Finite Impulse Response Filter Design Using Fuzzy Logic-Based Diversity-Controlled Self-Adaptive Differential Evolution

Te design of fnite impulse response (FIR) flters involves the estimation of efective flter coefcients, making the designed flter exhibit infnite stopband attenuation and have a fat-shaped passband. Te few conventional flter design methods such as impulse response truncation (IRT) and windowing technique exhibit undesirable characteristics owing to the Gibbs phenomenon, thus making them unsuitable for various practical complexities. Tis research work employs the fuzzy logic-based diversity-controlled self-adaptive diferential evolution algorithm (FLDCSaDE) for the design of FIR band stop (BS) and high pass (HP) flters. In order to validate the results of the proposed technique, various population-based evolutionary computing techniques such as the covariance matrix adaptation evolution strategy ( CMAES), diferential evolution (DE), self-adaptive diferential evolution (SaDE), and Jaya algorithm have also been applied for determining the efective flter coefcients. Te performance of the various algorithms has been analysed and compared based on the parameters such as stopband attenuation, passband attenuation, and ripples. Te simulation results show that the FLDCSaDE algorithm outperforms other evolutionary algorithms having 4% and 1.5% lower ripples than the SaDE algorithm for high pass and band stop flters, respectively. Experimental results depict that the performance of the fuzzy approach causes positioning and tracking accuracy obtained to be improved by 27% and the corresponding false positive rate (FPR) is substantially reduced to 0.11 from the mean amplitude value obtained from the fuzzy approach in the frequency response. Te frequency response obtained from the FLDCSaDE algorithm is close to the ideal response of the BS and HP FIR flters.


Introduction
A signal can be termed as a physical parameter that changes with space, time, or any other quantity. In general, signals are responsible for the movement of information which sprouts in almost every feld of science and technology. Signals are categorized into two sections, namely, discrete and continuous-time signals, depending on the nature of space and time. Given that most of the inherently occurring signals are continuous in nature (analog), an efective fltering method of analog signals is a necessity. One such method of fltering is through digital signal processing (DSP).
Filters are the most extensively used systems for processing signals that are digital in nature. Tis can be employed to obtain the needed output spectral attributes by altering the input signal spectrum. Digital flters play a critical role in the communication and processing of signals. Te main advantage of digital flters is higher reliability, accuracy, and lower tolerances with lower noises leading to increased efciency. Tey are robust to changes in characteristics which make them suitable for variable multiple applications such as voice signal synthesis and analysis [1], image processing [2], removing random noise from seismic data [3], and rectifying many biomedical signals [4] such as EEG, MRIs, and ECGs. Digital flters can be further classifed into IIR and FIR flters based on the response. FIR flters exhibit a linear phase which is inherently stable, nonrecursive in nature, and less sensitive in utilizing bounded word duration efects. Te abovementioned attributes make FIR flters more desirable despite the requirement of numerous coefcients as compared to infnite impulse response (IIR) flters where the memory requirement is high.
Tere are several traditional techniques available with the notion of designing FIR flters, among which the windowing approach is the most preferred technique. Selection of a suitable window function such as Hamming, Hanning, and Kaiser is dependent on several factors with respect to the desired frequency response such as variation in stopband attenuation, transition in width, and ripples in stopband and passband [5]. Frequency domain convolutions carried out in the windowing technique results in tapered rectangular edges, leading to ripples in passband and stopband and in turn limit the performance of the given FIR flter. Te obtained ripple is not uniformly distributed across a given rectangular window but in turn follows a pattern in which ripples decay as we move away from the path along the discontinuous points according to the side-lobe pattern observed from the window. Tolerance to given ripple behaviour can be observed by allowing more freedom to the stated ripple, thus reducing the complexity and improving the robustness of the given flter. However, in this way, the order of the flter gets reduced and it afects the transition from passband to stopband, thus deteriorating signal strength by interference [6].
In contrast to the traditional method, procedures involved in the digital FIR flter designing process could be conceived to be an optimization problem [7], alongside the intention to minimize the error function, which basically indicates an inconsistency in the flter designed from the intended response [8]. Classical methods such as the least square and gradient-based methods, optimizing L 1 and L 2 norms, can provide better passband response and high stopband attenuation, with minimal ripples [9,10]. However, these gradient-based methods can result only in local optimal solution and cannot support multiobjective optimization problems. Furthermore, these gradient descent methods have difculties in handling large dimension problems, due to the irregularities in the approximate gradients.
Due to the drawbacks of classical gradient-based methods, in recent times, many researchers have started to implement evolutionary and swarm-based algorithms to get global optimal solutions in designing digital FIR flters. Te use of evolutionary algorithms and intelligent swarm techniques are very well suited for handling nondiferentiable and multiobjective functions [11]. It is important in laying the foundation for understanding flter design with the use of diferent possible solution algorithms, namely, the search algorithms. Both the single objective and multiobjective formulations have their own set of advantages, wherein the single objective problems have lower constraints and complexity and the multiobjective formulation has better accuracy. Te adaptation of these formulations depends on the nature of the problem in consideration [12]. Many evolutionary and swarm techniques are developed in designing the FIR flter, and they are discussed as follows.
Te evolutionary process-based algorithms such as simulated annealing and genetic algorithms are implemented in FIR flter design [13]. In 2015, the simulated annealing method was employed to design the FIR flter with the objective of attaining a desired magnitude response of the flter [14]. Moreover, various variants of the genetic algorithms (GAs) such as real-coded GA and hybrid GA are also implemented to design the FIR flter [9,13,15].
Te swarm intelligence-based metaheuristic algorithms, one of which is called particle swarm optimization (PSO), was developed in the year 2016 [16]. Its greatest advantage is its simplicity in implementation and conceptual level, leading to highly utilized computational algorithms for the flter design. References [17][18][19][20]. Te linear phase FIR flter has been designed using PSO and GA by considering the feasible passband and stopband frequencies and the size of the passband and stopband ripples [21]. Craziness-based PSO was implemented in designing the 28 th and 36 th order band stop flter [22,23]. Te cat swarm optimization algorithm is applied to design the linear phase of the FIR flter to meet the desired frequency response characteristics [24]. Te linear phase multiband stop flter for the order of 40, 48, and 58 is developed using an improved cuckoo search PSO. Te adaptive cuckoo search algorithm (ACSA) is also developed to design a digital FIR flter design [25].
Further research through the years led to the development of much more refned techniques for estimating the flter coefcients. Te algorithms so found had to inherently displace the issue of global search space and replace it with the local search mechanism. One such algorithm was developed in 2005 and was called the covariance matrix adaptation evolution strategy (CMAES). It generates renewed population members by sampling from the probability distribution of a given search space. Te use of correlations between flter coefcients accelerates the convergence process because of the absence of derivatives [26]. However, when compared to diferential evolution (DE), the CMAES algorithm sufers from the nonstationary error that increases the noise component in the flter which if present in the magnitude response may remain undetected, thus decreasing the efciency of the flter. Te use of eigenvectors and covariance matrix to fnd the frequency response of the given flter increases the time complexity [27], especially for higher order matrix where the optimization of the flter coefcients becomes a complicated process.
Te Jaya algorithm solves the abovementioned problem as it is free of gradient, free of algorithm-specifc parameters, and the time complexity for flter optimization is much lesser for this algorithm. In terms of reducing mixed noise or ripples in the flter, Jaya outperformed other algorithms as shown in reference [28]. However, the convergence rate is not as efcient when compared to CMAES as its performance benchmark is lesser when applied to optimizing flter coefcients.
Furthermore, the FIR flter design has been designed by improving the candidate solution iteratively based on the evolutionary process using DE, proposed in 2020 [29]. Te major advancement in DE is fnding the true global minimum of a model search space from the coefcient set regardless of the initial parameter values [30], thus making the initialization process independent of the delay. In contrast to GA, the DE algorithm uses mutation operation, thus utilizing fewer control parameters for flter responses, namely, the magnitude responses for stopband and passband regions [31]. Low pass digital FIR flter parameters are identifed using the DE algorithm by minimizing the least mean square function [32]. Te FIR flter has been designed using DE in consideration of diferent word lengths, and the same is implemented using Cadence RTL Compiler (UMC 90 nm technology) [33]. However, there are inherent disadvantages of the decreasing search spaces because of the diversity of the possible movements in exploration. Te performance of DE sufers due to the predefned control parameters (CO and F) and mutation strategy. If the problem difers, mutation strategy and control parameters are to be fne-tuned for obtaining the consistent optimization performances. Due to this drawback of DE, many adaptive DE algorithms such as self-adaptive DE [34], DE [35], opposition-based DE [36], composite DE [37], random neighbours-based DE [38], and neighbourhood mutation and opposition-based learning (NBOLDE) [39] have been proposed recently. Te nuances of the diferent optimization algorithms are summarized in Table 1. Te basic understanding of the processes and parameters to determine the optimal flter coefcients were analysed. Te DE algorithm had an interesting approach to improving candidate solution and minimizing it using elitist replacement. Tis methodology of DE and SaDE is combined to avoid using the predefned parameters to improve the performance.
Tis paper proposes fuzzy logic-based diversitycontrolled self-adaptive diferential evolution (FLDCSaDE) for the frst time to design the FIR flter by overcoming the accuracy and performance limitations from the previous literature. Te superiority of the FLDCSaDE algorithm over other algorithms is analysed in terms of its ripple content, performance, attenuation, and others. Te algorithm overcomes the inherent problems of the evolutionary algorithm when designing the flter, namely, the dependency among the flter parameters, improper initialization of control parameters, and premature convergence. Te abovediscussed DE, CMAES, Jaya, SaDE, and FLDCSaDE algorithms are used in constrained and unconstrained real-life problems as their parameter requirements are much lesser than other algorithms, and hence, their computation makes their limitations negligible. FLDCSaDE employs fuzzy logic mainly for two purposes. One is to keep the diversity in the population, and the other is to obtain weights that would help narrow down the search space, thus fnding the optimal flter parameters at a much faster rate. Te designed flter possesses minimum magnitude error and ripples in passband and stopband. Te general design procedure involved in the design of the FIR flter using the evolutionary algorithm is discussed in Figure 1.
Although the conventional SaDE algorithm is used to generate success and failure rates based on four mutation strategies [40], it requires a total of K n strategies for choosing an optimal value for updating the CO value. Te use of fuzzy logic makes it feasible to update CO values based on the diversity of the population, thus surpassing the performance of SaDE and removing numerous types of noise and harmonics. Proper tuning of flter coefcients dynamically gives a way for better performance and at the same time improves the trade-of between adaptation and robustness. In case of improper tuning, the parameters increase the computational efort and complexity [39]. Te core notion of employing local search in the FLDCSaDE algorithm helps strike a balance between the two important driving forces imperative for any optimization algorithm which is exploitation and exploration. It is not required to look for optimal flter coefcients in each iteration since the search can be applied to any randomly chosen individual in the given population, and if the solution obtained is better than the one obtained in the previous generation, the individual is included in the given solution set, thus enhancing the time complexity. Tis helps obtain feasible FIR flter coefcients having an execution time much lesser than that of SaDE and lower ripple content, and thus, higher efciency shows that this algorithm can be applied to obtain the optimal FIR flter response.

Methodology
Te methodology for the implementation of the design of the fnite impulse response (FIR) flter has been listed as per the sequence in the following section: (1) Determination of objective function.
(2) Comparison of optimization algorithms. Tis research paper is organized beginning with the formulation of the flter design, objective function, and constraint parameters in Section 3. It is followed by an in-International Transactions on Electrical Energy Systems

FIR Filter Design
FIR-based flters are adopted for many practical applications because of their properties such as linear phase and inherent stability over IIR-based flters. At the forefront of designing an N th order FIR high pass flter which is optimal, the desired response of the flter H(e jω ) is compared with the ideal feedback H d (e jω ), specifed as follows: where the cut-of frequency of the desired flter is given by ω c . Utilizing the inverse discrete time-based Fourier transform (IDTFT) response of the given ideal flter and multiplying the result with a window function gives the impulse or delta response obtained for the flter h(n).
Taking the discrete-time Fourier transform (DTFT) for the given impulse response h(n) into account, the desired flter response H(e jω ) for the given FIR flter could be attained as follows: where the order of the given flter is given by N. Basically, for a linear phase type-I flter having coefcients which are symmetric and have bounded length, where 0 ≤ n ≤ N − 1, have the magnitude response, which is given by the following equation: where H r (e jω ) is the function with a real-valued response and M � (N − 1)/2. Trough the process of limiting H r (e jω ) with the response obtained for the ideal flter, the error function E(ω) can be found. By the defned L 2 norm given as the standard for designing a high pass flter, the error function E(ω) can be stated as follows: Tis paper aims at deriving the response for the stated flter through the utilization of the aforementioned error function E(ω) as an objective function in the optimization problem. Te main objective is to obtain a series of optimized symmetric coefcients that minimize the error function of the flter and to use those coefcients in designing a linear phase type-1 FIR flter for band stop and high pass modes. Te error function can be minimized by adopting any evolutionary algorithm to fnd the optimal flter coefcients for the design of the FIR band stop and high pass modes.

Evolutionary Algorithms
Tis section deals with the steps involved in the optimization process of various evolutionary algorithms applied in this paper.

Covariance Matrix Adaptation Evolution Strategy (CMAES). Nikolaus
Hansen developed an efcient evolution algorithm used as an optimizing tool for continuous search spaces with real objectives [41]. Tis algorithm was termed as the covariance matrix adaptation evolution strategy (CMAES). It is a derivative-free stochastic approach used specifcally for nonlinear problems [42], unlike other methods which are being based on derivatives failing to yield a solution due to various reasons such as improper search with sharp turns, numerous breaks in continuity, and local optimal solutions. Te CMAES algorithm is continuous in nature and can be used for various applications involving complex optimization numerical problems at a search region which is continuously nonconvex and nonlinear. It utilizes multivariate distribution samples N(m, C) which are constructed with the help of its mean value and its covariance matrix; it is symmetric, positive, and defnite in nature during the process of optimization. Tese samples can be taken into account for obtaining renewed members of the population. Te covariance matrix and the mean involving the given set of samples point out to the limits in the region of m ∈ R n and C ∈ R n×n , respectively. In the previous generation, by updating the value of "m," which corresponds International Transactions on Electrical Energy Systems to the translational displacement of the distribution, the possibility of obtaining the optimal solution is maximized. Te covariance matrix can be represented geometrically, with a distinct shape such as that of an isodensity ellipsoid. Te shape of this geometric representation, whose axis has length, corresponds to the respective eigenvalues of the covariance matrix. Te algorithm uses two diferent adaptation strategies, namely, CMA and step size adaptation. Te numerous steps used during the implementation of the algorithm are specifed as follows: Step 1. Sampling population Te search positions are found through the process of sampling a distribution consisting of multiple variables. For every search point belonging to the generation "g," we fnd the step size (σ (g) ), covariance matrix C (g) , and the mean value m (g) . From the abovementioned process, the new variables or individuals obtained are again sampled at the subsequent generation (g + 1) as follows: where X (g+1) k belongs to the g + 1 generation and is the k th ofspring belonging to the generation g + 1; population size is denoted by Ns.
Step 2. Recombination along with the selection process After completion of the sampling process, the new mean m (g+1) is obtained by selecting the top µ (weighted average) samples having the highest ftness among the given total population size (Ns). Te new mean becomes the weighted average of μ samples with the weight parameter w i and is given by the following equation: where μ ≤ Ns denotes the original population size, and subsequently, by equating w i � 1/μ, we calculate the mean for all μ samples and X (g+1) i: Ns denotes that among the given (Ns) sampling points, it is the i th best ranked individual.
Step 3. Adaptation of the covariance matrix C Tere are two more reliable but complex methods to update C, which are as follows: (i) Using Polyak's average to estimate the rank-µ update of C: It is based on updating the value of C using the previous history, (ii) Utilizing the evolution path-cumulation: Te second way is using an evolution path (p g c ), to log symbol information, (P c ) can be calculated using the standard conditions defned initially We can use P c to update the covariance matrix C: (iii) Combining two methods: Using the updated values of the covariance matrix from the two previous methods, we combine the values to derive the renewed formula for updating the fnal CMA for the covariance matrix (C (g+1) ) by using equations (7) and (10), with µ cov ≥ 1, rank-one update and weighting between rank-μ: Te eigen composition of the covariance matrix obeys the following: is given to the variance for the given optimized mass; c c � 4/(n + 4) is denoted as the learning ratio which can be used to fnd the combined efective step for the evolution path; µ cov defned a constraint for rank-μ update and rank-one weighting.
gives the learning rate of the updated covariance matrix C (g) , B (g) � n * n orthogonal matrix, and D (g) � n * n diagonal matrix. During the usage of population sizes for small ofspring, the rank-one update is found to be inefcient as it minimizes the evaluations of the functions [43].
Step 4. Controlling the given step-size For the purpose of updating the global step size (σ (g) ), we utilize the relation between the mean trajectory denoted by (p (g) σ ) [44]. Te evolution path for adaptive step size is computed by 6 International Transactions on Electrical Energy Systems where, c σ is defned as the learning for the given step size which is found from P σ , the path movement.
Step size (σ (g) ) is linked to the conjugate evolution path (p (g) σ ) and is given as follows: where I is an Identity matrix; N(0, I) denotes the normal distribution with unity covariance matrix and zero mean. E||N(0, I)|| defnes the expectation of the Euclidean normal distribution of N(0, I) with the distributed random vector initialized to zero p (0) σ � 0 initially. c σ � 10/(n + 20) denotes the timer horizon of evolution in the backward path and also c σ � μ eff + 2/n + μ eff + 3; the step-size damping factor is given by

Jaya Algorithm.
Te Jaya algorithm is a new populationbased metaheuristic optimization algorithm that is used to determine the optimal subset of features to improve the performance of the classifcation process. It combines the features of evolutionary algorithms and swarm-based intelligence. Te Jaya algorithm has the tendency to move to the best, i.e., it is closer to success, and avoids the worst solution obtained in the iteration [45]. Tis nature makes this algorithm victorious, and hence, the name Jaya is defned. It is derived from a Sanskrit word meaning "victory" [46]. Te advantage of the algorithm over other metaheuristic algorithms is that it requires only a few control parameters such as the maximum number of generations, population size, and the number of design variables that are common for all algorithms, and it is independent of algorithm-specifc parameters; therefore, it does not require extensive tuning, so we can avoid unwanted convergence and reduce the computational costs. Te primary objective of this algorithm is to minimize/ maximize an objective function f(x).
Te following steps are followed in the implementation of the Jaya algorithm: Step 1: initializing the decision variable (x i ), population size (N p ), and iteration number (T). Te decision variable is initialized by a value between the lower and upper bound ranges such that Step 2: creating a job preference vector V k,l � v 1,k,l , v 2,k,l , . . ., v i,k,l , . . ., v n,k,l , where V k,l is an ndimensional vector which represents a sequence of jobs in the kth schedule at the lth iteration and v i,k,l is the preference value assigned to an ith job in the kth schedule at the lth iteration. Te preference value is randomly generated with a uniform random number distribution as follows: where rand is a uniform function that generates a random number between 0 and 1 and i ∈ 1, 2, 3, · · · , n { }, k ∈ 1, 2, 3, · · · , N p , l ∈ 1, 2, 3, · · · , N p .
Step 3: the objective function f (x i ) for each solution is calculated, and the job preference vector is sorted in the ascending order based on the objective function values [47]. Te minimum and the maximum objective functions are determined, and their corresponding best preference vectors and the worst preference vectors are V k,lmin and V k,lmax , respectively.
Step 4: Updating the preference values of all vectors based on the preference values of (V k,lmin and V k,lmax ), respectively.
where V i,k,l is the preference value for the ith job in the kth schedule at the lth iteration.; V i,k,lmin and V i,k,lmax are the preference values of the ith job in kth schedule with the minimum and maximum values of the objective function.; and r1 and r2 are random values generated with the uniform distribution function U [0,1], which are used to achieve the right balance between the exploration and exploitation processes [48].
Step 5: converting the values to a new preference value and identifying the corresponding new objective function value Step 6: comparing the values of the new schedule with the previous schedule and updating it with the better solution Step 7: repeating the abovementioned process until an optimal solution is obtained and the current solution is replaced by the optimal solution Te viability and efciency of the Jaya algorithm make it suitable for real-world problems such as feature selection, image processing, designing PID controllers, and many other applications [49][50][51]. Te algorithm is used to optimize multiple-objective cases such as (i) minimizing the total operating cost, (ii) minimizing the system loss, and (iii) minimizing voltage deviation. In addition, the algorithm is investigated to satisfy the multiobjective cases such as (i) minimizing the total cost and system loss, (ii) minimizing the total cost and voltage deviation, and (iii) minimizing the system loss and voltage deviation.

Self-Adaptive Diferential Evolution (SaDE).
In the SaDE algorithm, the population is initialized randomly with NS (population size) target vectors, strategy probability, number of strategies available, and learning period (LP). Fitness for International Transactions on Electrical Energy Systems the population is evaluated for the parent selection process. Te mutation operator is enforcing a very objective vector in the existing period to bring about the evolution vector. Te crossover process is enforced on every combination of objective vectors and uses the respective mutated vector to create a preliminary vector combination. For ofspring generation, the SaDE algorithm mainly calculates (i) strategy probability and (ii) assigns control parameters to the process of obtaining preliminary vectors for each objective vector. Te choice of three control constants such as crossover rate (CO), mutation rate (M), and population size (NS) in successive generations highlights the SaDE algorithm from the conventional DE algorithm by providing randomness [52], which in turn helps enhance the measure of searching for exploitation and exploration.
Tus, the strategy candidate pool has been divided into four diferent strategies; the frst three strategies use the binomial-crossover operator, and the fourth strategy generates a trial vector without a solitary crossover.
(1) DE/rand/1/bin (ST1) (2) DE/rand-to-best/2/bin (ST2) (3) DE/rand-to-best/2/bin (ST3) (4) DE/current-to-rand/1 (ST4) In the SaDE technique, x r1 , x r2 , x r3 , x r4 , and x r5 are the distinct solutions for the current population. Population size (NS) is not needed to be tuned, and common values should be attempted based on the intricacy of the application under consideration. Te scaling factor, M, is frmly linked to the concurrent speed. Proper selection of CO results in enhancing the efciency of the optimization, while incorrectly selected values may worsen the efciency. Hence, for a given problem, there is a gradual adjustment in the prior values so that they are in the range and have entered the next generation successfully. CO is usually spread across a spectrum with a mean (COmk) and a standard deviation of 0.1 analogous to the kth strategy. In the beginning, (COmk) is assigned to be 0.5 for fundamental LP generation and is repeated for all strategies. Subsequently, after LP iterative generations, (COmk) is assigned to be the median of the successful CO values. Post evaluation of the latest obtained objective vectors, CO variables in (COmk) belonging to the previously defned generation are substituted with the numeric values found in the most recent generation for the k th outcome.
A single outcome for obtaining the objective vector is taken from one of the efcient strategies assigned to each target vector in order to obtain an objective vector. From the knowledge of previous candidate solutions, the SaDE algorithm adapts to the strategies for generating objective vectors and related control parameter settings by oneself during evolution. As a result, a better and more appropriate outcome for obtaining is found and the various arguments are found by adjusting the control parameters to match at various stages during the optimization problem [53]. Te selected control strategy is then enforced on the analogous output vector for creating an objective vector. After iteratively passing through the generations and calculating all the obtained objective vectors, the numeric value of objective vectors obtained in a particular strategy that can be passed and discarded in the subsequent generation is kept in the pass and fail memories, respectively, within the given learning period. Te chances of obtaining a unique outcome shall be modifed after each iterative generation with respect to the passed and failed ones.

Fuzzy Logic-Based Diversity Controlled Self-Adaptive Diferential Evolution (FLDCSaDE).
Tis methodology of diversity control is elicited through CO adaptation and executed by means of a fuzzy system using a feedback loop.
To bring about obligatory changes in CO, we calculated the mean for the Gaussian or normal distribution of the crossover rate (COm) on the basis of diversity. Implementation of these changes in the algorithm helps enhance the exploration attribute. Diversity in the population is controlled by changing the rate of crossover based on the necessity of the evolutionary process using the fuzzy system. To strike an equality between exploitation and exploration, there exist numerous techniques for obtaining appropriate validity in the evolutionary process and for diversity calculations. In the FLDCSaDE algorithm, for the purpose of obtaining the diversity among the population, we use the "distance-to-average-point" [54] estimate, which is given as follows: where the required dimension for the problem is given by D, the size of the population is given by NS for a population Pop, and the diagonal length in the search space defned in the range of S⊆R D is given by |L|. Te search area is defned by �������������� (x max − x min ) 2 where every search variable x lies at bounded limits given by x min < x < x max ; x j denotes j th value for the given average point x, whereas x ij indicates the j th numeric value for the i th independent individual. Te given "distance-to-average" measure needs the size of the population as a constraint, the limits across which the variable used can be searched, and the problem amplitude in terms of its dimensions. Tus, the aforementioned approach is found to be very efective in tackling complicated problems involving dynamic design characteristics [55]. Te fuzzy approach used in the system is interpreted by putting together a unique mapping, extending from the specifed input onto the output with the help of fuzzy logic [56]. Tis approach elucidates the numeric value defned for the input vector and with the help of a few laid-down regulations allocates a numeric value for the given output vector. Given COm adaptation in the DCSaDE strategy usually demands two components mentioned as follows: where COm s,k value is acquired by using the SaDE algorithm that usually takes the median defned for every COm d,k constituent involving diversity, and CO numeric value defned during the past generations of linear programming optimization LP is taken into consideration. Initially, the COm d,k value was obtained by using a set of if-then statements. If the diversity is above a certain level, then COm d,k is gradually decreased by a fxed factor; otherwise, it is gradually increased by a fxed factor to improve the overall crossover rate. Te problem with this method was determination of the factor by which to increase or decrease the COm d,k value. Larger values resulted in quicker settling to the maximum or minimum value of COm, which resulted in poor diversity control. Also, fne tuning of the COm value was not possible with the if-then statements.
Te fuzzy system-based adaptation causes a gradual change leading to better evolutionary process. Te fuzzy system usually takes into consideration the numeric value of COm immediately besides any user mediation based on the problem attributes. Te COm value should necessarily be carried out in accordance with the numerous stages in the evolution as well.
During the various stages of evolution, the crossover rate should be suitably modifed based on the search space. Te fuzzy system helps in the efective tuning process for the numeric value of COm, at the instant of many levels in the evolutionary development. Te adoption of COm can be carried out in a similar fashion for a feedback-based fuzzy controller, where the feedback is stated using the diversity deviation involving the current population. Diversity deviation or error is measured by involving deviation of population diversity of the current generation with respect to a reference or desired diversity level. Terefore, for COm k , variables which are predefned are adapted on the basis of fuzzy logic.
In the given task, a SISO (single input and single output) fuzzy logic-based system developed by Mamdani is used. Te input used for the fuzzy system involves deviation in the diversity at g th generation ∆err g is defned as given deviation in the diversity among the individuals of the population div g present at g th generation and the diversity taken as a ref- Te output obtained FS is interpreted as the deviation of the mean for the given normal distribution at the specifed numeric value of the crossover rate (∆COm g k ) under the k th strategy for the g th generation. For the given generation, the COm g k is obtained through COm Fuzzy variables for FS are established from output and input. Subfunctions of these variables are classifed into Positive (P), Negative (N), Zero (Z), Large Positive (LP), and Large Negative (LN). All of these are specifed with the help of functions of triangular membership. For obtaining defuzzifcation, we use the centroid approach. Towards the completion of every generation, calculation is carried out for obtaining reference diversity (ref) and current population diversity. Te deviation among the two terms is sent for the feedback control in the FS loop. Te FS makes note of all the important implementation changes in numeric value of COm k numeric value using the assistance of rule set, obtained from the SaDE approach for revising the crossover rate in the upcoming generation. Tis combined with International Transactions on Electrical Energy Systems success history-based adaptations of CO will be able to adopt the control parameter value depending on the search space of the functions and the stage of evolution during the evolutionary process. Te fuzzy approach is summarized in Figure 2. Te corresponding false positive rate (FPR) can be computed based on the ripple content which contributes towards false decision outputs. Lower the ripple obtained, the better is the frequency response and in-turn performance thus obtained.

Implementation of FLSaDE for FIR Filter Design.
Te general steps involved in the implementation of the abovementioned algorithm are summarized as follows. Te objective or ftness function used to minimize the error function is given in equation (4) for both band stop and high pass flters. Te abovementioned function is evaluated after each iteration in order to derive optimal coefcients for the given FIR flter. Te target error value (tolerance) that should be reached is set to 0.001, and the target objective value is given to be 1.000001. Te equality constraint for the given frequency response h(n) is bounded and given by h where N is the order of the given FIR flter.
Te various values of function parameters such as upper and lower bound limits and size of the given population are selected such that they are used in fnding and generating optimal coefcients of the given flter for minimizing the error function of high pass and band stop flters.
Step 1: values are initialized to the population size, N s � 20. Te lower and upper bound limits for the coefcients of the given flter are assigned −1 and +1, respectively, for both band stop and high pass flters. Te number of iterations N � 20 for improving the accuracy of the given search space. Te values of control parameters are initialized using the formulation rate M � 0.5 and the crossover rate CO � 0.5.
Step 2: we evaluate the strategy probability to the 4 strategies listed in the SaDE algorithm. Te learning period is set to 50, and the coefcients are divided into lower bound and upper bound values. Te fuzzy interference system (FIS) parameters are initialized. In the initialization, the inputs of the system are made fuzzy, followed by the fuzzy operators being applied to obtain the output.
Step 3: the parent population is evaluated. Te best value and its id are computed for the given generation. In order to rotate the given population to randomize, the following steps are performed: frst, the old population is saved, and then the current index of the pointer array is found. Second, the vector's locations are shufed, and their indices are rotated.
Step 4: we group the given population into strategies, and their diversity is calculated by fnding the error, length of the FIS, and mean CO values. From the mean, the optimal CO and M are computed. For the given bounded coefcients, boundary conditions are checked. After the given parent population is shufed, the child population is selected from the parent population. If the selected child is better than the previous generations, it is added to the list. For the given population, local search is applied for fnding flter coefcients.
Step 5: each function call is performed for each individual iteration, and the parameters are passed in column form. Te number of design variables is set to 11. Te maximum function evaluation is performed for 200 generations, and the population in each generation is updated based on the worst and best solutions obtained.
Step 6: for each iteration (or run), we fnd the optimal frequency (fbest) and flter coefcients (xbest). Te same procedure is repeated for all the algorithms used in the design of the FIR flter. Finally, the iterations are repeated until the tolerance error and target objective value are reached. After 20 iterations, the one with the minimum fbest value is treated as the best member of the given generation. Te values corresponding to the best are the optimal coefcients for the given algorithm.

High Pass Filter (HPF).
A comparison of the high pass flters designed using diferent evolutionary algorithms (EA) is made with regard to the set of attenuation values and the flter frequency response. Te optimum flter coefcients for the algorithms, namely, SaDE, FLDCSaDE, CMAES, Jaya, and DE, are depicted through Table 2.
Te ideal flter order is determined by the ability to achieve higher attenuation between stopband and passband, a fatter passband resulting in signifcantly minimal ripples, and narrow transition band. While lower order flters have minimal roll-of and complexity, they have a larger transition band, leading to irregular fltering action. In comparison to lower order flters, higher order flters also have disadvantages of their own. Tey have higher complexity which increases in the powers of 2 as we double the order. Hence, there is a need to determine the ideal flter order which encompasses the advantages of both the lower and higher order flters. In this paper, we have utilized the order of the flter with 21 coefcients and 20 iterations. We fnd that this 21 st order flter ofers a lower transition band than other higher order flters. Moreover, the flter gain obtained is higher and the error is minimal, making the flter closer to the ideal response [57]. Te frequency response for the optimized flter coefcients of each algorithm for both band Stop and high pass flters is obtained through the freqz function in MATLAB which is used to fnd the frequency response of the given digital flter as shown in Figure 3. We measure the attenuation values for the given flter by taking the absolute values of the given frequency response, converting the values to decibels, and utilizing the peaks obtained in stopband and passband to fnd the corresponding attenuation [8]. Also, for fnding ripples, the absolute values are taken as they are not converted to decibels. Te stopband and passband attenuation analogous to the distinct algorithms mentioned above are listed as follows. Te abovementioned algorithms form the basis for the algorithms discussed in the paper. Te performance of the various algorithms is discussed in the latter sections. Te following algorithms are found to be superior among all the other genetic and swarm optimization algorithms due to the fact that they are much more recent, and the limitations of the previous algorithms have been rectifed in the following algorithms.
In this paper, a high pass flter is designed to demonstrate the efectiveness of optimization techniques. Te specifcations for the high pass flter are order of the flter N � 21, cutof frequency ω c � 0.43, and the number of samples: 1000. Te objective function in equation (4) is computed for each and every step until optimal solution is obtained. Te limits assigned for the flter coefcients are found to lie within the range of −1 to +1. We analyze the parameters obtained from the frequency response of a high pass FIR flter for their performance, utilizing various evolutionary algorithms (EA), most importantly the stopband ripple, passband, and stopband attenuation. Tese parameters are tabulated in Table 3.
Based upon the magnitude response in dB obtained from the graphical representation of various algorithms such as SaDE, FLDCSaDE, Jaya, CMAES, and DE in Figure 3, we observe that the plot clearly depicts SaDE and FLDCSaDE having the highest negative magnitude in dB for a given normalized frequency. For more accurate visualization, we use stopband attenuation (A stop ) as recorded in Table 4 since it is based on the mean value from each iteration, thus making it a stable value obtained in any iteration rather than a single highest peak, followed by a subsequent lower value. Te flters are mainly characterized by their stopband attenuation [58,59]. From Table 3 Table 4, we fnd FLDCSaDE to outperform other algorithms in the region around the stopband, thus indicating the highest stable attenuation throughout the response. Moreover, we fnd that the passband attenuation is the lowest for FLDCSaDE (0.4330 dB), indicating that it takes the minimum time for computation. Lower time correlates to a lower number of parameters involved in the flter design. It is seen from the plot that all the algorithms produce almost the same magnitude of overshoot, but among them, FLDCSaDE delivers the minimum overshoot at the point where the ideal flter contains discontinuity [60]. CMAES and Jaya generate the maximum. SaDE as well as DE produces an intermediate overshoot [61,62]. It shows that FLDCSaDE type flters yield a level and smooth passband response; that is, they have a fat response at the passband. Tese considerations are studied and passed onto the calculation of the variance, highest, and mean values of the ripples obtained at the passband as in Table 5. Since the variance of the ripples in the passband for all the algorithms is close to 0, it states that the passband ripples of the algorithms for 20 runs are very close to each other. Te lower value of stopband ripple from Table 6 for FLDCSaDE indicates that the distortion and aliasing in the waveform are comparatively lower than the other algorithms.

Band Stop Filter (BSF).
In this section, we adopt a similar approach as in the design of a high pass FIR flter. One of the diferences in execution is that we have two crossover frequencies for the band stop flter when compared to one for the high pass flter. It is basically the inverse of the band pass flter, and these second order reject flters are designed to provide higher attenuation at and near the single crossover frequency with little attenuation at all other frequencies as shown in Figure 4. For any standard band stop flter, the highest frequency attenuated is about 10 to 100 times more than that obtained from the lowest frequency of attenuation. Te conventional band stop flter is designed because of its lower interference and increased performance; also, the amplitude of the reference level of the sideband is comparatively reduced, thus producing a wider stopband. Te band stop flter coefcients for various evolutionary algorithms are listed in Table 7.
Te design specifcations taken into consideration for the states' band stop flter are as follows: order for the given flter N � 20, cut-of frequency ω c1 � 0.38, ω c2 � 0.73, and number of samples: 1000. Te objective function in equation (4) has to be estimated after each step in order to come up with the solution which is most efective. Te limits assigned for the flter coefcients are found to lie within the range of −1 to +1. Similar to the high pass flter, we analyze the parameters of the frequency response of the band stop FIR flter for fnding their performance, utilizing various evolutionary algorithms (EA), most importantly the stopband ripple, passband, and stopband attenuation observed and tabulated in Table 8 [62].
From the normalized magnitude response representation in graphical form in Figure 4, for various algorithms such as SaDE, FLDCSaDE, Jaya, CMAES, and DE, we fnd that although the maximum flter stopband attenuation from Table 8 is favourable for DE with −26.0818 dB, the passband attenuation is high (1.03995 dB). Taking the mean value in Table 9 into consideration for the stopband attenuation, we fnd that FLDCSaDE has the highest negative value of −18.86844 dB, as seen from Table 9, which shows that it has the most stable response with the highest magnitude. Moreover, the response of DE is skewed, indicating a sole     Table 11 except for the Jaya algorithm are very close to each other, and therefore, these algorithms ofer high attenuation. Some ripples are observed in the stopband for FLDCSaDE, suggesting that the gain obtained is irregular in the passband region. But the fact that the average value of the ripple in Table 11 is the minimum for the FLDCSaDE algorithm suggests that it has better convergence and a stable response compared to other algorithms.

Conclusion and Future Scope
Trough the medium of this paper, performance for the designed fnite impulse response (FIR), high pass (HP), and band stop flter (BS) has been scrutinized. Te analysis is compared for various metaheuristic evolutionary optimization algorithms, namely, self-adaptive diferential evolution (SaDE), fuzzy logic-based diversity-controlled selfadaptive diferential evolution algorithm (FLDCSaDE), covariance matrix adaptation evolution strategy (CMAES), and Jaya algorithm, by taking the diferential evolution (DE) algorithm as a reference. Te FIR flter design aims at fnding optimal coefcients for the given flter such that it minimizes its ripple (both stopband and passband) and hence obtains a smoother response with a minimal relative absolute error, thus attaining a response closer to the ideal response. Te observed optimal solutions obtained for the FLDCSaDE algorithm were found to be the best when compared to other algorithms and satisfed the requirements of an efcient FIR flter in terms of performance and reduced ripple. At the same time, the algorithms are found to attain a high attenuation in the stopband, in-turn having a narrower transition band. Te superiority of the FLDCSaDE algorithm compared to other algorithms can be attributed to the fact that it ofers better diversity because it updates the crossover rate for the next generation. Tis helps in the retrieval of useful information from the given population and in obtaining a better solution. Te application of the given FLDCSaDE algorithm provides better tuning and performance. Te performance in terms of amplitude and ripples for both high pass and band stop flters has been appreciable. Te ripples in the order of 0.02 and 0.07 provide a smooth transition. In a practical sense, circuits with lower attenuation in the passband which extends from 100 MHz to infnity will cause the signals to experience an attenuation loss of approximately −10 dB which limits the appreciable bandwidth that can be used in the circuit. Te fact that utilizing diferent flter topologies can cause reduction in ripples may have an added disadvantage of undesired resonances and introduce amplifying elements. Te use of FLDCSaDE reduces the ripples to an order of 1, thus scaling the desired output and reducing the group delay. Although the fact that the fuzzy approach can be applied to any randomly generated population, there might be cases where the accuracy can be compromised, especially in the case of multimodal problems. In addition to it, one might need to manually feed the crossover rate. Even in these problems, the accuracy obtained is almost comparable to that obtained by the other algorithms. Moreover, the value of the crossover rate defned will be optimized post each iteration, so the crossover rate fed initially does not result in inaccuracy which makes it efcient. Te research can be extended to solve multidimensional and multiobjective problems and can be used to design a corresponding infnite impulse response (IIR) flter. Te usage of FIR flter design can be used to improve modulation parameters such as spectral efciency, clock synchronization, frequency masking, and latency, thus improving bandwidth diversifcation [63]. Te use of digital flters to improve energy efciency in industrial automation can be considered for future investigation. Te use of fuzzy-based optimization can further provide frequency sampling. Tis approach can be applied to the decision support system which can be utilized to control the diferential curve, thereby making it suitable for many nonlinear applications. Te usage of adaptive fuzzy systems can further add to the reasoning on the fuzzy patterns, thus enabling task processing and broadening the spectrum of functional intelligence. Tis provides a practical and an efective solution for various applications such as audio systems for sending and controlling the variable frequency components, biological instruments, and communication systems.