# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
As a result, \(f\) is an increasing function of both \(x\) and \(y\). The rest of the proof is as follows

\[=\sum_{r_{1}=r_{2}=r_{3}=r_{4}}^{K_{r}}\!\!\sum_{r_{3}=r_{4}}^{M }\!\!\sum_{r_{5}=1}^{K_{t}}h_{RIr_{1},r_{2}}\theta_{r_{3},r_{3}}h_{IRr_{4},r_ {5}}\!\!\left|r_{1}\right\rangle\!\!\left\langle r_{2}\right|\!\!\left\langle r _{3}\right|\!\!\left\langle r_{4}\right\rangle\!\!\left\langle r_{5}\right|\]8 (A54) \[=\sum_{r_{1}=r_{2}=r_{3}=r_{4}}^{K_{r}}\!\!\sum_{r_{3}=r_{4}}^{M }\!\!\sum_{r_{5}=1}^{K_{t}}h_{RIr_{1},r_{2}}\theta_{r_{3},r_{3}}h_{IRr_{4},r_ {5}}\!\!\left|r_{1}\right\rangle\!\!\left\langle r_{2}\right|\!\!\left\langle r _{3}\right|\!\!\left\langle r_{4}\right\rangle\!\!\left\langle r_{5}\right|\]9 (A55) \[=\sum_{r_{1}=1}^{K_{r}}\sum_{r_{3}=1}^{M}\sum_{r_{5}=1}^{K_{t}}h_{ RIr_{1},r_{3}}\theta_{r_{3},r_{3}}h_{IRr_{3},r_{5}}\!\!\left|r_{1}\right\rangle \!\!\left\langle r_{5}\right|\,. \tag{19}\]0\[=\sum_{r_{1}=1}^{K_{r}}\sum_{r_{3}=1}^{M}\sum_{r_{5}=1}^{K_{t}}h_{ RIr_{1},r_{3}}\theta_{r_{3},r_{3}}h_{IRr_{3},r_{5}}\!\!\left|r_{1}\right\rangle \!\!\left\langle r_{5}\right|\,. \tag{19}\]1 \[=\sum_{r_{1}=1}^{K_{r}}\sum_{r_{3}=1}^{M}\sum_{r_{5}=1}^{K_{t}}h_{ RIr_{1},r_{3}}\theta_{r_{3},r_{3}}h_{IRr_{3},r_{5}}\!\!\left|r_{1}\right\rangle \!\!\left\langle r_{5}\right|\,. \tag{19}\]2 (101) in which \[=\sum_{r_{1}=1}^{K_{r}}\sum_{r_{3}=1}^{M}\sum_{r_{5}=1}^{K_{t}}h_{ RIr_{1},r_{3}}\theta_{r_{3},r_{3}}h_{IRr_{3},r_{5}}\!\!\left|r_{1}\right\rangle \!\!\left\langle r_{5}\right|\,. \tag{19}\]3 holds because \[=\sum_{r_{1}=1}^{K_{r}}\sum_{r_{3}=1}^{M}\sum_{r_{5}=1}^{K_{t}}h_{ RIr_{1},r_{3}}\theta_{r_{3},r_{3}}h_{IRr_{3},r_{5}}\!\!\left|r_{1}\right\rangle \!\!\left\langle r_{5}\right|\,. \tag{19}\]4

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