Reflection probability in wireless networks with metasurface-coated environmental objects: an approach based on random spatial processes

Di Renzo, Marco; Song, Jian

doi:10.1186/s13638-019-1403-7

EURASIP Journal on Wireless Communications and Networking

Table 2 Auxiliary functions used in Theorem 1

From: Reflection probability in wireless networks with metasurface-coated environmental objects: an approach based on random spatial processes

Function definition
\(f\left ({\alpha,\xi } \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({\left [ {m\sin \alpha + \cos \alpha } \right ]\xi + z\sin \alpha } \right)\)
\(g\left ({\alpha,\omega } \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({\frac {{\left [ {m\sin \alpha + \cos \alpha } \right ]\left [ {\omega - z} \right ]}}{m} + z\sin \alpha } \right)\)
\(\Theta \! \!\left (\! \!{\alpha \!\left \|\! {\begin {array}{*{20}{c}} {{\mu _{1}}}&{{\mu _{2}}} \\ {{\mu _{3}}}&{{\mu _{4}}} \end {array}} \right.}\! \!\!\right)\! \!= \!\left [\! {{{\left ({\min \left \{ {{\mu _{1}},{\mu _{2}},1} \right \}} \right)}^{2}}\! -\! {{\left ({\max \left \{ {{\mu _{3}},{\mu _{4}},0} \right \}} \right)}^{2}}} \right ]H\!\left ({\min \left \{ {{\mu _{1}},{\mu _{2}},1} \right \} \!- \!\max \!\left \{ {{\mu _{3}},{\mu _{4}},0} \right \}}\! \right)\)
\({\theta _{1}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left \| {\begin {array}{*{20}{c}} {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times H\left (m \right)\)
\({\theta _{2}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left \| {\begin {array}{*{20}{c}} {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times \bar H\left (m \right)\)
\({\theta _{3}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left \| {\begin {array}{*{20}{c}} {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times H\left (m \right)\)
\({\theta _{4}}\left ({\alpha,{x_{{\text {Tx}}}},{x_{{\text {Rx}}}},{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right) = \Theta \left ({\alpha \left \| {\begin {array}{*{20}{c}} {f\left ({\alpha,\min \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\max \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \\ {f\left ({\alpha,\max \left ({{x_{{\text {Tx}}}},{x_{{\text {Rx}}}}} \right)} \right)}&{g\left ({\alpha,\min \left ({{y_{{\text {Tx}}}},{y_{{\text {Rx}}}}} \right)} \right)} \end {array}} \right.} \right) \times \bar H\left (m \right)\)

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