## 4 DUDA Scheme for Interference Avoidance
### Satellite Association  
Given a typical moment, a satellite may reside in the exclusion area for uplink, the exclusion area for both links, or the areas without interference avoidance. Utilizing the satellite's exact position, we can classify its positioning into a total of five distinct scenarios: (a) Both satellites reside entirely within the exclusion zone for both links. (b) Both satellites fall within the confines of the exclusion zone for downlink, with at least one satellite not overlapping within either the exclusion zone for both links. (c) One satellite is situated within the exclusion zone for both links, whilst the second is positioned within the normal zone. (d) One satellite remains in the exclusion zone for downlink, whilst the second is situated within the normal zone. (e) Both satellites are occupying positions within the normal zone. These distinctive (a)-(e) position-based situations align with the four access scenarios delineated by (32), as illustrated below:  
Case 1: Failed access  
\[\left\{\max\{\vartheta_{L_{0}},\vartheta_{H_{0}}\}\leq\vartheta_{0}^{U}\right\}, \tag{33}\]  
which emerges when (a) happens and both satellites in exclusion zone are not accessible stipulated by (32a). Case 2: UT connected to marked satellites solely in uplink,  
\[\left\{\left(\vartheta_{H_{0}}>\vartheta_{0}^{U}\right)\cup\left(\vartheta_{L _{0}}>\vartheta_{0}^{U}\right);\max\{\vartheta_{L_{0}},\vartheta_{H_{0}}\}< \vartheta_{0}^{D}\right\}, \tag{34}\]  
which corresponds with situation (b) exclusively. It is a distinctive characteristic of the DUDA platform as distinguished from DUCA, which does not permit individual link access. Case 3: Decoupling uplink and downlink access, i.e.,  
\[ \begin{split}&\left\{\vartheta_{0}^{U}<\min\{\vartheta_{L_{0}}, \vartheta_{H_{0}}\}\leq\vartheta_{0}^{D};\max\{\vartheta_{L_{0}},\vartheta_{ H_{0}}\}>\vartheta_{0}^{D}\right\}\cup\\ &\left\{\min\{\vartheta_{L_{0}},\vartheta_{H_{0}}\}>\vartheta_ {0}^{D};\left(\frac{Z_{L_{0}}(\hat{\vartheta}_{L_{0}})}{Z_{L_{0}}(\hat{ \vartheta}_{H_{0}})}>\frac{p_{H}}{P_{L}}\cap\frac{Z_{L_{0}}(\hat{\vartheta}_{ L_{0}})}{Z_{H_{0}}(\hat{\vartheta}_{H_{0}})}\leq 1\right)\right.\\ &\left.\cup\left(\frac{Z_{L_{0}}(\hat{\vartheta}_{L_{0}})}{Z_{H_{ 0}}(\hat{\vartheta}_{H_{0}})}\leq\frac{p_{H}}{P_{L}}\cap\frac{Z_{L_{0}}(\hat{ \vartheta}_{L_{0}})}{Z_{H_{0}}(\hat{\vartheta}_{H_{0}})}>1\right)\right\}. \end{split}  \tag{35}\]  
Situations (d) and (e) align with this case. In MSS, the significance of decoupling operation lies in two aspects. Firstly, it involves the disparity between the average power of received signals, a principle akin to that of THS. Secondly, situation (c) arises from interference avoidance, wherein certain satellites can solely receive uplink signals, necessitating the transmission of downlink signals through another satellite. Therefore, (35) comprises the union of two components, with the former describing situation (d) and the latter representing situation (e). For an exhaustive comprehension, we recommend consulting the derivations in [10; 12]. Case 4: UT connected to a satellite for both links, i.e.,  
\[ \begin{split}&\left\{\min\{\vartheta_{L_{0}},\vartheta_{H_{0}}\}< \vartheta_{0}^{U,I};\max\{\vartheta_{L_{0}},\vartheta_{H_{0}}\}>\vartheta_{0}^{D }\}\cup\\ &\left\{\min\{\vartheta_{L_{0}},\vartheta_{H_{0}}\}>\vartheta_{0}^ {D};\left(\frac{Z_{L_{0}}(\hat{\vartheta}_{L_{0}})}{Z_{L_{0}}(\hat{\vartheta}_ {H_{0}})}\leq\frac{p_{H}}{P_{L}}\cap\frac{Z_{L_{0}}(\hat{\vartheta}_{L_{0}})}{ Z_{H_{0}}(\hat{\vartheta}_{H_{0}})}\leq 1\right)\right.\\ &\left.\cup\left(\frac{Z_{L_{0}}(\hat{\vartheta}_{L_{0}})}{Z_{H_{ 0}}(\hat{\vartheta}_{H_{0}})}>\frac{p_{H}}{P_{L}}\cap\frac{Z_{L_{0}}(\hat{ \vartheta}_{L_{0}})}{Z_{H_{0}}(\hat{\vartheta}_{H_{0}})}>1\right)\right\}. \end{split}  \tag{36}\]  
The derivation of (36) accounts for situations (c) and (e). Situation (c) denotes the presence of an inaccessible satellite alongside a normal one, resulting in the UT communicating with a normal satellite for both links.