From: Spatial degrees-of-freedom in large-array full-duplex: the impact of backscattering
Case | \(d_{2}' = \text {min}\left \{d_{2}^{\mathsf {max}}, d_{\mathsf {sum}}^{\mathsf {max}} - d_{1}^{\mathsf { max}}\right \}\) |
---|---|
\({L}_{T_{1}}|\Psi _{T_{11}}| \geq {L}_{R_{1}}|\Psi _{R_{11}}|,\) | Â |
 | \(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\cup \Psi _{T_{12}}| \) |
 | \({\phantom {0000}}-2{L}_{R_{1}}|\Psi _{R_{11}}\cap \Psi _{R_{12}}| \}\) |
\({L}_{T_{2}}|\Psi _{T_{12}}| \geq {L}_{R_{1}}|\Psi _{R_{12}}|\) | Â |
\( {L}_{T_{1}}|\Psi _{T_{11}}| \geq {L}_{R_{1}}|\Psi _{R_{11}}|,\) | Â |
 | \(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\setminus \Psi _{T_{12}}|\) |
 | \(+ 2{L}_{R_{1}} |\Psi _{R_{12}}\setminus \Psi _{R_{11}}|\}\) |
\( {L}_{T_{2}}|\Psi _{T_{12}}| < {L}_{R_{1}}|\Psi _{R_{12}}|\) | Â |
\( {L}_{T_{1}}|\Psi _{T_{11}}| < {L}_{R_{1}}|\Psi _{R_{11}}|,\) | Â |
 | \(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\cup \Psi _{T_{12}}| \) |
 | \(+ 2{L}_{R_{1}}|\Psi _{R_{11}}\setminus \Psi _{R_{12}}| - 2{L}_{T_{1}}|\Psi _{T_{11}}| \}\) |
\( {L}_{T_{2}}|\Psi _{T_{12}}| \geq {L}_{R_{1}}|\Psi _{R_{12}}|\) | Â |
\( {L}_{T_{1}}|\Psi _{T_{11}}| < {L}_{R_{1}}|\Psi _{R_{11}}|,\) | Â |
 | \(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\setminus \Psi _{T_{12}}|\) |
 | \(+ 2{L}_{R_{1}}|\Psi _{R_{11}}\cup \Psi _{R_{12}}| - 2{L}_{T_{1}}|\Psi _{T_{11}}| \}\) |
\( {L}_{T_{2}}|\Psi _{T_{12}}| < {L}_{R_{1}}|\Psi _{R_{12}}|\) | Â |