# FPAA-Based Realization of Filters with Fractional Laplace Operators of Different Orders

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## Abstract

**:**

## 1. Introduction

- (a)
- Employment of fractional-order capacitors, which substitute the conventional (i.e., integer-order) capacitors in the topologies of integer-order standard filters. Due to the absence of commercial availability of such elements, they are approximated by RC schemes, such as the Foster and Cauer networks. The price paid for the offered quick design procedure is the absence of on-the-fly tuning of the filter’s characteristics, making it suitable only for cases of filters with pre-defined type and frequency characteristics [3,4,5].
- (b)
- Approximation of the fractional-order Laplace operators in (1) using appropriate tools, such as Continued Fraction Expansion, Oustaloup, Matsuda and Carlson [6], and then substitution of the resulting rational function approximations of these operators. The derived filter function is an integer-order polynomial ratio that can be implemented utilizing classical filter design techniques, such as cascade connection of elementary filter stages and multi-feedback structures. This design procedure is more complicated than the previous one, but it offers the advantage of electronic tuning capability of the derived filter structures, considering that active elements with electronically controlled characteristics (e.g., transconductance) are utilized [2,7,8,9,10,11]. However, this approach is efficient only for approximations of filter functions with a single fractional order. In the case of functions, such as the one in (1), the order of the derived rational function will be at least double the applied order of approximation [2].

- (a)
- Instead of individually approximating the fractional-order Laplace operators, the magnitude and phase frequency responses of the whole filter function in (1) are fitted by a suitable integer-order transfer function derived through the employment of specialized Symbolic Math Toolbox™ built-in functions.
- (b)
- The order of the derived rational function is always equal to the order of the employed approximation, and thus, this approach offers the advantage of minimizing the circuit complexity, compared to the aforementioned conventional approach.The paper is organized as follows: an investigation of the problem regarding the approximation of the general filter function following the conventional and the proposed procedures is discussed in Section 2, and a design example for comparison purposes is given in Section 3. The validity of the suggested procedure was experimentally evaluated using a field-programmable analog array (FPAA) device [12] and the results are presented in Section 4.

## 2. Proposed Design Procedure

- step #1:
- Extraction of the frequency response data of (1) within a specific frequency range of interest, a process that is performed using the frd built-in function.
- step #2:
- Approximation of the obtained data, based on the fitfrd built-in function, which forms the state-space model of the data for a given order of approximation $\left(n\right)$.
- step #3:
- Conversion of the model to an integer-order transfer function of the form in (3) using the ss2tf built-in function.

## 3. Design Example and Comparative Results

## 4. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AP | All pass |

BP | Band pass |

BS | Band stop |

CAB | Configurable analogue block |

FO | Fractional order |

FBD | Functional block diagram |

FLF | Follow the leader feedback |

FPAA | Field-programmable analog array |

HP | High pass |

IAE | Integral absolute of error |

LP | Low pass |

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**Figure 1.**Gain and phase frequency responses, derived from 5th-order Oustaloup (dashed-dotted lines) and 5th-order curve-fitting-based (solid lines) approximation methods, for the (

**a**) LP and HP, (

**b**) BP and BS and (

**c**) AP filter cases given in Table 2, along with the corresponding ideal responses (dashed lines).

**Figure 3.**FPAA-based realization of the functional block diagram in Figure 2.

**Figure 5.**Experimental gain and phase frequency responses (triangle symbols), along with the corresponding approximated (solid lines) and ideal (dashed lines) responses, for (

**a**) LP and HP, (

**b**) BP and BS and (

**c**) AP filter cases.

**Figure 6.**Screenshot of the Agilent 4395A spectrum analyzer, showing the gain frequency response during the measurement of the BP type of filter for orders $\left(\alpha ,\beta \right)=\left(0.7,0.4\right)$.

**Table 1.**Order of the functions in (3) for different types of filters using conventional approximation methods.

Filter | Order | |
---|---|---|

$\mathit{\alpha}\mathbf{+}\mathit{\beta}\mathbf{\in}\left(\mathbf{0}\mathbf{,}\mathbf{1}\right)$ | $\mathit{\alpha}\mathbf{+}\mathit{\beta}\mathbf{\in}\left(\mathbf{1}\mathbf{,}\mathbf{2}\right)$ | |

LP | $\left[p,q\right]=\left[2n,2n\right]$ | $\left[p,q\right]=\left[2n,2n+1\right]$ |

HP | $\left[p,q\right]=\left[2n+1,2n+1\right]$ | |

BP | $\left[p,q\right]=\left[2n,2n+1\right]$ | |

BS | $\left[p,q\right]=\left[2n+1,2n+1\right]$ | |

AP | $\left[p,q\right]=\left[2n+1,2n+1\right]$ |

Filter | Parameter | ||
---|---|---|---|

$\left(\mathit{\alpha},\mathit{\beta}\right)$ | $\left({\mathit{K}}_{\mathit{LP}},{\mathit{K}}_{\mathit{HP}},{\mathit{K}}_{\mathit{BP}}\right)$ | $\mathit{\tau}$ | |

LP | $\left(0.3,0.5\right)$ | $\left(1,0,0\right)$ | $22.91\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ |

$\left(0.3,0.8\right)$ | $69.97\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ | ||

HP | $\left(0.5,0.3\right)$ | $\left(0,1,0\right)$ | $1.106\phantom{\rule{0.166667em}{0ex}}\left({m}\mathrm{s}\right)$ |

$\left(0.8,0.3\right)$ | $362\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ | ||

BP | $\left(0.4,0.7\right)$ | $\left(0,0,1\right)$ | $186.7\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ |

$\left(0.7,0.4\right)$ | $135.7\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ | ||

BS | $\left(0.4,0.7\right)$ | $\left(1,1,0\right)$ | $205\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ |

$\left(0.7,0.4\right)$ | $123.5\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ | ||

AP | $\left(0.3,0.5\right)$ | $\left(-1,1,0\right)$ | $184.8\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ |

$\left(0.6,0.8\right)$ | $547.8\phantom{\rule{0.166667em}{0ex}}\left(\mathsf{\mu}\mathrm{s}\right)$ |

**Table 3.**Values of the coefficients in (3) derived from the 5th-order approximation of the initial function in (1) for various filter types.

Coefficient | Filter | |||||
---|---|---|---|---|---|---|

LP | HP | BP | BS | AP | ||

$\left(\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{5}\right)$ | $\left(\mathbf{0}.\mathbf{5},\mathbf{0}.\mathbf{3}\right)$ | $\left(\mathbf{0}.\mathbf{7},\mathbf{0}.\mathbf{4}\right)$ | $\left(\mathbf{0}.\mathbf{7},\mathbf{0}.\mathbf{4}\right)$ | $\left(\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{5}\right)$ | $\left(\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{8}\right)$ | |

${\mathit{C}}_{\mathbf{0}}$ | $1.812\times {10}^{20}$ | $2.383\times {10}^{15}$ | $4.68\times {10}^{16}$ | $6.689\times {10}^{17}$ | $-1.026\times {10}^{19}$ | $-1.146\times {10}^{19}$ |

${\mathit{C}}_{\mathbf{1}}$ | $4.361\times {10}^{17}$ | $1.811\times {10}^{14}$ | $1.101\times {10}^{15}$ | $5.767\times {10}^{15}$ | $-3.783\times {10}^{16}$ | $-6.374\times {10}^{15}$ |

${\mathit{C}}_{\mathbf{2}}$ | $1.317\times {10}^{14}$ | $8.018\times {10}^{11}$ | $2.705\times {10}^{12}$ | $7.219\times {10}^{12}$ | $-8.754\times {10}^{12}$ | $2.488\times {10}^{12}$ |

${\mathit{C}}_{\mathbf{3}}$ | $6.778\times {10}^{9}$ | $5.37\times {10}^{8}$ | $1.139\times {10}^{9}$ | $1.886\times {10}^{9}$ | $2.109\times {10}^{9}$ | $2.18\times {10}^{9}$ |

${\mathit{C}}_{\mathbf{4}}$ | $5.138\times {10}^{4}$ | $6.26\times {10}^{4}$ | $2.169\times {10}^{4}$ | $1.237\times {10}^{5}$ | $1.555\times {10}^{5}$ | $1.386\times {10}^{5}$ |

${\mathit{C}}_{\mathbf{5}}$ | $0.02145$ | $0.9617$ | $0.01929$ | $0.9802$ | $0.8324$ | $0.9842$ |

${\mathit{D}}_{\mathbf{0}}$ | $1.871\times {10}^{20}$ | $1.296\times {10}^{17}$ | $5.653\times {10}^{17}$ | $7.264\times {10}^{17}$ | $1.11\times {10}^{19}$ | $1.165\times {10}^{19}$ |

${\mathit{D}}_{\mathbf{1}}$ | $5.069\times {10}^{17}$ | $1.296\times {10}^{15}$ | $5.51\times {10}^{15}$ | $7.152\times {10}^{15}$ | $5.894\times {10}^{16}$ | $1.961\times {10}^{16}$ |

${\mathit{D}}_{\mathbf{2}}$ | $1.966\times {10}^{14}$ | $2.196\times {10}^{12}$ | $8.418\times {10}^{12}$ | $1.059\times {10}^{13}$ | $4.652\times {10}^{13}$ | $1.243\times {10}^{13}$ |

${\mathit{D}}_{\mathbf{3}}$ | $1.65\times {10}^{10}$ | $8.576\times {10}^{8}$ | $2.753\times {10}^{9}$ | $3.223\times {10}^{9}$ | $7.759\times {10}^{9}$ | $3.067\times {10}^{9}$ |

${\mathit{D}}_{\mathbf{4}}$ | $3.144\times {10}^{5}$ | $7.504\times {10}^{4}$ | $1.368\times {10}^{5}$ | $1.473\times {10}^{5}$ | $2.532\times {10}^{5}$ | $1.528\times {10}^{5}$ |

**Table 4.**Values of time constants and scaling factors in (4), calculated using the values of Table 3 and the design equations in (5).

Coefficient | Filter | |||||
---|---|---|---|---|---|---|

LP | P | BP | BS | AP | ||

$\left(\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{5}\right)$ | $\left(\mathbf{0}.\mathbf{5},\mathbf{0}.\mathbf{3}\right)$ | $\left(\mathbf{0}.\mathbf{7},\mathbf{0}.\mathbf{4}\right)$ | $\left(\mathbf{0}.\mathbf{7},\mathbf{0}.\mathbf{4}\right)$ | $\left(\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{5}\right)$ | $\left(\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{8}\right)$ | |

${\mathit{K}}_{\mathbf{0}}$ | $0.969$ | $0.018$ | $0.083$ | $0.921$ | $-0.924$ | $-0.984$ |

${\mathit{K}}_{\mathbf{1}}$ | $0.86$ | $0.14$ | $0.2$ | $0.806$ | $-0.642$ | $-0.325$ |

${\mathit{K}}_{\mathbf{2}}$ | $0.67$ | $0.365$ | $0.321$ | $0.682$ | $-0.188$ | $0.2$ |

${\mathit{K}}_{\mathbf{3}}$ | $0.41$ | $0.626$ | $0.414$ | $0.585$ | $0.272$ | $0.711$ |

${\mathit{K}}_{\mathbf{4}}$ | $0.163$ | $0.834$ | $0.158$ | $0.84$ | $0.614$ | $0.907$ |

${\mathit{K}}_{\mathbf{5}}$ | $0.021$ | $0.962$ | $0.019$ | $0.98$ | $0.832$ | $0.984$ |

${\mathit{\tau}}_{\mathbf{1}}$ | 3.18 ($\mathsf{\mu}$s) | 13.32 ($\mathsf{\mu}$s) | 7.31 ($\mathsf{\mu}$s) | 6.79 ($\mathsf{\mu}$s) | 3.95 ($\mathsf{\mu}$s) | 6.546 ($\mathsf{\mu}$s) |

${\mathit{\tau}}_{\mathbf{2}}$ | 19.05 ($\mathsf{\mu}$s) | 87.5 ($\mathsf{\mu}$s) | 49.7 ($\mathsf{\mu}$s) | 45.71 ($\mathsf{\mu}$s) | 32.63 ($\mathsf{\mu}$s) | 49.81 ($\mathsf{\mu}$s) |

${\mathit{\tau}}_{\mathbf{3}}$ | 83.92 ($\mathsf{\mu}$s) | 390.5 ($\mathsf{\mu}$s) | 327 ($\mathsf{\mu}$s) | 304.2 ($\mathsf{\mu}$s) | 166.8 ($\mathsf{\mu}$s) | 246.6 ($\mathsf{\mu}$s) |

${\mathit{\tau}}_{\mathbf{4}}$ | 387.9 ($\mathsf{\mu}$s) | 1.69 (ms) | 1.53 (ms) | 1.78 (ms) | 789.25 ($\mathsf{\mu}$s) | 634.1 ($\mathsf{\mu}$s) |

${\mathit{\tau}}_{\mathbf{5}}$ | 2.71 (ms) | 10 (ms) | 9.75 (ms) | 9.85 (ms) | 5.31 (ms) | 1.68 (ms) |

**Table 5.**Critical frequencies for the filter cases in Figure 5, along with the theoretical values given between parentheses.

Frequency | Filter | ||||
---|---|---|---|---|---|

LP | HP | BP | BS | AP | |

${f}_{c}/{f}_{peak}$ (kHz) | 1 (1) | 1 (1) | 1 (1) | 1 (1) | 1 (1) |

${f}_{low}$ (Hz) | – | – | 142.5 (140.7) | 173.8 (174.1) | – |

${f}_{high}$ (kHz) | – | – | 4.36 (4.32) | 3.48 (3.37) | – |

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**MDPI and ACS Style**

Kapoulea, S.; Psychalinos, C.; Elwakil, A.S.
FPAA-Based Realization of Filters with Fractional Laplace Operators of Different Orders. *Fractal Fract.* **2021**, *5*, 218.
https://doi.org/10.3390/fractalfract5040218

**AMA Style**

Kapoulea S, Psychalinos C, Elwakil AS.
FPAA-Based Realization of Filters with Fractional Laplace Operators of Different Orders. *Fractal and Fractional*. 2021; 5(4):218.
https://doi.org/10.3390/fractalfract5040218

**Chicago/Turabian Style**

Kapoulea, Stavroula, Costas Psychalinos, and Ahmed S. Elwakil.
2021. "FPAA-Based Realization of Filters with Fractional Laplace Operators of Different Orders" *Fractal and Fractional* 5, no. 4: 218.
https://doi.org/10.3390/fractalfract5040218