esa-sceva/satcom-chunk-collection huggingface dataset:
# Design of Parabolic Off-Axis Reflector Optical System for Large Aperture Single Star Simulators
## 5 Experimental Verification  
The primary mirror and secondary mirror were processed, and the finished physical images of the primary mirror and secondary mirror are shown in Figures 12a and 12b, respectively. The main mirror is made of microcrystalline glass. The supporting material and method of the main mirror are as follows: the back of the main mirror and the frame are bonded into a whole with non-stress adhesive, surrounded by polytetrafluoroethylene paper, and fixed with a gland. The material of the second mirror is fused quartz. The supporting material and method of the secondary mirror are as follows: the secondary mirror and the frame are isolated by polytetrafluoroethylene, and the gland is fixed. The reflector is coated with a custom high-reflection film, with a reflectance of up to 99% in the 400 nm to 900 nm spectral range.  
The parabolic off-axis reflection optical system is constructed in the laboratory environment, and the completed parabolic off-axis reflection optical system is shown in Figure 12c. Then, the optical system's wave aberration, parallelism, and focal length are measured.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Stray Light Suppression Diaphrag Name** & **Placement/mm** & **Calibre/mm** \\ \hline Stray light suppression diaphragm 1 & 364.34 & 40 \\ Stray light suppression diaphragm 2 & 1999.97 & 260 \\ Stray light suppression diaphragm 3 & 925.64 & 498 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Stray light suppression diaphragm position table.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Optics and Structural Components** & **Optical Properties** & **Surface Treatment** &
\begin{tabular}{c} **Surface Property** \\ **Parameters** \\ \end{tabular} \\ \cline{2-4}  & & **Reflex** & **Assimlate** \\ \hline Primary and secondary mirror surfaces & Actual reflection & High-reflective coating & 99\% & 1\% \\ Primary mirror mechanical surfaces & Darken & Blackening process & 0.05\% & 94.8\% \\ Secondary mirror mechanical surfaces & Darken & Blackening process & 0.001\% & 99.999\% \\ Mechanical surfaces of stray light suppression & Darken & Blackening process & 0.001\% & 99.999\% \\ diaphragms 1–3 & Darken & Blackening process & 0.001\% & 99.999\% \\ Internal surfaces of enclosures & Darken & Blackening process & 0.001\% & 99.999\% \\ Optical vibration isolation platform & Darken & Blackening process & 0.001\% & 99.999\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: Optical properties of optical and structural elements under actual conditions.
## 3 Results and Discussion
### Discussion of the Results in the Context of Previous Studies  
The results of this study confirm the effectiveness of ultrasonic exposure in reducing the smoke emissions of diesel engine exhaust gases, supporting previous findings on the use of physical methods for gas purification. In particular, the studies by [PERSON] et al. in 2021 and 2022 showed that ultrasound facilitates the coagulation of solid particles and reduces the concentration of CO\({}_{2}\), CO, and CH in the exhaust gases of both gasoline and diesel engines [43; 47; 48]. The present work complements these studies by showing that the efficiency of ultrasonic exhaust purification is strongly influenced by the emitter placement and power--two critical factors in optimizing the overall cleaning performance.  
One of the key distinctions of the presented study is the development and testing of a vertical ultrasonic muffler, whereas previous research has primarily focused on horizontal designs. The analysis of the results confirms that the vertical arrangement of ultrasonic emitters enhances the deposition of coagulated particles due to the force of gravity. This is in line with the theoretical expectations, as the vertical design minimizes the likelihood of re-entraining coagulated particles into the gas stream, unlike horizontal setups, where PM remains in a suspended state.  
Additionally, it was established that as the engine speed increases, the cleaning efficiency decreases, primarily due to the reduced time that the particles interact with the ultrasonic field. This confirms the findings made in the works of [PERSON] et al. in 2024, where it was noted that to maintain high cleaning efficiency at high speeds, an increase in the power of the emitters is necessary [44]. The results also prove that the use of multiple ultrasonic emitters with stepped placement is a more effective approach compared to single emitters, as demonstrated by the experimental studies of previous authors.  
Furthermore, research on the use of ultrasound can be extended along the direction proposed by [PERSON] et al. (2023), who demonstrated that ultrasonic waves can be used to blend fuels such as ethanol and diesel in real time. This method has been shown to significantly reduce smoke emissions at medium and full loads, with reductions reaching up to 74% compared to pure diesel [49].
## 2 Degradation of Materials in Space--Stability Issues on Mirrors
### Atomic Oxygen
However, LiF thin films are hygroscopic and exhibit reflectance degradation and increased scattering with age. Al high intrinsic reflectance extends beyond MgF\({}_{2}\) and LiF cutoff wavelengths down to \(\sim\)83 nm. However, Al reactivity in the presence of oxygen results in a dramatic FUV/EUV reflectance decrease, and no transparent material is available in nature to preserve reflectance to such a short wavelength. Al extended reflectance was not ignored by space instrument developers since early in the space era, and such an extension would be very beneficial for space observations due to the presence of important spectral lines in the extended region. There have been several proposals to make use of Al extended reflectance in space telescopes, mostly based on aluminization in a natural ultra-high vacuum at a high enough orbit [61]. In order to implement this technology, the degradation of the FUV reflectance of unprotected Al through controlled oxidation to O\({}_{2}\), H\({}_{2}\)O, and other species [62; 63] and to AO [64; 65] has been investigated. AO was found to be orders of magnitude more effective at degrading Al reflectance compared with the same doses of O\({}_{2}\). Non-protected Al mirrors have been also exposed to an LEO environment [50]; even though Al oxidation occurs rapidly, which happened right after the sample was taken out of the vacuum chamber in the lab, Al mirrors experienced further reflectance losses below 250 nm once in orbit, which was attributed to the greater reactivity of Al with AO compared to atmospheric O\({}_{2}\). In view of the sensitivity of bare Al reacting with AO, some procedure to significantly reduce the rate of impingement of oxidizing species must be developed, either based on the selection of high orbits [66] or through the use of some scheme that shields the mirrors from ambient oxygen [67; 68].
# Adaptive MPC Based Real-Time Energy Management Strategy of the Electric Sanitation Vehicles
## 2 Modeling and Energy Consumption Analysis on an Electric Sanitation Vehicle  
The electric sanitation vehicle studied in this paper is shown in Figure 1, which was equipped with two electric motors, named the working motor and the driving motor hereafter, as shown in Figure 2. The main parameters of the vehicle are listed in Table 1.  
Figure 1: Research object: the sanitation vehicle: (**a**) front view (**b**) side view.  
Figure 2: Structure of the electric sanitation vehicle.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Components** & **Description** \\ \hline \multirow{4}{*}{Electric motor 1: Driving motor} & Type: permanent magnet synchronous motor \\  & Rated/Maximum power: 100/184 kW \\  & Rated/Maximum speed: 1600/5200 rpm \\  & Rated/Peak torque: 600/1100 Nm \\ \hline \multirow{4}{*}{Electric motor 2: Working motor} & Type: permanent magnet synchronous motor \\  & Rated/Maximum power: 63/125 kW \\  & Rated/Maximum speed: 2000/6000 rpm \\  & Rated/Peak torque: 300/650 Nm \\ \hline \multirow{4}{*}{Automated mechanical transmission} & Gear ratio: [2.5, 1]; Average efficiency: 0.9 \\  & Final drive ratio: 4.88 \(\times\) 2 (planet set ring ratio) \\ \hline \multirow{2}{*}{Battery} & Nominal cell voltage: 3.2 V; Total cells: 384 \\  & Nominal voltage: 614.4 V; Capacity: 254 Ah \\ \hline \multirow{4}{*}{Vehicle} & Mass: 12,000/18,000 kg \\  & Frontal area: 8 m\({}^{2}\); Air resistance coefficient: 0.55 \\ \cline{1-1}  & Road resistance coefficient: 0.008; Tire radius: 0.506 m \\ \cline{1-1}  & Rotational mass conversion factor: 1.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main parameters of the sanitation vehicle insurance.  
The electric sanitation vehicle had two operating modes: One was driving mode at high speed, the other was working mode at low speed. For the case in the working mode, the driving motor and the working motor operated separately. Namely, the torque split was not allowed through the planet sets in the transmission. The clutch was kept closed, and the working motor drove three clean devices consisting of a high-pressure water pump, a powerful vacuum collector, and a sweeper to clean the road surface. Since the particular principles of the clean process were not the point, the energy consumed by the three devices was simplified as an integration equaling that of the working motor.
# MB-OFDM-UWB Based Wireless Multimedia Sensor Networks for Underground Coalmine: A Survey
###### Abstract  
Safety production of coalmines is a task of top priority which plays an important role in guaranteeing, supporting and promoting the continuous development of the coal industry. Since traditional wireless sensor networks (WSNs) cannot fully meet the requirements of comprehensive environment monitoring of underground coalmines, wireless multimedia sensor networks (WMSNs), enabling the retrieval of multimedia information, are introduced to realize fine-grained and precise environment surveillance. In this paper, a framework for designing underground coalmine WMSNs based on Multi-Band Orthogonal Frequency-Division Multiplexing Ultra-wide Band (MB-OFDM-UWB) is presented. The selection of MB-OFDM-UWB wireless transmission solution is based on the characteristics of underground coalmines. Network structure and design challenges are analyzed first, which is the foundation for further discussion. Then, key supporting technologies and open research areas in different layers are surveyed, and we provide a detailed literature review of the state of the art strategies, algorithms and general solutions in these issues. Finally, other research issues like localization, information processing, and network management are discussed.  
wireless multimedia sensor networks (WMSNs); underground coalmine; multi-band orthogonal frequency-division multiplexing (MB-OFDM); ultra-wide band (UWB); network management +
Footnote †: journal: Sensors
## 2 Influential Factors on the Performance of GSP
### Raw Materials and Admixtures
#### 2.2.1 Aggregate Gradation
Cantabro loss rises to a certain value, which indicates that the cohesion between aggregate in GSP is weakened. From a different perspective, [PERSON] et al., proposed that uniform gradation could conduct smoother void spaces and less stress concentration than continuous gradation. He compared two types of gradations: uniform gradation (one tier: 10-5 mm) and continuous gradation (three tiers: 13-10 mm, 10-5 mm, and 5-3 mm) [53]. It was found that GSP with uniform gradation was superior to that with continuous gradation in the properties of Marshall stability, splitting strength, compressive resilience, and low-temperature bending capacity [53]. [PERSON] et al., evaluated seven types of gradations using the hierarchical ranking strategy considering parameters as void content, permeability, abrasion resistance, and tensile strength of OAC; BSI-4% and Densiphalt12-4.5% were selected as the optimal gradation which had a main tier of 4.75 mm\(\sim\)12.5 mm [47]. In general, gradation with the main particle sizes in the scope of 4.75 mm\(\sim\)12.5 mm is appropriate to improve the performance of GSP to form a better structure (Figure 2). However, a conflict exists in previous hypotheses that mainly concerns coarse particles. For example, a given continuous gradation may also contain larger-tier aggregate. The reason may be that the influence of gravel morphology and fine aggregate is overlooked which needs further research [52]. In our opinion, the two viewpoints are all correct, though only the applying scope of these principles is different. The definitions for describing gradation types are difficult to quantify, due to the existence of few samples with untested properties, such as residual void rate and water permeability. Large particle size and uniform gradation can both contribute to the formation of evener and larger void space.
# TAEO-A Thermal Aware & Energy Optimized Routing Protocol for Wireless Body Area Networks
## 4 Proposed Methodology-TAEO
### Routing  
The proposed routing protocol is based on a multi-hop schema and it works by selecting a Data Forwarder (DF), who is responsible for the collection of packets from surrounding nodes and thereby forwarding them to sink for onward submission to a remote server.  
A node with less temperature and distance from the sink along with higher residual energy is selected as DF. The temperature of DF is estimated, and is only allowed for transmission if it is below the specified threshold; otherwise, the respective DF is suspended for upcoming few rounds, until its temperature is revamped to normal. As a multi-hop routing protocol, a neighbor node can also be selected as a forwarder, especially, in the case where the distance of DF to sink is comparatively greater than a node's direct distance to sink. In such a scenario, the node itself will directly forward data to Sink. Neighbor node's heat estimation will be performed likewise for Hot-Spot detection and avoidance.  
In order to find an increase in the temperature of human tissue upon exposure to RF communication, there is a need to calculate the SAR value of the tissue where the sensor is implanted. Equation (4) is used to calculate SAR. Temperature rise is estimated by using Equation (5) [10; 31]:  
\[\Delta T=(SAR)\frac{t}{C} \tag{5}\]  
where, t is the time duration for which tissue is exposed to electromagnetic radiation and C is the specific heat capacity.  
The TAEO protocol also controls the energy utilization to increase the network lifetime and stability period. It first specifies a distance value. Afterward, it calculates the distance in between six non-critical nodes and to sink. During transmission, the distance between the two nodes involved in the communication is compared to the specified distance value. If it is below the defined distance limit, it means that the two nodes are closely placed. Therefore, it utilizes less transmission power. Otherwise, for distant nodes, the packet is transported with maximum energy.
# Enhancing Kazakh Sign Language Recognition with BiLSTM Using YOLO Keypoints and Optical Flow
## 7 Conclusions  
This study proposed a two-stage framework for Kazakh Sign Language recognition, combining YOLOv8-Pose for spatial hand detection with a Bi-LSTM model for temporal sequence classification. The system handles both static and dynamic gestures.  
YOLOv8-Pose accurately detected hands and extracted 21 keypoints, which, along with bounding-box coordinates and optical flow vectors, formed informative input sequences. The usage of motion features allowed the model to distinguish between similar gestures, improving recognition performance. A structured dataset covering alphabetic and frequently used dynamic gestures was presented as a basis for further research into Kazakh Sign Language.  
The results demonstrate the effectiveness of combining real-time detection with temporal modeling, showing a new approach for developing communication tools for the deaf and hard of hearing communities in Kazakh.  
Conceptualization, Z.B., A.Y. and M.Z.; methodology, Z.B., Z.A. and M.Z.; software, Z.A., Z.Z. and M.Z.; validation, M.A., Z.Z. and M.Z.; formal analysis, M.A., Z.Z. and M.Z.; investigation, M.A., Z.Z. and M.Z.; resources, A.Y., Z.A., Z.B. and M.Z.; data curation, A.Y., M.A. and M.Z.; writing--original draft preparation, M.A., Z.Z. and M.Z.; writing--review and editing, M.A., Z.Z. and M.Z.; visualization, M.A., Z.Z. and M.Z.; supervision, Z.B. and M.Z.; project administration, Z.B. and M.Z.; funding acquisition, Z.B. and M.Z. All authors have read and agreed to the published version of the manuscript.  
This research has been funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. BR24992875).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available in article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.
## 5 Technique Comparison
### Ridge Method  
RR is a regularized version of LR, which addresses multicollinearity by introducing an L2 penalty on the size of the coefficients. This technique shrinks the regression coefficients, reducing their variance and stabilizing the estimates, especially when the dataset contains highly correlated predictors. RR has proven effective in modeling SPVS where environmental factors often exhibit collinearity, such as solar irradiance and temperature [10].  
It is worth noting that the estimators of the parameters of MLRM using the method of OLS are unbiased estimates and have the smallest variance compared with all the unbiased estimators of the MLRM using other methods. However, in many cases, the assumptions of the method of OLS are not met in the data to be modeled, which makes OLS estimates not as good as the above specifications. One of these important assumptions is the assumption of multicollinearity, that is, the independent variables of MLRM are strongly and significantly related to each other, and thus some other assumptions.  
In addition, and because of the above problem of multicollinearity, the matrix \((X^{T}\ X)^{-1}\) is singular and the OLS estimators for the MLRM cannot be obtained. Using the technique of RR, which is one of the regularization/shrinkage methods, the OLS estimators of the MLRM can be obtained [30; 31; 32]. In addition, the technique of RR is based on adding a Ridge parameter (\(\gamma\)) multiplied by the identity matrix \(\mathrm{I}\) to the \((X^{T}\ X)^{-1}\). This operation results in a new matrix, \((X^{T}\ X)^{-1}\ +\gamma I\), which is a nonsingular matrix. Now, the Sum of Residual (errors) Squares of Ridge Method (RSSR) is given by  
\[RSSR=\sum_{i=1}^{n}[y_{i}-\hat{y}]^{2}=\sum_{i=1}^{n}\{[-(\hat{\beta}_{0}\ +\sum_{j=1}^{p}\hat{\beta}_{j}\ x_{ji})]^{2}+\sum_{j=1}^{p}\gamma\hat{\beta}_{ j}^{2}\}, \tag{5}\]  
where \(\gamma\) is called the shrinkage factor. Then, the modified Ridge estimators of the parameters of MLRM using Lagrangian form are  
\[\hat{\beta}_{R}=argmin\sum_{i=1}^{n}\{[y_{i}-(\beta_{0}+\sum_{j=1}^{p}\beta_{ j}x_{ji})]^{2}+\sum_{j=1}^{p}\gamma\hat{\beta}_{j}^{2}\},\ \gamma\geq 0. \tag{6}\]  
subject to  
\[\sum_{j=1}^{p}\hat{\beta}_{j}^{2}<t, \tag{7}\]  
where \(t\) is a factor, such that \(t\geq 0\).  
It may be useful to give the final estimators \(\hat{\beta}_{R}\) using the following matrix form:  
\[\hat{\beta}_{R}=(X^{T}\ X+\gamma I)^{-1}\ X^{T}\ Y. \tag{8}\]
# Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory
## 2 Sensor Placement Method
### Derivation of Forward Kinematics of Industrial Robots
\begin{array}{c}n_{x}=-\ s_{6}(c_{4}s_{1}+s_{4}(c_{1}c_{2}c_{3}-c_{1}s_{2}s_ {3}))-c_{6}(c_{5}(s_{1}s_{4}-c_{4}(c_{1}c_{2}c_{3}-c_{1}s_{5}s_{3}))+s_{5}(c_ {1}c_{2}s_{3}+c_{1}s_{2}c_{3})\\ n_{y}=s_{6}(c_{4}c_{1}+s_{4}(s_{1}s_{2}s_{3}-c_{2}c_{3}s_{1}))+c_{6}(c_{5}(c_ {1}s_{4}-c_{4}(s_{1}s_{2}s_{3}-c_{2}s_{3}s_{1}))-s_{5}(c_{2}s_{3}s_{1}+c_{3}s_{ 1}s_{2})\\ n_{z}=s_{4}s_{6}(c_{2}s_{3}+c_{3}s_{2})-c_{6}(s_{5}(c_{2}c_{3}-s_{2}s_{3})+c _{4}c_{5}(c_{2}s_{3}+c_{3}s_{2}))\\ o_{x}=s_{6}(c_{5}(s_{1}s_{4}-c_{4}(c_{1}c_{2}c_{3}-c_{1}s_{2}s_{3}))+s_{5}(c_ {1}c_{2}s_{3}+c_{1}c_{3}s_{2}))-c_{6}(c_{4}s_{1}+s_{4}(c_{1}c_{2}c_{3}-c_{1}s_{ 2}s_{3}))\\ o_{y}=c_{6}(c_{1}c_{4}+s_{4}(s_{1}s_{2}s_{3}-c_{2}c_{3}s_{1}))-s_{6}(c_{5}(c_ {1}c_{4}-c_{4}(s_{1}s_{2}s_{3}-c_{2}c_{3}s_{1}))-s_{5}(c_{2}s_{3}s_{1}+c_{2}c_{ 3}s_{1}))\\ o_{z}=s_{6}(s_{5}(c_{2}c_{3}-s_{2}s_{3})+c_{4}c_{5}(c_{2}s_{3}+c_{2}c_{3}))+c _{6}s_{4}(c_{2}s_{3}+c_{2}c_{3})\\ a_{x}=c_{5}(c_{1}c_{2}s_{3}+c_{1}c_{3}s_{2})-s_{5}(s_{1}s_{4}-c_{4}(c_{1}c_{2}c _{3}-c_{1}s_{2}s_{3}))\\ a_{y}=s_{5}(c_{1}s_{4}-c_{4}(s_{1}s_{2}s_{3}-c_{2}c_{3}s_{1}))+c_{5}(c_{2}s_{1}s _{3}+c_{3}s_{1}s_{2})\\ a_{z}=c_{5}(c_{2}c_{3}-s_{2}s_{3})-c_{4}s_{5}(c_{2}s_{3}+s_{2}c_{3})\\ p_{x}=(3c_{1}c_{2})/5-(3s_{5}(s_{1}s_{4}-c_{4}(c_{1}c_{2}c_{3}-c_{1}s_{2}s_{3})) )/25+(3c_{5}(c_{1}c_{2}s_{3}+c_{1}c_{3}s_{2}))/25+(16s_{3}c_{1}c_{3}s_{2})/25 +(16s_{3}c_{1}s_{2})/25\\ p_{x}=(3c_{2}s_{1})/5+(3s_{5}(c_{1}s_{4}-c_{4}(s_{1}s_{2}s_{3}-s_{1}c_{2}c_{3})) )/25+(3c_{5}(s_{1}c_{2}s_{3}+s_{1}c_{3}s_{2}))/25+(16s_{1}s_{3}c_{2})/25\\ (16s_{1}s_{3}c_{2})/25+(16c_{3}s_{1}s_{2})/25\\ p_{z}=(16c_{2}c_{3})/25-(3s_{2})/5-(16s_{2}s_{3})/25+(3c_{5}(c_{2}c_{3}-s_{2}s_{3}) )/25-(3c_{4}s_{5}(c_{2}s_{3}+c_{3}s_{2}))/25\end{array}  \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Link\(\backslash\)Parameters** & \(\mathbf{a}_{i-1}\) & \(\mathbf{\kappa}_{i-1}\) & \(\mathbf{d}_{i}\) & \(\mathbf{\theta}_{i}\) \\ \hline
1 & 0 & 0 & 0 & \(\theta_{1}\) \\
2 & \(a_{1}\) & \(-\pi/2\) & 0 & \(\theta_{2}\) \\
3 & \(a_{2}\) & 0 & 0 & \(\theta_{3}\) \\
4 & 0 & \(\pi/2\) & \(d_{
# Deep Learning of GNSS Acquisition
## 2 GNSS Signal Model and Acquisition
### Benchmark Performance Using the Receiver Operating Characteristic Function  
The so-called Receiver Operating Characteristic (ROC) is a popular metric to assess the performance of any detector/classifier. An ROC is a plot of the detection probability (\(P_{d}\)) as a function of the false alarm probability (\(P_{fa}\)). More precisely, \(P_{d}=\mathbb{P}\{|\mathcal{C}_{i}(\tau,f_{d})|^{2}>\beta|\mathcal{H}_{1}\}\) is the probability of correctly detecting a GNSS signal given that it was present, while \(P_{fa}=\mathbb{P}\{|\mathcal{C}_{i}(\tau,f_{d})|^{2}>\beta|\mathcal{H}_{0}\}\) is the probability of detecting the signal given that it should have not been detected. Ideally, the aim is to have the classifier operate such that \(P_{d}\to 1\) and \(P_{fa}\to 0\).  
Theoretical ROC curves are well known for GNSS signal acquisition [10] and used to benchmark different algorithmic solutions. The remainder of this section provides a quick summary of the theoretical ROC used in coherent/noncoherent integration schemes. In this article, we use this theoretical ROC to assess the performance of our DNN-based solution against the best achievable performance under standard (i.e., non-data-driven) method.  
In order to calculate the ROC curves, first, one needs to calculate the \(P_{fa}\) and \(P_{d}\) probabilities. The value of the detection threshold \(\beta\) is typically computed for a given false-alarm probability, given by  
\[P_{fa,K}(\beta)=\exp\left(-\frac{\beta}{2\sigma_{n}^{2}}\right)\sum_{k=0}^{K-1 }\frac{1}{k!}\left(\frac{\beta}{2\sigma_{n}^{2}}\right)^{k} \tag{10}\]  
where \(K\) indicates the number of noncoherent integrations (i.e., averages of \(K\)-coherent integrations, as in (9)) considered (such that \(K=1\) in the absence of noncoherent integration) and \(\sigma_{n}^{2}=\frac{\sigma_{n}^{2}}{2N}\) is the variance of the in-phase and quadrature outputs.  
Then, the \(P_{d}\) can be calculated as a function of \(\beta\) as  
\[P_{d,K}(\beta)=Q_{K}\left(\sqrt{K\frac{\lambda}{\sigma_{n}^{2}}},\sqrt{\frac{ \beta}{\sigma_{n}^{2}}}\right) \tag{11}\]  
where \(\lambda=\alpha_{i}^{2}/4\) is the noncentrality parameter, and the generalized Marcum \(Q\)-function is defined as  
\[Q_{K}(a,b)=\frac{1}{a^{K-1}}\int_{b}^{+\infty}x^{K}\exp\left(-\frac{a^{2}+x^{ 2}}{2}\right)I_{K-1}(ax)dx\;, \tag{12}\]  
which allows for computation of the ROC curves.