Adaptive filtering: Issues, challenges, and best-fit solutions using particle swarm optimization variants

Citation: Khan, A.; Shafi, I.; Khawaja, S.G.; de la Torre Diez, I.; Flores, M.A.L.; Galvlán, J.C.; Ashraf, I. Adaptive Filtering: Issues, Challenges, and Best-Fit Solutions Using Particle Swarm Optimization Variants. Sensors 2023,23, 7710. https://doi.org/10.3390/s23187710 Academic Editor: Enrico Meli Received: 7 July 2023 Revised: 3 September 2023 Accepted: 3 September 2023 Published: 6 September 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). sensors Review Adaptive Filtering: Issues, Challenges, and Best-Fit Solutions Using Particle Swarm Optimization Variants Arooj Khan 1, Imran Shafi 1, Sajid Gul Khawaja 1, Isabel de la Torre Díez 2,* , Miguel Angel López Flores 3,4,5 , Juan Castañedo Galvlán 3,6,7 and Imran Ashraf 8,* 1College of Electrical and Mechanical Engineering, National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan; [email protected] (A.K.); [email protected] (I.S.); [email protected] (S.G.K.) 2Department of Signal Theory and Communications and Telematic Engineering, University of Valladolid, Paseo de Belén 15, 47011 Valladolid, Spain 3Research Group on Foods, Universidad Europea del Atlántico, Isabel Torres 21, 39011 Santander, Spain; [email protected] (M.A.L.F.); [email protected] (J.C.G.) 4Research Group on Foods, Universidad Internacional Iberoamericana, Campeche 24560, Mexico 5Instituto Politécnico Nacional, UPIICSA, Ciudad de Mexico 04510, Mexico 6Universidad Internacional Iberoamericana Arecibo, Arecibo, PR 00613, USA 7Department of Projects, Universidade Internacional do Cuanza, Cuito EN250, Bié, Angola 8Department of Information and Communication Engineering, Yeungnam University, Gyeongsan 38541, Republic of Korea *Correspondence: [email protected] (I.d.l.T.D.); [email protected] (I.A.) Abstract: Adaptive equalization is crucial in mitigating distortions and compensating for frequency response variations in communication systems. It aims to enhance signal quality by adjusting the characteristics of the received signal. Particle swarm optimization (PSO) algorithms have shown promise in optimizing the tap weights of the equalizer. However, there is a need to enhance the optimization capabilities of PSO further to improve the equalization performance. This paper provides a comprehensive study of the issues and challenges of adaptive filtering by comparing different variants of PSO and analyzing the performance by combining PSO with other optimization algorithms to achieve better convergence, accuracy, and adaptability. Traditional PSO algorithms often suffer from high computational complexity and slow convergence rates, limiting their effectiveness in solving complex optimization problems. To address these limitations, this paper proposes a set of techniques aimed at reducing the complexity and accelerating the convergence of PSO. Keywords: adaptive filtering; particle swarm optimization; bit error rate; signal quality 1. Introduction Particle swarm optimization (PSO) is a computational optimization technique inspired by the collective behavior of swarms. It was originally proposed by Kennedy and Eberhart in 1995 [ 1 ] and has since become a popular and effective method for solving various optimization problems [ 2 ]. PSO simulates the social behavior of a swarm of particles, where each particle represents a potential solution in the search space [ 3 ]. The particles move through the search space, adjusting their positions based on their own experience and the experiences of their neighboring particles. The objective is to find the optimal solution by iteratively updating the positions of the particles in search of better solutions. This study makes significant contributions to the field of adaptive equalization by exploring PSO techniques. Motivated by the need to enhance the optimization capabilities of PSO in communication systems [ 4 ], the research aimed to address the limitations of traditional PSO algorithms, such as slow convergence rates and high computational complexity [ 5 ]. The study investigated the combination of PSO with other optimization algorithms, adaptive mechanisms, multi-objective optimization, the constriction factor Sensors 2023,23, 7710. https://doi.org/10.3390/s23187710 https://www.mdpi.com/journal/sensors Sensors 2023,23, 7710 2 of 28 approach, and the dynamic neighborhood topology [ 6 ]. The primary research question driving this study is how to improve PSO for adaptive filters in terms of convergence, accuracy, and adaptability. By answering this question, the research provides valuable insights and recommendations to optimize the tap weights of adaptive filters, thereby enhancing signal quality and mitigating distortions in communication systems. The primary research questions driving this study are as follows: • RQ1: How can the optimization capabilities of PSO be further enhanced for adaptive filters in the context of equalization? • RQ2: How does the resemblance of PSO with algorithms such as the the least mean squares (LMS) and recursive least squares (RLS) contribute to the understanding and development of adaptive filters? • RQ3: What are the recent advancements in PSO algorithms, such as ring topology, dynamic multi-swarm PSO, and fully informed PSO, and how do they improve the performance of adaptive filtering? • RQ4: How does the dynamic neighborhood concept in PSO contribute to better exploration and exploitation of the search space? • RQ5: What are the benefits and challenges of hybridization techniques, such as hybrid PSO and cooperative PSO, in improving the optimization capabilities of PSO? • RQ6: What are the time and space complexity considerations of PSO algorithms, and how do they impact the scalability and efficiency of the optimization process? • RQ7: What are the limitations and challenges of PSO to achieve a better convergence rate? The study sought to explore and propose various techniques to improve the convergence, accuracy, and adaptability of PSO algorithms. The research investigated the combination of PSO with other optimization algorithms, the introduction of adaptive mechanisms, the application of multi-objective optimization, and the utilization of the constriction factor approach and dynamic neighborhood topology. By answering these research questions, the study aimed to provide insights and recommendations for optimizing the tap weights of adaptive filters using PSO in communication systems. This review is further divided into six parts. Section 2elaborates on the techniques used for adaptive equalization. PSO, its time complexity, and its resemblance to other optimization algorithms are discussed in Section 3. Section 4provides a comprehensive overview of PSO approaches used for adaptive filtering including the comparative analysis of PSO variants. Hybrid PSO, the best-fit PSO solution for adaptive filtering, is discussed in Section 5concerning its advantages and disadvantages. In the end, the conclusion and future directions are given in Section 6. Section 2is dedicated to addressing RQ1 and RQ2. The answers to RQ3 and RQ4 can be found in Section 3. Section 4presents the discussions regarding RQ5. Lastly, Section 5 delves into the responses to RQ6 and RQ7. 2. Techniques Used for Adaptive Equalization Adaptive equalization is a fundamental signal-processing technique utilized in numerous communication systems to improve the quality and reliability of transmitted data [ 7 ]. It serves as a crucial step in combating the detrimental effects of channel impairments, such as multipath propagation and frequency response variations, which can introduce intersymbol interference (ISI) and degrade the received signal quality [ 8 , 9 ]. The primary goal of adaptive equalization is to dynamically adjust the characteristics of the received signal to closely align with the desired signal [ 10 ], effectively mitigating distortions and restoring the fidelity of the transmitted data [ 11 ]. To achieve adaptive equalization, a diverse range of techniques has been developed [ 12 ], each with its own approach and advantages. These techniques are designed to adaptively modify the parameters or coefficients of the equalizer based on the characteristics of the channel and the received signal. By continuously monitoring and updating the equalizer, it can adapt to the changing conditions of the communication channel and optimize its performance accordingly. Sensors 2023,23, 7710 3 of 28 The field of adaptive equalization has witnessed significant advancements and innovation over the years [ 13 ], driven by the increasing demands for high-speed data transmission and reliable communication systems [ 14 , 15 ]. Researchers and engineers have explored various approaches including algorithmic optimization techniques, machine learning algorithms, and advanced signal-processing methods, to enhance the performance of adaptive equalization [ 16 , 17 ]. These techniques aim to strike a balance between computational complexity, convergence speed, and adaptability to different channel conditions, providing robust and efficient solutions for adaptive equalization in a wide range of applications [ 18 ]. The choice of adaptive equalization technique depends on several factors, such as the specific characteristics of the channel [ 19 ], the desired performance metrics [ 20 ], the available computational resources, and the trade-off between complexity and effectiveness [ 21 ]. As the field continues to evolve, researchers are constantly pushing the boundaries of adaptive equalization techniques, seeking novel approaches to address the challenges posed by emerging communication technologies and ever-changing channel conditions. By harnessing the power of adaptive equalization, communication systems can achieve higher data rates, improved spectral efficiency, and enhanced reliability, paving the way for seamless and efficient transmission of information in diverse environments. Adaptive equalization can be achieved using various techniques [ 22 ], each with its unique approach and advantages. Some of the techniques include LMS, RLS, PSO, genetic algorithms (GAs), and deep learning, which are discussed below. 2.1. Least-Mean-Squared Error LMS is an adaptive filtering algorithm widely used for adaptive equalization [ 23 ]. It aims to minimize the mean squared error between the desired signal and the filter output [ 24 ]. LMS updates the filter coefficients iteratively based on the instantaneous estimation error and the input signal [ 25 – 27 ]. In the context of adaptive equalization, LMS is employed to adjust the equalizer’s coefficients and compensate for distortions caused by the channel [ 28 ]. By continuously adapting the filter coefficients, LMS enables the equalizer to adapt to changing channel conditions and optimize its performance [ 29 ]. LMS is known for its simplicity and ease of implementation, making it a popular choice in various communication systems. 2.2. Recursive Least Squares RLS is another popular adaptive filtering algorithm used for adaptive equalization [ 30 ]. It recursively updates the filter coefficients based on the instantaneous estimation error and the input signal [ 31 ]. RLS utilizes a matrix inversion technique to achieve optimal filter updates [ 32 ]. In the context of adaptive equalization, RLS offers fast convergence and provides accurate filter estimation [ 33 ]. However, RLS has higher computational complexity and memory requirements compared to LMS [ 34 ]. Despite these limitations, RLS is preferred in applications that require rapid convergence and optimal filter updates [35]. 2.3. Particle Swarm Optimization PSO is a population-based stochastic optimization algorithm inspired by social behavior [ 36 ]. In the context of adaptive equalization, PSO is utilized to optimize the equalizer’s coefficients by iteratively exploring a multidimensional search space [ 37 ]. PSO works by simulating the movement of particles, where each particle represents a potential solution. By leveraging the best experiences of the swarm and their own experiences, particles dynamically adjust their positions in the search space to find optimal solutions [ 38 ]. PSO provides a global search capability, allowing it to handle complex and nonlinear optimization problems [ 39 ]. This makes PSO suitable for adaptive equalization tasks that require optimal filter coefficients and enhanced convergence [ 39 ]. The detailed analysis of the PSO algorithm and its variants for adaptive equalization is discussed in later sections. Sensors 2023,23, 7710 4 of 28 2.4. Genetic Algorithms The GA is an optimization technique inspired by the process of natural selection and genetics [ 40 ]. In the context of adaptive equalization, the GA is employed to evolve a population of candidate solutions towards the optimal solution [ 41 ]. The GA involves the use of selection, crossover, and mutation operators to iteratively improve the quality of solutions [ 42 ]. The GA can handle complex optimization problems and provides a diverse set of solutions [ 43 ]. By using appropriate genetic operators and fitness evaluation criteria, the GA can effectively optimize the equalizer’s coefficients for adaptive equalization. 2.5. Deep Learning Deep learning techniques, specifically deep neural networks, are increasingly used for adaptive equalization tasks [ 44 ]. Deep learning approaches involve training neural networks to learn the mapping between the received signal and the desired signal [ 45 ]. In the context of adaptive equalization, deep neural networks can model the complex and nonlinear relationship between the input signal and the equalized output [ 46 ]. By utilizing large amounts of training data and employing sophisticated network architectures, deep learning techniques can adapt to a wide range of channel characteristics and achieve superior equalization performance [ 47 , 48 ]. A deep learning approach requires significant computational resources, substantial training data, and careful regularization techniques to mitigate overfitting [ 49 ]. Table 1provides a comprehensive overview of the pros and cons of techniques used for adaptive equalization. Table 1. Techniques used for adaptive equalization. Technique Limitations Advantages LMS • Susceptible to getting stuck in local optima [50]. • Simplicity and ease of implementation [25]; • Low computational complexity [26]; • Achieves significant error reduction and convergence with reasonable computational resources. RLS • High computational complexity and memory requirements [34]; • Sensitive to numerical issues due to matrix inversion. • Fast convergence rate; • Provides optimal filter updates [31]; • Achieves improved convergence speed and provides accurate filter estimation. PSO • May suffer from premature convergence and lack of diversity [51]; • Requires fine-tuning of algorithm parameters. • Provides global search capability; • Can handle complex and nonlinear optimization problems; • Achieves optimal filter coefficients with enhanced convergence and improved equalization performance [38]. GA • Convergence speed may be slower compared to other algorithms; • Requires a suitable representation of solutions and the design of appropriate genetic operators. • Can handle complex optimization problems; • Provides a diverse set of solutions; • Achieves improved equalization performance with diverse and globally optimal solutions [43]. Deep Learning • Requires a large amount of training data [52]; • May suffer from overfitting. • Can adapt to complex and nonlinear channel characteristics; • Provides high flexibility in modeling the equalization process [46]; • Achieves superior equalization performance with accurate mapping of the input–output relationship. Sensors 2023,23, 7710 5 of 28 3. Particle Swarm Optimization 3.1. Standard PSO Algorithm PSO is an optimization method inspired by swarm behavior observed in nature, where a population of particles represents the optimization parameters [ 53 ]. These particles collectively search for optimal solutions within a multi-dimensional search space. The objective of the algorithm is to converge toward the best-possible values for each parameter [ 54 ]. The fitness of each particle is evaluated using a fitness function, which quantifies the quality of the particle’s solution estimate [ 5 , 55 ]. Each particle maintains two state variables: its position ( x(i) ) and velocity ( v(i) ), where i represents the iteration index. The fitness of a particle is determined by evaluating a cost function associated with its solution estimate. Through information sharing, each particle combines its own best solution with the best solution found by the entire swarm, adjusting its search pattern accordingly. This iterative process continues until an optimal solution is reached or a termination criterion is met. The equation of the standard PSO algorithm is given as vd k(i+1) = vd k(i) + c1.r1,k(i).(pd k−xd k(i)) + c2.r2,k(i).(gd−xd k(i)) (1) xd k(i+1) = xd k(i) + vd k(i+1)(2) The equation represents the velocity and position update mechanism in PSO. The velocity is updated by combining the particle’s previous velocity, the cognitive component based on its personal best solution, and the social component based on the global best solution found by the swarm. This combination allows the particle to maintain its momentum, explore its individual best solution, and be influenced by the overall best solution. The updated velocity is then used to update the particle’s position, determining its next location in the search space. The updated version of PSO having an inertia term is given below: vd k(i+1) = vd k(i).r1,k(i).(pd k−xd k(i)) + c2.r2,k(i).(gd−xd k(i)) (3) where c1 represents the cognitive term, c2 represents the social term, d is the dimension of the particles, Pk is the local best, g is the global best of the particle, and r1 and r2 are the random variables, their range lying between 0 and 1 [ 56 ], while the momentum of a particle is controlled by inertia, represented by w. When the inertia of the particle is zero, the model will only explore and become independent of past values. The convergence rate of the PSO algorithm refers to the speed at which the algorithm converges toward an optimal solution [ 57 ]. The convergence rate of PSO can be influenced by various factors, including problem complexity, population size, inertia weight, acceleration coefficients, and termination conditions [ 58 ]. The flow chart of the standard PSO algorithm mentioned in [59] is shown in Figure 1. PSO has the potential for fast convergence due to its ability to share information among particles in the swarm [ 60 ]. Collective knowledge sharing enables particles to converge towards promising regions of the search space [ 61 ]. However, the convergence rate of standard PSO can be affected by the balance between exploration and exploitation. If the exploration is too dominant, the algorithm may take longer to converge. On the other hand, if the exploitation is too dominant, the algorithm may converge prematurely to local optima [ 62 ]. To enhance the convergence rate, several strategies can be employed. One approach is to adaptively adjust the parameters of the algorithm during the optimization process. This includes modifying the inertia weight and acceleration coefficients to balance exploration and exploitation at different stages of the optimization [ 63 ]. Different variants of PSO algorithms exist in the literature, shown in Figure 2, to achieve better complexity and faster convergence. Sensors 2023,23, 7710 6 of 28 Figure 1. Flow chart of standard PSO algorithm. Figure 2. Variants of particle swarm optimization. 3.2. Resemblance of Artificial Intelligence and PSO PSO and artificial intelligence (AI) are two distinct computational approaches with both similarities and differences [ 64 ]. Both PSO and AI share the common goal of solving complex problems and optimizing system performance. They rely on algorithms and techniques to learn from data, make decisions, and improve overall performance. Furthermore, both PSO and AI have versatile applications across various domains, including optimization, pattern recognition, decision-making, and control systems. There are notable differences between PSO and AI. PSO is a specific optimization algorithm inspired by the collective behavior of bird flocks or fish schools [ 65 ]. It is a population-based metaheuristic algorithm that iteratively adjusts the positions of particles in search of the optimal solution. On the other hand, AI is a broader field encompassing various techniques, including but not limited to PSO, such as neural networks, genetic algorithms, and expert systems [ 66 ]. While PSO is primarily designed for optimization problems and focuses on finding the best solution within a given search space, AI encompasses a wider range of techniques. These techniques can include machine learning, natural language processing, robotics, and more. Sensors 2023,23, 7710 7 of 28 AI techniques can be applied to various problem domains [ 67 ], not necessarily limited to optimization. PSO operates based on the principles of collective intelligence and social behavior, where particles communicate and learn from each other to find the best solution. In contrast, AI approaches can involve learning from data, simulating human cognitive processes, or mimicking intelligent behavior using different algorithms and methodologies. 3.3. Resemblance with Least Mean Square and Recursive Least Squares The PSO, LMS, and RLS algorithms share certain resemblances in terms of their learning mechanisms and optimization objectives. PSO is an algorithm where particles within a swarm collectively explore and exploit the search space to find optimal solutions. Similarly, both the LMS and RLS algorithms are adaptive filtering techniques used in signal processing and parameter estimation [ 68 ]. They aim to adjust the internal parameters iteratively to minimize the error between the predicted and actual outputs. One resemblance between PSO, LMS, and RLS is their learning mechanism. In PSO, particles adjust their positions and velocities based on their individual experiences and the collective knowledge of the swarm [ 69 ]. This learning process allows particles to explore the search space and exploit promising regions. Similarly, in LMS and RLS, the algorithms update their weight vectors or coefficients based on the input data and the discrepancy between the predicted and actual outputs. This iterative learning mechanism in all three algorithms enables them to converge toward optimal solutions or parameter estimates. 3.4. Applications of PSO Due to advancements in and modifications of PSO, various applications have been found in the literature [ 70 ] due to its ability to efficiently search for optimal solutions. PSO can be applied to optimize mathematical functions with multiple variables. By exploring the search space, particles can locate the global minimum or maximum of a function. This application is particularly useful in fields such as engineering design, data analysis, and financial modeling. In image and signal processing, PSO has been employed for image and signal processing tasks [ 71 ]. It can optimize parameters in image reconstruction, denoising, feature extraction, and object recognition. PSO algorithms have shown promising results in optimizing parameters for imageand signal-processing techniques, enhancing the quality and efficiency of these processes [72]. PSO can be used to train the weights and biases of neural networks. It has been employed as an alternative to traditional optimization algorithms, such as backpropagation, to improve the training process and avoid local optima. PSO-based training algorithms can enhance the convergence speed and accuracy of neural networks, making them more effective in pattern recognition, classification, and prediction tasks. They are also used to solve optimization problems in power systems [ 70 ]. They can optimize various aspects such as power flow, unit commitment, economic dispatch, and capacitor placement. PSObased approaches enable efficient utilization of power resources, leading to improved power system operation, reduced costs, and enhanced stability. They can be used for feature selection in machine learning and data-mining tasks. By selecting a subset of relevant features, PSO helps in dimensionality reduction, improving classification accuracy and reducing computational complexity. This application is particularly useful in areas such as text mining, bioinformatics, and image recognition [73]. PSO is also used to optimize vehicle-routing problems, including route planning, delivery scheduling, and fleet management. By considering factors such as distance, capacity, and time constraints, PSO algorithms can determine efficient routes and schedules, minimizing transportation costs and improving logistics operations. Another application of PSO in electronics is in the optimization of antenna design [ 74 ]. Antennas are crucial components in wireless communication systems, and their performance greatly impacts signal reception and transmission. Moreover, in radar waveform design, PSO can optimize the characteristics of radar waveforms, such as pulse duration, modulation schemes, Sensors 2023,23, 7710 8 of 28 and frequency characteristics, to enhance target detection, resolution, and interference mitigation. By iteratively adjusting particle positions representing waveform parameters, PSO can efficiently explore the design space and converge on optimal solutions that maximize radar performance. This enables radar systems to improve their capabilities in detecting and tracking targets, reducing interference, and enhancing overall operational efficiency. 3.5. Time and Space Complexity of PSO In terms of time complexity, the main computational cost of PSO lies in evaluating the objective function for each particle in each iteration. The objective function represents the problem to be optimized and can vary in complexity depending on the problem domain. Therefore, the time complexity of PSO is closely related to the evaluation time of the objective function [ 75 ]. In each iteration, all particles need to evaluate their positions, update their personal bests and the global best, and adjust their velocities and positions. This process continues until a termination condition is met. The number of iterations required for convergence depends on various factors such as the problem complexity, the size of the search space, and the convergence speed of the swarm [ 76 ]. Generally, the time complexity of PSO is considered to be moderate, as it typically requires a reasonable number of iterations to converge to an acceptable solution [ 77 ]. The greater number of iterations leads to the requirement of large memory. PSO requires memory to store the positions, velocities, personal bests, and global best of each particle in the swarm [ 78 ]. The amount of memory required is proportional to the population size, which is typically determined by the problem being solved. PSO may also require memory to store auxiliary variables, such as acceleration coefficients and parameters controlling the swarm behavior. The space complexity of PSO is, therefore, determined by the memory requirements for storing the swarm’s state and other relevant variables. The space complexity is generally considered to be reasonable, as it scales linearly with the population size and does not depend on the size of the search space. Figure 3shows the improvements in PSO over time to quickly converge to the optimal solution. Figure 3. Convergence improvements of different variants of PSO. Sensors 2023,23, 7710 9 of 28 3.6. Recent Advancements in PSO for a Better Convergence Rate In recent years, significant advancements have been made for PSO, enhancing its performance and expanding its applications [ 73 ]. These advancements have focused on addressing various challenges and improving the algorithm’s effectiveness. One notable advancement is the development of techniques to handle large-scale optimization problems [ 79 ]. Researchers have devised parallel and distributed PSO algorithms, which utilize multiple computing resources to tackle computationally intensive tasks efficiently [ 80 ]. This advancement has opened the door to optimizing complex problems that were previously unfeasible with traditional PSO approaches. Another noteworthy development is the integration of PSO with machine learning techniques [ 81 ]. By combining PSO with algorithms such as neural networks or deep learning models, the optimization process becomes more robust, enabling the solution of intricate problems and improving prediction tasks. Additionally, self-adaptive PSO algorithms have emerged, allowing for dynamic adjustments of algorithm parameters during optimization. These algorithms utilize adaptive mechanisms to fine-tune parameters based on the particles’ performance, leading to improved convergence and solution quality. These advancements in PSO continue to push the boundaries of optimization capabilities, making it a valuable tool for tackling real-world challenges. A brief overview of the advancements in the PSO algorithm is presented in Table 2. Over time, different variants of the PSO algorithm have been introduced, which are discussed in the subsequent sections. Table 2. Advancements in the PSO algorithm. Year Advancement 1995 Introduction of PSO algorithm by Kennedy and Eberhart [1,82] 1997 Inclusion of inertia weight to balance exploration and exploitation [83] 1998 Exploration of PSO variants such as constriction factor approach [84] 1999 Incorporation of adaptive parameter settings for improved performance [85] 2001 Multi-objective PSO developed for handling optimization problems with multiple conflicting objectives [86] 2003 Hybridization of PSO with other metaheuristic or local search algorithms [87] 2004 Introduction of dynamic PSO variants to adapt to changing environments [88] 2006 Application of PSO in solving complex real-world problems, such as the optimization of neural networks and data clustering [89] 2008 Development of parallel and distributed PSO algorithms for enhanced computational efficiency [90] 2010 Integration of PSO with machine learning techniques for improved optimization and prediction tasks [91] 2012 Self-adaptive PSO algorithms introduced to dynamically adjust algorithm parameters during optimization [92] 2014 Improved PSO variants focusing on handling dynamic and uncertain environments [93] 2016 Application of PSO in feature selection, image processing, and bioinformatics problems [94] 2018 Exploration of hybrid PSO algorithms with deep learning models for enhanced optimization and decision-making [95,96] 2020 Advancements in multi-objective PSO algorithms for solving complex optimization problems with conflicting objectives [ 97 ] 2022 Development of PSO variants incorporating social-network-inspired behaviors for collective decision-making and coordination [98] 3.6.1. Ring Topology in Particle Swarm Optimization The ring topology in PSO is a variation of the algorithm where the particles are arranged in a circular ring structure instead of a fully connected network [ 99 ]. Figure 4 shows the settings of the ring topology in PSO. In this topology, each particle is only connected to its immediate neighbors, creating a cyclic structure [ 100 ]. In the ring topology, the communication and information sharing among particles are limited to the adjacent neighbors. This arrangement allows for a more-localized interaction, as each particle only Sensors 2023,23, 7710 16 of 28 Table 4. Convergence of hybrid PSO with different values of N. Number of Iterations MSE (dB) N= 10 MSE (dB) N= 20 MSE (dB) N= 40 MSE (dB) N= 60 0 20 20 20 20 50 −12 −15 −28 −29 100 −20 −23 −29 −30 200 −21 −25 −29.5 −30 300 −19.5 −24.5 −29 −30.5 400 −20.5 −25 −29.5 −31 500 −22 −26 −30 −31 The performance of least mean squares (LMS) degrades compared to hybrid PSO due to its limited ability to handle nonlinear and non-convex optimization problems. Hybrid PSO incorporates global search capabilities and adaptive techniques, providing better convergence and optimization results. In the conducted experiment, the performance of three different optimization techniques LMS, the PSO VCF, and HPSO was evaluated in the context of a digital communication channel. The experiment involved varying SNR levels to simulate different channel conditions. The SNR represents the ratio of signal power to noise power and serves as a key factor in determining the quality of communication in noisy environments. For each SNR level, the BER was measured using the three optimization techniques: LMS, the PSO VCF, and HPSO. Table 5shows the comparison of LMS, the PSO variable constriction factor (VCF), and HPSO. A lower BER signifies better channel performance, indicating fewer errors in received bits. The comparison of the BER values among the three techniques at different SNR levels provides insights into their respective abilities to mitigate errors and enhance communication quality. Table 5. BER performance of LMS and HPSO. SNR LMS PSO VCF HPSO 0 0.8922 0.8929 0.8929 2 0.8017 0.8019 0.8019 4 0.8402 0.8402 0.8402 6 0.8051 0.805 0.805131 8 0.74283 0.7428 0.7427 10 0.65015 0.6505 0.6505 12 0.5526 0.5527 0.5527 14 0.4026 0.4026 0.4026 The experiment revolved around evaluating four distinct PSO variants using the sphere function. The sphere function, a common optimization benchmark, calculates the sum of squared differences between the candidate solution and the optimal solution. The aim was to gauge the performance of these PSO variants in terms of mean function values across different dimensions, i.e., 30 and 60. Lower mean function values signify moreproficient optimization, thereby enabling a comparative analysis of the PSO techniques’ effectiveness in searching for optimal solutions. The experimental results demonstrated the superior performance of hybrid PSO compared to other variants of PSO in various optimization tasks. In a comparative study, different PSO variants, including standard PSO, adaptive PSO, and HPSO, were evaluated for their convergence speed and solution quality. The results revealed that HPSO outperformed the other variants in terms of both convergence speed and solution quality. HPSO demonstrated faster convergence, Sensors 2023,23, 7710 17 of 28 reaching the optimal or near-optimal solution more quickly compared to standard PSO and adaptive PSO. This was attributed to HPSO’s ability to balance exploration and exploitation through the combination of particle interactions and adaptive parameters. The solution quality achieved by HPSO was consistently superior to other variants. The algorithm’s hybrid nature, incorporating elements of both particle swarm optimization and local search techniques, allowed for better exploration of the search space, leading to improved solutions. HPSO effectively balanced global exploration to escape local optima with local exploitation to refine solutions, resulting in enhanced overall performance. Table 6shows the results of 30D particle convergence. Table 6. The 30D particles’ convergence comparison. f(x) fl f2 Value Mean Iterations Comp Mean Iterations Comp PSO 7.1e−2 500 100% 55.44 500 100% PSO-D 5.706e−53 500 100% 0 264 100% PSO-DE 6.35e−20 412 70.3% 293 66.90% DMS 0.71 (2%) 500 100% 37.97 500 100% DIMS-D 3.646e−54 500 100% 0 273 100% DAIS-DE 3.85e−20 392 69.88% 0 320 66.48% CL 1.056e−47 500 60% 0 312 60% CL-D 3.486e−51 500 60% 0 279 60% CEDE 7.19e−19 319 39.2% 0 258 37.63% HP 1.0486e−5 50 80% 29.56 500 80% HP-D 4.906e − 111 500 80% 0 87 80% HP-DE2 5.216e−15 55 3.99% 1.18e−13 227 12.80% Table 7shows the convergence performance using 60D particles. The findings suggested that HPSO is a robust and effective optimization algorithm, which can outperform other PSO variants in various applications. Its ability to strike a balance between exploration and exploitation, along with the integration of local search techniques provide HPSO a competitive advantage. The superior performance of HPSO makes it a promising choice for optimization tasks where fast convergence and high-quality solutions are desired, such as channel adaptive equalization in communication systems, where accurate estimation and compensation of channel distortion are crucial for reliable data transmission. The evaluated dimensions ranged from 10 to 100. Lower mean function values indicate better optimization performance. The experiment involved applying these PSO variants to the sphere function, a well-known optimization benchmark. The sphere function computes the sum of squared differences between a candidate solution and the optimal solution. This experiment aimed to compare the effectiveness of the PSO techniques in achieving optimal solutions within different dimensional spaces. Sensors 2023,23, 7710 18 of 28 Table 7. The 60D particles’ convergence comparison. f(x) fl f2 Value Mean Iterations Comp Mean Iterations Com PSO x x x 150.12 (98%) 500 100% PSO-D 1.266e−53 500 100% 0 273 100% PSO-DE 1.46e−19 446 70.29% 0 291 66.94% DMS x x x 164.86 (62%) 500 100% DMS-D 1.616e−54 500 100% 0.00% 269 100% DMS-DE 9.58e−20 420 69.90% 0% 304 66.49% CL 6.546e−44 500 60% 115% 500 60% CL-D 52 500 60% 0.00% 275 60% CEDE 1.42e−18 310 39.19% 0% 259 37.59% HP 0.16 500 80% 6905% 500 80% HP-D 6.566e−106 500 80% 0.00% 92 80% HP-DE2 9.9e−15 3.98% 1.52e−228 228 12.78% Table 8shows the mean function value (MFV) of different PSO algorithms. The simulation results consistently demonstrated the robustness and effectiveness of HPSO across various domains and problem types. Whether applied to engineering design optimization, function optimization, or other complex tasks, HPSO consistently outperformed the competing algorithms. These findings establish HPSO as a promising optimization approach that can provide significant benefits in terms of convergence speed and solution quality, making it an attractive choice for numerous real-world optimization problems. Table 8. Mean function value of different PSO algorithms. No.of Dimensions PSO DPSO CPSO HPSO 10 −185.37 −156.31 −300 −50.10100664 20 −81.951 −60.194 −296.087 −33.131 30 −48.781 −32.039 −281.737 −30.1019 40 −31.22 −19.417 −45.652 −27.6767 50 −18.537 −7.767 −3.9130 −26.4646 60 −18.537 −3.8835 −1.396 −26.2623 70 −13.659 −1.9418 −2.6628 −24.8481 80 −7.8049 −3.8835 −3.9138 −24.2424 90 −6.8293 0 −5.2173 −24.040 100 −4.878 0 0 −22.4242 The simulation results consistently demonstrated the robustness and effectiveness of HPSO across various domains and problem types. Whether applied to engineering design optimization, function optimization, or other complex tasks, HPSO consistently outperformed the competing algorithms. Tables 9and 10 present the MSE values under different signal-to-noise ratio (SNR) levels, which indicate the quality of the channel’s output signal. Lower MSE values signify better performance, indicating a closer approximation to the desired output. A negative MSE value can be an artifact of data representation or computation. Sensors 2023,23, 7710 19 of 28 Table 9. MSE performance for the linear channel. SNR LMS PSO-CCF PSO-VCF HPSO 0−0.0515 1.4433 5.051546 5.15464 50 −5.2062 −11.289 −13.9691 −18.0928 100 −8.6082 −11.753 −14.3299 −18.0928 150 −10.876 −11.907 −14.1237 −18.4021 200 −12.68 −11.598 −14.1237 −18.4536 250 −13.66 −11.804 −14.2268 −18.7113 300 −14.794 −11.804 −14.1237 −18.6598 350 −14.897 −11.959 −14.1237 −18.6598 400 −14.794 −11.753 −14.1753 −18.6598 450 −14.794 −11.753 −13.9691 −18.6082 500 −14.948 −11.959 −14.3814 −18.4021 Comparing the techniques, it is evident that, under the given SNR conditions, the PSO-CCF, PSO-VCF, and HPSO methods consistently outperformed the basic LMS method, showcasing their efficacy in optimizing the adaptive filtering process for a linear channel. Among these three advanced methods, HPSO tended to yield the lowest MSE, suggesting its potential to provide the best approximation to the desired signal under different SNR scenarios. These findings establish HPSO as a promising optimization approach that can provide significant benefits in terms of convergence speed and solution quality, making it an attractive choice for numerous real-world optimization problems. The experiment was conducted for a linear time-invariant (LTI) system and nonlinear digital channel model having a sphere and cubic function model. The LTI system showed a linear mapping between the input and output signals, while time invariance indicated that the system will produce the same output signals if an input is used now or T seconds later, except for the time delay. The results showed that HPSO had better performance as compared to all other techniques that exist in the literature. Table 10. MSE performance for the nonlinear channel. SNR LMS PSO-CCF PSO-VCF HPSO 0−0.06568 4.9835 5.02463 12.2088 50 −2.159 −4.41707 −5.6075 −8.31691 100 −3.2676 −4.499 −5.8928 −8.6452 150 −3.6319 −4.622 −5.894 −8.6042 200 −3.9244 −4.4589 −5.8535 −8.6863 250 −4.1297 −4.4170 −5.93592 −8.6065 300 −4.70443 −4.41785 −5.85384 −8.6042 350 −4.8275 −4.2939 −5.89628 −8.76025 400 −4.78296 −4.2527 −5.77185 −8.604 450 −4.786296 −4.37602 −5.689 −8.8095 500 −4.745 −4.25287 −5.6077 −8.6863 5. HPSO: Best-Fit Solution for Adaptive Filtering It can be seen from the above experiments that the performance of HPSO was superlative as compared to other optimization techniques. HPSO is an effective approach for Sensors 2023,23, 7710 20 of 28 channel adaptive equalization, leveraging its global search capability to optimize equalizer coefficients and enhance the performance of communication systems by mitigating the effects of channel distortion and inter-symbol interference. This section will elaborate on the advantages, issues, and challenges faced by the HPSO in the optimization of adaptive filters. 5.1. Advantages of HPSO 5.1.1. Exploiting Complementary Techniques Hybrid PSO allows for the integration of different optimization algorithms or problemsolving methods that excel in different aspects. By combining their strengths, hybrid PSOs can overcome the limitations of individual algorithms and achieve better performance. For example, hybridizing PSO with genetic algorithms can leverage the exploration capabilities of both algorithms, leading to improved diversity and convergence toward optimal solutions. 5.1.2. Enhanced Global and Local Search Hybrid PSO combines the global search ability of PSO with the local search capabilities of other techniques. This integration allows for efficient exploration of the search space, enabling the algorithm to quickly identify promising regions and converge towards optimal solutions. The hybrid approach benefits from the balance between global exploration and local exploitation, providing better search efficiency. 5.1.3. Adapting to Problem Characteristics Different problems have distinct characteristics, such as multimodality, nonlinearity, or constraints. Hybrid PSO can be customized by selecting appropriate hybridization techniques based on the problem at hand. For instance, if a problem exhibits multimodality, combining PSO with niching techniques can enhance the algorithm’s ability to locate multiple optima. By adapting to the problem characteristics, hybrid PSO increases its effectiveness and robustness across diverse optimization scenarios. 5.1.4. Handling Complex Constraints Many real-world optimization problems involve complex constraints that must be satisfied. Hybrid PSOs can integrate constraint-handling techniques to ensure the feasibility of solutions. By incorporating constraint-handling mechanisms such as penalty functions, repair operators, or constraint satisfaction techniques, hybrid PSO can effectively handle constraints and generate feasible solutions, even in challenging constraint optimization problems. 5.1.5. Domain-Specific Knowledge Incorporation Hybrid PSO allows for the incorporation of problem-specific knowledge or heuristics. This customization leverages domain expertise to guide the search process toward morepromising regions of the search space. By integrating problem-specific knowledge, hybrid PSO can effectively exploit the problem structure and reduce the search space, leading to faster convergence and improved solution quality. 5.1.6. Performance Versatility Hybrid PSO’s flexibility enables it to adapt to various problem types and domains. It can be tailored to different optimization objectives, such as continuous optimization, discrete optimization, multi-objective optimization, or dynamic optimization. The ability to combine different algorithms and techniques makes hybrid PSO versatile, allowing it to tackle a wide range of optimization challenges effectively. Hybrid particle swarm optimization (HPSO) stands out in dealing with complex optimization landscapes, which can be quite tricky due to the presence of multiple possible solutions and complex patterns. Due to its cooperative and adaptable nature, HPSO is Sensors 2023,23, 7710 21 of 28 particularly adept at exploring a wide range of potential solutions and skillfully adjusting its search strategy to navigate through intricate fitness landscapes. When faced with multiobjective optimization challenges, in scenarios where adaptive filtering requires finding a balance between conflicting objectives, such as fast convergence and precise tracking, HPSO’s strength lies in its ability to seamlessly integrate diverse optimization techniques. This integration enables HPSO to harmonize these contrasting objectives effectively. HPSO proves valuable in hybrid approaches, especially when the optimization task requires the integration of specific problem-solving strategies or domain expertise. This becomes particularly beneficial when traditional optimization methods struggle to handle complex problems due to their intricacy. HPSO demonstrates adaptability in dynamic environments by dynamically adjusting parameters, incorporating updates based on local neighborhoods, and creating multiple swarms, allowing it to stay in sync with evolving optimization needs. This makes it a wellsuited choice for scenarios where the underlying system’s characteristics change over time. In cases where substantial computational power is needed, HPSO can be parallelized across multiple processors or nodes, leading to faster optimization processes. This is particularly useful for tasks that require real-time processing capabilities. HPSO is highly versatile in handling various problem types and dynamic conditions with limited prior knowledge. Its hybrid nature strikes a perfect balance between exploring a broad range of solutions and refining them, making it particularly advantageous in adaptive filtering tasks. 5.2. Challenges and Limitations of HPSO Hybrid PSO offers numerous advantages, as discussed earlier. However, like any optimization approach, it also faces certain challenges and limitations that should be taken into consideration. 5.2.1. Algorithm Complexity Hybrid PSO introduces additional complexity due to the integration of multiple optimization techniques or problem-solving methods. Managing the interactions and parameter settings between different components can be challenging. The design and implementation of a hybrid PSO algorithm require careful consideration to ensure effective cooperation and avoid conflicts between the integrated components. 5.2.2. Hybridization Overhead Integrating different optimization techniques or problem-solving methods in hybrid PSO may increase computational overhead. The hybridization process requires additional computational resources, such as memory and processing power. The impact on computational efficiency should be carefully assessed, especially when dealing with large-scale optimization problems or real-time applications. 5.2.3. Algorithm Selection and Tuning The success of hybrid PSO heavily depends on selecting appropriate optimization techniques or problem-solving methods to hybridize. Identifying the most-suitable algorithms or methods for a given problem can be challenging. Moreover, the tuning of parameters becomes more complex in hybrid PSO, as it involves optimizing the parameters of both the PSO algorithm and the integrated techniques. This parameter-tuning process requires expertise and extensive experimentation. 5.2.4. Integration Compatibility Integrating different optimization techniques or problem-solving methods in hybrid PSO might encounter compatibility issues. Some methods may require specific problem representations or assumptions that are not easily integrated with others. Ensuring compatibility and smooth integration of different components can be a challenge and may require adaptations or transformations to make them compatible. Sensors 2023,23, 7710 22 of 28 5.2.5. Increased Sensitivity to Problem Characteristics Hybrid PSO’s performance can be sensitive to the problem characteristics and the choice of hybridization techniques. The effectiveness of hybrid PSO heavily relies on the compatibility and synergy between the integrated components and the problem at hand. In some cases, the hybrid approach may not provide significant improvements compared to standalone PSO or other individual techniques, particularly if the problem does not align well with the selected hybridization methods. 5.2.6. Limited Generalizability Hybrid PSO’s effectiveness may be problem-dependent, meaning that the success observed in one problem domain may not necessarily translate to other domains. The performance of a hybrid PSO is heavily influenced by the specific problem structure, objectives, and constraints. Consequently, the development of a hybrid PSO algorithm that performs well across diverse problem domains requires careful customization and adaptation to each specific problem. 5.2.7. Increased Development and Maintenance Effort Hybrid PSO requires additional effort in the development and maintenance stages. Combining multiple algorithms or methods necessitates expertise in those areas. As new optimization techniques emerge, the integration and evaluation of their compatibility with hybrid PSO may require continuous effort and expertise, making the development and maintenance of hybrid PSO algorithms more demanding. 6. Conclusions and Future Directions 6.1. Conclusions Advancements in HPSO for adaptive equalization are pivotal for addressing limitations and enhancing practical implementation. Research must target simplifying the algorithm while upholding performance, optimizing hybridization for reduced computational overhead and automating parameter selection for enhanced efficiency. Efforts should prioritize enhancing integration compatibility, robustness, and generalizability across diverse equalization schemes and problem contexts. The utilization of user-friendly frameworks and libraries could streamline development and foster HPSO adoption, ultimately leading to improved BER performance in real-world adaptive equalization applications. 6.2. Future Directions To tackle the algorithm complexity, one potential direction is to simplify HPSO. This involves analyzing the algorithm’s components and identifying areas where complexity can be reduced without compromising performance. Streamlining the algorithm can make it more accessible and easier to implement in practical scenarios, enabling wider adoption of HPSO for adaptive equalization. Addressing hybridization overhead is another crucial future direction. Researchers can explore methods to optimize the integration of different optimization techniques in HPSO. This optimization can minimize computational overhead by intelligently determining when and how to employ local search mechanisms. By optimizing the hybridization process, the overall efficiency of HPSO can be improved, making it more suitable for real-time adaptive equalization applications. Automated parameter selection is another promising direction to overcome the challenges associated with algorithm selection and tuning. By developing automated methods, such as metaheuristic optimization or machine learning algorithms, the task of selecting appropriate parameter values can be automated. This enables HPSO to adapt and optimize its parameters based on the specific adaptive equalization problem at hand, reducing the manual effort and subjectivity involved in parameter tuning. Integration compatibility is a significant limitation that can be addressed through future research efforts. Investigating ways to enhance the compatibility of HPSO with Sensors 2023,23, 7710 23 of 28 different equalization schemes and systems is essential. Developing adaptive mechanisms that seamlessly integrate HPSO with diverse equalization techniques and architectures can significantly improve its effectiveness and versatility in adaptive equalization tasks. Measuring the stability of HPSO is another important avenue for future work. We intend to incorporate Monte Carlo to measure the stability of HPSO during the convergence process by obtaining the standard deviation and mean value of the MSE curve. Enhancing the robustness and generalizability of HPSO is another crucial direction. Research can focus on reducing the algorithm’s sensitivity to problem characteristics and environmental conditions. By developing mechanisms to handle diverse channel conditions, noise levels, and signal variations, HPSO can become more reliable and applicable in real-world adaptive equalization scenarios. Efforts can be directed toward reducing the development effort required for HPSO implementation. This can involve the creation of user-friendly software frameworks, libraries, or toolkits that provide pre-defined implementations of HPSO for adaptive equalization. By simplifying the development process, researchers and practitioners can more readily adopt and utilize HPSO, accelerating its application and impact in the field of adaptive equalization. Author Contributions: Conceptualization, A.K., I.S.; methodology, S.G.K.; software, M.A.L.F., J.C.G.; validation, I.A.; formal analysis, A.K., S.G.K.; investigation, I.d.l.T.D., J.C.G.; resources, M.A.L.F.; data curation, I.S., S.G.K.; writing—original draft preparation, A.K., I.S.; writing—review and editing, I.A.; visualization, M.A.L.F., J.C.G.; supervision, I.A.; project administration, I.d.l.T.D.; funding acquisition, I.d.l.T.D. All authors have read and agreed to the published version of the manuscript. Funding: This study is supported by the European University of Atlantic. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. References 1. Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN’95-International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948. 2. Djaneye-Boundjou, O.S.E. Particle Swarm Optimization Stability Analysis. Doctoral’s Dissertation, University of Dayton, Dayton, OH, USA, 2013. 3. Shi, Y. Particle swarm optimization. IEEE Connect. 2004,2, 8–13. 4. Minasian, A.A.; Bird, T.S. Particle swarm optimization of microstrip antennas for wireless communication systems. IEEE Trans. Antennas Propag. 2013,61, 6214–6217. [CrossRef] 5. Banks, A.; Vincent, J.; Anyakoha, C. A review of particle swarm optimization. Part I: Background and development. Nat. Comput. 2007,6, 467–484. [CrossRef] 6. Parsopoulos, K.E.; Vrahatis, M.N. Multi-objective particles swarm optimization approaches. In Multi-Objective Optimization in Computational Intelligence: Theory and Practice; IGI Global: Hershey, PA, USA, 2008; pp. 20–42. 7. Vaseghi, S.V. Advanced Digital Signal Processing and Noise Reduction; John Wiley & Sons: Hoboken, NJ, USA, 2008. 8. Stojanovic, M.; Catipovic, J.A.; Proakis, J.G. Phase-coherent digital communications for underwater acoustic channels. IEEE J. Ocean. Eng. 1994,19, 100–111. [CrossRef] 9. Gustafsson, R. Combating Intersymbol Interference and Cochannel Interference in Wireless Communication Systems. Ph.D. Thesis, Blekinge Institute of Technology, Karlskrona, Sweden, 2003. 10. Stonick, J.T.; Wei, G.Y.; Sonntag, J.L.; Weinlader, D.K. An adaptive PAM-4 5-Gb/s backplane transceiver in 0.25µ m CMOS. IEEE J. Solid-State Circuits 2003,38, 436–443. [CrossRef] 11. Hamamreh, J.M.; Furqan, H.M.; Arslan, H. Classifications and applications of physical layer security techniques for confidentiality: A comprehensive survey. IEEE Commun. Surv. Tutor. 2018,21, 1773–1828. [CrossRef] 12. Liu, H.; Wang, Y.; Xu, C.; Chen, X.; Lin, L.; Yu, Y.; Wang, W.; Majumder, A.; Chui, G.; Brown, D.; et al. A 5-Gb/s serial-link redriver with adaptive equalizer and transmitter swing enhancement. IEEE Trans. Circuits Syst. I Regul. Pap. 2013 ,61, 1001–1011. [CrossRef] 13. Qureshi, S.U. Adaptive equalization. Proc. IEEE 1985,73, 1349–1387. [CrossRef] Sensors 2023,23, 7710 24 of 28 14. Katayama, M. Introduction to robust, reliable, and high-speed power-line communication systems. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 2001,84, 2958–2965. 15. Vahidi, V. Uplink data transmission for high speed trains in severe doubly selective channels of 6G communication systems. Phys. Commun. 2021,49, 101489. [CrossRef] 16. Rojo-Álvarez, J.L.; Martínez-Ramón, M.; Munoz-Mari, J.; Camps-Valls, G. Digital Signal Processing with Kernel Methods; John Wiley & Sons: Hoboken, NJ, USA, 2018. 17. Ali, S.; Saad, W.; Rajatheva, N.; Chang, K.; Steinbach, D.; Sliwa, B.; Wietfeld, C.; Mei, K.; Shiri, H.; Zepernick, H.J.; et al. 6G white paper on machine learning in wireless communication networks. arXiv 2020, arXiv:2004.13875. 18. Pelekanakis, K.; Chitre, M. Robust equalization of mobile underwater acoustic channels. IEEE J. Ocean. Eng. 2015 ,40, 775–784. [CrossRef] 19. Stojanovic, M.; Beaujean, P.P.J. Acoustic communication. In Springer Handbook of Ocean Engineering; Springer: Berlin/Heidelberg, Germany, 2016; pp. 359–386. 20. Esmaiel, H.; Jiang, D. Multicarrier communication for underwater acoustic channel. Int’L Commun. Netw. Syst. Sci. 2013,6, 361. 21. Arablouei, R.; Do˘gançay, K. Low-complexity adaptive decision-feedback equalization of MIMO channels. Signal Process. 2012 , 92, 1515–1524. [CrossRef] 22. Lucky, R. Techniques for adaptive equalization of digital communication systems. Bell Syst. Tech. J. 1966 ,45, 255–286. [CrossRef] 23. Dixit, S.; Nagaria, D. LMS adaptive filters for noise cancellation: A review. Int. J. Electr. Comput. Eng. (IJECE) 2017 ,7, 2520–2529. [CrossRef] 24. Bhotto, Z.A.; Antoniou, A. A family of shrinkage adaptive-filtering algorithms. IEEE Trans. Signal Process. 2012 ,61, 1689–1697. [CrossRef] 25. Khan, A.A.; Shah, S.M.; Raja, M.A.Z.; Chaudhary, N.I.; He, Y.; Machado, J.T. Fractional LMS and NLMS algorithms for line echo cancellation. Arab. J. Sci. Eng. 2021,46, 9385–9398. [CrossRef] 26. Zhu, Z.; Gao, X.; Cao, L.; Pan, D.; Cai, Y.; Zhu, Y. Analysis on the adaptive filter based on LMS algorithm. Optik 2016 , 127, 4698–4704. [CrossRef] 27. Ao, W.; Xiang, W.Q.; Zhang, Y.P.; Wang, L.; Lv, C.Y.; Wang, Z.H. A new variable step size LMS adaptive filtering algorithm. In Proceedings of the 2012 International Conference on Computer Science and Electronics Engineering, Hangzhou, China, 23–25 March 2012; Volume 2, pp. 265–268. 28. Atapattu, L.; Arachchige, G.M.; Ziri-Castro, K.; Suzuki, H.; Jayalath, D. Linear adaptive channel equalization for multiuser MIMO-OFDM systems. In Proceedings of the Australasian Telecommunication Networks and Applications Conference (ATNAC) 2012, Brisbane, QLD, Australia, 7–9 November 2012; pp. 1–5. 29. Reddy, B.S.; Krishna, V.R. Implementation of Adaptive Filter Based on LMS Algorithm. Int. J. Eng. Res. Technol. 2013,2, 1–4. 30. Douglas, S.C. Adaptive filtering. In Digital Signal Processing Fundamentals; Nova Science Publishers: Hauppauge, NY, USA, 2017. 31. Jaar, F.Y.S. Recursive Inverse Adaptive Filtering for Fading Communication Channels. Master’s Thesis, Eastern Mediterranean University (EMU)-Do˘gu Akdeniz Üniversitesi (DAÜ), Gazimagusa, Cyprus, 2019. 32. Khokhar, M.J.; Younis, M.S. Development of the RLS algorithm based on the iterative equation solvers. In Proceedings of the 2012 IEEE 11th International Conference on Signal Processing, Beijing, China, 21–25 October 2012; Volume 1, pp. 272–275. 33. Zhang, S.; Zhang, J. An RLS algorithm with evolving forgetting factor. In Proceedings of the 2015 Seventh International Workshop on Signal Design and its Applications in Communications (IWSDA), Bengaluru, India, 14–18 September 2015; pp. 24–27. 34. Claser, R.; Nascimento, V.H.; Zakharov, Y.V. A low-complexity RLS-DCD algorithm for Volterra system identification. In Proceedings of the 2016 24th European Signal Processing Conference (EUSIPCO), Budapest, Hungary, 29 August–2 September 2016; pp. 6–10. 35. Hakvoort, W.B.; Beijen, M.A. Filtered-error RLS for self-tuning disturbance feedforward control with application to a multi-axis vibration isolator. Mechatronics 2023,89, 102934. [CrossRef] 36. Wang, D.; Tan, D.; Liu, L. Particle swarm optimization algorithm: An overview. Soft Comput. 2018,22, 387–408. [CrossRef] 37. Bansal, S.; Ali, H.; Singh, S. Performing Adaptive Channel Equalization by Hybrid Differential Evolution Particle Swarm Optimization. Int. Res. Appl. Sci. Eng. Tech. 2014,2. 38. Ho, S.; Yang, S.; Ni, G.; Wong, H. A particle swarm optimization method with enhanced global search ability for design optimizations of electromagnetic devices. IEEE Trans. Magn. 2006,42, 1107–1110. [CrossRef] 39. Das, S.; Abraham, A.; Konar, A. Particle Swarm Optimization and Differential Evolution Algorithms: Technical Analysis, Applications and Hybridization Perspectives. In Advances of Computational Intelligence in Industrial Systems; Liu, Y., Sun, A., Loh, H.T., Lu, W.F., Lim, E.P., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; pp. 1–38. [CrossRef] 40. Rao, R.V.; Savsani, V.J.; Rao, R.V.; Savsani, V.J. Advanced Optimization Techniques; Springer: Berlin/Heidelberg, Germany, 2012. 41. Vikhar, P.A. Evolutionary algorithms: A critical review and its future prospects. In Proceedings of the 2016 International Conference on Global Trends in Signal Processing, Information Computing and Communication (ICGTSPICC), Jalgaon, India, 22–24 December 2016; pp. 261–265. 42. Qiu, T.; Liu, J.; Si, W.; Wu, D.O. Robustness optimization scheme with multi-population co-evolution for scale-free wireless sensor networks. IEEE/ACM Trans. Netw. 2019,27, 1028–1042. [CrossRef] 43. Kundu, D.; Nijhawan, G. Performance analysis of adaptive channel equalizer using LMS, various architecture of ANN and GA. Int. J. Appl. Eng. Res. 2017,12, 12682–12692. Sensors 2023,23, 7710 25 of 28 44. Xu, W.; Zhong, Z.; Be’ery, Y.; You, X.; Zhang, C. Joint neural network equalizer and decoder. In Proceedings of the 2018 15th International Symposium on Wireless Communication Systems (ISWCS), Lisbon, Portugal, 28–31 August 2018; pp. 1–5. 45. Wu, X.; Huang, Z.; Ji, Y. Deep neural network method for channel estimation in visible light communication. Opt. Commun. 2020 , 462, 125272. [CrossRef] 46. Mathews, A.B.; Mathews, A.B.; Agees Kumar, C. A Non-Linear Improved CNN Equalizer with Batch Gradient Decent in 5G Wireless Optical Communication. IETE J. Res. 2023, 1–13. [CrossRef] 47. Erpek, T.; O’Shea, T.J.; Sagduyu, Y.E.; Shi, Y.; Clancy, T.C. Deep learning for wireless communications. In Development and Analysis of Deep Learning Architectures; Springer: Berlin/Heidelberg, Germany, 2020; pp. 223–266. 48. Wang, T.; Wen, C.K.; Wang, H.; Gao, F.; Jiang, T.; Jin, S. Deep learning for wireless physical layer: Opportunities and challenges. China Commun. 2017,14, 92–111. [CrossRef] 49. Moradi, R.; Berangi, R.; Minaei, B. A survey of regularization strategies for deep models. Artif. Intell. Rev. 2020 ,53, 3947–3986. [CrossRef] 50. Tripathi, S.; Ikbal, M.A. Optimization of lms algorithm for adaptive filtering using global optimization techniques. Int. J. Comput. Appl. 2015,975, 8887. [CrossRef] 51. Wang, H.; Sun, H.; Li, C.; Rahnamayan, S.; Pan, J.S. Diversity enhanced particle swarm optimization with neighborhood search. Inf. Sci. 2013,223, 119–135. [CrossRef] 52. Najafabadi, M.M.; Villanustre, F.; Khoshgoftaar, T.M.; Seliya, N.; Wald, R.; Muharemagic, E. Deep learning applications and challenges in big data analytics. J. Big Data 2015,2, 1. [CrossRef] 53. Marini, F.; Walczak, B. Particle swarm optimization (PSO). A tutorial. Chemom. Intell. Lab. Syst. 2015,149, 153–165. [CrossRef] 54. Pal, S.K.; Rai, C.; Singh, A.P. Comparative study of firefly algorithm and particle swarm optimization for noisy nonlinear optimization problems. Int. J. Intell. Syst. Appl. 2012,4, 50. 55. Huang, C.L.; Dun, J.F. A distributed PSO–SVM hybrid system with feature selection and parameter optimization. Appl. Soft Comput. 2008,8, 1381–1391. [CrossRef] 56. Tuppadung, Y.; Kurutach, W. Comparing nonlinear inertia weights and constriction factors in particle swarm optimization. Int. J. Knowl.-Based Intell. Eng. Syst. 2011,15, 65–70. [CrossRef] 57. Eberhart, R.C.; Shi, Y. Comparing inertia weights and constriction factors in particle swarm optimization. In Proceedings of the 2000 congress on evolutionary computation. CEC00 (Cat. No. 00TH8512), La Jolla, CA, USA, 16–19 July 2000; Volume 1, pp. 84–88. 58. Zhan, Z.H.; Zhang, J.; Li, Y.; Chung, H.S.H. Adaptive particle swarm optimization. IEEE Trans. Syst. Man Cybern. 2009 ,39, 1362–1381. [CrossRef] 59. Urade, H.S.; Patel, R. Study and analysis of particle swarm optimization: A review. In Proceedings of the IJCA Proceedings on 2nd National Conference on Information and Communication Technology NCICT (4): 31–33 November 2011; pp. 1–5. 60. Du, K.L.; Swamy, M.; Du, K.L.; Swamy, M. Particle swarm optimization. In Search and Optimization by Metaheuristics: Techniques and Algorithms Inspired by Nature; IEEE: New York, NY, USA, 2016; pp. 153–173. 61. Das, S.; Konar, A.; Chakraborty, U.K. Two improved differential evolution schemes for faster global search. In Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation, Washington, DC, USA, 25–29 June 2005; pp. 991–998. 62. Qin, Q.; Cheng, S.; Zhang, Q.; Li, L.; Shi, Y. Particle swarm optimization with interswarm interactive learning strategy. IEEE Trans. Cybern. 2015,46, 2238–2251. [CrossRef] 63. Kuo, R.; Zulvia, F. Automatic clustering using an improved particle swarm optimization. J. Ind. Intell. Inf. 2013 ,1, 1–6. [CrossRef] 64. Ali Ghorbani, M.; Kazempour, R.; Chau, K.W.; Shamshirband, S.; Taherei Ghazvinei, P. Forecasting pan evaporation with an integrated artificial neural network quantum-behaved particle swarm optimization model: A case study in Talesh, Northern Iran. Eng. Appl. Comput. Fluid Mech. 2018,12, 724–737. [CrossRef] 65. Cuevas, E.; Fausto, F.; González, A.; Cuevas, E.; Fausto, F.; González, A. A swarm algorithm inspired by the collective animal behavior. New Advancements in Swarm Algorithms: Operators and Applications; Springer : Berlin/Heidelberg, Germany, 2020; pp. 161–188. 66. Islam, M.Z.; Sajjad, G.S.; Rahman, M.H.; Dey, A.K.; Biswas, M.A.M.; Hoque, A. Performance comparison of modified LMS and RLS algorithms in de-noising of ECG signals. Int. J. Eng. Technol 2012,2, 466–468. 67. Dunjko, V.; Briegel, H.J. Machine learning & artificial intelligence in the quantum domain: A review of recent progress. Rep. Prog. Phys. 2018,81, 074001. [PubMed] 68. Variddhisaï, T.; Mandic, D.P. On an RLS-like LMS adaptive filter. In Proceedings of the 2017 22nd International Conference on Digital Signal Processing (DSP), London, UK, 23–25 August 2017; pp. 1–5. 69. Mahmutoglu, Y.; Turk, K.; Tugcu, E. Particle swarm optimization algorithm based decision feedback equalizer for underwater acoustic communication. In Proceedings of the 2016 39th International Conference on Telecommunications and Signal Processing (TSP), Vienna, Austria, 27–29 June 2016; pp. 153–156. 70. Xu, J.; Yue, X.; Xin, Z. A real application of extended particle swarm optimizer. In Proceedings of the 2005 5th International Conference on Information Communications & Signal Processing, Bangkok, Thailand, 6–9 December 2005; pp. 46–49. 71. Arumugam, M.S.; Rao, M. On the improved performances of the particle swarm optimization algorithms with adaptive parameters, cross-over operators and root mean square (RMS) variants for computing optimal control of a class of hybrid systems. Appl. Soft Comput. 2008,8, 324–336. [CrossRef]