is essential for improving the actual performance of communication systems.

The multi-user detection problem results in a high level of complexity and difficulty because it involves multiple users transmitting data simultaneously, the presence of multiple-access interference and proximity effects, and the influence of channel loss and noise interference. In CDMA systems, the multi-user detection problem can be regarded as a Non-deterministic Polynomial-time (NP) complete problem, so intelligent optimization algorithms can be considered to find the optimal approximation estimation of the detector data bit information. However, intelligent optimization algorithms have advantages and disadvantages in different aspects. Optimization using only a single algorithm cannot satisfy the need for accurate solutions in complex environments. Therefore, efficient algorithms need to be designed and applied to address these complexities to improve the performance and efficiency of the system. In this paper, we consider the hybrid construction of new multi-user detection algorithms, which effectively enhances the solution accuracy and reliability of the detection by dynamically fusing the advantages of multiple algorithms.

In multi-user detection, global search capability, local search capability, robustness, and adaptability are crucial for the performance of optimization algorithms. Having a good local search capability can quickly converge the algorithm after discovering the potential optimal solution to improve the accuracy and stability of the solution. Having a certain degree of adaptability can adapt the algorithm to changes in different channel environments and multi-user situations. The algorithm should have a strong global search capability, able to search the potential solution space extensively in the search space to avoid falling into local optimal solutions. The algorithm needs to be robust to channel noise and interference to ensure stability and reliability in complex channel environments. A new nature-inspired intelligent optimization algorithm, Cheetah Optimizer (CO) [16], was proposed in 2022 by Mohammad Amin Akbari et al. The algorithm is excellent and adaptable in local search, being successfully implemented in various engineering problems, including power detection [17], wireless transmission [18], multi-system exploration [19], natural resource detection [20], and renewable resource management [21]. However, the cheetah optimization algorithm has obvious problems of slow convergence and it is prone to being stuck in local optima when dealing with the multi-user detection problem. Differential Evolution (DE) [22] is a typical auxiliary algorithm that has the characteristics of high global search capability, strong robustness, etc., but has poor convergence accuracy in local search. Based on the highly complementary nature of the above two algorithms in coping with the multi-user detection problem, this paper considers the dynamic fusion of these two algorithms to deal with the multi-user detection problem.

Firstly, according to the selection mechanism of the CO algorithm, the position update method is dynamically selected, and the contrarian learning strategy [23] is introduced to optimize the control parameters and individual update strategy of the CO algorithm to improve the searching accuracy. After that, the improved CO algorithm dynamically introduces the DE algorithm into the search process to improve the global search ability and convergence speed of the algorithm. Finally, a go-home strategy is introduced to return to the initial position to restart the next hunt if the leader fails to hunt or is unsuccessful in the hunting cycle. This improves the algorithm’s search diversity, ensuring that it does not prematurely converge to incorrect positions during optimization, thereby enhancing the accuracy and reliability of the results.

The meanings of abbreviations in the article can be found in

Table 1. A table of acronyms.

Abbreviation	Full From
HCO	Hybrid Cheetah Optimizer
CO	Cheetah Optimizer
BER	Bit Error Rate
NP	Non-deterministic Polynomial-time
DE	Differential Evolution
GA	Genetic Algorithm
GWO	Grey Wolf Optimization
GOA	Grasshopper Optimization Algorithm
WOA	Whale Optimization Algorithm

. This paper’s subsequent organization is as follows:

Section 2.1 outlines the robust multi-user detection model, and Section 2.2 and Section 2.3 briefly introduce the CO algorithm and DE algorithm. Section 2.4 and Section 2.5 introduce the improved cheetah optimization algorithm strategy and the improved complete detection algorithm is analyzed and designed. Section 3 demonstrates the accuracy and reliability of the algorithm by introducing various intelligent optimization algorithms and comparing them with the proposed algorithm in terms of initialization parameter settings, convergence performance, anti-noise performance, resistance to near and far effects, and communication capacity.

2. Methods

2.1. Robust Multi-User Detection Model

In complex channel environments, conventional multi-user detectors are not resistant to the adverse effects of disturbances such as impact noise on the detector and thus require a robust and improved design of the detector.

Assuming the system has P users with a pseudo-code period of T for each user, the multi-user detection model for a conventional CDMA communication system is

$$y\left(t\right)={\displaystyle \sum _{l=1}^P{\displaystyle \sum _{i=-M}^M{A}_l{B}_l}\left[i\right]s_l\left(t-iT-{\tau }_l\right)+\sigma n\left(t\right)}$$

(1)

where $A_l$ represents the amplitude, $B_l\left[i\right]$ denotes a sequence of information consisting of 1 or −1, $n\left(t\right)$ represents the channel noise, ${\tau }_l\in \left[0,T\right)$ is the size of the signal time delay for the $l$ user, and $s_l\left(t\right)$ denotes the pseudo-code sequence. Assuming the sampling period is equal to the width of a single code slice, the system output after sampling can be expressed as [13]

$$y=\mathit{SAB}+\mathit{\delta }\mathit{n}$$

(2)

where $\mathit{S}=\left[{\mathit{s}}_1\hspace{0.33em}{\mathit{s}}_2\hspace{0.33em}\dots \hspace{0.33em}{\mathit{s}}_P\right]$, $\mathit{A}=diag\left(A_1,A_2,\dots ,A_P\right)$, $\mathit{B}={\left[B_1,\dots ,B_P\right]}^T$, $\mathit{n}$ is the impact noise vector [13].

Non-Gaussian wireless transmission channels containing impact noise are commonly modeled with a steady distribution model, and the eigenfunction of the steady distribution model is denoted as

$$\phi (t)=\left\{\begin{array}{c}\exp \left\{jut-y{\left|t\right|}^{\alpha }\left[1+j\beta \mathrm{sgn}(t)\tan (\frac{\alpha \pi }{2})\right]\right\},\alpha \ne 1\\ \exp \left\{jut-y{\left|t\right|}^{\alpha }\left[1+j\beta \mathrm{sgn}(t)\frac{2}{\pi }\log \left|t\right|\right]\right\},\alpha =1\end{array}\right.$$

(3)

where $\alpha \in \left(0,2\right]$ is the eigenvector, the symmetry parameter is denoted by $\beta \in \left[-1,1\right]$, the dispersion coefficient is denoted by $\gamma >0$, and the location coefficient is denoted by $-\infty <u<\infty$ [24].

According to the literature [25], robust multi-user detection is obtained:

$$\hat{b}=\underset{b}{\arg \min }{\displaystyle \sum _{g=1}^M\rho \left(y_g-{\displaystyle \sum _{l=1}^P{s}_{gl}{A}_l{B}_l}\right)}$$

(4)

The Huber M estimator’s penalty function is formulated as

$$\rho \left(x_g\right)=\left\{\begin{array}{c}\frac{x_g^2}{2{z_g}^2},\hspace{0.33em}\left(\left|x_g\right|\le {\epsilon }_g{z}_g^2\hspace{0.33em},{\epsilon }_g=\frac{3}{2z_g}\right)\\ {\epsilon }_g\left|x_g^2\right|-\frac{{\epsilon }_g^2{z}_g^2}{2},\hspace{0.33em}\left|x_g\right|>{\epsilon }_g{z}_g^2\end{array}\right.$$

(5)

where ${\gamma }_g=\sqrt{2}{z}_g$, ${\gamma }_g$ denotes the noise divergence. Equation (4) is the robust multi-user detection expression; the detector can detect the abnormal wild values of the signal and effectively limit the magnitude size of the wild values, which plays an important role in shock resistance without destroying the original data.

2.2. Cheetah Optimization Algorithm

Through the simulation of the three main hunting strategies of cheetahs’ hunting selection mechanism, the cheetah optimization algorithm updates the population position.

As shown in Figure 1, the cheetah searches for prey by patrolling and scanning its surroundings. When prey is found, it sits in place, waits for the prey to approach, and then attacks. Cheetahs choose a hunting mechanism by evaluating prey conditions, size, and distance from the prey. Throughout the search process, it may abandon the hunt due to energy constraints, rapid escape of prey, etc., and go home to rest before starting a new hunt. Overall, the CO algorithm expresses the entire search process of the optimization algorithm by dynamically selecting the hunting strategy during the search cycle.

Set the Cheetah-initialization position in d-dimensional space [16]:

$$\begin{array}{cc}{X}_{i,j}=L{B}_j+rand\left(U{B}_j-L{B}_j\right)& i=1,2,\dots ,q; j=1,2,\dots ,d\end{array}$$

(6)

Here, $q$, $d$ denote the cheetah community and the search space dimension, respectively, and are initialized with the random number of [0, 1]. The location of the Cheetah-i in the j-dimensional space is represented by $X_{i,j}$. $U{B}_j$ represents the upper range, and $L{B}_j$ represents the lower range of the j-dimensional space.

Search strategy: during hunting, the cheetah dynamically selects a search mode to find prey based on the condition of the prey, the scope of coverage, and the cheetah’s condition. Set the cheetah to search within the d-dimensional condition, denoted as [16]

$$\begin{array}{cc}{X}_{i,j}^{t+1}=X_{i,j}^t+{\overline{r}}_{i,j}^{-1}\cdot {\alpha }_{i,j}^t& t=1,2,\dots ,T\end{array}$$

(7)

$${\alpha }_{i,j}^t=\left\{\begin{array}{c}0.001\frac{t}{T}(UB-LB)\\ 0.001\frac{t}{T}\left|{X_b}^t(i,j)-X^t(i,j)\right|+rand\end{array}\right.\begin{array}{c}k=1\\ k\ne 1\end{array}$$

(8)

where T is set as a hunting cycle, the Cheetah’s position in the jth dimension at t-th and t + 1st iterations is represented by $X_{i,j}^t$, $X_{i,j}^{t+1}$. $X_b^t$, $X^t$ is the t-th iteration leader position, the Cheetah-i position. ${\alpha }_{i,j}^t$ is the step size of Cheetah-i in the jth dimension at the t-th iteration, randomly adjusted by the distance between Cheetah-i and the other neighbors or leaders. ${\overline{r}}_{i,j}^{-1}$ represents a random number that is normally distributed. The algorithm in this process mainly adjusts the search space and flexibly handles the hunting situation by introducing random numbers ${\overline{r}}_{i,j}^{-1}$ and randomly adjusting the step size ${\alpha }_{i,j}^t$.

Sit-and-wait strategy (see Figure 1b): in the search mode, when the prey appears in the line of sight, it is very easy to spook the prey with small movements, and the cheetah will choose the sit-and-wait strategy to wait for the prey to approach. The mathematical representation is [16]

$$X_{i,j}^{t+1}=X_{i,j}^t$$

(9)

where T is a hunting cycle length, the Cheetah’s position in the jth dimension at t-th and t + 1st iterations is represented by $X_{i,j}^t$, $X_{i,j}^{t+1}$. The algorithm under the sit-and-wait strategy mainly waits for the prey position to be updated to a suitable distance to launch the attack through the constant judgment of the qualifying conditions.

Attack strategy: there are two modes of attack strategy: capture and full impact. During the implementation of this strategy, each cheetah dynamically updates its position according to the prey escape, neighboring cheetahs, or leader cheetah position, to find the optimal position for full impact. This can be formulated as [16]

$$X_{i,j}^{t+1}=X_{B,j}^t+{\overline{r}}_{i,j}\cdot {\beta }_{i,j}^t$$

(10)

$${\beta }_{i,j}^t={X_1}^t(i,j)-X^t(i,j)$$

(11)

In the j-dimensional space, $X_{i,j}^{t+1}$ represents the position of the Cheetah-i at the t + 1st iteration, and $X_{B,j}^t$ represents the position of the prey at t-th, i.e., the current optimal position in the community. At the t-th iteration, $X^t$ represents the position of Cheetah-i, and $X_1^t$ denotes the positions of the neighboring cheetahs. ${\overline{r}}_{i,j}$ is the Steering Factor, and ${\beta }_{i,j}^t$ is the table Interaction Factor; both parameter settings are used to map the interaction between cheetahs as well as to control the cheetahs to be swift enough to capture the prey flexibly in its escape path.

Going Home Strategy:

• If the cheetah hunt fails to converge, the CO algorithm then randomly selects the location of one cheetah as the prey location.
• In the absence of successful hunting actions for a period, the hunt is reset, prompting the cheetahs to return to their starting positions in preparation for the next hunt. The algorithm sets the condition that when the cheetah’s energy drops and the leader’s position remains unchanged, the cheetah will go home and update the leader’s position to prevent falling into local convergence. This strategy can enhance the algorithm’s ability to search globally.

The schematic diagram of the algorithm selection mechanism is shown in Figure 2. The red, purple, and blue lines represent the movements of cheetahs, neighboring cheetahs, and prey, respectively, during a complete hunting operation. In each permutation of the algorithm, each cheetah represents a separate hunting mode from the other. The energy of the cheetah is independent of the prey and related to time. The CO algorithm controls the energy by setting the hunting period and representing each decision variable as the alignment of a cheetah during the optimization process [16], which prevents the optimization from converging prematurely and improves the global search capability.

In the hunting process, the search and attack strategies are deployed randomly, but as the energy decreases over time, narrowing down the random range according to different situations and selecting the optimal strategy with timely and accurate judgment will lead to the best solution faster. Therefore, the algorithm controls the selection mechanism to dynamically change with energy changes by setting random numbers r1, r2, r3, r4, and H. The algorithm is designed to adjust the selection mechanism based on energy changes.

For r2 ≥ r3, the choice is made for the sit-and-wait strategy. For r2 ≤ r3, if H ≥ r4, the decision is made to enter the attack mode; otherwise, the execution of the search mode commences. The switching rate between the sit-and-wait strategy and the other two strategies is controlled by adjusting r3 to regulate the rate of change of the decision variables [16], thereby enhancing the success rate of hunting.

2.3. Differential Evolution Algorithm

DE, as an optimization algorithm driven by population intelligence, is distinguished by its robustness, adaptability, and capacity for global search. It emulates the natural process of individual genetic variation, crossover, and selection. DE is not only able to adapt to the problems of nonlinearity, high dimensionality, and multi-peak ability but also does not need the gradient information of the problem, which, taken together, makes it an evolutionary algorithm that can effectively cope with the problem of multi-user detection in a complex environment.

The basic idea of the DE algorithm is to update the individuals in the population by random mutation, crossover, and selection operations. Specifically, the DE algorithm includes the following steps:

• Initialize the population

Setting up a d-dimensional space with $M$ individuals, then the initialized population can be represented as

$$\begin{array}{cc}{E}_x(0)=E_{x,1}(0),E_{x,2}(0),\dots ,E_{x,d}(0)& x=1,2,\dots ,M\end{array}$$

(12)

The ith individual in the jth dimension is initialized as

$$\begin{array}{ccc}{E}_{x,y}(0)=L_y+rand(0,1)(U_y-L_y)& x=1,2,\dots ,M& y=1,2,3,\dots ,d\end{array}$$

(13)

Here, $U_y$ represents the upper range and $L_y$ represents the lower range in the j-dimensional space.

2.

Mutation

DE mainly operates on individuals for mutation by differential strategy; in a certain iteration, three individuals are randomly selected and mutated as follows

$$H_x(e)=E_{p1}(e)+F\times (E_{p2}(e)-E_{p3}(e))$$

(14)

Here, $E_{p1}(e)$, $E_{p2}(e)$, and $E_{p3}(e)$ are the randomly selected individuals for e iterations, respectively, F is the scaling factor of the variance, and the convergence of the local extreme points is inversely proportional to F.

3.

Crossover

Setting a certain fixed crossover probability, according to which the new mutant individuals undergo crossover with the original individuals, this operation significantly enriches the population’s diversity. Mathematically represented as

$$V_{x,y}=\left\{\begin{array}{l}{h}_{x,y}(e),rang(0,1)\le cr\hfill \\ E_{x,y}(e),else\hfill \end{array}\right.$$

(15)

where $cr$ is the crossover probability. In this scenario, $E_{x,y}(e)$ represents the original individual in the population, while $h_{x,y}(e)$ represents the individual undergoing the crossover operation.

4.

Selection

All individuals of the current generation corresponding to the crossover individuals and individuals of the original population are judged by their fitness value, and eugenic selection of well-adapted individuals as the next generation, in which all individuals $E_x(e+1)$ are not inferior $E_x(e)$. The mathematical representation of this part is

$$E_x(e+1)=\left\{\begin{array}{l}{V}_x(e),F(V_x(e))<F(E_x(e))\hfill \\ E_x(e),else\hfill \end{array}\right.$$

(16)

where $E_{x,y}(e)$ represents the original individual in the population, $V_x(e)$ is the crossover individual, $E_x(e+1)$ is the next generation individual, and $F(V_x(e))$ and $F(E_x(e))$ are the fitness values of the crossover and original population individuals, respectively.

2.4. Improved Cheetah Optimization Algorithm

From the multi-user detection simulation data of the CO algorithm, it can be observed that the multi-user detection problem is characterized by nonlinearity, high dimensionality, and multiple peaks, and the initial iteration of the CO algorithm cannot be quickly adapted to and quickly converge the range of optimization seeking. To tackle this issue, this paper proposes incorporating oppositional learning concepts to enhance the diversity of the initial cheetah population, promote the initial population homogenization, and accelerate the convergence of the algorithm’s pre-optimization range. The opposites learning idea [23]

$${\tilde{X}}_i^k=\gamma \times (l+u)-X_i^k$$

(17)

The value range of $X_i^k$ is $\left[l,u\right]$, $k=\alpha ,\beta ,\delta$, and $\gamma$ is a randomly generated value distributed uniformly within the range of (0, 1). Here, ${\tilde{\mathrm{X}}}_i^k$ and $X_i^k$ are employed as the initial population samples for the cheetah optimization algorithm.

2.4.1. Control Parameter Update Strategy

The key concept behind improving the cheetah optimization algorithm is to update the typical linear change mechanism strategy, which can better adapt to the algorithm’s nonlinear optimization process. This nonlinear change provides a guarantee for the global convergence of the algorithm. The improved cheetah optimization algorithm parameter update strategy is

$${\alpha }_{i,j}^t=\left\{\begin{array}{l}0.001\frac{t}{T}(UB-LB)\hfill \\ 0.001\frac{t}{T}\left|X_s+\frac{a_1}{a_1+a_2}\times [{X_b}^t(i,j)-{\overline{X}}_i^k]+\frac{a_2}{a_1+a_2}\times [X^t(i,j)-{\overline{X}}_i^k]-X^t(i,j)\right|+rand\hfill \end{array}\right.\begin{array}{c}k=1\\ k\ne 1\end{array}$$

(18)

$${\beta }_{i,j}^t=X_s+\frac{a_3}{a_3+a_2}\times [{X_1}^t(i,j)-{\overline{X}}_i^k]+\frac{a_2}{a_3+a_2}\times [X^t(i,j)-{\overline{X}}_i^k]$$

(19)

where $X_s$ represents a random number selected from the range of values denoted by $\left[UB,LB\right]$, $\left[UB,LB\right]$ represents the limiting boundary value of the algorithm individual, and ${\overline{X}}_i^k$ represents the initial population of individuals. $a_1$, $a_2$, $a_3$ represent the fitness values of the leader, Cheetah-i, and neighboring Cheetahs at the t-th iteration, respectively. $X_b^t$, $X^t$, $X_1^t$, are the leader position, Cheetah-i position, and neighbor Cheetah position at the t-th iteration.

Compared to traditional algorithms, in Equation (18), the new evolutionary position is obtained by using the information of the evolutionary vectors of the leader and Cheetah-i as well as the ratio of the fitness values, and the search strategy of adjusting the step size by the difference of the distance between the leader and the Cheetah-i is improved to dynamically adjusting the search step size by using the difference of the distance between the new evolutionary position and the Cheetah-i [16]. The improvement scheme is shown in Figure 3. The solid arrows represent the algorithm’s optimization direction, and the dashed lines represent the information on each cheetah’s evolution vector. The two solid line ranges represent the algorithm search space and the cheetah sitting area, respectively, and the two dotted dashed line ranges represent search asymptotic surfaces of the Cheetah gradually converging to the prey with the iteration before and after the change in the step size updating strategy. As can be seen from the figure, the improved step size adjustment strategy not only enhances the precision of the search but also significantly accelerates the convergence ability of the algorithm’s pre-search range.

In Equation (19), the updating of the optimal evolutionary position depends on the proportion of fitness values, using the information of the evolutionary vectors of neighboring cheetahs and Cheetah-i. The Interaction Factor, which is indicated by the disparity between the positions of the neighboring cheetah and Cheetah-i in the attacking strategy, is improved to the difference between the optimal evolutionary location and Cheetah-i’s location. The original Interaction Factor mainly controls the degree of movement toward the neighbor position through the difference between current and neighboring locations and guiding the search agent to move toward the neighbor position. The improved method not only considers the difference between the current position and the neighbor position but also adjusts it according to the proportion of its fitness value to better guide the degree of the search agent’s movement towards the neighbor position. Such a setup can better reflect the interaction between search agents, scientifically plan the prey escape route based on the fitness ratio data within the capture range, and improve the cheetah’s ability to accurately capture prey on escape routes.

2.4.2. Individual Location Update Strategy

This paper addresses the multi-user detection problem, which is a binary data information individual search for the optimal problem. The usual bit information is in the form of a combination of 1 and −1, but if the cheetah optimization algorithm is not modified and directly deals with the problem, it will lead to the algorithm search for optimal unity of the problem, thus restricting the algorithm to the scope of the search and increasing the solution error rate. Therefore, this paper aims to improve the position update strategy of the searching individual ${\overline{X}}_{i,j}^{t+1}$ in the Cheetah optimization algorithm

$$X_{i,j}^{{t+1}_{\prime }}=\frac{1}{1+\exp \left(-{\overline{X}}_{i,j}^{t+1}\right)}$$

(20)

$$X_{i,j}^{t+1}=\left\{\begin{array}{c}-1,\hspace{0.33em}\hspace{0.33em}rand>X_{i,j}^{{t+1}_{\prime }}\\ 1,\hspace{0.33em}\hspace{0.33em}otherwise\end{array}\right.$$

(21)

where ${\overline{X}}_{i,j}$ represents a random number selected from the range of values denoted by $\left[UB,LB\right]$. The function in Equation (21) serves to restrict the estimated position to [0, 1]. This is subsequently transformed into estimated information bits according to Equation (20). Improvements to the individual position update strategy can be used to solve problems such as local convergence due to the monolithic search for empty rooms when dealing with multi-user detection problems.

2.5. Robust Multi-User Detection Implementation

The fitness function of the multi-user detector in an impact noise channel environment is

$$F=1/\left({\displaystyle \sum _{g=1}^M\rho \left(r_g-{\displaystyle \sum _{l=1}^P{s}_{gl}{A}_l{B}_l}\right)}+P_{\epsilon }\right)$$

(22)

The fitness function value is directly related to the individual’s optimization performance.

The improved Cheetah optimization algorithm is dynamically fused with the DE algorithm into the new Hybrid Cheetah Optimizer (HCO) algorithm. The pseudo-code implementation diagram for the entire HCO algorithm is depicted in Algorithm 1.

Algorithm 1. The HCO algorithm

1 Define the problem data, dimension (d), Hunting cycle T, and the initial population 2 on size SearchAgents-no 3 Initialise: pCR = 0.5; beta_min = 0.2; beta_max = 0.5 4 Generate the initial population of cheetahs and evaluate the fitness of each cheetah 5 Initialize the population’s home, leader, and prey solutions 6 t$\leftarrow$0, it$\leftarrow$1; 7 Max-iter$\leftarrow$desired maximum number of iterations 8 While it$\le$Max-iter do 9  Select m (2$\le$m$\le$n) members of cheetahs randomly 10  for each member $\in$m do 11   Define the neighbor agent of member i 12   for each arbitrary arrangement j $\in$ (1, 2, ..., d) do 13    Calculate r, r_Check, H; 14    Update the step size using Equation (18) // Step length update 15    Interactors are updated according to Equation (19) 16    // Interaction factor renewal 17    r2,r3$\leftarrow$random numbers are chosen uniformly from 0 to 1 18    if r2$\le$r3 then 19     r4$\leftarrow$A random number is chosen uniformly from 0 to 3 20     if H $\ge$ r4 then 21      Calculate the new position of the member in arrangement using Equation 22      (10) // Attack 23     else 24      Calculate the new position of member in arrangement using Equation (7) 25      // Search 26     end 27    else 28     Calculate the new position of the member in arrangement using Equation 29     (9) // Sit-and-wait 30    end 31   end 32   Update the solutions of members and the leader 33  end 34  t$\leftarrow$t + 1; 35  for i = 1 i $\le$ SearchAgents-no i++ do 36   Three individuals were randomly selected in the rearranged individual orderM 37   The scaling factor betade is generated randomly, and the variant operation 38   produces the intermediate y 39   Performing a cross-selection operation to get a new individual 40  end 41  Update the solutions of members and the leader 42  if If the leader position doesn’t change for a time // Go back home 43  then 44   Implement the leave the prey and go back home strategy and change the leader 45   position 46   Substitute the position of the member by the prey position 47   t$\leftarrow$0 48  end 49  it$\leftarrow$it + 1 50  Update the prey (global best) solution 51 end

3. Results and Discussion

3.1. Initialization Parameter Test of the Algorithm

The experiments are based on the improved HCO algorithm for multi-user detection, using direct expansion system communication signals. The signal-to-noise ratio is set to 5 dB, and the stable distribution model is employed to model the complex impact noise. By determining the model parameters and superimposing the signals, signal data containing noise are generated. According to Equation (3) and the noise requirements of this paper, the specific model parameters are set as follows: set $\alpha$ to 1.6. $\beta$ uses the default value of 0. $\gamma$ is obtained by calculating the variance of the noise signal and the standard deviation of the signal, and the center of the distribution can be dynamically adjusted to adapt to different situations. $\mu$ uses the default value of 0. The experimental verification is carried out in the case of 10 users, with each user having the same signal power. The data signal has a length of 10,000 bits, and the loop iterates eight times. The relationship between the change in the initialization parameter and the BER is shown in Figure 4.

The simulation experiment results demonstrate that the optimal values of T1 and mod in the initialization parameter T = ceil(dim/10) $\times$ T1 of the algorithm are 0.52 and 0.34, which have the least effect on the BER. The simulation plot of the maximum and minimum values of the scaling factor (beta) of the hybrid algorithm indicates that the effect of the randomly generated scaling factor is optimal when the upper limit is in the vicinity of 0.3 and the lower limit is in the vicinity of 0.1, which has the least impact on the detection BER. Several parameters involved in this algorithm also have a very small impact on the BER, fluctuating within 0.01%, and demonstrate adaptive adjustment capability.

3.2. Performance Testing of the Algorithm

• Investigate the correlation between the algorithm’s BER and the iteration count in an α-stable noise environment.

To validate the advantages of the proposed HCO algorithm, the simulation experiment introduces the Genetic Algorithm (GA) [26], the classical evolutionary algorithm of DE [22], and the more advanced Grey Wolf Optimization algorithm (GWO) [27], Grasshopper Optimization Algorithm (GOA) [28], Whale Optimization Algorithm (WOA) [29], and CO [16] group intelligence optimization algorithms for performance comparison. Based on the analytical setup of the previous paper, the algorithms are set up to establish a multi-user detection study. According to the literature [13] on fusion strategy, the cheetah optimization algorithm introduced in the DE is replaced with JADE [30] (HCO-jade) to participate in the simulation experiments in the search process part, to compare and verify the fusion advantage of this paper’s algorithm. Under the aforementioned experimental conditions, the initialization parameters are set based on the optimal values obtained from the above experiments. Figure 5a illustrates the correlation between the algorithm’s iteration count and the BER.

Figure 5a demonstrates that the cheetah optimization algorithm, due to its homecoming characteristics, enriches the diversity of the population. However, the return home greatly affects the hunting efficiency. The hybrid HCO algorithm solves this problem well, and the HCO algorithm has converged at a very low BER when the number of algorithm iterations reaches eight. In contrast, the other optimization algorithms require around 25 iterations before convergence. As can be seen from Figure 5, according to the literature [13], the HCO-jade algorithm fusion strategy exhibits superior performance compared to other algorithms. However, its convergence performance is significantly lower than that of the HCO algorithm proposed in this paper.

2.

Analyze the relationship between the algorithm’s BER and the generalized SNR in an α-stable noise environment.

Similarly, other basic experimental conditions are set as above. The range of generalized SNR is 0–10, and the relationship between the SNR and BER of each algorithm is shown in Figure 5b.

Figure 5b demonstrates that with the continuous deterioration of the complex channel environment, the BER of each experimental algorithm is greatly affected. However, according to the figure, the HCO algorithm proposed in this paper has an obvious advantage over other algorithms in terms of noise immunity.

3.

Analyze the relationship between the BER of the algorithms and the near-far ratio under the α-stable noise environment.

Other basic experimental conditions are the same as above. Here, the signal-to-noise ratio (SNR) of the first user is set to 5 dB, and the SNR of the other users is the same and changes at the same time. Figure 5c illustrates how the Bit Error Rate (BER) relates to the Nearness/Farness Ratio.

Figure 5c demonstrates that as the proximity effect increases gradually, the BER of each algorithm in the complex channel environment also rises gradually. However, the comparison shows that the performance of the HCO algorithm surpasses that of the other algorithms, being least affected by the proximity effect and the most stable.

4.

Analyze the relationship between the BER of the algorithms and the number of users in the α-stable noise environment.

Neglecting the effect of the proximity ratio between the users, setting the SNR to 5 dB, and other basic experimental conditions as above, the relationship between the number of users and the BER analyzed in this experiment is shown in Figure 5d.

From Figure 5d, we observe that although the BER of the HCO algorithm grows from 0.039 to 0.042 as the system’s user count increases, the evolution of the BER of the HCO algorithm grows more gently with an increase in the number of users than the other algorithms. The BER is kept at the lowest level, indicating that the HCO algorithm is minimally affected by growth in the number of users and has the largest communication capacity.

4. Conclusions

Aiming to address the problems of poor anti-noise performance and low solving performance associated with traditional multi-user detectors under complex channel environments, this paper presents a new multi-user detection algorithm method based on the cheetah optimization algorithm, which firstly improves the control parameter and individual updating strategy of the CO algorithm according to the shortcomings of the CO algorithm’s adaptability. After that, dynamically introducing a differential evolutionary algorithm into the improved CO algorithm and leveraging the strengths of both algorithms to find the optimal solution not only improves the optimal solution ability of the multi-user detection algorithm under the complex channel environment but also greatly ensures population diversity throughout the algorithmic solution process. Through comparison simulations in a noisy environment, it is evident that our algorithm achieves fast convergence, with a BER lower than 0.02 after eight iterations. In contrast, most other algorithms converge only after 25 iterations, with a BER above 0.03. This demonstrates that our algorithm outperforms others in terms of resistance to noise and convergence. As the signal-to-noise ratio increases, the BER of other algorithms typically fluctuates between 0.12 and 0.04, while our algorithm’s BER fluctuates between 0.11 and 0.02. This consistently lower BER, coupled with a more significant decrease trend, demonstrates our algorithm’s clear advantage in noise resistance over others. As the near-to-far ratio increases, all algorithms are affected. Our algorithm’s BER fluctuates between 0.04 and 0.06, consistently lower than the range of 0.07 to 0.10 observed in other algorithms. This indicates our algorithm’s superior resistance to near-far effects and its stability. As the number of users increases, the BER of all algorithms rises. However, our algorithm exhibits a narrower BER fluctuation range of 0.035 to 0.063, compared to the range of 0.05 to 0.14 observed in other algorithms. This demonstrates not only our algorithm’s significantly lower BER but also its greater stability and larger capacity. The simulation demonstrates that our algorithm is least affected by user growth and has the largest capacity. In summary, our algorithm exhibits superior advantages over others in noise resistance, convergence, shock resistance, near and far resistance, and high throughput. It can not only improve the speed and accuracy of optimal position solving for multi-user detection but also realize the purpose of accurate solving in complex environments.

The next work plan is to optimize the multi-user detector in a more realistic simulation, placing the model into a problem closer to a real application scenario for testing and optimization. We aim to explore and address the system-influencing factors one by one. User mobility may affect the channel state and interference situation, which in turn affects the effectiveness of the optimization algorithm. Current experimental tests only consider the use of cooperative optimization to reduce the impact of user mobility, which is far from sufficient. User mobility as a significant influence factor in mobile communication systems deserves our focused research. Therefore, the next step is that we intend to take this part as the base point for large-scale optimization of the system model as well as the experiments and try to adapt to the changes in user mobility by searching for and trying to introduce the dynamic adaptive strategy and real-time updating of the model parameters, etc., and try to analyze the laws of user mobility in fixed scenarios and introduce the prediction mechanism to make optimization adjustments in advance, to improve the algorithms’ performance and stability in the complicated environments.