esa-sceva/satcom-chunk-collection huggingface dataset:
# Self-Recognition Grasping Operation with a Vision-Based Redundant Manipulator System
## 3 The Method of ISPF for Pose Transformation
### Manipulator Simplification and Parameters Definition  
Considering the configuration of a 7-DOF manipulator, it can be simplified into four parts according to three nodes, namely, the wrist, elbow, and shoulder, which are labeled as \(P_{1},P_{2},P_{3}\). The node of the base and the node of the end-effector are labeled as \(P_{0},P_{T}\) respectively. The simplified configuration is shown in Figure 3. In the figure, the possible position of \(P_{2}\) forms a circle, which is named as the locus circle of \(P_{2}\). The locus circle has the following properties:  
1. The center of the locus circle is located on the line between nodes \(P_{1}\) and \(P_{3}\).
2. The line between \(P_{1}\) and \(P_{3}\) is normal to the plane of locus circle.  
After simplification, the parameters of manipulator can be defined as follows:  
\(\mathbf{P}_{e}\): The position of the end-effector of the manipulator.  
\(\mathbf{R}_{e}\): The attitude matrix of the end-effector of the manipulator.  
\(\sum_{m}\): The \(m^{th}\) joint coordinate system, \(m=0,1,2,\dots,7,T,P_{i}\), where \(\sum_{0}\) represents the base coordinate system, \(\sum_{T}\) represents the coordinate system of the end-effector, and \(\sum_{P_{i}}\) represents the coordinate system at node \(P_{i}\), \(i=0,1,2,3,4,e\).  
\({}^{j}_{i}\mathbf{T}\): Transformation matrix between the \(j^{th}\) and \(i^{th}\) coordinate system.  
\(\mathbf{P}_{i}=\left(x_{i},y_{i},z_{i}\right)^{\mathrm{T}}\): The position coordinate of the \(i^{th}\) node, \(i=0,1,2,3,4,e\).  
\(\mathbf{o}=\left(\sigma_{1},\sigma_{2},\sigma_{3}\right)^{\mathrm{T}}\): The center of locus circle of \(P_{2}\).  
\(\mathbf{a}=\left(a_{1},a_{2},a_{3}\right)^{\mathrm{T}}\): Arbitrary vector in the plane of locus circle.  
\(\mathbf{b}=\left(b_{1},b_{2},b_{3}\right)^{\mathrm{T}}\): Vector in the plane of locus circle, fulfilling \(\mathbf{a}\bot\mathbf{b}\).  
\(\mathbf{n}=\left(n_{1},n_{2},n_{3}\right)^{\mathrm{T}}\): Normal vector of the plane of locus circle.  
\(d_{1,\rho}\): The distance between node \(P_{1}\) and the center of circle \(O\).  
\(d_{i,j}\): The distance between the \(i^{th}\) and \(j^{th}\) node.  
\(l_{i,j}\): The vector between the \(i^{th}\) and \(j^{th}\) node.  
According to the configuration of manipulator shown in Figure 3, the joint angles are divided into three groups, which are respectively derived in the following section.
## 3 Joint Optimization Algorithm Design
### SMP-Based Buoy-UAV Association Algorithm
According to the \(P_{u,k}\) obtained by the MAHPPO algorithm, the transmit power matrix of the buoys at time slot \(k\) can be expressed as  
\[\mathbf{P}=\left[ \begin{array}{cccc}P_{1,k}&\dots&P_{1,k}&\dots&P_{1,k}\\ \vdots&\ddots&&\therefore&\vdots\\ P_{u,k}&&P_{u,k}&&P_{u,k}\\ \vdots&\therefore&&\ddots&\vdots\\ P_{U,k}&\dots&P_{U,k}&\dots&P_{U,k}\end{array} \right]_{U\times M} \tag{26}\]Then, the preference matrix can be expressed as  
\[\mathbf{G}=\left[ \begin{array}{ccccc}g_{1,1,k}&\cdots&g_{1,m,k}&\cdots&g_{1,M,k}\\ \vdots&\ddots&&\cdots&\vdots\\ g_{u,1,k}&&g_{u,m,k}&g_{u,M,k}\\ \vdots&\therefore&&\ddots&\vdots\\ g_{U,1,k}&\cdots&g_{U,m,k}&\cdots&g_{U,M,k}\end{array} \right]_{U\times M} \tag{27}\]  
It is unfair to match the association relationship directly according to the above preference matrix. For example, the distance between UAV \(u\) and buoy \(m\) is further than that between UAV \(u^{\prime}\) and buoy \(m\), but the agent \(u\) chooses higher transmit power to make \(g_{u,m,k}>g_{u^{\prime},m,k}\), which causes buoy \(m\) to tend to be associated with UAV \(u\) and consume more energy of buoy \(m\). Hence, the transmit power of all buoys was set to be the same. Since the distances between the buoys and UAV \(u\) are different, there is no element value the same in any row. For buoy \(m\), the distances between the UAVs and buoy \(m\) are also different due to the limit of \(d^{\min}\), so there is no element value the same in any column, except in the case of \(P^{\prime}_{u,k}=0\). Hence, there must be \(u\
eq u*\) and \(m\
eq m*\) if there is \(g_{u,m,k}=g_{u*,m,k}\) in \(\mathbf{G}\). The maximum SNR is defined as \(g^{\max}\). Combined with the idea of the GS algorithm, we introduce Lemma 1. **Lemma 1**. : _The buoy-UAV with \(g^{\max}\) in \(\mathbf{G}_{U,M,k}\) is the best choice for each._  
Proof of Lemma 1.: Suppose \(U=2\) and \(M=3\) and the preference of the first and second UAVs is \(g_{1,1,k}>g_{1,2,k}>g_{1,3,k}\) and \(g_{2,1,k}>g_{2,3,k}>g_{2,2,k}\), respectively. The preferences of the three buoys are \(g_{2,1,k}>g_{1,1,k}\), \(g_{1,2,k}>g_{2,2,k}\), and \(g_{2,3,k}>g_{1,3,k}\). Hence, we have \(g^{\max}\)=\(g_{2,1,k}\). First, UAV 1 and UAV 2 both choose Buoy 1 according to their own preferences. However, Buoy 1 is selected to be associated with UAV 2 because of \(g_{2,1,k}>g_{1,1,k}\). The second row and first column of \(\mathbf{G}_{U,M,k}\) are deleted to obtain \(\mathbf{G}^{\prime}\). UAV 1 chooses the maximum value in \(\mathbf{G}^{\prime}\), which means it chooses to be associated with Buoy 2, and so on. The above result could be also obtained by changing the goal to the cost of the buoys. Hence, the SAA algorithm achieves stable matching while obtaining the global optimal solution and is shown in Algorithm 2.
## 4 Proposed Approach
### Motivation
#### 4.1.2 Reduction in Computational Complexity  
[PERSON] et al. [1] proposed the use of t-SNE before applying ML algorithms. t-SNE first develops a probability distribution using the dataset, such that it starts to assign conditional probability to all tuples, thus answering the question of classification in terms of probability. Given a tuple of a particular class, it is necessary to know what the distance around it is, or the neighborhood around it related to the same class. The following formula is used to make this calculation:  
\[p_{j|i}=\frac{\exp\Bigl{(}-\frac{\|x_{i}-x_{j}\|^{2}}{2\sigma_{i}^{2}}\Bigr{)}} {\sum_{k\
eq i}\exp\Bigl{(}-\frac{\|x_{i}-x_{j}\|^{2}}{2\sigma_{i}^{2}}\Bigr{)}} \tag{1}\]  
In the above formula, \(p_{j|i}\) is the probability of point \(j\) being selected as the neighbor of point \(i\) in the dataset. The reason for taking the exponent of the L2 norm divided by the Gaussian variance is simply to make sure that data points that are distant will have an extremely low probability of being selected. The value of \(\sigma\) is found using another variable called perplexity. The value of perplexity is usually taken between 5 and 50, and indicates how many neighbors should be considered between these two values. This value then leads to the selection of the optimum value for \(\sigma\) through an algorithm (the explanation is beyond the scope of this paper).  
The next step is to obtain a low-dimensional space in which the same distribution with the same number of points exists. For this, we use the t-distribution as it helps avoid the crowding problem, with a single degree of freedom making the above formula into the one below:  
\[q_{j|i}=\frac{\exp\bigl{(}-\|y_{i}-y_{j}\|^{2}\bigr{)}}{\sum_{k\
eq i}\exp \bigl{(}-\|y_{i}-y_{k}\|^{2}\bigr{)}^{\prime}} \tag{2}\]Afterwards, the gradient descent algorithm is used to calculate this particular distribution by optimizing the Kullback-Leibler divergence represented by \(KL\).  
\[C=\sum_{i}KL(P_{i}||Q_{i})=\sum_{i}\sum_{j}p_{j|i}log\frac{p_{j|i}}{q_{j|i}} \tag{3}\]  
where \(C\) is the cost. In order to minimize \(C\), the correct dimensions need to be selected in the values of \(y\). The derivative of the cost function then becomes as follows:  
\[\frac{\partial C}{\partial y_{i}}=2\sum_{j}(p_{j|i}-q_{j|i}+p_{i|j}-q_{ij})(y _{i}-y_{j}) \tag{4}\]  
The literature review explains the implications of using the t-SNE algorithm. It was decided, because of those implications, that applying it would not be of much benefit in the case of the datasets under consideration in this study. Therefore, removing t-SNE removes a whole computation layer used in [PERSON] et al. (2016), resulting in a more efficient algorithm. The values of \(y_{i}\) and \(y_{j}\) are the new low-dimensional representations of the old points \(i\) and \(j\), which were in a high-dimensional space.
## Proposed System
### Applicability  
Figure 9 shows an example of the Pb battery characteristics or other chemistry type by a CCCV charging. The proposed system can charge/discharge battery cell/pack with a pulse waveform or any other arbitrary waveform. The proposed system can configure the cycle duty of the pulse waveform from 10% up to 90%. Figure 9 illustrates the voltage responses for three types of batteries by a pulse current discharging.  
As a contribution from some of the authors of this paper, pulse waveform charging has been considered to extend battery life in EVs [13; 14; 15]. It clarifies the differences of the battery characteristics as the transient responses. Figure 10 shows the use of arbitrary waveform based on the sinusoidal waveform.  
Figure 11 shows the usage of an arbitrary waveform based on a dynamic discharging operation of the 3 series and 6 parallel configurations with the developed BMS circuit. This model tries to simulate the real behavior of an EV operation through the practical tests carried out  
Figure 10: Arbitrary waveform charging/discharging (Li-Ion 1 cell): (**a**) sinusoidal waveform, (**b**) another example of arbitrary waveform during discharging.  
Figure 9: Voltage responses of three type of rechargeable battery based on pulse current discharging of 50 mHz frequency.  
The capability of the arbitrary waveform operation and the scaled-down battery configuration of the proposed system gives the possibility to study and test the battery pack for the real application using any actual power waveform. The feature is suitable for the investigation of BSS not only for EVs but also for renewable energy generations because the system simulates the dynamic and various characteristics of the generators and loads. In the following section, two simulated examples of the proposed system for renewable generators are explained before the application to EV's BSS.
## 3 Results and Discussion
### Lipid Composition  
The fatty acid composition of the investigated flaxseed oil was established. The analysis of flaxseed oil revealed the presence of 20 fatty acids (100% of the total composition). The primary fatty acids were \(\alpha\)-linolenic acid (57.5%) followed by oleic (17.5%), linoleic (12.5%), palmitic (6.0%), and stearic acid (4.3%). The results of the fatty acid composition of flaxseed oil are presented in Table 1 and Figure 1.  
Flasseed oil is distinguished by a unique fatty acid composition, which is predominated by the presence of polyunsaturated \(\alpha\)-linolenic acid. The quantity of \(\alpha\)-linolenic acid (57.5%) is in agreement with the data presented earlier by [PERSON] et al. [37], where it is between 50.0% and 60.0%. The higher content of \(\alpha\)-linolenic acid was at the expense of the lower quantity of oleic acid (17.5%). The results concerning the amount of oleic acid correspond with the data from the literature [5; 9; 10; 11; 38; 39; 40; 41; 42], reporting it to be between 15.0% and 24.0%. In the investigated flaxseed oil, the position isomer of oleic acid (C\({}_{\text{11--18--1}}\)) was not detected, which was determined earlier in other studies [43]. The examined oil contained one of the essential components, \(\omega\)-6 linoleic acid (12.5%). These results are analogous to those reported by several other authors (between 10.0% and 20.0%) [11; 12; 37; 38; 40; 41; 42; 44]. The results of the palmitic acid content analysis in our study are in agreement with those reported by some other authors, where it is 5.0-7.0% [10; 11; 12; 37; 38; 40; 41; 42; 44; 45]. The content of stearic acid was insignificantly different from that reported in other studies, where it is in a range between 3.0% and 6.0% [9; 10; 12; 42]. The differences in the values of fatty acids probably depend on environmental factors and different solvents used to extract the oils. The remaining fatty acids in the examined oil were present in minimal amounts. The content of saturated, unsaturated, mono- and polyunsaturated fatty acids is presented in Figure 2.  
The studied oil was rich in unsaturated fatty acids (88.7%), with the majority being polyunsaturated (70.7%). The composition of the unsaturated fatty acids included linolenic, oleic, and linoleic acids, which play a significant role in the higher amounts of this fraction. The content of the monounsaturated fatty acids was about 18.0%. Saturated fatty acids (11.3%) are not typical for flaxseed oils. The ratio of saturated to unsaturated fatty acids was about 1:8. According to all previous studies by other authors, unsaturated fatty acids predominate in flaxseed oil. Biologically active components in lipids (unsaponifiable matter, sterols, and tocoperols) were determined in the oil, as well. Data on the content of the main biologically active components in the investigated oil are shown in Table 2 and Figure 3, and in Sup
## 5 Results and Discussions
### Network Throughput  
The network throughput for all network states is shown in Figure 6. The simulation results show how the provisioning of network slices is affected by the number of NSRs received in the network. It can be observed that, although the number of provisioned requests has decreased due to the influx of NSRs, the proposed DCB-NS scheme is still able to outperform the other two algorithms used for testing. The proposed scheme can allocate network resources efficiently by implementing node selection and mapping methods. The NVA and maximum data flow computation allow the NSA to select the ideal candidates for network slice creation and deployment. In addition, the goal of the NSA is to maximize its cumulative gain by considering the total resource utilization through the calculation of RIF values. This advantage, in turn, enables the optimal distribution of resources for all NSRs, considering the current network state, thus minimizing the number of unserved requests.  
Finally, a summary showing an average comparison of the proposed scheme's performance is provided in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Rewards** & **RE** & **Throughput** \\ \hline DCB-NS & 90\% & 77\% & 85\% \\ Thompson Sampling & 86\% & 63\% & 83\% \\ Epsilon-Greedy & 77\% & 60\% & 80\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: The proposed scheme’s average performance comparison with the Epsilon-Greedy and Thompson Sampling algorithms.  
Figure 5: Average Network Resource Efficiency for (**a**) Network State 1, (**b**) Network State 2, and (**c**) Network State 3.
## 1 Introduction  
The upcoming wireless networks, including 5G/6G and beyond [1, 2], are poised to deliver markedly improved data rates, decreased latency, and expanded network coverage in comparison to their predecessors. These advancements in wireless networks stem from new design principles that enable them to support an extensive array of connected devices concurrently, ensuring robust connectivity and efficient data exchange. This is particularly crucial for burgeoning Internet of Things (IoT) applications, involving the integration of billions of sensors and smart devices, as referenced in [3, 4, 5]. In ultra-dense networks (UDN) [6], the signaling communication, specifically control commands, constitutes a substantial portion of the overall network traffic. Moreover, the segregation of signaling and data infrastructures places a considerable burden on base stations, negatively impactingboth energy and spectrum efficiency [6]. Wireless ad-hoc networks emerge as a promising solution to alleviate these challenges.  
A wireless mobile ad-hoc network (MANET) [7] represents a decentralized form of wireless network architecture wherein devices establish direct communication with each other, bypassing the need for centralized controls at base stations or access points. In the realm of wireless mobile ad-hoc networks, users make use of unlicensed and shared spectrum resources. This not only reduces the signaling load on base stations but also facilitates a higher number of user connections to base stations, especially in ultra-dense networks (UDN) [8]. Nevertheless, the network's capacity is constrained by environmental uncertainties and resource limitations.  
Simultaneously, reconfigurable intelligent surfaces (RIS) [9; 10] represent a transformative technology in wireless communication and signal propagation, effectively mitigating the limitations of conventional wireless ad-hoc networks. RIS comprise a two-dimensional surface equipped with low-cost passive reflecting elements, which can be electronically and adaptively controlled to manipulate the phase, amplitude, and direction of incoming and outgoing electromagnetic waves. This capability significantly enhances the signal quality and coverage. RIS have gained considerable attention as one of the most promising techniques, attracting interest from both research communities and industrial enterprises [11; 12].
## 3 Results and Discussion
### Second-Order Statistics  
We adopted the types of -\(\rho\)UV, -\(\rho\)UW, and -\(\rho\)VW and nondimensionalized them using wall shear stress. Figure S1 is the turbulence statistical results of velocity field, Figure S2 is the profiles of the normal Reynolds stress intensity components in the central (y = 0) vertical x-z plane. The Reynolds stress averaged over time along the jet axis is shown in Figure 10. In total, Reynolds stress increases with an increasing number of crushing needles. The Reynolds stress is smaller at the potential core and larger at the shear layer. The Reynolds stress along the axis shows a slow upward trend at the potential core, soars at the shear layer, and finally decreases at the end of the jet. However, momentum transportation in the lateral and radial directions (\(\tau\)-UV) was obvious to be higher than that in other directions (\(\tau\)-UW, \(\tau\)-VW), and the differences were 2-5 times and 1-3 times, respectively. In summary, the main momentum transport of the jet along the axis is in the lateral and radial directions, then in the lateral and vertical directions, and the intensity in the radial and vertical directions was minor [8]. The Reynolds stress after adding the crushing needle is greater than the Reynolds stress without adding the crushing needle. Consequently, the main reason for the increase in Reynolds stress after adding the crushing needle was that the crushing needle boundary disturbed the flow in the lateral and radial directions. The jet will expand in these two directions and produce strong shear flow. The vertical direction is affected by factors such as gravity and the crushing needle boundary effect, resulting in a small vertical component of Reynolds stress. In addition, it indicates that the crushing needle may dominate the lateral mixing and entrainment of the jet [5].  
Figure 9: Average vertical velocity contour map: (**a**) Re = 32,639, n = 0. (**b**) Re = 32,639, n = 2. X is the lateral coordinate, and Z is the vertical coordinate.
## 2 Methodology for Calculating Transient Magnetic Field and Induced Voltage
### Formula Derivation of Transient Magnetic Field  
The transient magnetic field is described by [PERSON]'s equations. The solution of the transient magnetic field is based on the vector potential [19]  
\[\mathbf{B}=\
abla\times\mathbf{A} \tag{1}\]  
where \(\mathbf{B}\) is the magnetic flux density and \(\mathbf{A}\) is the vector potential. \(\mathbf{A}\) is expressed by the current density [5; 19]  
\[A(\mathbf{\rho},t)=\frac{\mu_{0}}{4\pi}\int_{V}\frac{\mathbf{I}(\mathbf{\rho}^{\prime},\,t- \frac{r}{v})}{r}\mathrm{d}V \tag{1}\]  
where \(\mu_{0}\) is the permeability of free space (\(4\pi\times 10^{-7}\) H/m), \(v\) is the velocity of light (\(3\times 10^{8}\)/m/s) and \(\mathbf{J}\) is the current density. The geometrical quantities in (2) are illustrated in Figure 3. According to Figure 1, the spatial locations of all the branches can be classified into three categories: tilted, vertical and horizontal. On a tilted branch shown in Figure 4, a differential carrying-current segment \(\mathrm{d}\mathbf{l}\) is taken into consideration. The differential current element \(\mathbf{J}(\rho^{\prime},\,t-r/v)\mathrm{d}V\) is written as \(i(\rho^{\prime},\,\phi^{\prime},\,z,\,t-r/v)\ \mathrm{d}\mathbf{l}\) in the cylindrical coordinates. From (2) the differential vector potential is given by [20]  
\[\mathrm{d}\mathbf{A}=\frac{\mu_{0}}{4\pi}\frac{i(\rho^{\prime},\phi^{\prime},z,\,t -r/v)}{r}\mathrm{d}\mathbf{l} \tag{3}\]  
Figure 4: Tilted branch.  
Figure 3: Geometrical quantities.  
Using \(\mathrm{d}I=\mathrm{d}l\mathrm{cos}\mathrm{V}\mathrm{e}_{\mathrm{\rho}}+\mathrm{d}l \mathrm{sin}\mathrm{V}\mathrm{e}_{\mathrm{z}}\) in (3) gives  
\[\begin{array}{l}\mathrm{d}A_{\mathrm{\phi}}=0\\ \mathrm{d}A_{\mathrm{\rho}}=\frac{\mathrm{\mu}_{0}\mathrm{d}l\mathrm{cos}\, \mathrm{\sqrt{\pi}}\,\frac{i(\rho^{\prime}\phi^{\prime},z^{\prime},\,t-r/v)}{4 \pi}e_{\mathrm{\rho}}\\ \mathrm{d}A_{z}=\frac{\mathrm{\mu}_{0}\mathrm{d}l\mathrm{sin}\,\mathrm{\sqrt{ \pi}}\,\mathrm{i}(\rho^{\prime}\phi^{\prime},z^{\prime},\,t-r/v)}{r}e_{z}\end{array} \tag{4}\]  
where the distance \(r\) is  
\[r=\sqrt{\rho^{2}+{\rho^{\prime}}^{2}-2\rho\rho^{\prime}\cos(\phi-\phi^{\prime} )+(z-z^{\prime})^{2}} \tag{5}\]  
Evaluating the curl of \(\mathrm{d}A\) for (1) gives  
\[\mathrm{d}B_{1}=\
abla\times\mathrm{d}A=\frac{1}{\rho}\frac{\partial(\mathrm{ d}Az)}{\partial\phi}\mathrm{e}_{\mathrm{\rho}}+\bigg{[}\frac{\partial( \mathrm{d}A\rho)}{\partial z}-\frac{\partial(\mathrm{d}Az)}{\partial\rho} \bigg{]}\mathrm{e}_{\mathrm{\phi}}-\frac{1}{\rho}\frac{\partial(\mathrm{d}A \rho)}{\partial\phi}\mathrm{e}_{z} \tag{6}\]  
Making a change of variables \(\eta=t-r/v\), the current derivatives with respective to \(\rho\), \(\phi\)
# An S/C/X-Band 4-Bit Digital Step Attenuator MMIC with 0.25 \(\upmu\)m GaN HEMT Technology
## 3 Measurement Results  
The 4-bit wideband digital step attenuator MMIC was fabricated with 0.25 \(\upmu\)m GaN HEMT technology on a GaN-on-SiC substrate. A microphotograph of the 4-bit GaN digital step attenuator MMIC is shown in Figure 5. The chip size is 2.45 \(\times\) 1.75 mm\({}^{2}\) including DC and RF pads. The DC and the digital control pads of the GaN digital step attenuator MMIC are bonded on the PCB, which includes the serial-to-parallel interface with a negative gate control voltage generator for the digital control voltages of \(-\)3.3 V(off-state) and 0 V(on-state). On-wafer measurement was performed to characterize the GaN digital step attenuator MMIC using a vector network analyzer and short-open-load-through (SOLT) calibration.  
Figure 4: Schematic of (**a**) the GaN SPDT and (**b**) the GaN DPDT switch.  
Figure 6 shows the measured insertion loss and normalized attenuation performance of the GaN digital step attenuator MMIC. The insertion loss of 2.8\(-\)8.3 dB is achieved in the reference state at DC-12 GHz. The attenuation range of 30 dB and the attenuation step of 2 dB are achieved with flat attenuation performance versus the frequencies. As shown in Figure 7, the input and output return losses of \(>\)12 dB are achieved from DC to 12 GHz. Figure 8 shows the measured RMS amplitude and phase errors. The measured RMS amplitude error is less than 1 dB and the RMS phase error is less than 8.5\({}^{\circ}\) at DC-12 GHz. The total DC power consumption is nearly 0 mW. A performance comparison of the previously published III-V digital step attenuator MMICs is summarized in Table 2.  
Figure 5: Microphotograph of the S/C/X-band 4-bit GaN digital step attenuator MMIC and test module.  
Figure 6: Measured (**a**) S\({}_{21}\), and (**b**) normalized attenuation of the GaN digital step attenuator MMIC.  
Figure 8: Measured (**a**) RMS amplitude error, and (**b**) RMS phase error of the GaN digital step attenuator MMIC.  
Figure 7: Measured (**a**) \(S_{11}\), and (**b**) \(S_{22}\) of the GaN digital step attenuator MMIC.