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