Author Contributions
Conceptualization, H.H. and A.K.; methodology, H.H. and A.K.; software, H.H. and K.A.-h.; validation, H.H. and K.A.-h.; formal analysis, H.H. and K.A.-h.; investigation, H.H., A.K. and K.A.-h.; resources, H.H. and K.A.-h.; data curation, H.H. and K.A.-h.; writing—original draft preparation, H.H. and A.K.; writing—review and editing, A.K. and K.A.-h.; visualization, H.H. and K.A.-h.; supervision, A.K.; project administration, A.K. and K.A.-h. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
ACO | Ant colony optimization |
Alg. | Algorithm |
CRO | Chemical reaction optimization |
DAPSO | Dynamic adjustable particle swarm optimization |
DSP | Digital signal processing |
dB | Decibel unit |
ECG | Electrocardiogram |
EEG | Electroencephalogram |
FIR | Finite impulse response |
GA | Genetic algorithm |
Gen. | Generation |
IIR | Infinite impulse response |
PSO | Particle swarm optimization |
SLL | Side lobe level |
Appendix B. Analysis of Random Solutions for Crossover and Mutation
This section compares random seed solutions (no fixed seed) using crossover and mutation functions. For each function, the sub-functions are analyzed, and figures are shown. Four figures show the SLL and how the SLL varies. In each caption, the SLL worst case and best case are specified with mean and median values, followed by the SLL solutions.
Figure A1,
Figure A2,
Figure A3 and
Figure A4 show the filter’s magnitude frequency response with ten solutions for crossover/heuristic, crossover/two-point, mutation adapt-feasible, and mutation/Gaussian, respectively. By analyzing the mean and median values, we observe that mutation/adapt-feasible is the best, followed by mutation/Gaussian, then crossover/two-point, and then the worst case is crossover/heuristic.
Figure A1 depicts ten solutions for the crossover/heuristic sub-settings. The worst case is SLL: −19.0941 dB, and the best case is SLL: −25.5974 dB. Mean = −21.7458 dB. Median = −21.8616 dB. Ten SLL solutions = [−19.0941, −22.1809, −25.5974, −22.0978, −20.8282, −21.9670, −21.7562 −19.2285, −21.4284, −23.2790].
Figure A1.
Minimization of the SLL. Different colors resembles ten solutions for crossover/heuristic.
Figure A2 shows the minimization of the SLL. Different colors resemble ten solutions for crossover/two-point. Worst case SLL: −28.8705 dB. Best case SLL: −31.7864 dB. Mean = −30.8800 dB. Median = −30.9479 dB. Ten SLL solutions = [−31.1692, −31.7864, −31.5800, −30.6253, −31.3180, −30.6638, −28.8705, −30.7265, −31.6496, −30.4102].
Figure A2.
Minimization of the SLL. Different colors resembles ten solutions for crossover/two-point.
Figure A3 depicts the minimization of the SLL. Ten (10) solutions for mutation/adapt-feasible. Worst case SLL: −30.7847 dB. Best case SLL: −31.7460 dB. Mean = −31.3768 dB. Median = −31.3983 dB. Ten SLL solutions = [−31.2148, −31.1336, −30.7847, −31.5693, −31.6100, −31.6331, −31.4939, −31.3027, −31.2802, −31.7460].
Figure A3.
Minimization of the SLL. Different colors resembles ten solutions for mutation/adapt-feasible.
Figure A4 displays the minimization of the SLL. Ten (10) solutions for mutation/Gaussian. Worst case SLL: −29.1817 dB. Best case SLL: −32.0514 dB. Mean = −30.9744 dB. Median = −31.2508 dB. Ten SLL solutions = [−29.8972, −31.7044, −31.2590, −31.4947, −29.1817, −31.2425, −31.0011, −32.0514, −31.8377, −30.0746].
Figure A4.
Minimization of the SLL. Different colors resembles ten solutions for mutation/Gaussian.
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Figure 1.
Research methodology flow chart.
Figure 2.
Structure of a direct-form discrete-time FIR of order M.
Figure 3.
General structure of a (GA) for digital filter applications.
Figure 4.
A normalized filter gain vs relative frequency of a population for 20 filters (chromosomes). Each filter is visualized in differnt colours. The best filter is shown in red with minimum side lobe level (SLL). Relative or normalized frequency is in logarithmic scale. The filter order is N = 19.
Figure 5.
The the best filter based on the minimum SLL selected from the initial population. The SLL is at −16.166 dB. Relative or normalized frequency is in logarithmic scale. Filter order is (N) = 19.
Figure 6.
Minimization of the SLL. The magnitude spectrum shows two algorithm settings, each with two functions, as described in the legend.
Figure 7.
Illustration of algorithm setting and analysis four categories convergence speed of the SLL minimization. Four algorithm-setting categories are depicted in the figure.
Figure 8.
Minimization of the SLL (max gen. 100 and 200).
Figure 9.
Convergence speed of the SLL minimization. For max generations of 100, the fitness stops exactly at 100 generations, while, for the three other cases, the generations stop at 193 generations. The lines for the four categories are not clear because they have the same convergence in this case. Please refer to
Table 2 for detailed numbers.
Figure 10.
Minimization of the SLL. For population size increasing from 100 to 300, the SLL decreases from −30.7976 dB to −31.5810 dB, with a slight difference of less than 1 dB.
Figure 11.
Convergence speed of the SLL minimization.
Figure 12.
Minimization of the SLL, changing the function tolerance from to , shifting the SLL from −30.1490 dB to −29.9617 dB with faster convergence. Where the base e = 10 in the figure.
Figure 13.
Convergence speed of the SLL minimization of different tolerance options. Where the base e = 10 in the figure.
Figure 14.
Ideal ECG signal for a normal person.
Figure 15.
Example of one second of the noisy ECG signal.
Figure 16.
The magnitude spectrum of the noisy ECG signal. The ECG signal is contaminated with a 60 Hz signal from the power line. The sampling rate is 720 samples/s. Half sampling is at 360 Hz.
Figure 17.
Filtered ECG spectrum magnitude using GA Gaussian mutation.
Figure 18.
One second at the beginning of the filtered signal showing complete removal of the artifact, 60 Hz power signal interference with smoothed ECG signal using Gaussian mutation.
Figure 19.
Comparison between GA FIR filter and conventional FIR for low-pass filters generated by Matlab.
Table 1.
Algorithm setting evaluation results.
Alg. Setting | No. | Sub-Function Solver Setting | SSL (dB) | Gen. No. Convergence |
---|
Crossover | 1 | Heuristic | −19.0941 | 71 |
Crossover | 2 | Two-point | −30.1599 | 193 |
Mutation | 1 | Adapt-Feasible | −30.4694 | 235 |
Mutation | 2 | Gaussian | −30.8327 | 236 |
Table 2.
Population setting evaluation results.
Solver Option | No. | Sub-Function Solver Setting | SSL (dB) | Gen. No. Convergence |
---|
Max gens | 1 | 100 | −30.0219 | 100 |
Max gens | 2 | 200 | −30.1599 | 193 |
Stall time limit | 1 | 2 s | −30.1599 | 193 |
Stall time limit | 2 | 4 s | −30.1599 | 193 |
Table 3.
Runtime limits.
Pop. Setting | No. | Sub-Function Solver Setting | SSL (dB) | Gen. No. Convergence |
---|
Pop. Size | 1 | 100 | −30.7976 | 253 |
Pop. Size | 2 | 300 | −31.5810 | 235 |
Initial. Pop | 1 | Initial Pop. () | −31.3811 | 274 |
Initial. Pop | 2 | Initial Pop. () | −31.3141 | 202 |
Table 4.
Tolerance evaluation results.
Solver. Options | No. | Sub-Function Solver Setting | SSL (dB) | Gen. No. Convergence |
---|
Fun. tolerance | 1 | | −29.9617 | 92 |
Fun. tolerance | 2 | | −30.1490 | 148 |
Fitness limit | 1 | 0.01 | −26.3296 | 14 |
Fitness limit | 2 | 0.001 | −30.1599 | 193 |
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