esa-sceva/satcom-chunk-collection huggingface dataset:
_Appl. Sci._**2015**, \(5\), 1235-1251; doi:10.3390/app5041235  
_Applied sciences_  
**ISSN 2076-3417**  
www.mdpi.com/journal/applsci  
_Article_  
**Agent Based Fuzzy T-S Multi-Model System and Its Applications**  
**[PERSON]**  
School of Control and Computer Engineering, North China Electric Power University,  
Beijing 102206, China; E-Mail: [EMAIL_ADDRESS]; Tel.: +86-156-5239-1347.  
Academic Editor: [PERSON]  
_Received: 19 August 2015 / Accepted: 5 November 2015 / Published: 16 November 2015_  
**Abstract:** Based on the basic concepts of agent and fuzzy T-S model, an agent based fuzzy T-S multi-model (ABFT-SMM) system is proposed in this paper. Different from the traditional method, the parameters and the membership value of the agent can be adjusted along with the process. In this system, each agent can be described as a dynamic equation, which can be seen as the local part of the multi-model, and it can execute the task alone or collaborate with other agents to accomplish a fixed goal. It is proved in this paper that the agent based fuzzy T-S multi-model system can approximate any linear or nonlinear system at arbitrary accuracy. The applications to the benchmark problem of chaotic time series prediction, water heater system and waste heat utilizing process illustrate the viability and the efficiency of the mentioned approach. At the same time, the method can be easily used to a number of engineering fields, including identification, nonlinear control, fault diagnostics and performance analysis.  
**Keywords:** agent; fuzzy logic; multi-model system; nonlinear modeling; approximation characteristic; application
# What Is a Causal Graph?
## 2 Probabilistic DAG  
The most basic way of interpreting a DAG is as a representation of qualitative probabilistic relations of conditional independence between its variables. Such a situation occupies the lowest rung of the \"ladder of causation\" [19]. The semantics governing such a representation, while precise, are not totally straightforward, being described by either of two logically equivalent criteria, known as \"\(d\)-separation\" [20; 21] and \"moralisation\" [22]. Here, we describe moralisation, using the specific DAG \(\mathcal{D}\) of Figure 1 for illustration.  
Suppose we ask: Is the specific conditional independence property  
\[(B,R)\perp\!\!\!\perp(G_{1},Y_{1})\mid(A,N)\]  
(read as \"the pair \((B,R)\) is independent of \((G_{1},Y_{1})\), conditional on \((A,N)\)\"; see [23]) represented by the graph? To address this query, we proceed as follows:  
1. **Ancestral graph.** We form a new DAG \(\mathcal{D}^{\prime}\) by removing from \(\mathcal{D}\) every node that is not mentioned in the query and is not an ancestor of a mentioned node (i.e., there is no directed path from it to a mentioned node), as well as any arrow involving a removed node. Figure 2 shows the result of applying this operation to Figure 1.  
2. **Moralisation.** If in \(\mathcal{D}^{\prime}\) there are two nodes that have a common child (i.e., each has an arrow directed from it to the child node), but they are not themselves joined by an arrow--a configuration termed \"immorality\"--then an undirected edge is inserted between them. Then, every remaining arrowhead is removed, yielding an undirected graph \(\mathcal{G}^{\prime}\). In our example, this yields Figure 3.
3. **Separation.** Finally, in \(\mathcal{G}^{\prime}\), we look for a continuous path connecting an element of the first set in our query (here \((B,R)\)) to an element of the second set (here \((G_{1},Y_{1})\)) that does not contain an element of the third set (here \((A,N)\)). If there is no such path, we conclude that the queried conditional independence property is represented in the original DAG. Since this is the case in our example, the property \((B,R)\perp\!\!\!\perp(G_{1},Y_{1})\mid(A,N)\) is indeed represented in Figure 1.
# Condition Rating Prediction Using an Interactive Deterioration Model Development Package
## 4 Development of Deterioration Models
### Stochastic Deterioration Models Using Weibull Distribution  
A stochastic process has been developed to model the uncertainty in the estimation of the sojourn time, which represents the duration of a bridge element remaining at a particular condition rating [4; 30; 52]. The probability of survival \(S_{i}(t)\) indicates that a bridge at condition rating \(i\) remains in the same condition state when the sojourn time \(T_{i}\) exceeds \(t\) years, and is modeled using the cumulative distribution function of Weibull distribution, as follows:  
\[S_{i}(t)=\exp\{-\Lambda_{i}(t)\} \tag{8}\]In Equation (8), \(\Lambda_{i}(t)\) is defined as the integrated value of the hazard rate, \(\lambda_{i}(t)\), which can be modeled using Weibull distribution as follows:  
\[\lambda_{i}(t)=\frac{\eta_{i}}{\theta_{i}}\bigg{(}\frac{t}{\partial_{i}}\bigg{)} ^{\eta_{i}-1}\exp\bigg{\{}-\bigg{(}\frac{t}{\partial_{i}}\bigg{)}^{\eta_{i}} \bigg{\}},\ t>0,\ \theta_{i}>0,\ \eta_{i}>0,\ i=2,\ \ldots,\ 9 \tag{9}\]  
In Equation (9), \(\eta_{i}\) and \(\theta_{i}\) are the shape and scale parameters, respectively, which can be estimated using a curve-fitting mechanism. As an example, the distribution of Wyoming bridges regarding age belonging to each condition rating was modeled using Weibull distribution, as shown in Figure 3. The higher condition rating showed that the shape of the Weibull function generally skewed right more, and vice-versa. The sojourn time, \(T_{i}\), is defined as a mean of the estimated Weibull distribution and can be calculated by  
\[\mathrm{E}(T_{i})=\eta_{i}\Gamma\bigg{(}1+\frac{1}{\theta_{i}}\bigg{)} \tag{10}\]  
In Equation (10), \(\Gamma(.)\) denotes the gamma function. For each subset of the bridge inventory, a maximum of nine sojourn times can be estimated for all condition states, except the \"0 (failure)\" condition, indicating the average duration for which the bridges remain in a particular condition state. Third-order polynomial functions are implemented in the DMDP to develop deterioration models from these discretized data.
# Negative Poisson's Ratio Lattice Structure with Chiral and Re-Entrant Properties
## 4 Conclusions  
NPR structures are widely used in aerospace, sports, and medical fields because they are lightweight and have high strength, impact resistance, and seismic resistance. This study proposed a novel NPR structure based on ancient Chinese window grills. The structure combined chiral and re-entrant structures. Static tensile tests were conducted to verify the NPR effect of the structure, and finite element numerical simulation experiments were conducted to study changes in the NPR effect of the structure under changing geometric parameters. The following conclusions were obtained:  
1. The ancient Chinese window lattice structure exhibited an NPR effect under static tension.  
2. The deformation mechanism was explored. The unit cell rotated when subjected to longitudinal tension and the concave part opened outward under tensile force. The overall structure exhibited outward expansion at the macro level.  
3. As the deflection angle increased, the re-entrant structure weakened the NPR effect. As the length ratio increased, the NPR effect increased sharply, decreased sharply, and, finally, slowly increased.  
4. The strongest NPR effect of the structure was observed at a length ratio of 0.343 and deflection angles of 0\({}^{\circ}\)-20\({}^{\circ}\), and the NPR ratio reached \(-\)0.6.  
It can be envisioned that the proposed design strategy can be extended to any other structure that combines re-entrant and chiral properties, providing insights into the development of a new class of architected metamaterials with potential applications in construction engineering, sports protection, aerospace, biomedical sensors, and semiconductors. In the next step, we plan to study the low-speed impact resistance of the Chinese ancient window grill structure to explore the energy dissipation characteristics of the NPR structure.  
**Author Contributions:** [PERSON]. ([PERSON]) performed the conceptualization, data curation, formal analysis, and investigation. [PERSON] was responsible for the methodology, validation, and editing. [PERSON] completed the writing. [PERSON] conducted the experiments. [PERSON] provided the experimental platform. [PERSON]. ([PERSON]) conceived the idea. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data are contained within the article.  
**Acknowledgments:** The author appreciates the support of the 3D printing team at China Three Gorges University.  
**Conflicts of Interest:** The authors declare that they have no known competing financial interest or personal relationship that could have appeared to influence the work reported in this paper.
# Identify Road Clusters with High-Frequency Crashes Using Spatial Data Mining Approach
###### Abstract  
This paper develops a three-step spatial data mining approach to directly identify road clusters with high-frequency crashes (RCHC). The first step, preprocessing, is to store the roads and crashes in a spatial database. The second step is to describe the conceptualization of road-road and crash-road spatial relationships. The spatial weight matrix of roads (SWMR) is constructed to describe the conceptualization of road-road spatial relationships. The conceptualization of crash-road spatial relationships is established using crash spatial aggregation algorithm. The third step, spatial data mining, is to identify RCHC using the cluster and outlier analysis (local Moran's I index). This approach was validated using spatial data set including roads and road-related crashes (2008-2018) from Polk County, IOWA, U.S.A. The findings of this research show that the proposed approach is successful in identifying RCHC and road outliers.  
Keywords:data mining; road network; traffic crash; road clusters with high-frequency crashes (RCHC); spatial weight matrix (SWM); local Moran's I index; cluster identifying +
Footnote †: journal: Applied
# Mixed Reality Enhanced User Interactive Path Planning for Omnidirectional Mobile Robot
###### Abstract  
This paper proposes a novel control system for the path planning of an omnidirectional mobile robot based on mixed reality. Most research on mobile robots is carried out in a completely real environment or a completely virtual environment. However, a real environment containing virtual objects has important actual applications. The proposed system can control the movement of the mobile robot in the real environment, as well as the interaction between the mobile robot's motion and virtual objects which can be added to a real environment. First, an interactive interface is presented in the mixed reality device HoloLens. The interface can display the map, path, control command, and other information related to the mobile robot, and it can add virtual objects to the real map to realize a real-time interaction between the mobile robot and the virtual objects. Then, the original path planning algorithm, vector field histogram* (VFH*), is modified in the aspects of the threshold, candidate direction selection, and cost function, to make it more suitable for the scene with virtual objects, reduce the number of calculations required, and improve the security. Experimental results demonstrated that this proposed method can generate the motion path of the mobile robot according to the specific requirements of the operator, and achieve a good obstacle avoidance performance.  
Keywords:path planning; mixed reality; omnidirectional mobile robot; VFH* +
Footnote †: journal: Applied Sciences
# Integration of BIM and GIS for the Digitization of the Built Environment
## 1 Introduction
The development of GIS and BIM platforms has been driven by a variety of factors, but with the evolution of the AECO sector and the emergence of smart cities, the integration of BIM in the geospatial context has become increasingly strategic. Despite the evident advantages, this paradigm is confronted with a number of challenges, including the management of spatial scale, the definition of data layers, the representation of geometry, and the storage and access of data. Another challenge is the semantic discrepancy between BIM and GIS. Based on the available literature, it reveals that one of the principal challenges is the adaptation of standards, given that the structure and information in the two schemes differ considerably [31].
# Structure-Property Relation of Trimethyl Ammonium Ionic Liquids for Battery Applications
## 2 Materials and Methods
### Viscosity  
The temperature dependent dynamic viscosities were determined using a MCR 301 rheometer ([PERSON], Graz, Austria) with a cone-plate setup on a vibration-isolated table. The cone used was a CP50-1 with 49.95 mm diameter and a cone angle of 1\({}^{\circ}\). The gap size between the cone tip and plate was 0.101 mm. The rheometer was calibrated using a general viscosity standard with nominal viscosity of 129 mPa s (Paragon scientific, Prenton, UK). All measurements were conducted under flow of dry nitrogen to avoid the uptake of moisture. For the investigation of the \(T\)-dependence, viscosity values from 25 to 100 \({}^{\circ}\)C were measured in 5 \({}^{\circ}\)C steps. Prior to each measurement, the temperature was allowed to equilibrate for at least 20 min with deviation less than \(\pm 0.01\)\({}^{\circ}\)C during the measurements. For each temperature, the shear-dependent viscosity was recorded for shear-rate values ranging from 5 s\({}^{-1}\) to 80 s\({}^{-1}\).
## 5 Performance Evaluation
### Synthetic Data
#### 5.2.1 Node Contribution and Active Node Ratio (ANR)  
We conduct simulations for two scenarios: Scenario 1 with 500 nodes and Scenario 2 with 1000 nodes deployed over an area of size \(800\times 800\)\(m^{2}\). Figure 9 shows how the structure fidelity change over ANR and the corresponding statistic is given in Table 2. As we can see, the results that the fidelity monotonically increases with ANR are similar to those in Figure 6. However, it is obvious that there are differences between the two curves in Figure 9. For Scenario 1 the initial 183 active nodes accounting for \(36.6\%\) of 500 nodes contribute \(90.03\%\) of the whole fidelity, while for Scenario 2 the initial 174 active nodes only accounting for \(17.4\%\) of 1000 nodes also provide \(90.01\%\) of the whole fidelity. On the other hand, we can see from the last column in Table 2 that \(35\%\) nodes for Scenario 2 and \(18\%\) nodes for Scenario 1 do not really need to work when \(1\%\) distortion is permitted. Because of the higher node redundancy, we conclude that the large scale dense WSNs have an advantage in the sleeping node ratio (SNR) by using our data collection approach.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{500 nodes} & Number of active nodes & 1–183 & 184–300 & 301–410 & 411–500 \\  & Nodes contribution rate & \(90.03\%\) & \(7.09\%\) & \(2.35\%\) & \(0.53\%\) \\  & ANR & \(36.6\%\) & \(23.4\%\) & \(22\%\) & \(18\%\) \\ \hline \multirow{3}{*}{1000 nodes} & Number of active nodes & 1–174 & 175–358 & 359–650 & 651–1000 \\  & Nodes contribution rate & \(90.01\%\) & \(6.05\%\) & \(3.02\%\) & \(0.92\%\) \\ \cline{1-1}  & ANR & \(17.4\%\) & \(18.4\%\) & \(29.2\%\) & \(35\%\) \\ \hline \end{tabular}
\end{table}
Table 2: Nodes contribution rate with different active node rate.  
Figure 9: Structure fidelity vs. ANR.
## 6 Sensitivity Analysis of Kinematic Reliability
### Reliability Sensitivity Analysis  
The sensitivity of the kinematic reliability of the mechanism in the combined direction to the mean matrix \(\mu\) and variance matrix \(D\) of each input error random variable are  
\[\frac{\partial R}{\partial\mu^{T}}=\frac{\partial R}{\partial\beta}\left( \frac{\partial\beta}{\partial\mu_{g}}\frac{\partial\mu_{g}}{\partial\mu^{T}} +\frac{\partial\beta}{\partial D_{g}}\frac{\partial D_{g}}{\partial\mu^{T}}\right) \tag{35}\]  
\[\frac{\partial R}{\partial D^{T}}=\frac{\partial R}{\partial\beta}\left( \frac{\partial\beta}{\partial\mu_{g}}\frac{\partial\mu_{g}}{\partial D^{T}} +\frac{\partial\beta}{\partial D_{g}}\frac{\partial D_{g}}{\partial D^{T}}\right) \tag{36}\]where:  
\[\left\{\begin{array}{l}\frac{\partial R}{\partial\beta}=\varphi(\beta) \text{: }\frac{\partial\beta}{\partial\mu_{g}}=\frac{1}{\sqrt{D_{g}}}\\ \\ \frac{\partial\beta}{\partial D_{g}}=-\frac{u_{g}}{2}D_{g}^{-}\frac{3}{2}\\ \\ \frac{\partial\mu_{g}}{\partial\mu^{T}}=\left[\frac{\partial\mu_{g}}{\partial \mu_{1}}\text{, }\frac{\partial\mu_{g}}{\partial\mu_{2}},\cdots,\text{ }\frac{\partial\mu_{g}}{\partial\mu_{NR}} \right]^{T}\\ \\ \frac{\partial\mu_{g}}{\partial D^{T}}=\left[\frac{\partial\mu_{g}}{\partial D_ {1}}\text{, }\frac{\partial\mu_{g}}{\partial D_{2}},\cdots,\text{ }\frac{\partial\mu_{g}}{\partial D_{NR}} \right]^{T}\\ \\ \frac{\partial D_{g}}{\partial\mu^{T}}=\left[\frac{\partial D_{g}}{\partial \mu_{1}}\text{, }\frac{\partial D_{g}}{\partial\mu_{2}},\cdots,\text{ }\frac{\partial D_{g}}{\partial\mu_{NR}} \right]^{T}\\ \\ \frac{\partial D_{g}}{\partial D^{T}}=\left[\frac{\partial D_{g}}{\partial D_ {1}}\text{, }\frac{\partial D_{g}}{\partial D_{2}},\cdots,\text{ }\frac{\partial D_{g}}{\partial D_{NR}} \right]^{T}\end{array}\right. \tag{37}\]  
Recall that: in this paper, the response surface function as shown in Equation (1) is used to approximate the real response of the system. Although the response surface function includes the information of the first, second and cross terms described in Equation (1) of the input variables, sensitivity analysis uses the linear realization of input variables essentially to dilute the effects of nonlinearity and coupling errors caused by parallel mechanisms. Therefore, sensitivity analysis on reliability in this paper is mainly based on the above assumptions for theoretical research.
## 3 Assembly of Substructure with Non-Matching Interfaces
### Assembly of Impulse-Based Substructures with Non-Matching Interfaces
\begin{bmatrix}\mathbf{G}_{u}^{(s)}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{b}^{(s)}\\ \mathbf{u}_{i}^{(s)}\end{bmatrix} =\overline{\mathbf{G}}_{u}^{(s)}\mathbf{u}^{(s)} \tag{18}\]  
where \(\mathbf{I}\) is the identity matrix and \(\overline{\mathbf{G}}_{u}^{(s)}\) is defined as  
\[\overline{\mathbf{G}}_{u}^{(s)}= \begin{bmatrix}\mathbf{G}_{u}^{(s)}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}\end{bmatrix}  \tag{19}\]  
Substituting Equations (18) and (17) into Equation (15), the non-matching interface system-governing equation is rewritten as  
\[\left\{ \begin{array}{l}\mathbf{u}_{n}^{(s)}=\sum\limits_{i=0}^{n-1}\mathbf{H}_{n-i}^ {(s)}\Big{[}\hat{f}_{i}^{(s)}+\mathbf{f}_{i+1}^{(s)}+\mathbf{G}_{f}^{(s)}\mathbf{B}^{(s) \text{T}}\mathbf{\lambda}_{i}+\mathbf{G}_{f}^{(s)}\mathbf{B}^{(s)\text{T}}\mathbf{\lambda}_{i +1}\Big{]}\frac{\text{A}t}{2}\\ \sum\limits_{s=1}^{N_{s}}\mathbf{B}^{(s)}\overline{\mathbf{G}}_{u}^{(s)}\mathbf{u}_{n}^{(s )}=\mathbf{0}\end{array} \right. \tag{20}\]  
Similar to ordinary IBS, by solving Equation (20), \(\mathbf{\lambda}_{n}\) at time \(t_{n}\) can be found as  
\[\mathbf{\lambda}_{n}=-\left(\sum\limits_{s=1}^{N_{s}}\mathbf{B}^{(s)}\overline{\mathbf{G} }_{u}^{(s)}\mathbf{H}_{1}^{(s)}\mathbf{G}_{f}^{(s)}\mathbf{B}^{(s)^{T}}\right)\sum\limits_ {s=1}^{N_{s}}\mathbf{B}^{(s)}\overline{\mathbf{G}}_{u}^{(s)}\hat{\mathbf{u}}_{n-1}^{(s)} \tag{21}\]  
where  
\[ \begin{array}{l}\hat{\mathbf{u}}_{n-1}^{(s)}=\sum\limits_{i=0}^{n-2}\mathbf{H}_{n-1 }^{(s)}\Big{[}\hat{f}_{i}^{(s)}+\hat{f}_{i+1}^{(s)}+\mathbf{G}_{f}^{(s)}\mathbf{B}^{(s )^{T}}(\mathbf{\lambda}_{i}+\mathbf{\lambda}_{i+1})\Big{]}\\ +\mathbf{H}_{1}^{(s)}\Big{(}\hat{f}_{n-1}^{(s)}+\hat{f}_{n}^{(s)}+\mathbf{G}_{f}^{(s)} \mathbf{B}^{(s)^{T}}\mathbf{\lambda}_{n-1}\Big{)}\end{array}  \tag{22}\]  
Substituting \(\mathbf{\lambda}_{n}\) into Equation (20) can result in \(\mathbf{u}_{n}^{(s)}\). Then, \(\mathbf{u}_{n}^{\prime}\) and \(\mathbf{u}_{n}^{\prime\prime}\) can be obtained through the Newmark-\(\beta\) scheme  
\[\left\{ \begin{array}{l}\mathbf{u}_{n}^{\prime}{}^{(s)}=\frac{\gamma}{\beta d}( \mathbf{u}_{n}-\mathbf{u}_{n-1})+\left(1-\frac{\gamma}{\beta}\right)\mathbf{u}_{n-1}^{ \prime}+\left(1-\frac{\gamma}{2\beta}\right)d\mathbf{u}_{n-1}^{\prime\prime}\\ \mathbf{u}_{n}^{\prime\prime}{}^{(s)}=\frac{1}{\beta d^{2}}(\mathbf{u}_{n}-\mathbf{u}_{n-1 })-\frac{1}{\beta d}\mathbf{u}_{n-1}^{\prime}-\left(\frac{1}{2\beta}-1\right)\mathbf{u }_{n-1}^{\prime\prime}\end{array} \right. \tag{23}\]  
\(\gamma=0.5\) and \(\beta=0.25\) for default. We call this substructure method non-matching interfaces IBS (NMI-IBS for short).
## 4 Massive MIMO System Model
### OFDM-Massive MIMO  
Combining the OFDM into massive MIMO as shown in Figure 2. Each user transmits their data into an IFFT-N points block in the uplink, which inserts a cyclic prefix (CP) to the data block before sending it through the channel. CP is a component of the block's end that is added to the block's beginning. The received signal of each BS antenna is sent via an FFT-N points block at the BS after the CP is removed, and the resulting signals are given to the detection methods such as zero forcing or minim square error MMSE. Figure 3 represents  
Figure 1: An uplink massive MIMO system.  
the flowchart for the proposed OFDM Massive MIMO system to achieve a desired PSNR and SSIM values. The transmitted signal data from each user is represented as N point of IFFT [56]  
\[\mathbf{x}_{k,i}=\frac{1}{N}\sum_{k=1}^{K}\sum_{n=0}^{N-1}\mathbf{x}_{k,n}e^{ \frac{\left|2\pi m\right|}{N}}, \tag{12}\]  
where \(\mathbf{x}_{k,i}\) is the \(i\)-th component of the vector transmitted from the \(k\)-th antenna. After adding CP the frequency domain representation of the received signal can be expressed as:  
\[\begin{split}\mathbf{\hat{x}}_{m,j}&=\frac{1}{N} \sum_{m=0}^{M-1}\sum_{k=1}^{K}\sum_{n=0}^{N-1}\mathbf{Y}_{k,m,n}e^{\frac{\left| 2\pi m\right|}{N}}\\ &=\frac{1}{N}\sum_{q=0}^{N-1}\sum_{m=0}^{M-1}\sum_{k=1}^{K}\sum_ {n=0}^{N-1}\mathbf{H}_{k,m,n}\mathbf{x}_{m,q}e^{\frac{\left|2\pi m(i-q)\right|} {N}}+\eta_{m,j},\end{split} \tag{13}\]  
where \(\eta_{m,i}\) is the \(i\)-th element of the zero-mean complex white Gaussian noise with variance added to the \(m\)-th antenna of BS. Implementing OFDM-DWT with massive MIMO system can be addressed. The signal transmission from each user \(K\) is defined as:  
\[\mathbf{x}_{k,n}=\sum_{k=1}^{K}\sum_{d=0}^{\infty}\sum_{j=-\infty}^{\infty} \mathbf{x}_{k,j,d}\Psi(2^{-j}n-d). \tag{14}\]  
After the signal passes through the channel and receiver, the resulting receiver signal can be represented as:  
\[\mathbf{\hat{X}}_{m,j,d}=\sum_{k=1}^{K}\sum_{d=0}^{\infty}\sum_{j=-\infty}^{ \infty}\mathbf{x}_{k,m,j,d}\Psi(2^{-j}n-d)\mathbf{H}_{k,m,j,d}+\eta_{m,j,d}. \tag{15}\]  
On replacing FFT with FrFT, the receiver signal can be mathematically written as:  
\[\mathbf{\hat{X}}_{m,j,d}=\sum_{k=1}^{K}\sum_{d=0}^{\infty}\sum_{j=-\infty}^{ \infty}\mathbf{x}_{k,m,j,d}F_{a}(m,n)\mathbf{H}_{k,m,j,d}+\eta_{m,j,d}. \tag{16}\]  
All of the above equations represent the OFDM with different design methods such as FFT, DWT and Frft, with different parameters.  
Figure 2: Block diagram of OFDM Massive MIMO systems in the uplink.
## 6 NR Tracking Performance Analysis for Different NR and Receiver Configuration  
This section evaluates the impact of different NR signal and receiver configurations on tracking accuracy. The analysis results offer systematic insights for LEO OFDM tracking loop.