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Table 4 Inner precoders

From: Two-level frame precoding with non-cooperative gateways

\( \mathbf {T}_{m} = \sqrt {t_{m}} \left (\mathbf {B}_{m}^{H} \mathbf {A}_{m} \mathbf {B}_{m} + \gamma _{m} \mathbf {B}_{m}^{H} \mathbf {B}_{m} \right)^{-1} \mathbf {B}_{m}^{H} \sum _{i =1}^{N_{U}} {\mathbf {H}_{m,m}^{{[i]} \, H }} \)

\( {}\gamma _{m} = \frac {k \cdot N_{U}}{P_{m}}, \, \mathbf {A}_{m} \triangleq \sum _{i =1}^{N_{U}} {\mathbf {H}_{m,m}^{{[i]} \, H }} {\mathbf {H}_{m,m}^{[i]}} + \mathbf \Sigma _{m} \)

NU=NG and the design is made for each group of users

Cooperative\(\mathbf \Sigma _{m} \approx \sum _{\substack {p=1 \\ p\neq m}}^{M} \sum _{i =1}^{N_{U}} {\mathbf {H}_{p,m}^{{[i]} \, H }} {\mathbf {H}_{p,m}^{[i]}}{\phantom {\!\!0}}\)

 

Autonomous\(\mathbf \Sigma _{m} \approx N_{U} \sum _{\substack {p=1 \\ p\neq m}}^{M} \mathbb {E} [\mathbf {H}_{p,m}^{H} \mathbf {H}_{p,m}]\)