esa-sceva/satcom-chunk-collection huggingface dataset:
###### Abstract  
To control the large deformation that occurs in deep shaft-surrounding rock, the post-peak strain-softening characteristics of deep jointed rock mass are discussed in detail. An equivalent post-peak strain-softening model of jointed rock mass is established based on continuum theory and the geological strength index surrounding rock grading system, and numerical simulations are performed using FLAC\({}^{\text{3D}}\) software. The convergence-constraint method is used to analyze the rock support structure interaction mechanism. A composite support technique is proposed in combination with actual field breakage conditions. During the initial support stage, high-strength anchors are used to release the rock stress, and high-stiffness secondary support is provided by well rings and poured concrete. This support technology is applied in the accessory well of a coal mine in Niaoshan, Heilongjiang, China. The stability of the surrounding rock support structure is calculated and analyzed by comparing the ideal elastic-plastic model and equivalent jointed rock mass strain-softening model. The results show that a support structure designed based on the ideal elastic-plastic model cannot meet the stability requirements of the surrounding rock and that radial deformation of the surrounding rock reaches 300 mm. The support structure designed based on the equivalent join strain-softening model has a convergence rate of surrounding rock deformation of less than 1 mm/d after 35 days of application. The surrounding rock deformation is finally controlled at 140 mm, indicating successful application of the support technology.  
shaft support technology; deep weak plutons; strain-softening model; convergence-constraint method; numerical simulation +
Footnote †: journal: Applied sciences  
Article Investigation of Deep Shaft-Surrounding Rock Support Technology Based on a Post-Peak Strain-Softening Model of Rock Mass
# A Secure and Efficient Three-Factor Authentication Protocol in Global Mobility Networks
## 4 Cryptanalysis of [PERSON] et al.'s Protocol
### Masquerade Attack  
If a malicious adversary \(MU_{a}\) can attempt to impersonate a legitimate user, \(MU_{a}\) can easily generate the message \(M_{1}=\{U,V,W\}\) of the legitimate user. As discussed in Section 2.1, \(MU_{a}\) obtains the mobile device of \(MU\) and extracts the stored secret parameters in it. In addition, \(MU_{a}\) intercepts the message exchanged over a public channel. Finally, \(MU_{a}\) performs the masquerade attack and its detailed procedures.  
**Step 1:**: A \(MU_{a}\) calculates \(R_{MU}=U\oplus R\), \(V=(C_{MU}\oplus h(R)||ID_{FA})\oplus R_{MU}\) and \(W=(U||K||C_{MU}\oplus h(R))\). Then, \(MU_{a}\) generates a random number \(R_{a}\). After that, \(MU_{a}\) computes \(U_{a}=R\oplus R_{a}\), \(V_{a}=(C_{MU}\oplus h(R)||ID_{FA})\oplus R_{a}\) and \(W_{a}=(U_{a}||K||C_{MU}\oplus h(R))\) and sends \(M_{1a}=\{U_{a},V_{a},W_{a}\}\) to the \(FA\).
**Step 2:**: After obtaining the \(M_{1a}=\{U_{a},V_{a},W_{a}\}\), the \(FA\) selects a random number \(R_{FA}\) and encrypts \(E_{KFH}(M_{1},R_{FA})\) using a shared secret key. Then, the \(FA\) sends \(M_{2}=\{ID_{FA},E_{KFH}(M_{1a},R_{FA})\}\) to the \(HA\).
**Step 3:**: Upon reception of \(M_{2}=\{ID_{FA},E_{KFH}(M_{1a},R_{FA})\}\), the \(HA\) decrypts \(D_{KFH}(E_{KFH}(M_{1},R_{FA}))\) and computes \(a=h(a)\), \(g^{a}\) mod \(p\), \(R_{a}^{*}=V_{a}\oplus\) ((\(g^{a}\) mod \(p||ID_{FA}\)) and \(R^{*}=U_{a}\oplus R_{a}^{*}\). Then, the \(HA\) checks whether \(R^{*}\stackrel{{?}}{{=}}R\). After that, \(HA\) computes \(W_{a}^{*}\)(\(U_{a}||K||\) (\(g^{a}\) mod \(p\))) and checks whether \(W^{*}\stackrel{{?}}{{=}}W\). Finally, \(HA\) computes \(SK=h(g^{a}\) mod \(p)\)\(\oplus R_{a}\oplus R_{FA}\) and sends \(M_{3}=\{E_{KFH}(SK)\}\) to the \(FA\).
**Step 4:**: After obtaining the \(M_{3}=\{E_{KFH}(SK)\}\), the \(FA\) decrypts \(D_{KFH}(E_{KFH}(SK))\) and computes \(X=h(SK||R_{FA})\), then sends \(M_{4}=\{X,R_{FA}\}\) to the \(MU_{a}\).
**Step 5:**: Upon reception of \(M_{4}=\{X,R_{FA}\}\), the \(MU_{a}\) computes the \(SK^{*}=C_{MU}\oplus h(R)\oplus R_{a}\oplus R_{FA}\), \(X^{*}=h(SK^{*}||R_{FA})\) and checks whether \(X^{*}\stackrel{{?}}{{=}}X\). If it is correct, \(MU_{a}\) computes the \(SK\)  
\(MU_{a}\) obtains the session key between \(MU_{a}\) and \(FA\) and performs mutual authentication successfully. As a result, [PERSON] et al.'s protocol [16] is insecure against the masquerade attacks.
# Secure and Transparent Space Exploration Data Management Using a Hybrid Blockchain Model
## 4 Implementation and Experimental Analysis
### Performance Evaluation and Discussion
There is, of course, room to improve the efficiency of ZKP generation and to possibly parallelize verification for multiple data points. As ZKP technology matures, we expect this overhead to diminish further, making the case for using ZKPs in such systems even stronger. Overall, our evaluation shows that the hybrid blockchain approach is practical for enhancing data security and integrity in collaborative space missions, given the typical scale of data and required trust level. It does require some compromises in performance and added complexity in setup, but the benefits in terms of reliability, trustworthiness, and tamper resistance are significant for mission-critical contexts. Figure 4 compares the average synchronization delay between the Oracle-based approach and the proposed CCIP-based method, clearly highlighting the superior real-time synchronization performance of our proposed approach. Figure 4: Data synchronization delay comparison between Oracle-based and CCIP-based integration methods. The proposed CCIP-based approach significantly reduces average delay from 22.5 s to 1.5 s, enabling near real-time synchronization across blockchains.
## 2 Experimental Section
### Double Raschel GCCM Characteristics Evaluation  
In order to analyze the impact resistance of double raschel GCCMs, a free fall test was conducted. For the free fall test, a weight of 5.98 kg was dropped from 0.25 m, 0.5 m, and 1.0 m to observe whether or not it was destroyed. In order to quantitatively analyze the impact energy delivered to each sample, the impact performance of the concrete before and after double raschel reinforcement was analyzed using an impact strength tester. The impact strength tester used was the IT504 from Tinius Olsen, and was measured according to the test standards of ASTM D256 (Izod). The evaluation of the double raschel GCCM's characteristics involved measuring its bending strength. Following the curing of the double raschel GCCM in accordance with ASTM C 1185, specimens, each with dimensions of 127 mm in width and 300 mm in length, were placed on supports spaced at 254 mm intervals. The bending test was conducted by applying pressure to the center of the specimen. The bending strengths of the double raschel GCCM were measured using Zwick's Zoo5 Universal Testing Machine (Zwick Roell GmbH & Co. KG, Ulm, Germany). For the concrete's physical properties, the standard curing period is typically 28 days. In this study, bending strength was evaluated at different curing periods (10 days, 30 days) using fast-setting cement. To assess the durability of double raschel GCCCMs under environmental conditions, bending strength was evaluated in seawater and freezing and thawing conditions, as well as in acidic and alkaline solutions. After immersing the double raschel GCCM in seawater for 30 days, the bending strength was measured according to the ASTM C 1185 test standard. Considering the susceptibility of concrete to shrinkage cracks upon hardening, particularly under the cyclic conditions of freezing and thawing, a durability analysis was conducted under the assumption of seasonal changes from winter to spring. Freeze-thaw conditions were replicated for 30 days at alternating temperatures of 15 \({}^{\circ}\)C and room temperature, resembling the conditions of a domestic environment. The bending strength was assessed in accordance with the ASTM C 1185 test standard. The double raschel GCCM, cured for 10 days, was immersed in aqueous solutions of both pH 4 (acidic) and pH 12 (alkaline). Following treatment at 50 \({}^{\circ}\)C for 56 days, the bending strength was measured in accordance with the ASTM C 1185 test standard.
## 3 Proposed System Model
### Computation Offloading Model at the Edge Server  
The computation time consumption of task \(i\) on the edge server can be expressed as (7), where \(t^{edge}\) is the computation time consumption of task \(i\) in the edge server, \(C_{i}\) is the required resource for accomplished task \(i\), and \(V_{i}^{r}\) is the computation resource provided by the MEC server to accomplish task \(i\). \(V_{i}^{r}\) is measured in Gigahertz (GHz). According to [41], it is generally thought that the download data rate is relatively higher and the size of the computational result is typically smaller than input data, so the downlink delay and energy utilization of this stage are ignored.  
\[cost^{off}=\ t^{off}+e^{off} \tag{6}\]  
\[t^{edge}=\ \frac{C_{i}}{V_{i}^{r}} \tag{7}\]  
Energy computation consumption at the edge server can be expressed as (8), where \(e^{edge}\) denotes the energy amount for compute task \(i\) at the edge server, \(P_{j}^{r}=z^{r}\left(V_{i}^{r}\right)^{k^{r}}\) is the computation power consumption [42; 43] with effective switched capacitance \(z^{r}\geq 0\) and positive constant \(k^{r}\geq 1\)).  
\[e^{edge}=\ \frac{P_{j}^{r}C_{i}}{V_{i}^{r}} \tag{8}\]  
The computation cost for computing task \(i\) at the edge server can be denoted as (9) and the total computation cost for offloading task \(i\) at edge server as (10), where \(cost^{off\_total}\) is the total cost for offloading the task from the user device to the edge server without considering energy consumption at the edge server due to the fact that the edge server always connects to grid power.  
\[cost^{edge}=\ t^{edge}+e^{edge} \tag{9}\]  
\[cost^{off\_total}=\ cost^{off}+\ t^{edge} \tag{10}\]
## 1 Introduction
We observed acceptable results regarding the group delay and system fidelity factor. The structure of this paper is as follow: in Section 2, the design and configuration of the antenna is presented: Section 3 consists of a parametric study of the DGS slot; in Section 4, and antenna simulation and measurement results are discussed; Section 5 describes the time-domain performance of the antenna; Section 6 presents a comparison of the results of the proposed work with previous works; finally, Section 7 presents the conclusion.
## 4 Deep RL-Assisted Resource Allocation
### Proposed DQL-Based Approach
#### 4.2.3 Reward Function
```
1:Initialize the current environment \(E\) (all available CSI)
2:\(B_{\mathcal{U}_{i}}\gets B\%\) Initialize usable sub-bands for all users
3:Initialize replay memory \(D\) to capacity \(M\)
4. Initialize the state space \(S\) and action space \(A\)
5:Initialize train DQN \(Q\) and target DQN \(\hat{Q}\) with weights \(\theta\) & \(\theta^{-}\), respectively. 6:\(\{BS_{1},BS_{2},BS_{1},\ldots,BS_{k}\}\leftarrow BS\), where \(BS\in B_{\mathcal{U}_{i}}\)
7:for each episode do
8: Allocate a free RB with higher SIR for \(\mathcal{U}_{i}\) in each cell. 9:for\(t=1,\ldots,\infty\)do
10:\(SIN_{i}\) % calculate the \(SINR\) for \(\mathcal{U}_{i}\) in the network
11: Use the throughput of each user as state space \(S_{i}{}^{(t)}\)
12:for\(i=1,\ldots,K\)do
13: Select a random action \(a_{k}^{(t)}\) from \(A\) with probability \(\epsilon\) for user \(k\). 14: Otherwise choose \(a_{k}=\operatorname*{argmax}_{a}\!Q^{k}\left(s_{i}{}^{(t)},a_{i}{}^{(t)}\right)\)
15:endfor
16: Execute action \(a_{k}=[a_{1},a_{2},\ldots,a_{k}]\) and observe reward \(r_{t}\%\) calculate \(w_{i}^{t}R_{i}{}^{(t)}\) the throughput for entire system. 17: State transition happens \(s_{t+1}\) & move from \(s_{t}\) to \(s_{t+1}\)
18: Store all transition tuples \(\left(s^{(t)},a^{(t)},R^{(t+1)},s^{(t+1)}\right)\) in \(D\). 19: Sample mini-batch from \(D\) and train it at each time \(t\), \(D^{(t)}\). 20: Optimize the loss function between learning target \(\hat{Q}\) and train \(Q\)-function using tools of stochastic gradient descent algorithm, (15). 21: Update the target DQN \(\hat{Q}\) to be equal to train DQN \(Q\). 22:endfor
23:endfor
```  
**Algorithm 1** DQL-Based Resource Allocation  
After the transition of the environment due to the change of CSI and the actions taken, the agent stores all the transition tuple \(\left(s^{(t)},a^{(t)},R^{(t+1)},s^{(t+1)}\right)\) in a replay memory. At each step, a mini-batch of accumulated experience is sampled randomly from memory. The mini batch is adopted to train the DQN and a stochastic gradient-descent step is used to update the action-value DQN parameters. The objective is to minimize the sum-squared error based on (15) at each time step \(t\). The target DQN function \(\hat{Q}\) is initialized by duplicating the parameters of the training DQN function \(Q\) and after a certain amount of time the two DQNs are updated to clone the parameters of the training DQN. The process of training the DQN is repeated until the parameters converge.
## 4 Performance Evaluation
### Simulation Configuration  
In the simulation, the VDES network consists of a shore station and multiple ship stations located in a coastal area. To evaluate the performance of SOCA under various maritime environments, we consider static and dynamic ship scenarios, as illustrated in Figure 4.  
In both scenarios, all ship stations are assumed to be randomly positioned within a \(25\times 50\)-nautical-mile (NM) area. The number of ship stations in the simulation varies from 5 to 50, and the communication range for both the shore station and ship stations is set to 50 NM. Additionally, it is assumed that each ship station generates \(n_{pkt}\) VDE-TER data packets of uniform size at the beginning of every VDES frame and transmits them the shore station via VDE-TER.  
In the static ship scenario, it is assumed that all ship stations maintain the same sailing speed, resulting in identical values for RI, Rr, NI, and SI across all ship stations. The RI, which determines the transmission period of AIS message, is set to 2 or 6, depending on the speed of ship stations, as shown in Table 1[19]. The simulations are conducted separately for two cases: RI set to 2 with an assumed ship speed of 30 knots, and RI set to 6 with an assumed ship speed of 20 knots. In the dynamic ship scenario, it is assumed that the speed of each ship station varies depending on its distance from the shore station: 10 knots within 15 NM, 20 knots between 15 and 35 NM, and 30 knots beyond 35 NM. Consequently, the parameters \(RI\), \(Rr\), \(NI\), and \(SI\) vary across stations. Note that the values of \(Rr\), \(NI\), and \(SI\) are updated based on the changes in \(RI\) and are calculated by  
\[Rr=60/RI, \tag{10}\]  
\[NI=2250/Rr, \tag{11}\]  
\[SI=0.2\times NI. \tag{12}\]  
Figure 4: Example of ship scenarios: (**a**) static ship scenario and (**b**) dynamic ship scenario.  
The performance of SOCA is compared to that of the existing VDES slot allocation scheme in terms of aggregate throughput, average latency, collision ratio, and packet delivery ratio. The simulation was repeated 100 times, and the detailed simulation parameters are listed in Table 2.
## 3 Data Sets and Methods
### Data Set Description  
There are InSAR images over the target area, including three InSAR pairs L band ALOS PALSAR-2, a few X band Terra X SAR, and COSMO Skymed only around the Ulsan city area. L band PALSAR-2 is more coherent over all landcover types and X band SARs provide optimized resolution to observe the central city area. However, none of them, except Sentinel-1 SAR, have suitable temporal coverages to investigate two separate earthquakes with a time series analysis approach. The optimal data set for this study was Sentinel-1 SAR imagery, which is freely available on the Sentinel data hub. It is acquired by two SAR-satellite constellations that are composed of Sentinel 1A, which was launched in April 2015, and Sentinel 1B, which was launched in April 2016. The characteristics of C band wavelength and Interferometric Wide-swath mode (IW) operation with Terrain Observation and Progressive Scans SAR (TOPSAR) imaging allow for more precise monitoring of the target area with relatively minor ionospheric errors and wide coverage (Swath \(>\) 250 km), together with moderate spatial resolution (20 m in azimuth and a 4 m range) [42]. The biggest merit of the Sentinel-1 constellation is its short revisiting time (14 days); thus, the image graph for time series analysis consisting of 60 ascending modes and 86 descending modes can be readily composed for a three-year time period as shown in Figure 2. The other image characteristics of Sentinel-1 imagery are summarized in Table 1. InSAR observations covering the temporal coverage to monitor the long-term deformation trends of pre/co/postseismic phases with sufficient spatial resolution and extent was found to be feasible throughout multiple Sentinel-1 image sets.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Ascending Mode** & **Descending Mode** \\ \hline Image number & 60 & 86 \\ \hline Master image & 2017.02.19. & 2017.01.08. \\ \hline Time coverage & 2015.05.25–2018.06.08. & 2014.11.02–2018.06.26. \\ \hline Heading angle (deg) & \(-\)13.0523 & \(-\)166.969 \\ \hline Incidence angle (deg) & 39.189 & 39.213 \\ \hline Relative Orbit & 54 & 61 \\ \hline Acquisition time & 18:22 KST (GMT+9) & 06:24 KST \\ \hline \hline \end{tabular}
\end{table}
Table 1: Characteristics of employed Sentinel-1 images.
# A New Methodology for Estimating Surface Albedo in Heterogeneous Areas from Satellite Imagery
## 4 Methods
### Atmospheric Correction in Planetary Albedo
#### 4.1.2 V02 Method  
The V02 methodology uses a completely different atmospheric transmission (\(\tau_{SW}\)) equation (Equation (10)). It is calculated assuming clear sky and relatively dry conditions using a relationship based on the elevation [54]:  
\[\tau_{SW}=0.75+2\times 10^{-5}\times Z_{est}\, \tag{10}\]  
being  
\(Z_{est}\): elevation above sea level from the most relevant weather station (m).
# Removal of Metals and Dyes in Water Using Low-Cost Agro-Industrial Waste Materials
## 1 Introduction
It is worth noting that the biomaterials obtained 99.3% and 96.8% metal removal, respectively, the former being the most efficient. Likewise, [PERSON] and [PERSON] [14] evaluated the biomass of pineapple peels in slow pyrolysis processes at different temperatures and its application in removing Cr (VI). The carbon prepared at a temperature of 350 \({}^{\circ}\)C obtained the best adsorption capacity of 41.67 mg/g, in comparison with the others, and the chemical adsorption with covalent interactions between the metal ions and the functional groups of the surface of the biomaterial, as the most relevant removal mechanism in the process. On the other hand, [PERSON] et al. [15] carried out studies on the adsorption capacity of sugarcane bagasse treated with potassium hydroxide (KOH) to remove MB. The study of the process was carried out as a batch, analyzing variables such as contact time, pH, adsorbent dose, and the initial concentration of the solution, the presence of a large number of characteristic functional groups of this type of lignocellulosic materials, such as the -OH, C-H group, and the C=O carbonyl group. In addition, the spectrum results obtained after adsorption established that the adsorption occurred largely thanks to these functional groups. Regarding the adsorption studies, the optimal variables of the process were found to be 0.1 g of dose, an equilibrium time of 1.5 h, a pH range between 5 and 9, and the initial concentration of methylene blue of 100 ppm. Finally, the maximum adsorption capacities obtained were 186.67 mg/g for the untreated bagasse and 195.44 mg/g for the bagasse treated with KOH. With this, the importance of implementing modifications in the lignocellulosic biomass treatment processes that will be applied to remove contaminants such as Cr (VI) and MB is denoted. Thus, the present study is dedicated to using wheat bran as raw material for the synthesis of biochar for its use in the removal of methylene blue and hexavalent chromium from aqueous solutions. The adsorbents were characterized using Fourier transform infrared spectroscopy (FT-IR), scanning electron microscopy (SEM), pH point of zero charge, and Boehm titration. The parameters of the adsorption system were optimized to attain the maximum removal efficiency of methylene blue and hexavalent chromium. The adsorption process mechanisms were examined by fitting the experimental data to isotherms and kinetic models.
# Optimized Deployment of Generalized OCDM in Deep-Sea Shadow-Zone Underwater Acoustic Channels
## 3 GOCDM Underwater Acoustic Communication System
### Properties of GOCDM
 \tag{13}\]  
Next, extract all elements from \(\mathbf{H}_{K}\) with row indices \([\mathrm{mod}(m,M),\mathrm{mod}(m,M)+M,\ldots,\allowbreak\mod(m,M)+(N-1)M]\) and column indices \([\mathrm{mod}(n,M),\mathrm{mod}(n,M)+M,\ldots,\mathrm{mod}(n,M)\allowbreak+(N- 1)M]\), and arrange them in sequence to form a new matrix \(\mathbf{\widetilde{H}}_{N}\), where the element in the \(a\)-th row and \(b\)-th column of \(\mathbf{\widetilde{H}}_{N}\) can be expressed as follows:  
\[\mathbf{\widetilde{H}}_{N}(a,b)=\mathbf{H}_{K}(\mathrm{mod}(m,M)+(a-1)M, \mathrm{mod}(n,M)+(b-1)M), \tag{14}\]  
\(\mathbf{\widetilde{H}}_{N}\) is also a circulant matrix, which allows (13) to be simplified as follows:  
\[\mathbf{\hat{H}}_{K}(m,n)=\mathbf{\Phi}_{N}\Big{(}\mathrm{ceil}\Big{(}\frac{ m}{M}\Big{)},\cdot\Big{)}\mathbf{\widetilde{H}}_{N}\mathbf{\Phi}_{N}^{ \mathrm{H}}\Big{(};\mathrm{ceil}\Big{(}\frac{n}{M}\Big{)}\Big{)}. \tag{15}\]  
According to the linear convolution properties of the Fresnel transform described in Section 2, we obtain the following:  
\[ \begin{array}{ll}\mathbf{\hat{H}}_{K}(m,n)&=\mathbf{\widetilde{H}}_{N}\Big{(} \mathrm{ceil}\Big{(}\frac{m}{M}\Big{)},\mathrm{ceil}\Big{(}\frac{n}{M}\Big{)} \Big{)}\\ &=\mathbf{H}_{K}\Bigg{(}\mod(m,M)+\Big{(}\mathrm{ceil}\Big{(}\frac{m}{M}\Big{)} -1\Big{)}M,\\ &\mod(n,M)+\Big{(}\mathrm{ceil}\Big{(}\frac{n}{M}\Big{)}-1\Big{)}M\end{array}  \Bigg{)}\\ &=\mathbf{H}_{K}(m,n).\end{array} \tag{16}\]Therefore, the GOCDM modulation and demodulation matrix through the ideal multipath channel can be expressed as follows:  
\[\hat{\mathbf{H}}_{K}=(\mathbf{\Phi}_{N}\otimes\mathbf{I}_{M})\mathbf{H}_{K} \Big{(}\mathbf{\Phi}_{N}^{\mathrm{H}}\otimes\mathbf{I}_{M}\Big{)}=\mathbf{\Psi }_{K}\mathbf{H}_{K}\mathbf{\Psi}_{K}^{\mathrm{H}}=\mathbf{H}_{K}. \tag{17}\]  
From the above expression, it can be observed that the GOCDM modulation method retains the same properties as the OCDM modulation method under circulant matrices. This indicates that, for ideal multipath channels, the proposed GOCDM modulation method inherits the multipath resistance performance of OCDM. Similarly, based on the modulation and demodulation matrices of OCDM with circulant matrices, the linear convolution properties of the GOCDM transform can be derived. Assume three matrices \(\mathbf{S}\), \(\mathbf{H}\), and \(\mathbf{\Psi}\), where \(\mathbf{S}\) and \(\mathbf{H}\) are circulant matrices. By modulating the convolution of \(\mathbf{s}\) and \(\mathbf{h}\) with \(\mathbf{\Psi}\), we obtain the following:  
\[ \begin{array}{ll}\hat{\mathbf{r}}&=\mathbf{\Psi}\mathbf{r}=\mathbf{\Psi} \mathbf{H}\mathbf{\Psi}^{\mathrm{H}}\mathbf{\Psi}\mathbf{s}=\mathbf{\Psi} \mathbf{S}\mathbf{\Psi}^{\mathrm{H}}\mathbf{\Psi}\mathbf{h}\\ &=\mathbf{H}\mathbf{\Psi}\mathbf{s}=\mathbf{H}\mathbf{s}\\ &=\mathbf{S}\mathbf{\Psi}\mathbf{h}=\mathbf{S}\hat{\mathbf{h}},\end{array}  \tag{18}\]  
Thus, the GOCDM demodulation matrix also preserves the linear convolution properties.