Design of filter coefficients for optimal factored truncated cascade FIR filter using optimization algorithm

Abstract

Signal filtering acts as one of the basic requirement of communication networks for the removal of unwanted features from the signal. The design of appropriate digital filter requires the selection of optimal filter coefficients for the generation of desired frequency response with reduced hardware complexity. This paper proposes a hybrid optimization algorithm named as Brain Storm- Grey Wolf Optimizer (BSGWO) algorithm for the selection of filter coefficients in the design of factored truncated cascade FIR filter. The proposed algorithm is the hybridization of the optimization algorithms, namely Brain Storm Optimization (BSO) and Grey Wolf Optimizer (GWO). The input signal is interpolated initially for the formation of an intermediate signal using the FIR filter. Then, the factored truncated cascade filter is developed for the interpolation of the signal. After designing the filter coefficients, the optimal selection of the filter coefficients is performed using the proposed BSGWO algorithm. The original filter is developed with the use of the least square estimation and the new filter is developed using the proposed algorithm that tunes the filter coefficients. The performance of the proposed system is analyzed using the metrics, such as fitness, Mean Absolute Error (MAE), magnitude, and the number of components. The proposed method produces minimum fitness, MAE, magnitude and number of components of 0.05, 0.0155, $- 96.0 dB$ and 3372, respectively that shows the effectiveness of the proposed method.

Keywords

FIR filter filter coefficients frequency response cascade filter and least square estimation

1. Introduction

Design of digital filters has been leading the way to various intensive research works. Digital filters are classified as two groups, named as finite impulse response filters (FIR) and infinite impulse response filters (IIR) on the basis of the unit impulse response length [6]. IIR filters possess various advantages, including the usage of the design outcomes of analog filter including the parameters related to the design of the filter. However, the limitations are showed differently, like nonlinear phase and recursive structure that leads to the making of feedback loop unstable, and so forth [19]. On comparison, FIR digital filters possess the characteristics with strict arbitrary amplitude and linear phase that makes the filters highly stable. The FIR digital filters are easier to be designed as compared to that of the IIR digital filter design [15]. In addition, the communication channels involved in data transmission and image processing must possess the linear phase characteristics. In other words, the FIR digital filters can be used in various practical applications, such as in remote sensing, defence [9] and medicine [7, 8, 18], image processing, sensors [36], and so on [28]. Variable digital filters (VDFs) are the type of digital filters with spectral characteristics that are tuneable, with reduced hardware complexity.

The VDFs involving less complexity on hardware are of great interest in software defined radios (SDRs), specifically in green communication that aims on energy effective wireless communication components and techniques. Variable FIR filters are of great interest in most areas including wireless communication, biomedical signal processing [25], and speech and audio processing, because of its coefficient sensitivity and stability. However, the FIR filters have increased complexity in hardware, and thus needs maximum power of operation [24]. The recent decade has tolerated a constant effort over the design of low power FIR filters, where the multiplier-less design approach is the one among them [17]. Multiplier-less design leads to finite precision filters, where the full-fledged multipliers created the most power consuming elements in case of the digital signal processing (DSP) hardware [16], and thus are completely eliminated. The filter coefficients with finite precision filters are quantized with any signed power of two (SPT) representations that shows the multipliers with shift and add operations. Canonic signed digit (CSD) provides an efficient number representation with reduced number of non-zero terms (NZTs). The consumption of power and hardware resource utilization is reduced for multiplier-less filters as compared to that of the continuous coefficient filters [4].

Cascaded filters are designed due to the fact that they are very efficient than that of the standard filters. Adhoc and least-squares procedures are currently used in the design of these filters. However, these design procedures are not capable to provide precise control of the frequency response to efficiently satisfy the filter specifications. The peak-constrained least-squares (PCLS) concept in designing the cascaded filters allows for direct control of filter’s frequency response, and permits for the trade-off in case of peak stop-band gain for stop-band energy. Filters in the presence of quadratic cost functions and linear constraints are modelled with the use of GME algorithm [1], and the filters in the presence of nonlinear constraints are modelled using the RGME algorithm [30]. A hybrid anti-aliasing filter consists of an analog filter to band limit the analog signal that is cascaded with a digitizer and a digital filter to provide the frequency response compensation of the resulting digital signal. The advantage of this type of filtering arrangement is the reduction of complexity associated with the analog filter. The digital filter offers a narrower transition band for the signal as compared to that offered by the analog filter. The digital filter is capable of compensating the amplitude and phase distortions in the analog filter. The digital filter can either be a nonlinear phase FIR filter or a digital IIR filter, and the optimization procedure resulting from this design problem is nonlinear and needs the use of the Recursive Generalized Multiple Exchange (RGME) algorithm for solving it [14, 31].

1.1. Research questions

How to design a filter coefficient for optimal factored truncated cascade FIR filter with reduced hardware complexity?

How to generate the optimal values for the design of the filter by tuning filter coefficients?

How to perform an optimal pairing using an algorithm to provide a systematic method for the accomplishment of sequencing?

How to rectify the unexpected magnitude response that occurs occasionally by using the proposed method?

1.2. Motivation and problem formulation

In many modern signal processing systems, digital filters are considered as the basic structures due to its higher order practical filters at low cost with higher noise immunity. Designing an appropriate digital filter requires the selection of optimal filter coefficients with reduced hardware complexity. In the previous works, different features from the signal were defined for designing filter. However, it requires the selection of optimal filter coefficients for the generation of desired frequency response with reduced hardware complexity. Also, for better design, the convergence speed should be considerably better to use in real time applications and also the error and fitness should be minimum. This paper processes a hybrid optimization algorithm named BSGWO for the selection of filter coefficients in the design of factored truncated cascade FIR filter.

1.3. The contribution of the paper

Proposed BSGWO algorithm for tuning the filter coefficients: The proposed BSGWO algorithm is the hybridization of BSO and GWO, where the update rule of BSO is altered with the use of the GWO algorithm. The proposed BSGWO algorithm tunes the filter coefficients optimally in such a way that to design the FIR filter optimally with reduced hardware complexity. It result with minimum fitness, MAE, magnitude and number of components and produce effective results. With the incorporation of BSO with GWO, it provides high local optima avoidance and better convergence rate and enhances the performance in real word problems. Also, BSGWO has the advantage of global exploration.

1.4. The organization of the paper

Section 1 depicts the introduction to the need of filter design and Section 2 details the various existing methods and their challenges in the filter design. Section 3 depicts the proposed method of FIR filter design and the results of the proposed method are discussed in Section 4. Finally, Section 5 provides the conclusion of the paper.

2. Literature survey

In this section, the literature survey of various methods used for the filter design is presented, and the challenges of the existing methods are discussed. Peng Shao et al. [28] developed the Refracting opposite learning (refrpso), which provided faster convergence and enhanced accuracy, but cannot be used in real life. T. Bindima and Elizabeth Elias [4] designed a Multi-Objective Artificial Bee Colony (MOABC) optimization algorithm that minimized the resource utilization of hardware and the consumption of power drastically. The drawback of this method was the inability to handle the band pass and band reject filters. Judhisthir Dash et al. [10] modelled a method, known as Hybrid Firefly Differential Evolution (HFDE) algorithm with increased accuracy and satisfactory time of execution. The limitation of this method was that the convergence was very poor. Jacek Wodecki et al. [35] developed a Progressive genetic algorithm, which did not required any prior knowledge about the signal, and performed effective in case of multi dimensional nonlinear environment. The method failed due to the fact that it suffered from kurtosis, and thus produced false results. Luis M. San-José-Revuelta and Juan Ignacio Arribas [26] developed an optimisation technique known as flowers pollination algorithm (FPA). It improved the frequency characteristics of filters produced by the conventional methods, but the convergence speed was very low that was considered as the major drawback of this method. Apoorva Aggarwal et al. [2] designed the L1-norm based real-coded genetic algorithm (L1-RCGA) that led to considerable improvement in the application area of digital signal processing, but cannot be used in case of two dimensional filters. Shubhendu Kumar Sarangi et al. [27] modelled an efficient design of digital FIR high pass and band stop filters with the use of an adaptive cuckoo search algorithm (ACSA) that performed better in terms of magnitude response with optimum pass-band and high stop-band attenuation, but the convergence speed was very slow because of the involvement of more number of parameters. Apoorva Aggarwal et al. [3] developed the swarm intelligence (SI) based evolutionary computing method for the determination of optimal solutions, but mutation may cause drift in solutions. The features and the challenges of the existing works are described in Table 1.

Table 1
Review of existing approaches

Author [citation] Methodology Features Challenges

Peng Shao et al. [28] refrPSO
It has better performance on practical FIR digital filter design and optimization in digital signal processing systems.

provided faster convergence and enhanced accuracy

It cannot be used in real life.

T. Bindima and Elizabeth Elias [4] MOBAC
It minimized the resource utilization of hardware and the consumption of power drastically.

It has an inability to handle the band pass and band reject filters

Judhisthir Dash et al. [10] HFDE
It results with increased accuracy and satisfactory time of execution.

It has poor convergence.

Jacek Wodecki et al. [35] Progressive genetic algorithm
It is fully data-driven and does not require any prior knowledge of the signal.

It provides robustness by avoiding misguided direction of evolution.

It failed due to the fact that it suffered from kurtosis, and thus it may produce false results.

Luis M. San-José-Revuelta and Juan Ignacio Arribas [26] FPA
It suppresses more power of the unwanted signal and perfectly fits the transition bands.

It has low convergence speed.

Apoorva Aggarwal et al. [2] L1-RCGA
It has the significant improvement in digital signal pro-cessing application.

It cannot be used in case of two dimensional filters.

Shubhendu Kumar Sarangi et al. [27] ACSA
It performed better in terms of magnitude response with optimum pass-band and high stop-band attenuation.

It has very poor convergence speed.

Apoorva Aggarwal et al. [3] cuckoo search,particle swarm and real-coded genetical gorithm
It leads to more flexibility in designing the FIR filter with the absence of fine-tuning of parameters.

Mutation may cause drift in solutions.

Srivatsan. K [Proposed] Brain Storm- Grey Wolf Optimizer (BSGWO) algorithm
It produces minimum MAE, magnitude and number of components and produce effective results.

The method needs improvement in their design objectives including the throughput and required area, etc

Author [citation]	Methodology	Features	Challenges
Peng Shao et al. [28]	refrPSO	It has better performance on practical FIR digital filter design and optimization in digital signal processing systems. provided faster convergence and enhanced accuracy	It cannot be used in real life.
T. Bindima and Elizabeth Elias [4]	MOBAC	It minimized the resource utilization of hardware and the consumption of power drastically.	It has an inability to handle the band pass and band reject filters
Judhisthir Dash et al. [10]	HFDE	It results with increased accuracy and satisfactory time of execution.	It has poor convergence.
Jacek Wodecki et al. [35]	Progressive genetic algorithm	It is fully data-driven and does not require any prior knowledge of the signal. It provides robustness by avoiding misguided direction of evolution.	It failed due to the fact that it suffered from kurtosis, and thus it may produce false results.
Luis M. San-José-Revuelta and Juan Ignacio Arribas [26]	FPA	It suppresses more power of the unwanted signal and perfectly fits the transition bands.	It has low convergence speed.
Apoorva Aggarwal et al. [2]	L1-RCGA	It has the significant improvement in digital signal pro-cessing application.	It cannot be used in case of two dimensional filters.
Shubhendu Kumar Sarangi et al. [27]	ACSA	It performed better in terms of magnitude response with optimum pass-band and high stop-band attenuation.	It has very poor convergence speed.
Apoorva Aggarwal et al. [3]	cuckoo search,particle swarm and real-coded genetical gorithm	It leads to more flexibility in designing the FIR filter with the absence of fine-tuning of parameters.	Mutation may cause drift in solutions.
Srivatsan. K [Proposed]	Brain Storm- Grey Wolf Optimizer (BSGWO) algorithm	It produces minimum MAE, magnitude and number of components and produce effective results.	The method needs improvement in their design objectives including the throughput and required area, etc

2.1. Challenges

The various challenges involved in this research are detailed below as,

The algorithms like Chebyshev-type approximations [23] were used in the optimization of digital filters, as it exhibit enhanced performance on the design of FIR digital filter compared to the conventional methods, but the accurate estimation of the boundary frequency of stop band and pass band is not so easy using this method [28].

Min-max algorithms have the tendency of getting struck at the local minima solution. In addition, it possesses slow speed of convergence and accuracy [28].

Various classical optimization techniques were designed for the accurate linear phase filter modelling. The usual gradient-based optimization method is inadequate in the optimization of the multimodal, non-differentiable, nonlinear objective function while estimating the global minimum solutions [10, 11].

In hierarchical genetic algorithm (HGA) [32] and in hybrid Taguchi genetic algorithm (HTGA) [33], the linear phase response error is not taken into consideration that may lead to large distortion [13].

The Parks–McClellan (PM) algorithm [23] is an iterative method to obtain the best Chebyshev FIR filter. The aim of this technique is to reduce the error in stop and pass bands, but is not very attractive as stated in [10].

3. Proposed method of factored truncated cascade filter based FIR filter design using the hybrid BSGWO optimization algorithm

The proposed method aims to design the FIR digital filter due to its attractive application in different fields, such as wireless communication, processing of speech, and processing of biomedical signal, and so on. The drawbacks associated with the existing methods is rectified using the method of optimal factored truncated cascade FIR filter design [20] using a newly developed optimization algorithm. The filter coefficients required for the design of the FIR filter are selected by proposed BSGWO, which is the integration of the BSO and the GWO algorithms. This algorithm is developed to enhance the performance of existing BSABC algorithm used for the selection of filter coefficients. The fitness of the proposed BSGWO is developed depending on filter design and frequency response. The proposed method of optimal filter design is depicted in Fig. 1.

Fig. 1.

Proposed method of optimal filter design.

Implementation of an FIR filter as the cascade of subfilters acts as a challenging approach in designing efficient digital filters [5]. The concept was found in existing DSP work [5] with the use of the natural factors with the order two each related to a complex-conjugate zero-pair of the filter. The cascade of a prefilter and a second subfilter, or efficient similar subfilters, is found to be a useful method to reduce the coefficient sensitivity and the hardware complexity of the filter. An algorithm for the optimal factoring of FIR filter transfer functions was developed in [21] to find the optimal pairings on the unit circle in the z-plane, which provides a systematic method for the accomplishment of sequencing. The optimal pairing algorithm can be improved to include the capability of combining two off-unit-circles of either 2nd-order or 4th-order factors that are typically possessed with the passband ripples of the filter with each other or, pairing one of them with the on-unit-circle 2nd-order factors, to obtain the 4th, 6th or 8th-order linear-phase factors.

Fig. 2.

Structure of optimally factored truncated filter.

The resulting optimally factored cascade of factors is shown in Fig. 2. The figure consists of a cascade of 1st, 2nd, 4th, 6th, and 8th-order stages, followed by a post-stage shifter and a truncation operation to offer the capability of adjusting the non-uniform data-path word length. The post-filter multiplier δ allows the designer to choose the filter gain to expected level. A stage with higher order may become a part of act as a section of cascade with further factor pairings for the improvement of the stages of filter, in order to rectify the unexpected magnitude response that occur occasionally.

3.1. Development of filter coefficients with the concept of existing least square estimation

The conventional design model of the FIR filter is developed with the use of the existing LSE technique, which solves the linear simultaneous equations by utilizing the matrix inversion method [34]. The filter possess the capability of smoothening that is used to smoothen the input and produce the expected output $O (θ)$ using type-1 linear FIR filter, where transfer function has $C_{a} = I_{k - a}$ coefficients, with $k - a$ as an even order. The expected response is represented as, $O (θ)$ , and the amplitude response of $A (w)$ is indicated as, P. The amplitude response is expressed as, $\begin{matrix} (1) & P = \int_{α} {[O (θ) - C_{0} . e^{j θ}]}^{2} . \frac{d θ}{π} \end{matrix}$ where, $C_{0}$ indicates the magnitude response, $O (θ)$ represents the desired response, α is the area between the region $0 ⩽ θ ⩽ π$ , and the term $A_{0} . e^{j θ}$ is expressed as, $\begin{array}{l} (2) & C_{0} . (e^{j θ}) = \sum_{a = 0}^{k} I_{a, n} e^{- j (a - d) θ} \\ (3) & C_{0} . (e^{j θ}) = \sum_{a = 0}^{p} g_{a} \times cos a θ \\ (4) & g_{a} = \{\begin{matrix} 2 A_{\frac{k}{2} - a}; & a \neq 0 \\ A_{\frac{k}{2}}; & a = 0 \end{matrix} \end{array}$ where, $I_{a, n}$ is an element of the solution vector, k is the order number, and $p = \frac{k}{2}$ . The major objective of LSE is to solve the objective function expressed in Eq. (1), where the term α exceeds the transition band. The important parameter in the minimization of the objective function is $g_{a}$ , which is obtained by solving the linear equations with the matrix inversion method and is given as, $\begin{matrix} (5) & g = G^{- 1} . l \end{matrix}$ where, $g = {(g_{0} g_{1} ... g_{p})}^{s}$ , l and G always depends on $θ_{1}$ and $θ_{2}$ . The need of $d θ$ in the Eq. (1) shows that the trade-off among the accuracies of pass band and stop band is assured. This section develops a platform to know the design of digital FIR filters with the use of the conventional LSE method. However, the design of filter is not optimally tuned to satisfy the necessary specifications using smaller filter length. Thus, optimal tuning is linked in the factored truncated cascade filter-based FIR filter design using the proposed BSGWO algorithm.

3.2. Factored truncated cascade-based design of FIR filter with the proposed BSGWO

The proposed BSGWO is developed to obtain the filter coefficients available in the Factored truncated cascade based FIR filter. The proposed BSGWO is obtained from the integration of BSO [29] and GWO [22] algorithms for the determination of the global filter coefficients. The update equation of the BSO algorithm is modified with the update relation of the GWO algorithm in such a way to obtain the benefits of both the algorithms. The proposed algorithm has the capability to converge optimally at the local exploitation and exploration phases. Due to the advantages of BSO and GWO, these two are hybridized in the proposed method. GWO is selected as it capable of providing high local optima avoidance and better convergence rate. It is capable of providing enhanced performance in case of unknown and challenging search area, and can be used in real problem and can offer solutions to both unconstrained and constrained problems. The reason for the selection of BSO is the advantage in terms of global exploration.

BSO acts as an effective algorithm to optimize the individuals depending on the brain storming characteristics of human. This optimization algorithm provides the capability of transforming the solutions that are present in the solution space to the objective space. BSO produces the global optimal solutions via iterating the individuals that are provided as two groups, namely elite group and normal group. The new individual is produced with the random selection of an individual with the addition of a random number. The selection of individuals in random undergoes two major steps and it depends on five probabilities. In the first step, a random number is produced between the range of 0 and 1 and is compared with each probability as the selection of random individual relies on probability. A cluster, or any two clusters, or any individual from a cluster, or any one individual from the randomly selected cluster is selected in random for the production of the new individual. After the generation of the new population, the fitness for the new solution is estimated and compared with the fitness of the existing solution to choose the solution with better fitness for update. The iteration is continued until obtaining the global optimal solutions and until replacing all the existing populations with a new population. The GWO algorithm works based on the social and hunting characteristics of the grey wolves. Various tests were carried out to analyze the improved performance GWO based on convergence, local optima avoidance, exploration, and exploitation.

3.2.1. Solution encoding

The solution encoding aims to represent the optimal solution obtained using the proposed BSGWO algorithm. The length of the solution relies on the count of truncated cascade filter and total filter coefficients, and hence the dimension of solution is indicated as, $[1 \times L \times k]$ , where k is the number of filter coefficients and L is the number of structures.

3.2.2. Designing of the objective function

The intention of the objective function is the minimization of the function depending on two main factors, which is expressed as, $\begin{matrix} (6) & R = R_{1} + R_{2} \end{matrix}$ where, $R_{1}$ and $R_{2}$ indicates the fitness functions that depends on hardware utilization and frequency response. The objective of the fitness R is the minimization of the objective function with minimum hardware utilization and minimum frequency response. The fitness based on hardware utilization relies on the count of the adders and subtractors present in the new design in such a way that the hardware utilization is at its minimum. The new filter design with optimality is designed with the proposed BSGWO algorithm and expression of fitness depending on hardware utilization is as follows, $\begin{matrix} (7) & R_{1} = \frac{I_{BS - GWO} + J_{BS - GWO}}{I J_{LSE} + I J_{BS - GWO}} \end{matrix}$ where, $I_{BS - GWO}$ indicates adders available in the proposed filter design and $J_{BS - GWO}$ indicates subtractors available in the proposed filter design. $I J_{LSE}$ represents the count of adders and subtractors available in design of filter obtained using LSE method and $I J_{BS - GWO}$ represents the total adders and subtractors available obtained with the proposed BSGWO algorithm. The fitness depending on hardware utilization must be capable of assuring a minimum value to make the design of the filter effective. The second constraint of fitness $R_{2}$ depending on frequency response is expressed as, $\begin{matrix} (8) & R_{2} = \frac{1}{F} [\sum_{b = 1}^{F} \frac{β_{LSE} - β_{BS - gwo}}{β_{max} - β_{min}}] \end{matrix}$ where, $β_{max}$ and $β_{min}$ represents the maximum and minimum frequency values, respectively. F represents the total number of frequency scales, $β_{LSE}$ indicates the frequency characteristics of old LSE-based filter design technique, and $β_{BS - GWO}$ indicates the frequency characteristics of novel BSABC-based technique of filter design. The proposed method guarantees the need of less hardware to design the filter without causing any effect to the response of original frequency.

3.2.3. Proposed BSGWO algorithm for designing the factored truncated cascade-based design of FIR filter

The aim of the proposed BSGWO algorithm is the optimal selection of filter coefficients to provide effective filter design. The proposed BSGWO algorithm has the capability to converge optimally at the local exploitation and exploration phases. The proposed BSGWO is the hybridization of the GWO update equation in the BSO algorithm such that the capability to converge optimally at the local exploitation and exploration phases is obtained using GWO.

Algorithmic steps of the proposed BSGWO algorithm : The algorithmic steps involved in the proposed BSGWO algorithm are detailed as below,

i) Initialization : The first step is the initialization of individuals involved in optimal solution generation. Consider c number of individuals in the search space that is represented as, $K_{c}$ ; $(1 ⩽ i ⩽ c)$ .

ii) Cluster formation : The next step is the clustering of c number of populations as r number of clusters.

iii) Estimation of fitness of individuals in cluster : The next step is the estimation of individuals of a cluster to select the best individual. The estimation of the fitness depends on Eq. (6).

iv) Ranking of individuals : The fitness is evaluated on the basis of the constraints, namely resource utilization and frequency response. Based on maximum fitness value, the individuals are ranked to select the best solution to perform the process of optimization.

v) Cluster center selection in random : The cluster centers are chosen in random based on probability in this step. The random number is generated from 0 to 1 and is compared with the probability $ω_{1}$ . If the random number produced is less than the probability, then the cluster center is selected in random for the production of the new solution using Eq. (13).

vi) Update of new individuals by proposed BSGWO algorithm : The update of the new individuals is by the proposed BSGWO algorithm, where the update equation of BSO is modified with the update equation of the GWO. The update equation of GWO is expressed as, $\begin{matrix} (9) & {\vec{Q}}_{x}^{y + 1} = {\vec{Q}}_{v}^{y} - \vec{U} (\vec{H} ∙ {\vec{Q}}_{v}^{y} - {\vec{Q}}^{y}) \end{matrix}$ where, y indicates the present iteration, $\vec{U}$ and $\vec{H}$ indicates the coefficient vectors, $Q_{v}^{y}$ represents prey position, and $\vec{Q}$ represents the position vector of grey wolf. ${\vec{Q}}_{v}$ is modified based on BSO and is then integrated with the expression of GWO as below, $\begin{array}{l} (10) & {\vec{Q}}_{x}^{y + 1} = {\vec{Q}}_{v}^{y} (1 - \vec{U} ∙ \vec{H}) + \vec{U} ∙ {\vec{Q}}^{y} \\ (11) & {\vec{Q}}^{y} = \frac{1}{\vec{U}} ({\vec{Q}}_{x}^{y + 1} - {\vec{Q}}_{v}^{y} (1 - \vec{U} ∙ \vec{H})) \end{array}$

The standard equation of BSO is expressed as, $\begin{matrix} (12) & {\vec{Q}}_{v}^{y + 1} = {\vec{Q}}^{y} + γ \times f (η, ε) \end{matrix}$ where, $Q_{v}^{y + 1}$ indicates the $v^{th}$ solution in ${(y + 1)}^{th}$ iteration and $Q^{y}$ indicates the solution in $y^{th}$ iteration. The weight of Gaussian random value is expressed using the coefficient denoted as γ. The Gaussian random function is represented as $f (η, ε)$ . Equation (12) is modified using Eq. (11) as, $\begin{matrix} (13) & {\vec{Q}}_{v}^{y + 1} = \frac{1}{\vec{U}} ({\vec{Q}}_{x}^{y + 1} - {\vec{Q}}_{v}^{y} (1 - \vec{U} ∙ \vec{H})) + γ \times f (η, ε) \end{matrix}$

The above equation is the update equation of the proposed BSGWO algorithm that estimates the optimal coefficients for tuning the factored truncated cascade-based FIR filter. The individuals of the selected clusters are chosen and combined to produce the new solution using Eq. (13) that tunes the filter coefficients optimally to assure optimal FIR filter design.

vii) Fitness estimation of newly generated solution: The fitness of new candidate solution is evaluated and compared to existing one. The new individual replaces the existing value, only if it is found to be better.

viii) Termination: The steps ii) to viii) are repeated till the replacement of all the individuals in the solution. After the completion of c number of replacements, the algorithm is terminated. The pseudocode of proposed BSGWO algorithm is shown in Algorithm 1.

Algorithm 1

Pseudocode of the proposed BSGWO algorithm

4. Results and discussion

This section provides the results and discussion of the proposed BSGWO method of selection of optimal filter coefficients for Factored Truncated Cascade FIR Filter and its effectiveness in design of filter using the comparative analysis.

4.1. Experimental setup

The proposed method is implemented in MATLAB with the PC installed using Windows 10 OS in the presence of 4 GB RAM and 64-bit OS.

4.2. Evaluation metrics

The proposed BSGWO method is analysed using the metrics, such as fitness, Mean Absolute Error (MAE), magnitude, and the number of components.

i) Fitness : The fitness function depends on hardware utilization and frequency response. The fitness based on hardware utilization depends on the number of adders and sub tractors, and the fitness based on frequency response depends on minimum and maximum frequencies.

ii) MAE : It is defined as the rate of difference among two continuous variables or an average of two absolute errors.

iii) Magnitude : Magnitude is a function of frequency, where all the values are produced as the magnitude of complex frequency response value in that frequency.

iv) Number of components : For a filter design to be effective, the number of components used must be less. The resulting optimally-factored truncated cascade-filter for the example shown in Fig. 2 with reduced hardware complexity is shown in Fig. 3.

Fig. 3.

Factored truncated cascade for the achievement of lower hardware complexity.

4.3. Comparative methods

The performance analysis of the proposed filter design is compared to the conventional Particle Swarm Optimization (PSO) [28], Artificial Bee Colony (ABC) [12], improved Artificial Bee Colony (CABC) [12], and Accelerated Artificial Bee Colony algorithm (AABC) [12], Multi-Objective Artificial Bee Colony (MOABC) Optimization [4], Brain Storm- Artificial Bee Colony (BSABC), BSO [29] and GWO [22].

4.4. Comparative analysis of the proposed method

In this section, the comparative study of the proposed BSGWO and the conventional systems of filter design is detailed on the basis of the metrics, namely fitness, Mean Absolute Error (MAE), magnitude, and the number of components.

4.4.1. Analysis based on the filter order as 4

Figure 4 shows the comparative analysis with the filter order of 4 based on the performance metrics. Figure 4.a shows the analysis based on fitness value with respect to the various iterations. The fitness of the methods is 1 for all the comparative methods initially and with the increasing number of iterations, the fitness decreases. When the iteration number is 1000, the fitness value of the methods AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and the proposed BSGWO is 0.526, 0.493, 0.555, 0.506, 0.217, 0.1, 0.347, 0.326, and 0.05, respectively. Figure 4.b shows the analysis based on the MAE value with respect to various iterations. The MAE at the iteration number 250 using the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO is 0.369, 0.191, 0.364, 0.242, 0.047, 0.02, 0.107, 0.187, and 0.015, respectively.

Figure 4.c depicts the analysis based on magnitude with respect to the various frequencies. When the frequency is 0.56 Hz, the magnitude of the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is $- 0.0047 dB$ , $- 0.7749 dB$ , $- 2.2586 dB$ , 0 dB, $- 0.9398 dB$ , $- 0.9847 dB$ , 0 dB, $- 0.890 dB$ , and $- 0.9853 dB$ , respectively. The effective method uses less number of components to design the filter without causing any deviation in the waveform of filter output. Figure 4.d shows the analysis based on the number of components used with respect to different iterations. When the iteration is 250, the number of filter components used by the methods AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO is 3866, 3775, 3959, 3816, 3606, 3400, 3598, 3456, 3394, and 3394, respectively. It is clear that the proposed BSGWO uses very less number of components to design the filter as compared to the other methods.

Fig. 4.

Analysis with the filter order 4 based on the a) fitness, b) MAE, c) magnitude, and d) number of components required.

Fig. 5.

Analysis with the filter order 3 based on the a) fitness, b) MAE, c) magnitude, and d) number of components required.

4.4.2. Analysis based on the filter order as 3

Figure 5 shows the comparative analysis with the filter order of 3 based on the performance metrics. Figure 5.a shows the analysis based on fitness value with respect to the variation in number of iterations. The fitness of the methods is 1 for all the comparative methods initially however, the fitness of the methods decrease with the increase in number of iterations. Though the fitness decreases upon the iteration count, it is essential to report that the proposed BSGWO outperformed the existing methods with a minimal fitness measure throughout the iterations. When the iteration number is 1000, the fitness value of the methods AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and the proposed BSGWO is 0.5616, 0.5375, 0.5064, 0.5263, 0.2246, 0.1, 0.2584, 0.3434, and 0.05, respectively. Figure 5.b shows the analysis based on the MAE value with respect to variation in number of iterations. The MAE at the iteration number 250 using the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO is 0.2248, 0.3373, 0.083, 0.2035, 0.0782, 0.02, 0.1266, 0.0569, and 0.0155, respectively.

Figure 5.c depicts the analysis based on magnitude with respect to the various frequencies. When the frequency is 0.58 Hz, the magnitude of the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is $- 0.0037 dB$ , $- 4.7742 dB$ , $- 6.2687 dB$ , 0, $- 0.0931 dB$ , $- 0.5204 dB$ , $- 0.0755 dB$ , $- 0.5194 dB$ , and $- 0.5204 dB$ , respectively. The effective method uses less number of components to design the filter without causing any deviation in the waveform of filter output. Figure 5.d shows the analysis based on the number of components used with respect to different iterations. The number of components used by the methods AABC, CABC, MOABC, ABC, PSO, and BSABC, BSO, GWO, and BSGWO at the iteration number 250 is, 3828, 3872, 3747, 3746, 3654, 3424, 3611, 3711, and 3420, respectively. It is clear that the proposed BSABC uses very less number of components to design the filter as compared to the other methods.

Fig. 6.

Analysis with the filter order 5 based on the a) fitness, b) MAE, c) magnitude, and d) number of components required.

4.4.3. Analysis based on the filter order as 5

Figure 6 shows the comparative analysis with the filter order of 5 based on the performance metrics. Figure 6.a shows the analysis based on fitness value with respect to the variation in number of iterations. The fitness of the methods is 1 for all the comparative methods initially that decreases with the increase in the iteration count. When the iteration number is 1000, the fitness value of the methods AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and the proposed BSGWO is 0.5406, 0.5207, 0.5587, 0.4999, 0.2202, 0.1, 0.1909, 0.3726, and 0.05, respectively. Figure 6.b shows the analysis based on the MAE value with respect to different iterations. The MAE at the iteration number 250 using the methods, such as AABC, CABC, MOABC, ABC, PSO, BSO, GWO, and BSABC, and BSGWO is 0.1477, 0.1767, 0.3163, 0.3544, 0.0354, 0.02, 0.1887, 0.1501, and 0.0155, respectively.

Figure 6.c depicts the analysis based on magnitude with respect to the various frequencies. When the frequency is 0.58 Hz, the magnitude of the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is $- 0.0037 dB$ , $- 0.0274 dB$ , $- 2.1977 dB$ , $- 3.4313 dB$ , $- 0.2104 dB$ , $- 0.3903 dB$ , $- 0.2157 dB$ , $- 0.1868 dB$ , and $- 0.3903 dB$ , respectively. The effective method uses less number of components to design the filter without causing any deviation in the waveform of filter output. Figure 6.d shows the analysis based on the number of components used with respect to variation in iterations. The number of components used by the methods AABC, CABC, MOABC, ABC, PSO, and BSABC, BSO, GWO, and BSGWO at the iteration number 250 is 3844, 3827, 3911, 3849, 3636, 3404, 3804, 3591, and 3398, respectively. It is clear that the proposed BSABC uses very less number of components to design the filter as compared to the other methods.

4.4.4. Analysis based on the filter order as 64

Figure 7 shows the comparative analysis with the filter order of 64 based on the performance metrics. Figure 7.a shows the analysis based on fitness value with respect to the variation in number of iterations. The fitness of the methods is 1 for all the comparative methods initially decrease with the increase in number of iterations. When the iteration number is 1000, the fitness value of the methods AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and the proposed BSGWO is 0.5406, 0.5207, 0.5587, 0.4999, 0.2202, 0.1, 0.1909, 0.3726, and 0.05, respectively. Figure 4.b shows the analysis based on the MAE value with respect to variation in number of iterations. The MAE at the iteration number 250 using the methods, such as AABC, CABC, MOABC, ABC, PSO, BSO, GWO, and BSABC, and BSGWO is 0.3653, 0.3653, 0.3653, 0.3653, 0.2544, 0.1629, 0.1463, 0.1075, and 0.0365, respectively.

Figure 7.c depicts the analysis based on magnitude with respect to the various frequencies. When the frequency is 0.58 Hz, the magnitude of the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is $- 0.0037 dB$ , $- 1.2263 dB$ , $- 2.0732 dB$ , $- 4.6036 dB$ , $- 0.9658 dB$ , $- 0.0917 dB$ , $- 0.0029 dB$ , $- 0.9132 dB$ , and $- 0.0917 dB$ , respectively. The effective method uses less number of components to design the filter without causing any deviation in the waveform of filter output. Figure 7.d shows the analysis based on the number of components used with respect to variation in number of iterations. The number of components used at the iteration number 250, by the methods AABC, CABC, MOABC, ABC, PSO, and BSABC, BSO, GWO, and BSGWO is 3844, 3827, 3911, 3849, 3636, 3414, 3804, 3591, and 3398, respectively. It is clear that the proposed BSABC uses very less number of components to design the filter as compared to the other methods.

4.5. Discussion

The comparative discussion of the filter design methods are depicted in Table 2. The methods are analyzed based on Fitness, MAE, Magnitude, and the number of components. The fitness of the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is 0.526, 0.493, 0.555, 0.506, 0.217, 0.1, 0.347, 0.326, and 0.05, respectively. The MAE of the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is 0.355, 0.426, 0.6340.394, 0.084, 0.02, 0.2590, 0.4576, and 0.015, respectively. The magnitude of the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is $- 114.9 dB$ , $- 30.1 dB$ , $- 45.6 dB$ , $- 26.7 dB$ , $- 41.1 dB$ , $- 97.179 dB$ , $- 91.469 dB$ , $- 38.937 dB$ , and $- 96.0 dB$ , respectively. Similarly, the number of components used by the methods, such as AABC, CABC, MOABC, ABC, PSO, BSABC, BSO, GWO, and BSGWO, is 3866, 3775, 3959, 3816, 3606, 3400, 3598, 3456, and 3394, respectively. Thus, it is evident that the proposed BSGWO method produces better results and outperforms the existing methods of filter design in terms of the considered metrics. The proposed method has reduced hardware complexity and provides high local optima avoidance and better convergence rate. Also, produces global optimal solutions by iterating the individuals and enhance performance. Thus, the result for the proposed method is higher when compared with the existing methods.

Fig. 7.

Analysis with the filter order 64 based on the a) fitness, b) MAE, c) magnitude, and d) number of components required.

Table 2

Comparative analysis of the filter design methods

Metrics	AABC	CABC	MOABC	ABC	PSO	BSABC	BSO	GWO	Proposed BSGWO
Fitness	0.526	0.493	0.555	0.506	0.217	0.1	0.347	0.326	0.05
MAE	0.355	0.426	0.634	0.394	0.084	0.02	0.2590	0.4576	0.015
Magnitude (dB)	$- 114.9$	$- 30.1$	$- 45.6$	$- 26.7$	$- 41.1$	$- 97.179$	$- 91.469$	$- 38.937$	$- 96.0$
Number of components	3866	3775	3959	3816	3606	3400	3598	3456	3394

5. Conclusion

In recent years, artificial evolutionary techniques are used in the design of digital filters due to its enhanced performance. In general, these filters are used in various digital system networks to process the signals in two or more channels. The optimal selection of filter coefficients is required for FIR filter design with reduced hardware complexity. Thus, a hybrid optimization algorithm known as Brain Storm- Grey Wolf Optimizer (BSGWO) is proposed for the efficient selection of the filter coefficients. The proposed algorithm is the combination of Brain Storm Optimization (BSO) and Grey Wolf Optimizer (GWO) algorithms. Accordingly, input signal is initially interpolated to form an intermediate signal with the use of the FIR filter. Then, the factored truncated cascade filter is developed to interpolate the signal. After designing the filter coefficients, the optimal filter coefficients are selected using the proposed BSGWO algorithm. The original filter is obtained using the least square estimation and new filter is obtained using the proposed algorithm, which tunes the filter coefficients. The performance analysis is performed with respect to fitness, MAE, magnitude, and number of components. The proposed method produces minimum fitness, MAE, number of components and magnitude of 0.05, 0.0155, and 3372, $- 96.0 dB$ , respectively. The method needs improvement in their design objectives including the throughput and required area etc., which should be analyzed for practical applications, thus in future an intelligent optimization techniques will be employed to address this issue.

Conflict of interest

None to report.

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