Optimized FIR Filter using Genetic Algorithms: A Case Study of ECG Signals Filter Optimization

1. Introduction

Digital filters are used in many different fields to remove noise, split multiple signals [1], missing data imputation [2], and enhance the quality of received signals [3,4]. In recent years, there has been a lot of research into the use of Genetic Algorithms to optimize the design of digital filters. A Genetic algorithm [5] is a metaheuristic optimization algorithm that can be used to find optimal, or near optimal, solutions to problems that are too computationally expensive and are difficult to solve using traditional methods. In the case of digital filter optimization, the goal is to find a set of coefficients that optimizes the performance of the filter using some performance metric.

Electrocardiography (ECG) is a well-known and reliable tool that is widely used in the medical field to analyze and diagnose the condition of the heart. This analysis is very crucial for the early detection of heart diseases. ECG signals are very weak signals and hence susceptible to disturbances in the measurement environment. Using digital filters to remove unwanted parts of these signals is very critical for the performance and accuracy of the algorithms and software tools that are used to detect abnormal heart conditions [6].

The side lobe level (SLL) [7] is a measure of the noise that is produced by the filter outside of its passband. A low side lobe level indicates that the filter is good at removing noise. The genetic algorithm works by iteratively generating new populations of filters, each of which is a slight variation of the previous population. The filters in each population are evaluated based on their side lobe level, and the best filters are selected to create the next generation. This process continues until a filter with a sufficiently low side lobe level is found.

The genetic algorithm approach to digital filter optimization has several advantages over traditional methods [8,9]. First, it is not limited to finding local minima, which can be a problem with traditional methods. Second, it can be used to find solutions to problems that are too complex for traditional methods. Third, it is relatively easy to implement. This approach is a promising technique that has the potential to significantly enhance the design of digital filters used in different fields, including biomedical engineering.

In this paper, a GA-based digital filter is proposed to optimize the coefficients of FIR filters for ECG signals. The search space for the filter is defined by 20 coefficients. Finding the optimal values of these coefficients is practically extremely large due to the high non-linearity of the frequency response of the filters. The objective, or fitness, function to be optimized is the minimization of the side lobe level ($SLL$). As discussed earlier in this section, the $SLL$ is a measure of the noise that is produced by the filter outside of its passband. A low $SLL$ indicates that the filter is good at removing noise. The actual coefficients of the filter are mapped to the genes of a chromosome. Each coefficient represents a gene, while the whole coefficients represent a chromosome. The genetic algorithm works by iteratively generating new populations of filters, each of which is a slight variation of the previous population. The filters in each population are evaluated based on their SLL, and the best filters are selected to create the next generation. This process continues until a filter with a sufficiently low SLL is found.

The remainder of the paper is organized as follows. Section 2 presents the related work. Section 3 explain the research methodology. Section 4 highlight and compare different options of the proposed methodology. At the same time, Section 5 discuss the results for the case study of using $GA$ to filter ECG signal. Eventually, Section 6 concludes the paper.

2. RELATED WORK

Meta-heuristic optimization algorithms such as Genetic Algorithms ($GA$) [10], Ant Colony Optimization ($ACO$) [11], and Chemical Reaction Optimization ($CRO$) [12] are used to solve different optimization problems. These algorithms have been extensively used in recent years to implement digital filters in a wide range of applications [13].

The authors in [14] proposed a comprehensive fitness function for an infinite impulse response ($IIR$) digital filter design with six terms, including a low delay parameter. Low delay filters are preferred since they achieve quasi-real-time processing. Other parameters can be included in the fitness function to meet linear phase, minimum phase and stability constraints.

The authors in [15] proposed a method to detect breast malignancies. The filter bank that has been used to obtain the filter response is the Maximum Response Eight Filter Bank. Genetic Algorithm and Linear Discriminant Analysis have been used for feature selection and feature reduction, respectively.

An optimal fourth-order Butterworth active low-pass filter design by the hybrid intensified current search ($HICuS$) algorithm was proposed in [16]. With AI-based modern optimization, the HICuS algorithm is claimed to be one of the most efficient metaheuristic optimizers.

The authors in [17] presented a differential evolution algorithm that uses a combination of rectangular and polar coordinates. The algorithm was developed using thirteen commonly used numerical optimization test functions. The algorithm was then applied to design an infinite impulse response ($IIR$) digital filter.

A digital filter design problem that involves multiple, often conflicting filter parameters, has been presented in [18]. Hence an optimization algorithm is needed. A low pass finite impulse response (LPFIR) filter optimal design has been achieved using Particle Swarm Optimization (PSO) and Dynamic Adjustable PSO (DAPSO). Other recent works in [19,20,21,22,23] presented design of FIR with variable multiple stop band, coefficients optimization in adaptive equalization and FIR design using arithmetic optimization and genetic algorithms.

3. METHODOLOGY

The research methodology is described in four stages as depicted in Figure 1. Stage one, FIR filter design and coefficient generation. Stage two uses a genetic algorithm to map filter coefficient. Stage three, predict the best solution of the filter coefficient using a genetic algorithm. Stage four evaluates the genetic algorithm prediction using impulse response.

Stage one (FIR filter design) : The filtering process uses a direct form discrete-time FIR filter of order M. The top part is an M-stage delay line with $M+1$ taps shown in Figure 2. Each unit delay is a $z^{-1}$ operator in Z-transform notation. For a causal discrete-time FIR filter of order M, each value of the output sequence is a weighted sum of the most recent input values as in Equation 1.

$$y\left[n\right]=\sum _{i=0}^M{b}_{i}x[n-i]$$

(1)

where $x\left[n\right]$ and $y\left[n\right]$ are the discrete input and output signals respectively. M is the filter order, and $b_i$ is the value of the impulse response at the $i^{th}$ instant for M of an M-order FIR filter.

The frequency response $H\left(z\right)$ is expressed in terms of the Z-transform transfer function as shown in Equation 2:

$$H\left(z\right)=\frac{Y\left(z\right)}{X\left(z\right)}=\frac{{\sum }_{k=0}^M{b}_k{z}^{-k}}{1+{\sum }_{k=1}^N{a}_k{z}^{-k}}$$

(2)

Ideally, we would like to have M negligible for small computation with a target frequency response given by the ideal low-pass filter as follow in Equation 3:

$$H\left(\omega \right)=\left\{\begin{array}{cc}\hfill 1,& if\left|\omega \right|\le |{\omega }_c|\hfill \\ \hfill 0,& if{\omega }_c\le \left|\omega \right|\le \pi \hfill \end{array}\right.$$

(3)

Stage two (Map and process FIR coefficients using $GA$) : The general structure of $GA$ is depicted in Figure 3. An initial population of 20 chromosomes, each one containing 20 genes, was created randomly in the range [0,1]. The genes represent the filter coefficients. The initial set was passed to a fitness function for frequency response evaluation and side lobe level ($SLL$) estimation. The selection is done for the minimum $SLL$. If the condition of the minimum $SSL$ level does not meet the desired threshold, the $GA$ selects new genes or filter coefficient randomly using crossover and re-evaluates the fitness function $SSL$.

Stage three (Predict the best solution) : The mapped filter coefficients are filtered out by using $GA$ to select the best solution among the coefficients. Figure 4 depicts 20 frequency responses for the population and emphasizes the best solution shown in red with a thick solid line. Figure 5 illustrates the frequency response in magnitude for the best solution extracted from the initial population. The coefficients of the best solution are shown below. The $SLL$ is shown in Figure 5 with -16.1660 $dB$.

Stage four (Evaluate the best solution) : Finally, the best solution is evaluated using the impulse response to select the minimum $SSL$. This step is important to make sure the best solution is optimized in the desired frequency bandwidth and is at the extreme minimum level for other frequencies. The $GA$ has some randomness in the process, and this step is essential as a final quality control measure for choosing the best solution. Figure 5 depicts the impulse response coefficient of the best solution extracted from the initial population given by $b\left[n\right]$ coefficients. In this example, Filter Coeffs ($b\left[n\right]$) = [0.5009 0.4457 0.6175 0.5180 0.4395 0.8174 0.7331 0.9891 0.8428 0.7717 0.6087 0.9935 0.4332 0.3967 0.4869 0.6107 0.3800 0.0383 0.8333 0.7669]. The best $SLL$ from the initial population is -16.1660 $dB$.

4. Comparative analysis of the GA algorithm options for FIR filter optimization

The analysis of the $GA$ algorithm is formulated as an optimization problem. The core of an optimization problem is based on the minimization of a fitness function. This latter has to be set meticulously to achieve our objective of reaching a minimum $SSL$. The fitness function has input variables given by the filter coefficients and the resolution of the frequency response of the filter. The objective is to minimize the maximum of the $SLL$ as an output. This output is supplied to the genetic algorithm ($GA$) for minimization.

Detailed results and analysis for the study that includes 16 cases) are listed and discussed in Section 4.1,Section 4.2, Section 4.3 and Section 4.4, with addition to detailed information in Appendix A and Appendix B. These Sections summarize the comprehensive study options. Each section show results on using optimization solver functions with sub-functions for each $GA$ algorithm option. Results are given by the $SLL$ and the filter coefficients, along with graphs on convergence and frequency response. The $GA$ is set with different options in order to study the effect of each parameter on the result of the minimization. The $GA$ uses other options given by the solver. Options are algorithm settings, Section 4.1, population settings in Section 4.2, run time limits in Section 4.3, and tolerance in Section 4.4.

The fitness function based on estimating the maximum of the $SLL$ of the normalized discrete frequency response is shown in the following sections. For each solver option from the four alternatives, a graph compares the four $SLL$ cases, and another graph compares the convergence. The $GA$ results compared with the conventional $FIR$ results generated by Matlab. Where the seed of the random number in the Matlab generator is fixed.

4.1. Algorithm Settings Results and Analysis

The study of algorithm setting has two categories. Cross-over function and mutation function. The cross-over function has two options heuristic and cross-over two points. At the same time, the mutation options are adapt-feasible and Gaussian.

Figure 6 shows the gain with respect to normalized frequency for cases (1-4); see Appendix A. The lowest performance in Figure 6 is given by the heuristic function. The crossover 2-points, Gaussian mutation and adapt-feasible mutation, have similar performance up to 1 $dB$ margin.

At the same time, for the above-mentioned four cases, the convergence speeds are different, as shown in Figure 7. Considering the $SLL$ and the filter coefficients are listed for each case, see Appendix A. The crossover heuristic has the best performance since it converges in 71 generations, followed by the crossover 2-pts with 193 generations, and then comes the mutation with adapt-feasible, followed by the Gaussian mutation with 235 generations and 326 generations, respectively.

The results of various options of the Algorithm setting are listed in Table 1. The population size is 200 for all cases. By analyzing Figure 6 and Figure 7, we can observe the optimal solution is given by the Crossover with a two-point function since it achieves an $SLL$ close to the best solution with moderate convergence.

Further analysis and experiment results for mutation and cross-over results are presented in the Appendix B

4.2. Population Settings Results and Analysis

The population setting has two solver options. Max gens and Stall time limit. Max gens were evaluated using two sub-functions, settings 100 and 200. While the Stall time sub-functions are set as 2 and 4 seconds. Table 2 summarizes the result for various options on the configurations of the GA- population setting category. Each solver option is studied, with two sub-functions each. The $SSL$ values are presented in $dB$.

In Figure 8, when max generation is set to 100, the $SLL$ stops at -30.0219, while the $SLL$ stops at $SLL$ of -30.1599 $dB$ at generation 193 before the max generation of 200. For a stall time limit of 2 and 4 seconds, the $SLL$ converges at the same value with the same filter coefficients.

Figure 9 shows the convergence speed of the $SLL$. By analyzing Figure 8 and Figure 9, we can observe that increasing the number of generations or stall time limit will not improve the $SLL$ if the fitness converges before reaching the set values.

4.3. Run Time Limits Results and Analysis

The run-time limit has two categories. The population size and initial population. Population size has to sub-functions solver settings the 100 and 300. In contrast, the initial population has two solvers setting $x_0$ and $x_1$; see Appendix A for more details. The $SSL$ convergence of the number of population settings category is depicted in Figure 11 with respect to the population sample of different generations number 202, 235, 253 and 274, as listed in Table 3. The initial value $x_1$, population size 300, population size 100 and initial $x_0$, respectively. Changing the population size or initial condition will shift the convergence speed. Increasing population size will decrease the convergence iterations.

Figure 10 shows the default value of the $GA$ is 200 chromosomes. When the population is set to 300, the $SLL$ is almost constant when the initial condition is varied. It changes from -31.3811 $dB$ to -31.3141 $dB$ with a difference of 0.067 $dB$.

By analyzing Figure 10 and Figure 11, we can observe the effect of changing the population size and the change of the initial condition on the algorithm convergence for different setting categories.

4.4. Tolerances Results and Analysis

The tolerance option has two categories, function tolerance and fitness limit. The function tolerance solver setting is set to the $e^x$, and the fitness limit study selects 0.01 and 0.001 as two different decimal sub-function solvers. Table 4 summarizes the tolerance option setting of $GA$ algorithm. The table also lists different solver options and different generation numbers. Altering the fitness limit from 0.01 to 0.05 change the $SLL$ from -30.1599 $dB$ to -26.3296 $dB$; see Figure 12 for more details. For instance, $e^x$ in figures corresponds to ${10}^x$ (base 10, not base e=2.71). For the convergence speed Figure 13, When the function tolerance decreases from $1.e^{-5}$ to $1.e^{-4}$, the convergence decreases from 148 gen. to 92 gen. When the fitness limit varies from 0.01 to 0.05, the convergence gains high speed from 193 genenerations to 14 generations.

By analyzing Figure 12 and Figure 13, we can observe the effect of changing the function tolerance and the fitness limit on both the $SLL$ and the convergence speed (number of generations required for the stop criteria) as depicted in the figures captions.

5. ECG Filtering Results Discussion

This section presents a case study for the optimized FIR filter coefficients using GA. The validation of the proposed algorithm is performed on an ECG signal where an existing noise is to be removed, and the comparison of both time and frequency signals is carried out. The signal was then analyzed to assess the quality of the filtered signal. Comprehensive results are shown below in different figures.

Figure 14 depicts the ideal ECG signal for a normal person. The figure shows the main components of the ECG, which are the P-wave, QRS complex and T-wave. The ideal signal act as a reference while removing the noise to make sure all components are preserved.

The case study discusses a noisy ECG signal contaminated with 60 $Hz$ noise and other artifacts. The study removes the artifact from a noisy ECG signal using $GA$. The 60 $Hz$ noise from the power line is the common source of interference that affect the ECG signal. It has a considerable effect as its amplitude is significant to interfere with the low amplitude ECG signal, which is in the range of Millivolts ($mV$) in most cases. Furthermore, there are different type of artifacts and interference that affects ECG signal like body movement and external electromagnetic waves that are generated from other devices, mainly from medical imaging devices at hospitals.

As illustrated in Figure 15, the noise effect is significant on P-wave and T-wave as they have lower amplitude than the QRS complex of the ECG. Furthermore, there is an artifact at the beginning of the signal; this artifact is typically generated from motion or improper electrode attachments. The noise in the signal misleads the medical doctor while identifying the patient’s case.

To analyze the source of the noise on the complex structure of the ECG signal, we need to analyze the magnitude and frequency response of the spectrum. This is important to identify the signal that has a dominant spectrum over the typical range of ECG.

Figure 16 is the magnitude spectrum displaying the level of noise in $dB$ with respect to the frequency in $Hz$. Visualizing the noisy signal in the frequency domain help to identify the dominant noisy signal on the original data.

Figure 17 is the spectrum of the filtered signal. The filtered ECG using $GA$ with Gaussian mutation category which shows an attenuation of noise with more than 30 $dB$. At half sampling rate, the signal is at -70 $dB$. The interference signal of 60 $Hz$ is attenuated with more than 30 $dB$. The analysis is for case 3; see Appendix A for details.

The final filtered ECG is shown in Figure 18 with respect to the time domain. Comparing Figure 15 and Figure 18 shows the excellent filtering behaviour of the Gaussian mutation GA filter. The ECG signal in the Figure 18 may be different from the ideal ECG in Figure 14 as the real recording has some variations for normal cases between different patients.

Figure 19 shows the difference between $GA$ and standard FIR filter frequency response. The figure shows standard FIR attenuation gain as -40 $dB$ compared to -31 $dB$ for the GA. The GA FIR presents a sharp passband relative to the standard FIR magnitude for the same order, even though the relative cut-off frequency is set to 0.01 (1%).

6. Conclusions

In this study, a genetic algorithm is used to comprehensively analyze the synthesis of digital low-pass filters. Sixteen cases are analyzed, and results are shown to demonstrate the effectiveness of the method. The fitness function is derived from the frequency response of the finite impulse response low-pass filter, and the inputs of the fitness are the coefficients of the filter, which start with a random sequence, and the frequency resolution of the frequency response. The output is the side lobe level that has to be minimized for the best filter synthesis. The different options were analyzed, and their effects were shown in both graphs and summarized. A Gaussian filter was used to perform digital filtering of a noisy ECG real recording. The filter shows enhanced results in suppressing different kinds of noise.