# On the Design of Power Law Filters and Their Inverse Counterparts

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- The stability of the PLF approximant in S-K method [54] is governed by the stability of the mother filter function. While this guarantees a stable rational approximant, however, the zeros of the filter transfer function may lie on the right-half of s (RHS)-plane. For example, while the zeros of the BPPLFs and BSPLFs designed using the S-K method lie on the left-half s-plane; however, the LPPLF and HPPLF models for $\alpha =0.7$ have zeros located at {+183.0053, –23.6922, –8.6177, –1.7485} and {+0.0055, –0.5718, –0.1161, –0.0422}, respectively. It may be noted that for the S-K method-based LPPLFs and HPPLFs with $\alpha \in [0.51,0.99]$, one zero lies on the RHS-plane. Consequently, inverting such a transfer function will lead to an unstable inverse-LPPLF (ILPPLF) and inverse-HPPLF (IHPPLF) model. In contrast, the proposed approach can guarantee the generation of stable designs for both PLFs and IPLFs. Hence, this paper also presents the design of IPLFs that has not yet been reported in the literature.
- (ii)
- The modeling accuracy of the proposed optimal PLF approximants, as justified by the Mean Absolute Relative Error (MARE) metric, is significantly better (particularly for the LP, HP, and BP-types) in comparison to the S-K method.

## 2. Proposed Technique

## 3. MATLAB Simulations and Performance Analysis

#### 3.1. Statistical Analyses and Performance Evaluation

#### 3.1.1. Comparisons Based on the MARE Metric

#### 3.1.2. Comparisons Based on the Wilcoxon Rank-Sum Hypothesis Test

#### 3.2. Comparison with the Literature

## 4. Experimental Validation

- Step 1:

- Step 2:
- Select the desired center frequency for the filter. For instance, a center frequency of 1 kHz is used here.
- Step 3:
- Set the values of R, R${}_{\mathrm{F}}$, ${R}_{\mathrm{out}}$, and ${R}_{\mathrm{in}}$. Note that the resistor ratio ${R}_{\mathrm{out}}/{R}_{\mathrm{in}}$ helps in gain adjustment.
- Step 4:
- Step 5:
- Choose the nearest values of the passive components from the E24 industrial series for the resistors and the E12 series for the capacitors. The passive components required to implement the proposed PLFs and IPLFs are presented in Table 5 and Table 6, respectively. For better accuracy, ${R}_{\mathrm{out}}$ was selected from the E48 series for the IHPPLF.

#### 4.1. Measurement Results for the PLFs

#### 4.2. Measurement Results for the IPLFs

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Optimal coefficients of the designed PLFs based on different objective functions (${f}_{k}$).

Filter | ${\mathit{f}}_{\mathit{k}}$ | $\mathit{\alpha}$ | X |
---|---|---|---|

0.3 | [0.0237 6.5086 157.6053 608.9508 435.0749 55.8440 461.1371 792.3060 435.0843] | ||

${f}_{1}$ | 0.5 | [0.0000 1.0000 3.3454 3.9298 1.6952 4.0523 6.5467 5.1288 1.6952] | |

0.7 | [0.0000 0.0935 14.6833 215.5569 363.5909 56.1673 406.3852 575.4828 363.5909] | ||

0.3 | [0.0251 6.1852 130.3929 454.1981 337.5340 48.9649 357.4187 596.5829 337.5048] | ||

LPPLF | ${f}_{2}$ | 0.5 | [0.0000 1.0000 4.3404 5.8369 2.7806 5.0470 9.1598 7.8036 2.7806] |

0.7 | [0.0000 0.0923 14.9358 225.8841 382.2588 57.9809 426.5128 604.2660 382.2596] | ||

0.3 | [0.0226 6.7236 168.2873 653.3916 495.0099 59.0935 493.5963 863.3283 495.0150] | ||

${f}_{3}$ | 0.5 | [0.0000 1.0000 9.8083 16.3840 8.7772 10.5143 23.5806 22.5905 8.7772] | |

0.7 | [0.0000 0.0913 15.5822 238.8673 406.6527 60.8165 451.6448 641.4255 406.6529] | ||

0.3 | [1.0002 7.5918 14.8771 1.8950 0.0121 8.0206 18.1103 9.0732 0.3853] | ||

${f}_{1}$ | 0.5 | [1.0000 9.3030 14.9498 7.7159 0.0000 10.0097 21.7838 20.4089 7.7158] | |

0.7 | [1.0000 0.5868 0.038258 0.000224 0.0000 1.5767 1.1100 0.1493 0.0024555] | ||

0.3 | [1.0001 12.8022 22.9435 3.1493 0.0213 13.2325 28.3618 14.3874 0.6643] | ||

HPPLF | ${f}_{2}$ | 0.5 | [1.0000 19.5428 32.3707 16.4524 0.0000 20.2498 46.4427 44.0274 16.4512] |

0.7 | [0.9995 9.9045 2.9823 0.0395 0.0000 10.8723 13.5286 7.2042 0.3404] | ||

0.3 | [1.0000 1.3285 0.3407 0.01354 0.0000453 1.7526 1.0021 0.1194 0.00201] | ||

${f}_{3}$ | 0.5 | [1.0000 2.6111 2.5477 0.9238 0.0000 3.3182 4.6441 3.2008 0.9238] | |

0.7 | [1.0000 6.2426 2.1947 0.0444 0.0000 7.2319 8.8913 4.9099 0.2952] | ||

0.3 | [0.2099 18.7044 113.0789 18.4473 0.2033 38.0356 117.7061 37.7091 0.9723] | ||

${f}_{1}$ | 0.5 | [0.0687 9.0727 69.2331 9.0209 0.0678 29.8104 73.4942 29.7361 0.9906] | |

0.7 | [0.0194 4.3195 40.6182 4.2579 0.0189 22.5306 43.6451 22.4438 0.9807] | ||

0.3 | [0.2127 18.0508 102.8481 18.0055 0.2118 36.1575 106.9896 36.1009 0.9960] | ||

BPPLF | ${f}_{2}$ | 0.5 | [0.0703 8.6453 59.6221 8.6040 0.0697 27.2035 63.2085 27.1331 0.9931] |

0.7 | [0.0200 4.1789 35.4722 4.1789 0.0200 20.5126 38.2202 20.5126 1.0000] | ||

0.3 | [0.2176 17.2313 87.4857 17.2161 0.2167 33.5267 89.6188 33.5101 0.9971] | ||

${f}_{3}$ | 0.5 | [0.0727 8.6573 56.5588 8.6576 0.0727 26.6767 58.9923 26.6771 1.0001] | |

0.7 | [0.0202 4.2911 36.7490 4.2911 0.0202 21.2939 39.0462 21.2940 1.0000] | ||

0.3 | [0.9999 0.7280 2.0449 0.7280 0.9999 1.1484 2.1611 1.1484 1.0000] | ||

${f}_{1}$ | 0.5 | [0.9999 0.6126 2.0260 0.6126 0.9999 1.3146 2.2312 1.3146 1.0000] | |

0.7 | [0.9999 0.4551 2.0072 0.4551 0.9998 1.4395 2.2719 1.4395 0.9999] | ||

0.3 | [1.0006 19.0360 14.7799 21.6636 7.6187 19.4913 22.0923 24.9053 7.6180] | ||

BSPLF | ${f}_{2}$ | 0.5 | [0.9999 0.6374 2.0280 0.6374 1.0001 1.3406 2.2471 1.3407 1.0001] |

0.7 | [0.9999 0.4853 2.0080 0.4852 0.9998 1.4714 2.2974 1.4713 0.9999] | ||

0.3 | [1.0000 18.3224 15.0720 21.3661 7.3968 18.7480 22.5398 24.5052 7.3968] | ||

${f}_{3}$ | 0.5 | [1.0000 14.5925 9.3187 15.8700 4.9779 15.3015 19.2474 19.3917 4.9779] | |

0.7 | [1.0001 24.7915 14.1559 26.3442 10.5783 25.7848 38.2286 36.8248 10.5803] |

## Abbreviations and Symbols

BP | Band-Pass |

BPPLF | Band-Pass Power Law Filter |

BS | Band-Stop |

BSPLF | Band-Stop Power Law Filter |

CFOA | Current Feedback Operational Amplifier |

FFT | Fast Fourier Transform |

FO | Fractional-Order |

HP | High-Pass |

HPPLF | High-Pass Power Law Filter |

IBPPLF | Inverse Band-Pass Power Law Filter |

IBSPLF | Inverse Band-Stop Power Law Filter |

IHPPLF | Inverse High-Pass Power Law Filter |

ILPPLF | Inverse Low-Pass Power Law Filter |

IPLF | Inverse Power Law Filter |

LP | Low-Pass |

LPPLF | Low-Pass Power Law Filter |

MARE | Mean Absolute Relative Error |

PLF | Power Law Filter |

SD | Standard Deviation |

SFDR | Spurious-Free Dynamic Range |

THD | Total Harmonic Distortion |

${a}_{k}$ | Numerator coefficients of the proposed approximant |

$\alpha $ | Fractional order |

${b}_{k}$ | Denominator coefficients of the proposed approximant |

${\mathsf{\Delta}}_{N}$ | Hurwitz determinants |

${f}_{\mathrm{H}}$ | Half-power frequency |

${f}_{\mathrm{H},\mathrm{high}}$ | High half-power frequency |

${f}_{\mathrm{H},\mathrm{low}}$ | Low half-power frequency |

H | Decision index for Wilcoxon rank-sum test |

${iter}_{\mathrm{m}}$ | Maximum count of loop |

L | Total number of sampled data points |

N | Order of the proposed approximant |

${\omega}_{max}$ | Upper bound of bandwidth |

${\omega}_{min}$ | Lower bound of bandwidth |

${\omega}_{0}$ | Pole frequency |

$\omega $ | Angular frequency |

$p\text{-val}$ | p-value of Wilcoxon rank-sum test |

Q | Quality factor |

${s}^{\alpha}$ | Fractional-order Laplacian operator |

X | Vector of design variables |

${X}_{\mathrm{best}}$ | Best optimal vector of design variables |

## References

- Monje, C.A.; Chen, Y.; Vinagre, B.M.; Xue, D.; Feliu-Batlle, V. Fractional-Order Systems and Controls: Fundamentals and Applications; Springer Science & Business Media: New York, NY, USA, 2010. [Google Scholar]
- Petráš, I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation; Springer Science & Business Media: New York, NY, USA, 2011. [Google Scholar]
- Cafagna, D. Fractional calculus: A mathematical tool from the past for present engineers [Past and present]. IEEE Ind. Electron. Mag.
**2007**, 1, 35–40. [Google Scholar] [CrossRef] - Tavazoei, M.S.; Tavakoli-Kakhki, M.; Bizzarri, F. Nonlinear fractional-order circuits and systems: Motivation, a brief overview, and some future directions. IEEE Open J. Circuits Syst.
**2020**, 1, 220–232. [Google Scholar] [CrossRef] - Yazgac, B.G.; Kirci, M. Fractional differential equation-based instantaneous frequency estimation for signal reconstruction. Fractal Fract.
**2021**, 5, 83. [Google Scholar] [CrossRef] - Muresan, C.I.; Birs, I.R.; Dulf, E.H.; Copot, D.; Miclea, L. A review of recent advances in fractional-order sensing and filtering techniques. Sensors
**2021**, 21, 5920. [Google Scholar] [CrossRef] [PubMed] - Jain, M.; Rani, A.; Pachauri, N.; Singh, V.; Mittal, A.P. Design of fractional order 2-DOF PI controller for real-time control of heat flow experiment. Eng. Sci. Technol.
**2019**, 22, 215–228. [Google Scholar] [CrossRef] - Freeborn, T.J.; Critcher, S. Cole-impedance model representations of right-side segmental arm, leg, and full-body bioimpedances of healthy adults: Comparison of fractional-order. Fractal Fract.
**2021**, 5, 13. [Google Scholar] [CrossRef] - Elwakil, A.S. Fractional-order circuits and systems: An emerging interdisciplinary research area. IEEE Circuits Syst. Mag.
**2010**, 10, 40–50. [Google Scholar] [CrossRef] - Radwan, A.G.; Soliman, A.M.; Elwakil, A.S. First-order filters generalized to the fractional domain. J. Circuits Syst. Comput.
**2008**, 17, 55–66. [Google Scholar] [CrossRef] - Wang, S.F.; Chen, H.P.; Ku, Y.; Lin, Y.C. Versatile tunable voltage-mode biquadratic filter and its application in quadrature oscillator. Sensors
**2019**, 19, 2349. [Google Scholar] [CrossRef] [Green Version] - Krishna, B.T. Studies on fractional order differentiators and integrators: A survey. Signal Process.
**2011**, 91, 386–426. [Google Scholar] [CrossRef] - Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F.M. Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2000**, 47, 25–39. [Google Scholar] [CrossRef] - El-Khazali, R. On the biquadratic approximation of fractional-order Laplacian operators. Analog Integr. Circuits Signal Process.
**2015**, 82, 503–517. [Google Scholar] [CrossRef] - AbdelAty, A.M.; Elwakil, A.S.; Radwan, A.G.; Psychalinos, C.; Maundy, B.J. Approximation of the fractional-order Laplacian s
^{α}as a weighted sum of first-order high-pass filters. IEEE Trans. Circuits Syst. II Express Briefs**2018**, 65, 1114–1118. [Google Scholar] [CrossRef] - Shah, Z.M.; Kathjoo, M.Y.; Khanday, F.A.; Biswas, K.; Psychalinos, C. A survey of single and multi-component fractional-order elements (FOEs) and their applications. Microelectron. J.
**2019**, 84, 9–25. [Google Scholar] [CrossRef] - Kartci, A.; Agambayev, A.; Herencsar, N.; Salama, K.N. Series-, parallel-, and inter-connection of solid-state arbitrary fractional-order capacitors: Theoretical study and experimental verification. IEEE Access
**2018**, 6, 10933–10943. [Google Scholar] [CrossRef] - Zhang, L.; Kartci, A.; Elwakil, A.; Bagci, H.; Salama, K.N. Fractional-order inductor: Design, simulation, and implementation. IEEE Access
**2021**, 9, 73695–73702. [Google Scholar] [CrossRef] - Tsirimokou, G.; Kartci, A.; Koton, J.; Herencsar, N.; Psychalinos, C. Comparative study of discrete component realizations of fractional-order capacitor and inductor active emulators. J. Circuits Syst. Comput.
**2018**, 27, 1850170. [Google Scholar] [CrossRef] - Kartci, A.; Agambayev, A.; Farhat, M.; Herencsar, N.; Brancik, L.; Bagci, H.; Salama, K.N. Synthesis and optimization of fractional-order elements using a genetic algorithm. IEEE Access
**2019**, 7, 80233–80246. [Google Scholar] [CrossRef] - Adhikary, A.; Shil, A.; Biswas, K. Realization of foster structure-based ladder fractor with phase band specification. Circuits Syst. Signal Process.
**2020**, 39, 2272–2292. [Google Scholar] [CrossRef] - Koton, J.; Kubanek, D.; Dvorak, J.; Herencsar, N. On systematic design of fractional-order element series. Sensors
**2021**, 21, 1203. [Google Scholar] [CrossRef] [PubMed] - Ali, A.S.; Radwan, A.G.; Soliman, A.M. Fractional order Butterworth filter: Active and passive realizations. IEEE J. Emerg. Sel. Top. Circuits Syst.
**2013**, 3, 346–354. [Google Scholar] [CrossRef] - Freeborn, T.; Maundy, B.; Elwakil, A.S. Approximated fractional order Chebyshev lowpass filters. Math. Prob. Eng.
**2015**, 2015. [Google Scholar] [CrossRef] - Freeborn, T.J.; Elwakil, A.S.; Maundy, B. Approximated fractional-order inverse Chebyshev lowpass filters. Circuits Syst. Signal Process.
**2016**, 35, 1973–1982. [Google Scholar] [CrossRef] - Kubanek, D.; Freeborn, T.J.; Koton, J.; Dvorak, J. Validation of fractional-order lowpass elliptic responses of (1+ α)-order analog filters. Appl. Sci.
**2018**, 8, 2603. [Google Scholar] [CrossRef] [Green Version] - Mahata, S.; Saha, S.; Kar, R.; Mandal, D. Optimal integer-order rational approximation of α and α+ β fractional-order generalised analogue filters. IET Signal Process.
**2019**, 13, 516–527. [Google Scholar] [CrossRef] - Freeborn, T.J.; Maundy, B.; Elwakil, A.S. Field programmable analogue array implementation of fractional step filters. IET Circuits Dev. Syst.
**2010**, 4, 514–524. [Google Scholar] [CrossRef] - Maundy, B.; Elwakil, A.S.; Freeborn, T.J. On the practical realization of higher-order filters with fractional stepping. Signal Process.
**2011**, 91, 484–491. [Google Scholar] [CrossRef] - Psychalinos, C.; Tsirimokou, G.; Elwakil, A.S. Switched-capacitor fractional-step Butterworth filter design. Circuits Syst. Signal Process.
**2016**, 35, 1377–1393. [Google Scholar] [CrossRef] - Tsirimokou, G.; Psychalinos, C.; Elwakil, A.S. Fractional-order electronically controlled generalized filters. Int. J. Circuit Theory Appl.
**2017**, 45, 595–612. [Google Scholar] [CrossRef] - Mahata, S.; Kar, R.; Mandal, D. Optimal approximation of fractional-order systems with model validation using CFOA. IET Signal Process.
**2019**, 13, 787–797. [Google Scholar] [CrossRef] - Radwan, A.G.; Fouda, M.E. Optimization of fractional-order RLC filters. Circuits Syst. Signal Process.
**2013**, 32, 2097–2118. [Google Scholar] [CrossRef] - Hélie, T. Simulation of fractional-order low-pass filters. IEEE/ACM Trans. Audio Speech Lang. Process.
**2014**, 22, 1636–1647. [Google Scholar] [CrossRef] - Said, L.A.; Ismail, S.M.; Radwan, A.G.; Madian, A.H.; El-Yazeed, M.F.A.; Soliman, A.M. On the optimization of fractional order low-pass filters. Circuits Syst. Signal Process.
**2016**, 35, 2017–2039. [Google Scholar] [CrossRef] - Kubanek, D.; Freeborn, T. (1+ α) fractional-order transfer functions to approximate low-pass magnitude responses with arbitrary quality factor. AEU-Int. J. Electron. Commun.
**2018**, 83, 570–578. [Google Scholar] [CrossRef] - Kubanek, D.; Freeborn, T.; Koton, J. Fractional-order band-pass filter design using fractional-characteristic specimen functions. Microelectron. J.
**2019**, 86, 77–86. [Google Scholar] [CrossRef] - Mahata, S.; Kar, R.; Mandal, D. Optimal approximation of asymmetric type fractional-order bandpass Butterworth filter using decomposition technique. Int. J. Circuit Theory Appl.
**2020**, 48, 1554–1560. [Google Scholar] [CrossRef] - Freeborn, T.J. Comparison of (1+α) fractional-order transfer functions to approximate lowpass Butterworth magnitude responses. Circuits Syst. Signal Process.
**2016**, 35, 1983–2002. [Google Scholar] [CrossRef] - Soni, A.; Sreejeth, N.; Saxena, V.; Gupta, M. Series optimized fractional order low pass Butterworth filter. Arab. J. Sci. Eng.
**2020**, 45, 1733–1747. [Google Scholar] [CrossRef] - Mahata, S.; Herencsar, N.; Kubanek, D. Optimal approximation of fractional-order Butterworth filter based on weighted sum of classical Butterworth filters. IEEE Access
**2021**, 9, 81097–81114. [Google Scholar] [CrossRef] - Mahata, S.; Kar, R.; Mandal, D. Optimal modelling of (1+ α) order Butterworth filter under the CFE framework. Fractal Fract.
**2020**, 4, 55. [Google Scholar] [CrossRef] - Yousri, D.; AbdelAty, A.M.; Radwan, A.G.; Elwakil, A.S.; Psychalinos, C. Comprehensive comparison based on meta-heuristic algorithms for approximation of the fractional-order laplacian s
^{α}as a weighted sum of first-order high-pass filters. Microelectron. J.**2019**, 87, 110–120. [Google Scholar] [CrossRef] - Adhikary, A.; Choudhary, S.; Sen, S. Optimal design for realizing a grounded fractional order inductor using GIC. IEEE Trans. Circuits Syst. I Regul. Pap.
**2018**, 65, 2411–2421. [Google Scholar] [CrossRef] - Soni, A.; Gupta, M. Analysis and design of optimized fractional order low-pass Bessel filter. J. Circuits Syst. Comput.
**2021**, 30, 2150035. [Google Scholar] [CrossRef] - Tugnait, J.K.; Li, T. Blind detection of asynchronous CDMA signals in multipath channels using code-constrained inverse filter criterion. IEEE Trans. Signal Process.
**2001**, 49, 1300–1309. [Google Scholar] [CrossRef] - Mouchtaris, A.; Reveliotis, P.; Kyriakakis, C. Inverse filter design for immersive audio rendering over loudspeakers. IEEE Trans. Multimed.
**2000**, 2, 77–87. [Google Scholar] [CrossRef] - Yuce, E.; Minaei, S. New CCII-based versatile structure for realizing PID controller and instrumentation amplifier. Microelectron. J.
**2010**, 41, 311–316. [Google Scholar] [CrossRef] - Ansari, R.; Kahn, D.; Macchi, M.J. Pitch modification of speech using a low-sensitivity inverse filter approach. IEEE Signal Process. Lett.
**1998**, 5, 60–62. [Google Scholar] [CrossRef] - Bhaskar, D.R.; Kumar, M.; Kumar, P. Fractional order inverse filters using operational amplifier. Analog Integr. Circuits Signal Process.
**2018**, 97, 149–158. [Google Scholar] [CrossRef] - Hamed, E.M.; Said, L.A.; Madian, A.H.; Radwan, A.G. On the approximations of CFOA-based fractional-order inverse filters. Circuits Syst. Signal Process.
**2020**, 39, 2–29. [Google Scholar] [CrossRef] - Khalil, N.A.; Said, L.A.; Radwan, A.G.; Soliman, A.M. Multifunction fractional inverse filter based on OTRA. In Proceedings of the 2019 Novel Intelligent and Leading Emerging Sciences Conference (NILES), Giza, Egypt, 28–30 October 2019; Volume 1, pp. 162–165. [Google Scholar] [CrossRef]
- Srivastava, J.; Bhagat, R.; Kumar, P. Analog inverse filters using OTAs. In Proceedings of the 2020 6th International Conference on Control, Automation and Robotics (ICCAR), Singapore, 20–23 April 2020; pp. 627–631. [Google Scholar] [CrossRef]
- Kapoulea, S.; Psychalinos, C.; Elwakil, A.S. Power law filters: A new class of fractional-order filters without a fractional-order Laplacian operator. AEU-Int. J. Electron. Commun.
**2021**, 129, 153537. [Google Scholar] [CrossRef] - Kapoulea, S.; Psychalinos, C.; Elwakil, A.S.; Tavazoei, M.S. Power-law compensator design for plants with uncertainties: Experimental verification. Electronics
**2021**, 10, 1305. [Google Scholar] [CrossRef] - Kapoulea, S.; Elwakil, A.S.; Psychalinos, C.; Al-Ali, A. Novel double-dispersion models based on power-law filters. Circuits Syst. Signal Process.
**2021**, 40, 5799–5812. [Google Scholar] [CrossRef] - Ogata, K. Modern Control Engineering; Prentice Hall: Hoboken, NJ, USA, 2010. [Google Scholar]
- Devore, J.L. Probability and Statistics for Engineering and the Sciences; Cengage Learning: Belmont, CA, USA, 2011. [Google Scholar]
- Senani, R.; Bhaskar, D.; Singh, A.K.; Singh, V.K. Current Feedback Operational Amplifiers and Their Applications; Springer: New York, NY, USA, 2013. [Google Scholar]
- Mahata, S.; Herencsar, N.; Kubanek, D.; Kar, R.; Mandal, D.; Goknar, C.I. A fractional-order transitional Butterworth-Butterworth filter and its experimental validation. IEEE Access
**2021**, 9, 129521–129527. [Google Scholar] [CrossRef]

**Figure 2.**MARE comparison using boxplots for the (

**a**) LPPLF, (

**b**) HPPLF, (

**c**) BPPLF, and (

**d**) BSPLF. Note: The most accurate model (lowest MARE${}_{\mathrm{min}}$) for each case is indicated in blue.

**Figure 3.**(

**a**) Magnitude and (

**b**) phase–frequency responses of the proposed LPPLFs (dashed blue) and ILPPLFs (dashed green) as compared to the theoretical ones (solid black).

**Figure 4.**(

**a**) Magnitude and (

**b**) phase–frequency responses of the proposed HPPLFs (dashed blue) and IHPPLFs (dashed green) as compared to the theoretical ones (solid black).

**Figure 5.**(

**a**) Magnitude and (

**b**) phase–frequency responses of the proposed BPPLFs (dashed blue) and IBPPLFs (dashed green) as compared to the theoretical ones (solid black).

**Figure 6.**(

**a**) Magnitude and (

**b**) phase–frequency responses of the proposed BSPLFs (dashed blue) and IBSPLFs (dashed green) as compared to the theoretical ones (solid black).

**Figure 7.**Percentage improvement in absolute relative error of the proposed PLFs with respect to [54].

**Figure 10.**Comparisons between the theoretical (solid black) and experimentally (dashed blue) obtained (

**a**) LP, (

**b**) HP, (

**c**) BP, and (

**d**) BS filter frequency responses of the PLFs $(\alpha =0.5)$.

**Figure 11.**Input–output waveforms observed in oscilloscope for the proposed (

**a**) LPPLF $(\alpha =0.5)$ with an input frequency of ${f}_{\mathrm{H}}=1.11$ kHz, (

**b**) HPPLF $(\alpha =0.5)$ with an input frequency of ${f}_{\mathrm{H}}=738$ Hz.

**Figure 12.**Input–output waveforms observed in oscilloscope for the proposed BPPLF $(\alpha =0.5)$ with an input frequency of (

**a**) ${f}_{0}=1.066$ kHz, (

**b**) ${f}_{\mathrm{H},\mathrm{low}}=353.9$ Hz, and (

**c**) ${f}_{\mathrm{H},\mathrm{high}}=3.214$ kHz.

**Figure 13.**Input–output waveforms observed in oscilloscope for the proposed BSPLF $(\alpha =0.5)$ with an input frequency of (

**a**) ${f}_{\mathrm{H},\mathrm{low}}=614$ Hz and (

**b**) ${f}_{\mathrm{H},\mathrm{high}}=1.47$ kHz.

**Figure 14.**FFT spectrums (experimental) of the proposed filters ($\alpha =0.5$) pertaining to the (

**a**) LPPLF, (

**b**) HPPLF, (

**c**) BPPLF-a, (

**d**) BPPLF-b, (

**e**) BPPLF-c, (

**f**) BSPLF-a, and (

**g**) BSPLF-b.

**Figure 15.**Comparisons between the theoretical (solid black) and experimentally (dashed blue) obtained (

**a**) LP, (

**b**) HP, (

**c**) BP, and (

**d**) BS filter frequency responses of the IPLFs $(\alpha =0.5)$.

**Figure 16.**Input–output waveforms observed in oscilloscope for the proposed (

**a**) ILPPLF $(\alpha =0.5)$ with an input frequency of ${f}_{\mathrm{H}}=1.212$ kHz, (

**b**) IHPPLF $(\alpha =0.5)$ with an input frequency of ${f}_{\mathrm{H}}=887.8$ Hz.

**Figure 17.**Input–output waveforms observed in oscilloscope for the proposed IBPPLF $(\alpha =0.5)$ with an input frequency of (

**a**) ${f}_{0}=967$ Hz, (

**b**) ${f}_{\mathrm{H},\mathrm{low}}=330.3$ Hz, and (

**c**) ${f}_{\mathrm{H},\mathrm{high}}=2.919$ kHz.

**Figure 18.**Input–output waveforms observed in oscilloscope for the proposed IBSPLF $(\alpha =0.5)$ with an input frequency of (

**a**) ${f}_{\mathrm{H},\mathrm{low}}=650.3$ Hz and (

**b**) ${f}_{\mathrm{H},\mathrm{high}}=1.482$ kHz.

**Figure 19.**FFT spectrums (experimental) of the proposed inverse filters ($\alpha =0.5$) pertaining to the (

**a**) ILPPLF, (

**b**) IHPPLF, (

**c**) IBPPLF-a, (

**d**) IBPPLF-b, (

**e**) IBPPLF-c, (

**f**) IBSPLF-a, and (

**g**) IBSPLF-b.

**Table 1.**Transfer functions and frequency response expressions of the theoretical PLFs (${\omega}_{0}$: pole frequency, and Q: quality factor).

Type | Transfer Function | Magnitude | Phase |
---|---|---|---|

Low-pass | ${H}_{\mathrm{D}}^{\mathrm{LP}}\left(s\right)={\left(\frac{{\omega}_{0}^{2}}{{s}^{2}+\frac{{\omega}_{0}}{Q}s+{\omega}_{0}^{2}}\right)}^{\alpha}$ | $\frac{1}{{\left[1+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{4}+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}\xb7\left(\frac{1}{{Q}^{2}}-2\right)\right]}^{\alpha /\phantom{\alpha 2}\phantom{\rule{0.0pt}{0ex}}2}}$ | $-\alpha \xb7{tan}^{-1}\left[\frac{\left(\frac{\omega}{{\omega}_{0}}\right)\xb7\frac{1}{Q}}{1-{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}}\right]$ |

High-pass | ${H}_{\mathrm{D}}^{\mathrm{HP}}\left(s\right)={\left(\frac{{s}^{2}}{{s}^{2}+\frac{{\omega}_{0}}{Q}s+{\omega}_{0}^{2}}\right)}^{\alpha}$ | $\frac{{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2\alpha}}{{\left[1+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{4}+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}\xb7\left(\frac{1}{{Q}^{2}}-2\right)\right]}^{\alpha /\phantom{\alpha 2}\phantom{\rule{0.0pt}{0ex}}2}}$ | $\alpha \xb7\left\{\pi -{tan}^{-1}\left[\frac{\left(\frac{\omega}{{\omega}_{0}}\right)\xb7\frac{1}{Q}}{1-{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}}\right]\right\}$ |

Band-pass | ${H}_{\mathrm{D}}^{\mathrm{BP}}\left(s\right)={\left(\frac{\frac{{\omega}_{0}}{Q}s}{{s}^{2}+\frac{{\omega}_{0}}{Q}s+{\omega}_{0}^{2}}\right)}^{\alpha}$ | $\frac{\frac{1}{{Q}^{\alpha}}\xb7{\left(\frac{\omega}{{\omega}_{0}}\right)}^{\alpha}}{{\left[1+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{4}+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}\xb7\left(\frac{1}{{Q}^{2}}-2\right)\right]}^{\alpha /\phantom{\alpha 2}\phantom{\rule{0.0pt}{0ex}}2}}$ | $\alpha \xb7\left\{\frac{\pi}{2}-{tan}^{-1}\left[\frac{\left(\frac{\omega}{{\omega}_{0}}\right)\xb7\frac{1}{Q}}{1-{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}}\right]\right\}$ |

Band-stop | ${H}_{\mathrm{D}}^{\mathrm{BS}}\left(s\right)={\left(\frac{{s}^{2}+{\omega}_{0}^{2}}{{s}^{2}+\frac{{\omega}_{0}}{Q}s+{\omega}_{0}^{2}}\right)}^{\alpha}$ | $\frac{{\left[1-{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}\right]}^{\alpha}}{{\left[1+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{4}+{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}\xb7\left(\frac{1}{{Q}^{2}}-2\right)\right]}^{\alpha /\phantom{\alpha 2}\phantom{\rule{0.0pt}{0ex}}2}}$ | $-\alpha \xb7{tan}^{-1}\left[\frac{\left(\frac{\omega}{{\omega}_{0}}\right)\xb7\frac{1}{Q}}{1-{\left(\frac{\omega}{{\omega}_{0}}\right)}^{2}}\right]$ |

Filter | Index | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ||

LPPLF | Min | 0.0097 | 0.0100 | 0.0081 | 1.11 $\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $1.54\times {10}^{-4}$ | 0.0072 | 0.0073 | 0.0068 |

Max | 2.6271 | 2.3788 | 1.4640 | 2.9436 | 1.8702 | 1.9990 | 2.7724 | 2.7217 | 1.5765 | |

Mean | 0.1644 | 0.1960 | 0.0647 | 0.1095 | 0.0379 | 0.0740 | 0.1174 | 0.1410 | 0.0491 | |

SD | 0.4655 | 0.4563 | 0.1877 | 0.4031 | 0.1910 | 0.3262 | 0.3996 | 0.3878 | 0.1707 | |

HPPLF | Min | 0.0371 | 0.0380 | 0.0081 | $2.55\times {10}^{-4}$ | $2.06\times {10}^{-4}$ | $1.20\times {10}^{-5}$ | 0.0068 | 0.0415 | 0.0308 |

Max | 3.2533 | 3.0948 | 4.8429 | 4.7061 | 5.3711 | 5.3368 | 5.7460 | 2.2993 | 7.6727 | |

Mean | 0.2869 | 0.1752 | 0.5542 | 0.3599 | 0.1516 | 0.1725 | 0.3906 | 0.1994 | 0.2710 | |

SD | 0.5413 | 0.3953 | 0.9991 | 1.0042 | 0.6603 | 0.6509 | 0.8299 | 0.3567 | 0.9709 | |

BPPLF | Min | 0.0850 | 0.0822 | 0.0785 | 0.0786 | 0.0758 | 0.0735 | 0.0568 | 0.0557 | 0.0540 |

Max | 7.2483 | 5.6081 | 6.5838 | 4.9242 | 3.0704 | 8.2371 | 3.2517 | 7.1117 | 3.9668 | |

Mean | 0.5429 | 0.4603 | 0.3324 | 0.4141 | 0.3461 | 0.3522 | 0.3440 | 0.2805 | 0.2651 | |

SD | 0.9948 | 0.9361 | 0.7146 | 0.7051 | 0.5823 | 0.8862 | 0.6418 | 0.8186 | 0.5066 | |

BSPLF | Min | 0.0148 | 0.0564 | 0.0427 | 0.0133 | 0.0123 | 0.0438 | 0.0101 | 0.0090 | 0.0385 |

Max | 5.9545 | 7.6578 | 8.9775 | 5.6246 | 8.2135 | 7.5578 | 7.2166 | 7.0028 | 7.5591 | |

Mean | 0.4435 | 0.6127 | 1.5306 | 0.5749 | 0.7712 | 1.2279 | 0.8967 | 0.8219 | 1.4433 | |

SD | 0.9858 | 1.5685 | 2.1678 | 1.0939 | 1.5658 | 1.6760 | 1.6799 | 1.4094 | 1.8164 |

Filter | Index | ${\mathit{f}}_{1}$ vs. ${\mathit{f}}_{2}$ | ${\mathit{f}}_{1}$ vs. ${\mathit{f}}_{3}$ | ${\mathit{f}}_{2}$ vs. ${\mathit{f}}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ | ||

LPPLF | p-val | $3.3\times {10}^{-8}$ | 0.9173 | $8.0\times {10}^{-9}$ | $9.7\times {10}^{-4}$ | 0.0911 | 0.1443 | $1.5\times {10}^{-6}$ | 0.0013 | $1.4\times {10}^{-8}$ |

H | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | |

HPPLF | p-val | 0.7369 | 0.0303 | $3.7\times {10}^{-6}$ | 0.5551 | 0.7966 | $1.1\times {10}^{-8}$ | 0.2087 | 0.6574 | $6.1\times {10}^{-6}$ |

H | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | |

BPPLF | p-val | 0.2503 | 0.0211 | $8.8\times {10}^{-6}$ | 0.0474 | 0.0417 | 0.0322 | 0.1626 | 0.1416 | 0.8902 |

H | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | |

BSPLF | p-val | $1.0\times {10}^{-6}$ | $6.6\times {10}^{-4}$ | $1.1\times {10}^{-4}$ | 0.0032 | 0.0294 | 0.0276 | 0.0087 | 0.1739 | 0.0708 |

H | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |

**Table 4.**Comparison of the most accurate proposed PLFs with the published literature [54] in terms of the MARE metric.

Filter | Reference | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ |
---|---|---|---|---|

LPPLF | [54] | 0.0154 | $2.68\times {10}^{-4}$ | 0.0410 |

Proposed | 0.0081 | $1.11\times {10}^{-4}$ | 0.0068 | |

HPPLF | [54] | 0.0158 | $9.39\times {10}^{-5}$ | 0.0410 |

Proposed | 0.0081 | $1.20\times {10}^{-5}$ | 0.0068 | |

BPPLF | [54] | 0.0851 | 0.0884 | 0.0747 |

Proposed | 0.0785 | 0.0735 | 0.0540 | |

BSPLF | [54] | 0.0181 | 0.0138 | 0.0098 |

Proposed | 0.0148 | 0.0123 | 0.0090 |

**Table 5.**Values of the passive components for the realization of the proposed PLFs $(\alpha =0.5)$ (Note: All resistances are in $\mathrm{k}\mathsf{\Omega}$, and all capacitances are in nF).

Filter | ${\mathit{R}}_{\mathbf{out}}$ | ${\mathit{R}}_{\mathbf{in}}$ | R | ${\mathit{R}}_{\mathbf{F}}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

LPPLF | 10 | 10 | 10 | 10 | ∞ | 39 | 20 | 13 | 10 | 3.9 | 10 | 22 | 47 |

HPPLF | 10 | 10 | 10 | 10 | 10 | 13 | 18 | 36 | ∞ | 4.7 | 12 | 22 | 56 |

BPPLF | 10 | 10 | 10 | 10 | 130 | 30 | 10 | 30 | 130 | 0.56 | 6.8 | 33 | 390 |

BSPLF | 10 | 10 | 10 | 10 | 10 | 20 | 11 | 20 | 10 | 12 | 10 | 27 | 22 |

**Table 6.**Values of the passive components for the realization of the proposed IPLFs $(\alpha =0.5)$ (Note: All resistances are in $\mathrm{k}\mathsf{\Omega}$, and all capacitances are in nF).

Filter | ${\mathit{R}}_{\mathbf{out}}$ | ${\mathit{R}}_{\mathbf{in}}$ | R | ${\mathit{R}}_{\mathbf{F}}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ILPPLF | 30 | 20 | 20 | 20 | 0.1 | 5.1 | 10 | 15 | 20 | 0.039 | 2.2 | 6.8 | 18 |

IHPPLF | 11.5 | 10 | 200 | 200 | 200 | 160 | 110 | 56 | 0.43 | 0.33 | 0.82 | 2.2 | 390 |

IBPPLF | 51 | 51 | 51 | 51 | 3.6 | 16 | 51 | 16 | 3.6 | 0.027 | 0.47 | 22 | 390 |

IBSPLF | 51 | 51 | 51 | 51 | 51 | 24 | 47 | 24 | 51 | 4.7 | 1.0 | 10 | 2.2 |

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## Share and Cite

**MDPI and ACS Style**

Mahata, S.; Herencsar, N.; Kubanek, D.
On the Design of Power Law Filters and Their Inverse Counterparts. *Fractal Fract.* **2021**, *5*, 197.
https://doi.org/10.3390/fractalfract5040197

**AMA Style**

Mahata S, Herencsar N, Kubanek D.
On the Design of Power Law Filters and Their Inverse Counterparts. *Fractal and Fractional*. 2021; 5(4):197.
https://doi.org/10.3390/fractalfract5040197

**Chicago/Turabian Style**

Mahata, Shibendu, Norbert Herencsar, and David Kubanek.
2021. "On the Design of Power Law Filters and Their Inverse Counterparts" *Fractal and Fractional* 5, no. 4: 197.
https://doi.org/10.3390/fractalfract5040197