esa-sceva/satcom-chunk-collection huggingface dataset:
# Aging Performance and an Improved Evaluation Method for PE80 and PE100 Pipelines for Urban Gas
## 4 Aging Rate Calculation Model
### The Modified Calculation Model  
There is a relation between the activation energy, pressure, and frequency factor in the Arrhenius equation [12]. By substituting it into Equation (3), Equation (6) can be finally obtained, considering the effects of pressure and temperature.  
\[K=Bexp\left[-\frac{E_{0}}{RT}-\frac{\alpha\frac{P_{g}}{P_{f}}}{T}+\beta\frac{ P_{g}}{P_{f}}\right] \tag{6}\]  
Based on the aging test results, the constant values for the life calculation of the PE80 DN63 and PE80 DN110 pipes are shown in Table 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **PE80 DN63** & **PE80 DN110** \\ \hline \(e_{\text{n}}/\text{mm}\) & 5.8 & 10 \\ \hline \(A\) & 0.970515076 & 0.981791678 \\ \hline \(E_{0}/\text{J}\text{-mol}^{-1}\) & 67,611.62464 & 71,869.73828 \\ \hline \(B\) & 1,889,022.391 & 5,392,124.386 \\ \hline \(\alpha\) & 46.2374 & 107.85093 \\ \hline \(\beta\) & 0.22897 & 0.38765 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Wall thickness and calculation constants of PE80 DN110 and PE80 DN63.  
Figure 9: Arrhenius plot of ln\(K\) and \(1/T\) of PE80 DN110.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Pressure/MPa**} & \multicolumn{3}{c}{\(K\)} \\ \cline{2-4}  & **PE80 DN63** & **PE100 DN63** & **PE80 DN110** \\ \hline
0 & 2.19655 \(\times\) 10\({}^{6}\) & 9.02178 \(\times\) 10\({}^{7}\) & 1.10495 \(\times\) 10\({}^{6}\) \\ \hline
0.2 & 2.48867 \(\times\) 10\({}^{6}\) & 9.68853 \(\times\) 10\({}^{7}\) & 1.15063 \(\times\) 10\({}^{6}\) \\ \hline
0.4 & 2.93855 \(\times\) 10\({}^{6}\) & 1.05662 \(\times\) 10\({}^{6}\) & 1.21285 \(\times\) 10\({}^{6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: \(K\) of different PE pipes at room temperature (23 \({}^{\circ}\)C).  
As the wall thickness of the pipes increases, the material constant \(A\) and activation energy \(E_{0}\) of PE80 DN110 and PE80 DN63 at standard pressure are basically unchanged. However, the values of \(B\), \(\alpha\), and \(\beta\) change significantly as the wall thickness changes. Due to the influence of the wall thickness on the reaction rate \(K\), it is necessary to consider the wall thickness while calculating the aging rate. In Equation (6), \(B\), \(\alpha\), and \(\beta\) can be expressed as functions of the wall thickness \(e\). Combined with Equation (2), an improved life prediction equation can be established as Equation (7), which includes the effects of temperature, pressure, and wall thickness. According to the calculation constants in Table 5, the functions of different wall thicknesses can be simplified. In subsequent research, more experimental data should be supplemented to improve the accuracy of the life prediction method.  
\[t=\frac{\ln\frac{AF_{0}}{F}}{B(e)\exp\left[-\frac{E_{0}}{RT}-\frac{\alpha(e) \frac{P_{g}}{F}}{T}+\beta(e)\frac{P_{g}}{T_{g}}\right]} \tag{7}\]
# Robust DOA Estimation Using Multi-Scale Fusion Network with Attention Mask
## 5 Experiments and Discussion
### Experiments
#### 5.4.1 Performance Comparison  
On the simulation dataset generated in Section 5.1, MF-AMnet and baseline models were trained and tested. We kept the same settings such as the number of units in C-Conv and the activation function. Experimental results are shown in Table 1.  
As a traditional algorithm, the Acc5\({}^{\circ}\) of DPD-MUSIC was less than 50% in the experiment, indicating that the performance of DPD-MUSIC was poor in the case of strong interference. In particular, its performance was severely affected by the interference of reverberation. Similar results were obtained by SHPM-R and SHPM-CR, with SHPM-CR performing slightly better, indicating that the CVNN models the signal more accurately. Atthe same time, using SHC as a complex-valued input resulted in a smaller input dimension, which fundamentally reduced the network parameters and saved the computational overhead present in all complex-valued methods. MS used the real and imaginary inputs of SHC, extracted features on their own branches, concatenated them together, and output them via FC. Its performance was not as good as the above two methods, probably because the independent processing of the real and imaginary parts did not make better use of the information. Both the real and imaginary parts contained unique and related azimuth and elevation components; treating them as two separate features weakens the ability to represent the complete information. As complex-valued models, the performance of CV-CNN and CVRAEN were effectively improved. The MAE values of both methods were less than 3\({}^{\circ}\), indicating that CVNN had a more accurate estimation effect than RVNN. The higher Acc5\({}^{\circ}\) of CVRAEN may be due to the presence of residual structure, which enabled more accurate gradient descent and more stable network training. SADOAnet and SDL perform similarly and slightly worse than the complex-valued approaches like CV-CNN, indicating that the design of the estimation network is important. Among them, MF-AMnet still shows the best performance, because it employs effective multi-scale processing after the sparsity representation of the signal using the attention mask, which allows for better feature extraction on the sparse signal. The MAE was reduced to 2.029, indicating that MF-AMnet had a good estimation accuracy. The Acc5\({}^{\circ}\) of MF-AMnet exceeded 90% and reached 94.07%, which was about 3% higher than the previous methods, indicating that our method had good practical significance.  
Figure 3 shows the RMSEs for all methods at different SNRs and RT60s. The RMSEs of all methods decreased with increasing SNR and increasing RT60, showing a consistent trend. Each method performs worse under the highest reverberation and the strongest noise. Among them, MF-AMnet maintained a consistently low RMSE all the time, with an average RMSE of less than 3\({}^{\circ}\). This shows that the performance of MF-AMnet was very stable and had strong robustness and generalization even in the presence of interference.
# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 3 Distribution Diffusion Multi-Distribution Filtering of IMM
### DDMD Filtering of IMM  
To achieve accurate tracking of maneuvering targets, an initial requirement is that the kinematic model accurately represents the target's true motion. In this paper, the tracking of maneuvering targets is a central issue, driving the investigation of an appropriate and readily calculable method that enables prompt transitions between possible dynamic models when tracking targets undergo rapid changes. Multi-model estimation using hybrid systems provides a powerful adaptive approach for estimating the state of systems exhibiting changing structure or parameters. The core concept involves performing model selection at each time instant to represent the system's current operating mode, followed by fusing the filtering estimates generated by each model. To derive an optimal state estimate for Markov jump linear system with distribution nodes, the conditional diffusion average PDFs of \(\mathcal{M}=\{r_{k}|r_{k}=12\cdot\cdot\cdot m\}\) is given by  
\[\bar{p}(x_{k}|Z_{k})=\sum_{i=1}^{m}\bar{p}(x_{k}|\mathcal{M}_{k}^{i},Z_{k})\Pr \{\mathcal{M}_{k}^{i}|Z_{k}\} \tag{15}\]  
where \(\mathcal{M}_{k}^{i}\) means the i possible model at instant k. For a specific system, e.g., system (2), the total probability theorem is usually used to yield m filters utilized different model in \(\mathcal{M}\) running in parallel. In this paper, it can be described as follows:  
\[\bar{p}(x_{k}|Z_{k})=\sum_{i=1}^{m}\bar{p}(x_{k}|r_{k}=i,Z_{k})\Pr\{r_{k}=i|Z_ {k}\} \tag{16}\]  
where \(\Pr\{r_{k}=i|Z_{k}\}\) means the probability of the model i at instant k, \(p(x_{k}|r_{k}=i,Z_{k})\) means the model-conditioned posterior PDFs which can be derived via the Bayes rule as follows:  
\[\bar{p}(x_{k}|r_{k}=i,Z_{k})=\frac{\bar{p}(Z_{k}|r_{k}=i,x_{k})\bar{p}(x_{k}| r_{k}=i,Z_{k-1})}{\bar{p}(Z_{k}|r_{k}=i,Z_{k-1})} \tag{17}\]  
where  
\[p(x_{k}|r_{k}=i,Z_{k-1})\]  
means the prior probability density function which can be derived via the total probability theorem \[\bar{p}(x_{k}|r_{k}=i,Z_{k-1})\] \[=\sum_{j=1}^{m}\bar{p}(x_{k}|r_{k}=i,r_{k-1}=j,Z_{k-1}) \tag{18}\] \[\Pr\{r_{k-1}=j|r_{k}=i,Z_{k-1}\}\]  
where  
\[\bar{p}(x_{k}|r_{k}=i,r_{k-1}=j,Z_{k-1})=\] \[\int\bar{p}(x_{k}|r_{k}=i,x_{k-1},Z_{k-1})\bar{p}(x_{k-1}|r_{k-1}= j,Z_{k-1})dx_{k-1}\]  
and  
\[\Pr\{r_{k-1}=j|r_{k}=i,Z_{k-1}\}\] \[=\frac{\Pr\{r_{k}=i|r_{k-1}=j\}\Pr\{r_{k-1}=j|Z_{k-1}\}}{\sum_{l=1 }^{m}\Pr\{r_{k}=i|r_{k-1}=l\}\Pr\{r_{k-1}=l|Z_{k-1}\}}\]
## 2 Problem Formulation
### Reinforcement Learning  
#### 2.1.1 Preliminary Knowledge  
In RL, an agent learns an optimized policy through interacting with the environment. This process can be formulated as Markov Decision Processes (MDPs). An MDP can be defined as a 4-tuple \((S,A,T,R)\), where \(S\) represents the state space, \(A\) denotes the action space, transition probability \(T:S\times A\mapsto S\) denotes the state transition probability, and \(R\) represents the reward function. Specifically, at time step \(t\), the agent faces a state \(s_{t}\in S\), and it follows the policy \(\pi(a_{t}\mid s_{t}):s_{t}\mapsto a_{t}\) to choose an action \(a_{t}\in A\). Then the agent receives a reward \(r_{t}\), which could evaluate the performed action. After that, the agent reaches the next state \(s_{t+1}\). The aforementioned process will break when the terminal condition is satisfied.  
The return \(G_{t}=\sum_{k=0}^{\infty}\gamma^{k}r_{t+k}\) is the total accumulated rewards at state \(s_{t}\) with discount factor \(\gamma\in(0,1]\), which is a hyperparameter to adjust the impact of actions in the future.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Definition** \\ \hline \(\pi\) & policy \\ \(r_{t}\) & reward at time \(t\) \\ \(o_{t}\) & observation at time \(t\) \\ \(\mathcal{O}\) & observation space \\ \(s_{t}\) & ground-truth state at time \(t\) \\ \(\mathcal{S}\) & ground-truth state space \\ \(\phi(s)\) & feature vector representing state \(s\) \\ \(\Phi\) & state representation space \\ \(V_{\pi}(s)\) & value of state \(s\) under policy \(\pi\) \\ \(Q_{\pi}(s,a)\) & value of taking action \(a\) in state \(s\) under policy \(\pi\) \\ \(J\) & The objective function (expected total reward) \\ \(G_{t}\) & return (accumulated rewards) following time \(t\) \\ \(\theta\) & parameter vector of policy \(\pi_{\theta}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the notations in this paper.  
Figure 2: The unsupervised SRL scheme for DRL. The SRL network learns a mapping from original input observations to the embedded features.  
The goal of RL algorithms is to maximize the expected return \(R_{t}\) by updating the policy \(\pi\). The value function of state \(s_{t}\) under policy \(\pi\) is usually defined as:  
\[V^{\pi}(s_{t})=\mathbb{E}[G_{t}|s_{t}=s]. \tag{1}\]  
Similarly, the value of selected action \(a_{t}\) in state \(s_{t}\) under policy \(\pi\) is defined as:  
\[Q^{\pi}(s_{t},a_{t})=\mathbb{E}[G_{t}|s_{t}=s,a], \tag{2}\]  
which is called action-value function. The advantage function estimates the average advantage value of selected action \(a_{t}\) in state \(s_{t}\) under policy \(\pi\) and it is defined as:  
\[A^{\pi}(s_{t},a_{t})=Q^{\pi}(s_{t},a_{t})-V^{\pi}(s_{t}). \tag{3}\]
# Research on Vehicle-to-Vehicle MIMO Wireless Channels in Various Tunnels
## 3 Channel Analysis for V2V MIMO in Tunnels
### Cross Correlation Functions of V2V MIMO Channel  
It is well known that the propagation links in V2V wireless channels have different correlation properties in different time, space, and frequency domains. In the existing literature, the authors are inclined to adopt the space-time-frequency cross-correlation functions (STF CCFs) to study the correlation properties of propagation links in the V2V channels in the time, space, and frequency domains. In light of this, we follow this research solution in the following parts. In the proposed channel model, the CCF between the TVTF \(H_{kl}(f^{\prime},t)\) and \(H_{k^{\prime}l^{\prime}}(f^{\prime},t^{\prime})\) for MIMO antenna arrays can be expressed as [29]  
\[\rho_{kl,k^{\prime}l^{\prime}}(\delta_{T},\delta_{R},v^{\prime},\tau) = E\{H_{kl}^{*}(f^{\prime},t)\cdot H_{k^{\prime}l^{\prime}}^{*}( f^{\prime}+v^{\prime},t+\tau)\} \tag{29}\] \[\times \rho_{kl,k^{\prime}l^{\prime\prime}}^{Dif}(\delta_{T},\delta_{R},v^{\prime},\tau)+\rho_{kl,k^{\prime}l^{\prime}}^{LoS}(\delta_{T},\delta_{R},v ^{\prime},\tau)\] \[+ \rho_{kl,k^{\prime}l^{\prime\prime}}^{Spe}(\delta_{T},\delta_{R},v^{\prime},\tau)\,,\]  
calculated from \(D_{T}^{(l)}(x,y,z)\) and \(D_{R}^{(k)}(x,y,z)\). By using Equations (13) and (14), the distances \(D_{T}^{(l)}(x,y,z)\) and \(D_{R}^{(k)}(x,y,z)\) can be expressed as  
\[D_{T}^{(l)}(x,y,z) = D_{T}(x,y,z)-(M_{T}-2l+1)\frac{\delta_{T}}{2} \tag{39}\] \[\times \left[\ \cos\phi_{T}\cos\beta_{T}(x,y,z)\cos\left(\gamma_{T}-\alpha_{ T}(x,y,z)\right)+\sin\phi_{T}\sin\beta_{T}(x,y,z)\ \right],\]  
\[D_{R}^{(k)}(x,y,z) = D_{R}(x,y,z)-(M_{R}-2k+1)\frac{\delta_{R}}{2} \tag{40}\] \[\times \left[\ \cos\phi_{R}\cos\beta_{R}(x,y,z)\cos\left(\gamma_{R}-\alpha_{ R}(x,y,z)\right)+\sin\phi_{R}\sin\beta_{R}(x,y,z)\ \right],\]  
where \(D_{T}(x,y,z)\) and \(D_{R}(x,y,z)\) account for the distances from the scatter coordinate \((x,y,z)\) to the vehicle transceivers, respectively. Referring to the derivations of TVTF of SET, it is not difficult to derive the model of SCT and SRT. The CCF of tunnels can be obtained by  
\[\rho_{kl,k^{\prime}l^{\prime}}^{Dif}(\delta_{T},\delta_{R},v^{ \prime},\tau) = \frac{1}{2R(x_{R}-x_{T})(c_{R}+1)}\times\int_{-R}^{R}\int_{x_{T}} ^{X_{R}}d_{kk^{\prime}}(x,y)c_{l^{\prime}}(x,y) \tag{41}\] \[\times e^{j2\pi\left[f(x,y)\tau-\tau_{M}^{(xx)}\
u^{\prime}\right]}dxdy\,.\]  
Based on the aforementioned derivations of the space CCFs, temporal ACFs, frequency CCFs, and Doppler PSDs, we are able to investigate the propagation characteristics of the proposed V2V channel model for tunnel scenarios in the time, space, and frequency domains. This provides us with diversified perspectives for investigating the physical properties of the LoS and NLoS propagation links. In the following section, we will investigate the correlation properties of the propagation links with respect to the Rician factor, the motion properties of the transmitter and receiver.
## 2 Materials and Methods
### Analytical Methods  
Samples were taken at appropriate intervals for the quantification of biomass, glucose, glycerol, and VFAs concentration, intracellular lipids content, and long-chain fatty acids (LCFA) composition. Biomass concentration was quantified by optical density (\(\lambda\) = 600 nm) and converted to cellular dry weight (g\(\cdot\)L\({}^{-1}\)) by a previously constructed calibration curve. Glycerol, glucose, and VFAs (propionate, butyrate, and acetate) concentrations were quantified by HPLC in an Aminex HPX-87H column (300 mm \(\times\) 7.8 mm, 8 \(\upmu\)m particle size) at 60 \({}^{\circ}\)C and coupled with UV and RI detectors. The mobile phase was H\({}_{2}\)SO\({}_{4}\) 5 mM at a flow rate of 0.5 mL\(\cdot\)min\({}^{-1}\).  
Intracellular lipids were measured in freeze-dried cells (10 mg) after extraction with chloroform and methanol (2 mL, 1:1 \(v/v\)), and quantified by the phosphovanillin colorimetric method [14]. Freeze-dried cells and solvents were vortex-mixed for 3 min, and 250 \(\upmu\)L of the mixture was heated at 100 \({}^{\circ}\)C until the evaporation of the solvents. Then, 100 \(\upmu\)L of H\({}_{2}\)SO\({}_{4}\) 98% were added to the solution and heated at 100 \({}^{\circ}\)C for 15 min. After cooling to room temperature, 2.4 mL of phosphovanillin reagent (1.2 g\(\cdot\)L\({}^{-1}\) vanillin dissolved in orthophosphoric acid 85%) were added, and the mixture rested for 15 min allowing the reaction between the phosphovanillin solution and the lipids, producing a pink color. The final absorbance was measured at 490 nm and converted to lipids concentration (g\(\cdot\)L\({}^{-1}\)) by a calibration curve using olive oil as standard.  
The LCFA composition of intracellular lipids was measured in freeze-dried cells after extraction with chloroform, and derivatization into their fatty acid methyl esters (FAME) with a mixture of methanol-sulfuric acid (85:15, \(v/v\)), using pentadecanoic acid (C15:0) as internal standard [14]. FAME in the organic phase was analyzed by GC using a capillary column (TRACSIL TRWAX 30 m \(\times\) 0.25 mm \(\times\) 0.25 mm) and an FID detector. Helium at a flow rate of 1 mL\(\cdot\)min\({}^{-1}\) was used as the carrier gas, and the temperatures of the injector and the detector were 220 \({}^{\circ}\)C and 250 \({}^{\circ}\)C, respectively. For total separation of FAME, a temperature gradient (50 \({}^{\circ}\)C for 2 min, followed by an increase of 10 \({}^{\circ}\)C\(\cdot\)min\({}^{-1}\) up to 225 \({}^{\circ}\)C, which was maintained by 10 min) was used. The identification of FAME was performed by the comparison of the retention times with a mixture of FAME standards. The relative amount of each FAME (\(\%\), \(w/w\)) was defined as the ratio between its concentration (g\(\cdot\)L\({}^{-1}\)) and the total FAME detected in the sample.
## 3 Results and Discussion
### Reusability Studies with P3 and P4 Polymers in Acetamiprid Degradation  
Recoverability experiments were conducted for polymers P3 and P4 using the IWWTP water sample 1. The samples were adjusted at pH = 10 and the time reaction was set at 300 min. This assay aims to determine the number of reaction cycles the polymeric photosensitizer can perform before experiencing any degradation. Cycle 2 was conducted without irradiation as a control. The results of the reusability experiments demonstrated that both heterogeneous photocatalysts, P3 and P4, retained over 99% conversion efficiency across ten catalytic cycles when tested for the photo-oxidative degradation of Acetamiprid. Additionally, the absence of leaching was verified by analyzing the absorption spectra of the corresponding supernatants following the second catalytic cycle. Figure 9 presents a time-based graph of cycles in which polymer P4 was used over 10 reaction cycles in the photo-oxidation of Acetamiprid without showing any degradation.
## 2 Proposed Method
### Improved Dual-Stage Two-Phase Model Based on Attention Mechanism  
#### 2.2.1 Notation and Problem Statement  
(1) Notation  
Assume that there are \(N\) sites for all sites, given all sites' data, each site contains \(n\) exogenous series and a target series.  
Within the window size \(T\) of the all sites, the \(k\)-th (\(k\leqslant n\)) exogenous series is represented by  
\[x^{k}=(x_{1}^{k},x_{2}^{k},\ldots,x_{T}^{k})^{T}\in R^{T}. \tag{3}\]  
All exogenous series is represented by  
\[X=(x_{1},x_{2},\ldots,x_{T})^{T}\in R^{n\times T}(x_{k}=x^{k}). \tag{4}\]  
The target series is represented by  
\[Y=(y_{1},y_{2},\ldots,y_{T})^{T}\in R^{T}. \tag{5}\]  
Then a concatenation of target series and the output of the first phase attention can be represented by  
\[Z=(z_{1},z_{2},\ldots,z_{T})^{T}\in R^{(n+1)\times T}. \tag{6}\]  
Then, the future value of the target series can be represented by  
\[\hat{Y}=(y_{T+1},y_{T+2},\ldots,y_{T+\tau})^{T}\in R^{\tau}, \tag{7}\]  
where \(\tau\) is the time step to be predicted.  
(2) Problem statement  
Given the exogenous series and target series of all sites, the exogenous series and target series of the \(e\)-th site in all sites are  
\[(x_{1}^{(e)},x_{2}^{(e)},\ldots,x_{T}^{(e)}),x_{t}^{(e)}\in R^{n} \tag{8}\]  
Figure 2: The framework of the proposed DSTP-FC model.  
and  
\[(y_{1}^{(c)},y_{2}^{(c)},\ldots,y_{T}^{(c)}),y_{t}^{(e)}\in R. \tag{9}\]  
The model aims to predict the value of \(\tau\) time steps in the future, which can be expressed as:  
\[y_{T+1},y_{T+2},\ldots,y_{T+\tau}=F(y_{1}^{(e)},\ldots,y_{T}^{(e)},x_{1}^{(e)}, \ldots,x_{T}^{(e)}),e\leqslant N \tag{10}\]  
where \(F(\cdot)\) is the nonlinear function to be learned.
## 4 Manufacturing Experiment
### Manufacturing Microstrip Antenna  
A conformal microstrip antenna array containing four units was designed in this study, as shown in Figure 14. The dielectric constant and dielectric loss tangent of the substrate are 4.4 and 0.02, respectively. A microstrip line was used to feed the microstrip antenna unit. The antenna unit in the array has a dimension of \(12.4\times 10\) mm. Figure 14 shows the antenna manufactured by arrayed nozzle printing.  
The microstrip arrayed antenna was tested using a vector network analyzer (R&S ZNC). The frequency responses obtained from the simulation and experimental measurements are shown in Figure 15. The return loss of the antenna array at 6 GHz was measured to be -15.6 dB. The return loss in the bandwidth from 5.85 to 6.1 GHz was measured to be -10 dB. This value satisfies the performance requirements. As shown in the simulation and experimental results, the working frequency of the antenna is almost consistent with the original design. This feature indicates a highly accurate dimension of the antenna units during the printing process. The reduction in the return loss may be attributed to insufficient precision when printing extremely thin microstrip lines. This deficiency results in a small loss in the feed network. The difference between the simulation and the measurement results may be caused by the non-uniform thickness of the conductive pattern induced by the flow of the printing liquid.
# Dynamic Single-Electron Transistor Modeling for High-Frequency Capacitance Characterization
## 3 Simulation Results
### Characteristics of the SET-Based Amplifier
Actually, each terminal of SET should be connected to a constant-voltage source or a capacitor sufficiently larger than \(C_{\Sigma}\) so that the SET can be regarded as an independent circuit element [13], and this time the \(C_{\text{L}}\) is set to 30 aF (=10 \(C_{\Sigma}\)). As a consequence, the cutoff frequency \(f_{\text{C-CL}}\) of 53.1 MHz (=\(\frac{1}{4\pi R_{\Sigma}C_{\text{L}}}\)) consisting of SET output resistance and \(C_{\text{L}}\) dominates the low-frequency part of the frequency response. Figure 11 compares the frequency responses of SET-based amplifier with a load capacitor, simulated by the proposed transient model and the steady-state model [13]. From both curves we can see that at low frequencies voltage gain is equal to the slope of the descending side of CD, and the phase is inverted. The gain starts to decrease at around \(f_{\text{C-CL}}\), and continues to decrease in proportional to \(1/f\), but further drop is observed by the transient model at around \(f_{\text{C}}\) where transistor action is lifted. Such a drop cannot be described by the steady-state model because the model itself is frequency-independent. The phase starts to decrease at around \(f_{\text{C-CL}}\) and becomes saturated at 90 deg in case of the steady-state model, but it continues to decrease in the case of the transient model and finally gets saturated at \(-90\) deg, reflecting the transition from inverting to non-inverting operations at high frequencies above the \(f_{\text{C}}\) as observed in Figure 8, which also cannot be expressed by the steady-state model. These results clearly demonstrate the importance of the proposed transient model in accurately describing the high-frequency behavior of the real circuits. Figure 10: A circuit diagram of the SET-based inverting amplifier with constant-current load and load capacitance \(C_{\text{L}}\). Note that the relatively low \(f_{\text{C-CL}}\) and \(f_{\text{C}}\) could be increased by three orders of magnitude at the highest by decreasing the tunneling resistance to the level of the resistance quantum [32]. There is also room for decreasing the capacitance parameters by size reduction to increase the cutoff frequencies.