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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

\(d_{2}' = \text {min}\left \{d_{2}^{\mathsf {max}}, d_{\mathsf {sum}}^{\mathsf {max}} - d_{1}^{\mathsf { max}}\right \}\)

\({L}_{T_{1}}|\Psi _{T_{11}}| \geq {L}_{R_{1}}|\Psi _{R_{11}}|,\)

 
 

\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\cup \Psi _{T_{12}}| \)

 

\({\phantom {0000}}-2{L}_{R_{1}}|\Psi _{R_{11}}\cap \Psi _{R_{12}}| \}\)

\({L}_{T_{2}}|\Psi _{T_{12}}| \geq {L}_{R_{1}}|\Psi _{R_{12}}|\)

 

\( {L}_{T_{1}}|\Psi _{T_{11}}| \geq {L}_{R_{1}}|\Psi _{R_{11}}|,\)

 
 

\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\setminus \Psi _{T_{12}}|\)

 

\(+ 2{L}_{R_{1}} |\Psi _{R_{12}}\setminus \Psi _{R_{11}}|\}\)

\( {L}_{T_{2}}|\Psi _{T_{12}}| < {L}_{R_{1}}|\Psi _{R_{12}}|\)

 

\( {L}_{T_{1}}|\Psi _{T_{11}}| < {L}_{R_{1}}|\Psi _{R_{11}}|,\)

 
 

\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\cup \Psi _{T_{12}}| \)

 

\(+ 2{L}_{R_{1}}|\Psi _{R_{11}}\setminus \Psi _{R_{12}}| - 2{L}_{T_{1}}|\Psi _{T_{11}}| \}\)

\( {L}_{T_{2}}|\Psi _{T_{12}}| \geq {L}_{R_{1}}|\Psi _{R_{12}}|\)

 

\( {L}_{T_{1}}|\Psi _{T_{11}}| < {L}_{R_{1}}|\Psi _{R_{11}}|,\)

 
 

\(\text {min}\{d_{2}^{\mathsf {max}},\ 2{L}_{T_{2}}|\Psi _{T_{22}}\setminus \Psi _{T_{12}}|\)

 

\(+ 2{L}_{R_{1}}|\Psi _{R_{11}}\cup \Psi _{R_{12}}| - 2{L}_{T_{1}}|\Psi _{T_{11}}| \}\)

\( {L}_{T_{2}}|\Psi _{T_{12}}| < {L}_{R_{1}}|\Psi _{R_{12}}|\)

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