esa-sceva/satcom-chunk-collection huggingface dataset:
# A Novel Multi-Task Performance Prediction Model for Spark
## 4 Experimentation
### Comparative Modeling and Experimental Metrics  
We selected the baseline models kNN, LR, SVC, RF, GBDT, and XGB for regression prediction as comparison models. Comparison is made with the proposed MHAC prediction model. In the stage of testing the neural-network-based prediction model, in order to make an assessment of the performance of the prediction model, the experimental evaluation metrics of the prediction model we selected are the mean absolute percentage error (MAPE), the common root mean square error (RMSE), the mean absolute error (MAE), and the R-Square (\(R^{2}\)). Among them, the formula for MAPE is shown in Equation (20), the formula for RMSE is shown in Equation (21), the formula for MAE is shown in Equation (22), and  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Application Name** & **Input Data Size** & **Step** \\ \hline Pi & 1–50 & 10,000 \\ PageRank & 1–50 & 10,000 Pages \\ WordCount & 1–50 & 100 MB \\ LR & 1–50 & 100 MB \\ \hline \hline \end{tabular}
\end{table}
Table 1: This is the table of input data sizes used by the Spark applications.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Description** \\ \hline spark.num.executors & Number of executors \\ spark.executor.cores & CPU cores per executor \\ spark.executor.memory & Memory size per executor \\ spark.driver.memory & Total amount of memory allocated to the Driver driver \\ spark.driver.cores & Number of CPU cores allocated to the Driver driver process \\ spark.executor.instances & Number of instances of executing programs \\ spark.default.parallelisim & Default number of Tasks per stage \\ spark.memory.fraction & Proportion of memory used for execution and storage \\ spark.task.cpus & Number of CPU cores allocated to each Task \\ spark.shuffle.memoryFraction & Percentage of memory occupied by the Shuffle process \\ spark.shuffle.file.buffer & Buffering of write files during Shuffle \\ spark.reducer.maxSizeInFlight & The size of the buffer during a Shuffle read \\ spark.reducer.maxReqSizeShuffleToMem & Maximum value of the data buffer \\ \hline \hline \end{tabular}
\end{table}
Table 2: This is the table of parameter information.  
the formula for \(R^{2}\) is shown in Equation (23). The smaller MAPE, RMSE, and MAE are, the larger \(R^{2}\) is, and then the smaller the prediction error of the model is.  
\[\text{MAPE}=\frac{100\%}{n}\sum_{i=1}^{n}\frac{|y_{i}-y_{i}^{\prime}|}{y_{i}} \tag{20}\]  
\[\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{n}(y_{i}^{\prime}-y_{i})^{2}} \tag{21}\]  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}|y_{i}-y_{i}^{\prime}| \tag{22}\]  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-y_{i}^{\prime})^{2}}{\sum_{i=1}^{n}(y_{i}- \bar{y})^{2}} \tag{23}\]  
where \(N\) is the number of sample treatments; \(y_{i}^{\prime}\) is the predicted values for Spark execution; \(y_{i}\) is the actual value; and \(\bar{y}\) is the average value of \(y\).
## 4 Applications
### Single Fiber Spectroscopy  
Diffuse optical techniques gained interest in the biomedical world thanks to their capability to monitor non-invasively biological processes. On the other hand, there are applications (such as photodynamic therapy [124, 125, 126], monitoring of the healing process after surgery or deep hemorrhage, etc.) where there is the need to implant an optical fiber. To lessen the impact of the implant, several solutions have been recently studied such as the use of bioresorbable fibers [127, 128, 129, 130]. On the other hand, the use of a single fiber for both light injection and collection would decrease the burden of the implant thus being more attractive for many clinical applications. Similarly, to the non-contact measurements, the main limitation of this approach is given by the small \(\rho\) (in this case exactly null) requiring the rejection of the strong burst of early photons. In Ref. [130], [PERSON] and co-workers adopted the TG technique to reject the large amount of reflected/scarcely diffused photons and reconstruct curves with  
Figure 11: Relative changes with respect to the baseline of oxy-(O\({}_{2}\)Hb) and deoxy-hemoglobin (HHB) in the contralateral hemisphere during a finger tapping exercise. On the (**a**,**c**) results obtained using the TG approach while on the (**b**,**d**) with a standard one (i.e., not gated). Rows represents the \(\rho\) (**a**,**b**): 6 mm; (**c**,**d**): 30 mm [115].  
high dynamic range (seven orders of magnitude). They showed that, in the null \(\rho\) configuration, the change in the tail due to a difference in the absorption coefficient with respect to the homogeneous background medium can be extracted from signals at five orders of magnitude below the peak; thus making the TG acquisition fundamental to retrieve the medium optical properties. Figure 12 shows the results of the linearity measurements of the absorption coefficient obtained by [PERSON] and co-authors. Generally speaking, the absorption coefficient is properly retrieved over a wide range (\(\mu_{a}\) between 0.05 and 0.25 cm\({}^{-1}\)) notwithstanding the (fixed) value of the scattering coefficient (\(\mu_{s}\)\({}^{\prime}\) = 10 cm\({}^{-1}\) for first and third series--plus symbols and green dots in Figure 12--and \(\mu_{s}\)\({}^{\prime}\) = 15 cm\({}^{-1}\) for the second one--crosses--in Figure 12). A slight offset (0.01 cm\({}^{-1}\)) can be noticed and the authors attributed it to the TG reconstruction of the DTOF or to the memory effect. For absorption values larger than 0.2 cm\({}^{-1}\) underestimation of the value can be found. On the other hand, the authors showed that the addition of scattering particles (orange dots points in Figure 12) is not affecting the retrieval of the absorption coefficient which stays constant. The retrieval of the scattering coefficient is, conversely, very difficult at null \(\rho\) using TD measurements since for exactly \(\rho\) = 0 the DTOF gets independent of \(\mu_{s}\).
## 3 Materials and Methods
### Analysis of Phenolic Compounds
#### 3.3.2 HPLC-DAD Analysis of Flavonoids  
Analysis of the onion extracts was performed using an Agilent 1200 HPLC system (Agilent Technologies, Santa Clara, CA, USA). Samples were separated by a LUNA C18 column (250 \(\times\) 4.6 mm i.d., 5 \(\upmu\)m particle size) maintained at 27 \({}^{\circ}\)C. The flow rate was 0.8 mL/min. The mobile phase consisted of (A) water (pH = 3.2 HCOOH) and (B) 100% acetonitrile. The following binary gradient was used: 0-3 min, 5%-12% B; 3-8 min, 12%-14% B; 8-25 min, 14%-20% B; 25-35 min, 20%-35% B; 35-45 min, 35%-60% B, 45-60 min, 60%-100% B. UV/Vis spectra were recorded in the 190-600 nm range and the chromatograms were acquired at 260, 280, 330 and 350 nm. Identification of the quercetin glucosides was based on spectra, standards and literature data, while identification of other individual phenols was carried out using their retention times, spectroscopic and literature data. Quantification of individual phenolic compounds was directly performed by HPLC-DAD using a five-point regression curve (R\({}^{2}\)\(\geq\) 0.998) in the range 0-30 \(\upmu\)g on the basis of quercetin rutinoside standard. Quercetin derivatives were determined at 350 nm using rutin as reference compound. In all cases, actual concentrations of the derivatives were calculated after applying corrections for differences in molecular weight. Three samples were collected from each site so as to express the analytical results as an average with its standard deviation.
# Modeling and Performance Analysis of Satellite Network Moving Target Defense System with Petri Nets
## 2 Background
### Petri Nets
Although the dynamic behavior of the model will be affected by the time parameter, the introduction of time reduces the state space and enhances the description ability of Petri Nets. Therefore, in this paper, SPN is used to analyze the performance of satellite networks. Figure 2: Model elements of Petri Nets.
## 5 Conclusions  
This work proposes a competitive fronthaul solution for distributed antenna systems. Detailed design information and the reasons behind our design trade-offs are included. The system has good signal qualities for both the downlink and uplink data paths. Time-interleaving multiple sigma-delta modulated baseband signals into one bitstream allows  
Figure 12: RRU phase difference versus time and its distribution.  
for the transmission of non-return-to-zero (NRZ) signals over fiber. It also keeps the remote unit complexity low while improving the optical bit-rate efficiency by 60% compared to our previous work.  
The downlink and uplink performance satisfies the 3 GPP error vector magnitude (EVM) requirement for 256-QAM (3.5%): the EVM of 40.96 MHz-bandwidth OFDM signals centered around 3.6 GHz is 2.765% (\(-\)31.17 dB) and 2.648% (\(-\)31.54 dB), respectively. (Optical path: 100 m multi-mode fibers; wireless path: electrical back-to-back.) Additionally, without an extra reference clock signal provided for synchronization, the network guarantees fixed transmit and sampling timing offsets between all transceivers and synchronizes the carrier and sampling frequencies between remote units. The synchronism is thoroughly validated with measurements in different domains. The real-time oscilloscope observation shows that the two remote units have the same carrier frequency and the standard deviation of the relative jitter is 9.43 ps during the 20-second measurement period.  
In summary, as cell-free massive MIMO gains more attention, the proposed network architecture has a high potential for its good signal quality, guaranteed synchronism, and scalability.  
Conceptualization, H.L., P.D. and G.T.; hardware design (including implementation and verification), C.-Y.W., H.L., J.V.K. and C.M.; software design (offline signal processing), C.-Y.W.; system validation and experiment methodology, C.-Y.W., H.L. and J.V.K.; experiment result analysis, C.-Y.W.; resources, P.D. and G.T.; writing--original draft preparation, C.-Y.W.; writing--review and editing, H.L., J.V.K., C.M., P.D. and G.T.; supervision, G.T.; project administration, P.D. and G.T.; funding acquisition, P.D. and G.T. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the ERC Advanced Grant ATTO (No. 695495), EU H2020 ERC-POC BI-SDMoF (No. 839200), and EU H2020 Int5 Gent (No. 957403) project.  
All results have been provided in the manuscript. The generated signals for performance measurements and part of the collected raw data are available on request.  
The authors would like to thank EM-group of IDLab, Ghent University-imec, for the design and fabrication of the antennas used in the experiments.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
## 4 ASL Extensions
### Data Attribute's Formulas  
The first extension added to ASL was the support for specifying formulas defined at the data attribute level. Formulas are expressions that are evaluated at different moments of the application's runtime (e.g., when a record is saved in the database) to compute a value of a corresponding data attribute (or field) based on the value(s) of other data attribute(s).  
As shown in Figure 4, Genio supports the specification of different types of formulas. Upon analysis, we proposed two types of formulas: arithmetic and details formulas.  
**Arithmetic formula**. An arithmetic formula supports the specification of basic arithmetic operations (e.g., addition, subtraction, multiplication, or division) to compute the value of its associated data attribute. For instance, considering a data attribute \"Value With VAT\" of the data entity \"InvoiceLine\", an arithmetic formula defined by \"formula arithmetic (e_InvoiceLine.valueWithoutVAT\" e_Product.VATValue)\" can be used to calculate the value (including VAT) of the invoice line.  
**Details formula**. A details formula allows the specification of operations on related data entities of a given data entity. For example, considering a data attribute \"Total  
Figure 9: Partial ASL metamodel focused on actors and use cases (UML notation).  
lines\" of the data entity \"Invoice\", a details formula defined by \"formula details: count (e_InvoiceLine)\" can be used to calculate the total number of lines of an invoice.  
To implement the rigorous definition of such formulas, we add a simple yet flexible expression language [19]. The syntax of this language is similar to other expression languages, such as the one in Microsoft Excel [40]. Spec. 4 shows an example usage of this expression language to specify both arithmetic and details formulas in an improved specification of the \"Invoice\" data entity.  
**Spec. 4.** Example usage of formulas for the ASL specification of the \"Invoice\" data entity.
# Indole-3-Acetic Acid Action in Outdoor and Indoor Cultures of _Spirulina_ in Open Raceway Reactors
## 2 Materials and Methods  
### Microalga and Phytohormone  
_Spirulina_ LEB 18 was used for the experiments. This microorganism is sourced from the strain bank of the Laboratory of Biochemical Engineering at the Federal University of Rio Grande (FURG)--Brazil. The microalga was isolated at Lagoa Mangueira, at latitude 33\({}^{\circ}\)31\({}^{\prime}\)08\({}^{\prime\prime}\) S and longitude 53\({}^{\circ}\)22\({}^{\prime}\)05\({}^{\prime\prime}\) W [27].  
For supplementation of the microalgal cultures, an auxin-type phytohormone was employed. The commercial indole-3-acetic acid (IAA) (98%, Sigma-Aldrich, Cotia, SP, Brazil) was introduced into _Spirulina_ cultures from a 1000 ppm stock solution [15].
## 4 Advancements in Industry 4.0 and Pandemic Management
### Internet of Things (IoT)
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Study** & **Focus** & **AI/ML Technologies** \\ \hline
[40] & COVID-19 diagnosis using radiological images & Deep convolutional neural networks (CNN) \\ \hline
[41] & Smart factories and process automation & IoT, artificial intelligence (AI), cyber-physical systems, machine learning (ML) \\ \hline
[42] & AI in manufacturing for offensive and defensive uses & Security principles, detection techniques \\ \hline
[43] & Forecasting for manufacturing jobs & BD-I4 (big data and Industry 4.0) \\ \hline
[50] & “Smartization” of manufacturing industries & Data fission, ML strategies \\ \hline
[44] & Deep learning architectures for time series data & Long short-term memory (LSTM), convolutional neural networks (CNN), recurrent neural networks (RNN), stacked LSTM autoencoders \\ \hline
[45] & DDoS attacks in Industry 4.0 & Fuzzy-logic-based detection methods \\ \hline
[46] & Time series anomaly detection & Neural graph networks, gated recurrent units (GRU) \\ \hline
[47] & Unsupervised anomaly detection in production & WaterGAN (WGAN), encoder CNN \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparative summary of research on AI and ML integration in Industry 4.0 applications. Together, these papers shed light on the potential of IoT in Industry 4.0 and offer practical solutions to address challenges in implementing IoT technologies in various industrial contexts. The research presented in this group contributes to advancing IoT's transformative capabilities, revolutionizing industries, and enhancing operational efficiency. Its overview is provided in Table 2.
# Enhanced Localisation and Handwritten Digit Recognition Using ConvCARU
## 4 Experimental Results and Discussion
### Performance Comparison
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{_Model_} & \multirow{2}{*}{**SSIM\(\uparrow\)**} & \multicolumn{2}{c}{**PSNR\(\uparrow\)**} & \multirow{2}{*}{**MSE\(\downarrow\)**} & \multirow{2}{*}{**FVD\(\downarrow\)**} & \multicolumn{2}{c}{**Memory**} \\  & & **(dB)** & & & **Usage (GB)** \\ \hline _ConvLSTM_[(64)] & 0.873 & 26.46 & 0.055 & 185.2 & 9.7 \\ _PhyDNet_[(65)] & 0.885 & 27.52 & 0.048 & 175.4 & 12.3 \\ _PredRNN_[(66)] & 0.887 & 27.83 & 0.046 & 172.1 & 10.5 \\ _CrevNet_[(67)] & 0.888 & 27.99 & 0.045 & 172.1 & 11.5 \\ _MIM_[(68)] & 0.892 & 24.29 & 0.041 & 169.5 & 13.7 \\ _SimVP_[(12)] & 0.893 & 28.34 & 0.043 & 168.2 & 11.2 \\ _MotionRNN_[(69)] & 0.899 & 28.46 & 0.042 & 159.7 & 10.1 \\ _SwitLSTM_[(70)] & 0.891 & 26.55 & 0.040 & 158.6 & 11.5 \\ _DMVFN_[(14)] & **0.902** & 29.11 & **0.039** & 156.3 & 10.8 \\ \hline _Proposed Model_ & 0.901 & **29.31** & 0.041 & **155.5** & 10.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A performance comparison with SOTA methods is presented using the SSIM, PSNR, MSE, and FVD metrics.
# Deep Learning Method for Fault Detection of Wind Turbine Converter
## 1 Introduction
Regarding fault detection and diagnosis of wind turbine converters, the methods applied are generally categorized as model-driven-, data-driven- and signal processing-based; see a short review provided below.
# An Effective Adaptive Combination Strategy for Distributed Learning Network
## 1 Introduction  
It is generally beneficial to exploit diffusion strategies for distributed parameter estimation issues over adaptive networks [1, 2, 3, 4, 5, 6]. Specifically, the diffusion least-mean-square (LMS)-based methods have already been used in many contexts, such as biological behavior modeling [7, 8], distributed detection [9], distributed localization [10] and target tracking and escaping from predators [11], where scalability, robustness, and low power consumption are the desirable features [1]. For the diffusion strategy, each node of the network is allowed to receive the intermediate estimates from its neighboring nodes to improve the accuracy of its local estimate. Such cooperation enables each node to leverage the spatial diversity of noise profile over the entire network. From this point of view, the performance of distributed diffusion methods can be further enhanced by using suitable combination weights (combiners).  
There have been several static combination rules [1, 12], e.g., Metropolis rule, Laplacian rule, Uniform rule and Relative-degree rule. However, these static combiners are designed based solely on the topology of network, so they would generally be unadjustable to adapt to the spatial variation of signal and noise statistics. To address this problem, many studies resort to the adaptive combination (AC) strategies [13, 14, 15, 16, 17, 18, 12], most of which are developed for the adapt-then-combine (ATC) diffusion LMS algorithm [1].  
Based on the minimum variance unbiased estimation (MVUE), the classic AC strategy [12] outperforms existing static combiners when applying onto the diffusion LMS algorithms. An optimal adaptive combination scheme is derived by estimating the variances of the measurement noises adaptively [13]. Simulation results validate the superior steady-state performance of the diffusion LMS algorithm regarding the optimal combiners in [13], as compared to previous static and adaptive combiners. Based on the adaptive combination rule in [13], an optimal combination rule regarding the channel distortion is also proposed [16]. To achieve both the accelerated convergence rate and good steady-state network performance, the combination switching mechanisms [14, 15] are proposed, i.e.,static combination scheme in the converging stage and AC scheme when approaching the steady state.  
In addition, a decoupled adapt-then-combine (D-ATC) algorithm is proposed, for, the least-squares (LS)-based AC scheme is developed [17; 18], which could achieve rather approximate performance as the ATC algorithm with the classic AC in homogeneous networks.
# A One-Class-Based Supervision System to Detect Unexpected Events in Wastewater Treatment Plants
## 3 Experiments and Results
### Results
\begin{table}
\begin{tabular}{r c c c c c c c c} \hline
**Dev** & **Preproc** & **OP (\%)** & **k** & **TP** & **TN** & **FP** & **FN** & **F1 (\%)** \\ \hline
5 & NoNorm & 0 & 1 & 39.8 & 0.0 & 37.0 & 0.2 & 68.2 \\
15 & NoNorm & 0 & 5 & 39.9 & 0.0 & 37.0 & 0.1 & 68.3 \\
25 & Zscore & 0 & 3 & 40.0 & 0.0 & 37.0 & 0.0 & 68.4 \\
35 & Norm & 0 & 29 & 39.4 & 1.0 & 36.0 & 0.6 & 68.3 \\
45 & Norm & 0 & 12 & 39.7 & 0.7 & 36.3 & 0.3 & 68.4 \\
55 & Zscore & 15 & 9 & 32.3 & 18.8 & 18.2 & 7.7 & 71.4 \\
65 & Zscore & 10 & 9 & 34.3 & 17.3 & 19.7 & 5.7 & 73.0 \\
75 & Zscore & 10 & 13 & 33.9 & 20.2 & 16.8 & 6.1 & 74.8 \\
85 & Zscore & 10 & 17 & 33.0 & 24.1 & 12.9 & 7.0 & 76.8 \\
95 & Zscore & 10 & 13 & 33.3 & 25.1 & 11.9 & 6.7 & 78.2 \\ \hline \end{tabular}
\end{table} 
Table 4: Results and configuration for the best K-means classifier depending on anomaly percentage deviation. \begin{table}
\begin{tabular}{r c c c c c c c} \hline
**Dev** & **Preproc** & **RP** & **OP (\%)** & **TP** & **TN** & **FP** & **FN** & **F1 (\%)** \\ \hline
5 & Norm & 0 & 5 & 35.9 & 37.0 & 0.0 & 4.1 & 94.6 \\
15 & Norm & 0 & 0 & 38.7 & 37.0 & 0.0 & 1.3 & 98.3 \\
25 & Zscore & 0 & 0 & 38.7 & 37.0 & 0.0 & 1.3 & 98.3 \\
35 & Norm & 0 & 0 & 38.8 & 37.0 & 0.0 & 1.2 & 98.5 \\
45 & Zscore & 0.001 & 0 & 38.8 & 37.0 & 0.0 & 1.2 & 98.5 \\
55 & Norm & 0.001 & 0 & 38.9 & 37.0 & 0.0 & 1.1 & 98.6 \\
65 & Norm & 0.001 & 0 & 39.0 & 37.0 & 0.0 & 1.0 & 98.7 \\
75 & Zscore & 0.003 & 0 & 39.1 & 37.0 & 0.0 & 0.9 & 98.9 \\
85 & Zscore & 0.003 & 0 & 38.9 & 37.0 & 0.0 & 1.1 & 98.6 \\
95 & Zscore & 0.001 & 0 & 39.0 & 37.0 & 0.0 & 1.0 & 98.7 \\ \hline \end{tabular}
\end{table} 
Table 3: Results and configuration for best Gaussian classifier depending on anomaly percentage deviation.
## 3 Fractances with Tunable Factor's Order
### Equivalent Circuit through Rational Approximation  
First of all, the methodology for representing a fractance in terms of equivalent integer order circuit is illustrated. Several methodologies have been proposed in the literature, also corresponding to different kind of circuits' topologies [34].  
Let us consider the expression of the fractance \(Z(\omega)=1/F\ 1/(\mathrm{j}\omega)^{\alpha}\) and let us try to find a rational approximation \(G(\omega)\) as:  
\[G(s)=\frac{A(j\omega/b_{1}+1)(j\omega/b_{2}+1)\ldots(j\omega/b_{n}+1)}{(j \omega/a_{1}+1)(j\omega/a_{2}+1)\ldots(j\omega/a_{n}+1)} \tag{4}\]where \(A\), \(a_{i}\), and \(b_{i}\) are all positive numbers. Such an expression consists of n first-order damp elements and n first-derivative elements. Each couple of damp and derivative elements, separately considered, represent a piecewise linear function. So, if we consider the operator \(Z(\omega)\), we can define a tolerance threshold \(2\delta\) and require that the piecewise linear is included in the upper and lower boundaries, as shown in Figure 1.  
The approximation of the fractional characteristic will be acceptable in a certain bandwidth, depending by the accepted tolerance \(\delta\) and by the order \(n\) of the rational function \(G(\omega)\). It is worth to define as \(\omega_{0}\) the central frequency of the approximation and as \(Z_{0}\) the magnitude of the impedance for \(\omega=\omega_{0}\). Taking \(n\) as odd value and starting from the central point \((\omega_{0},Z_{0})\), it is possible to graphically calculate the coefficients \(a_{i}\) and \(b_{i}\) as:  
\[a_{(n+1)/2}=\omega_{0}10^{\delta/20\alpha} \tag{5}\]  
\[a_{i+1}=a_{i}10^{\delta/(10\alpha(1-\alpha))} \tag{6}\]  
\[b_{i+1}=a_{i}10^{\delta/(10(1-\alpha))} \tag{7}\]  
\[A=Z_{0}(1/a_{1})^{\alpha}10^{\delta/20} \tag{8}\]  
Once the parameters of (4) have been identified, it is possible to expand the rational approximation as:  
\[G(s)=A_{0}+\frac{A_{1}}{j\omega/a_{1}+1}+\ldots+\frac{A_{n}}{j\omega/a_{n}+1} \tag{9}\]  
where the parameters \(A_{0}\)... \(A_{n}\) can be calculated as:  
\[A_{0}=A(a_{1}a_{2}\ldots a_{n})/(b_{1}b_{2}\ldots b_{n}) \tag{10}\]  
\[A_{i}=\left.(\omega-\omega_{i})G(s)/a_{i}\right|_{\omega=\omega_{i}} \tag{11}\]  
The single first order rational functions composing (9) correspond, in terms of circuits, to a series of \(R\)-\(C\) parallels where:  
\[R_{i}=A_{i} \tag{12}\]  
\[C_{i}=1/(a_{i}A_{i}) \tag{13}\]  
In Figure 2, the obtained equivalent circuit is represented, corresponding to a low-pass filter with constant phase.  
Figure 1: Example of amplitude behavior of the operator \(Z(\omega)\) and the piecewise function.