## 3 Proposed Algorithms  
### The BCD Method for the CP Scheme  
The transmission and reflection phase shifts of STAR-RIS's CP scheme are coupled, introducing complex nonlinear constraints. This coupling complicates the optimization problem and hinders the direct application of traditional optimization methods. The BCD framework is widely used to solve non-convex problems, which can guarantee an approximately optimal solution by solving multiple convex subproblems derived from the original non-convex problem in chunks. This is particularly important for the CP scheme. Therefore, we propose a BCD-based method to solve the CP scheme. We decompose the variables into two blocks, i.e., \(\{\mathbf{w}_{l}\}\) and \(\{\boldsymbol{\theta}_{T},\boldsymbol{\theta}_{R}\}\), and perform iterative optimization in an alternating form. Specifically, the sub-problem of optimizing the reflection and transmission coefficients (i.e., \(\{\boldsymbol{\theta}_{T},\boldsymbol{\theta}_{R}\}\)) is addressed using the PDD framework. In this framework, the solved subproblems are transformed into augmented Lagrangian (AL) problems by incorporating penalty terms. The variables and the introduced penalty factors are optimized iteratively until convergence is achieved. To start with, we first decouple problem (6a) in tractable subproblem, and iteratively derive solutions for these subproblems. First of all, in the vicinity of point \(\left\{\mathbf{w}_{l}^{*},\boldsymbol{\theta}_{l}^{*}\right\}\), we have constants \(\{d_{l}^{1},d_{l}^{2},d_{l}^{3},d_{l}^{4}\}\), given by \[d_{l}^{1} =1\Big{/}\Big{(}\Big{|}\Big{(}\mathbf{h}_{Bl}^{H}+\boldsymbol{\theta }_{l}^{{}^{H}H}\mathbf{H}_{l}\Big{)}\mathbf{w}_{l}^{*}\Big{|}^{2}+\sigma^{2} \Big{)}, \tag{7a}\] \[d_{l}^{2} =\frac{d_{l}^{1}\Big{|}\Big{(}\mathbf{h}_{Bl}^{H}+\boldsymbol{ \theta}_{l}^{{}^{H}H}\mathbf{H}_{l}\Big{)}\mathbf{w}_{l}^{*}\Big{|}^{2}}{\Big{|} \Big{(}\mathbf{h}_{Bl}^{H}+\boldsymbol{\theta}_{l}^{{}^{H}H}\mathbf{H}_{l} \Big{)}\mathbf{w}_{l}^{*}\Big{|}^{2}+\Big{|}\Big{(}\mathbf{h}_{Bl}^{H}+ \boldsymbol{\theta}_{l}^{{}^{H}H}\mathbf{H}_{l}\Big{)}\mathbf{w}_{l^{\prime} }^{*}\Big{|}^{2}+\sigma^{2}},\] (7b) \[d_{l}^{3} =\Big{|}\Big{(}\mathbf{h}_{l}^{H}+\boldsymbol{\theta}_{l}^{{}^{H }G}\mathbf{{}_{l}}\Big{)}\mathbf{w}_{l}^{*}\Big{|}^{2}\Big{/}\Big{(}\Big{|} \Big{(}\mathbf{h}_{l}^{H}+\boldsymbol{\theta}_{l}^{{}^{H}G}\mathbf{{}_{l}} \Big{)}\mathbf{w}_{l}^{*}\Big{|}^{2}+\sigma^{2}\Big{)},\] (7c) \[d_{l}^{4} =-\Big{|}\Big{(}\mathbf{h}_{l}^{H}+\boldsymbol{\theta}_{l}^{{}^{ H}G}\mathbf{{}_{l}}\Big{)}\mathbf{w}_{l^{\prime}}^{*}\Big{|}^{2}+\sigma^{2}, \tag{7d}\]  
Based on the given equations and neglecting the constant terms, around the specified point \(\{\mathbf{w}_{l}^{*},\boldsymbol{\theta}_{l}^{*}\}\), \(R_{l}^{SEC}\) can be reformulated as follows:  
\[R_{l}^{SEC} =d_{l}^{1}2\mathfrak{R}\Big{\{}\mathbf{w}_{l}^{H}\Big{(}\mathbf{h }_{Bl}+\mathbf{H}_{l}^{H}\boldsymbol{\theta}_{l}\Big{)}\Big{(}\mathbf{h}_{Bl}^ {H}+\boldsymbol{\theta}_{l}^{{}^{H}H}\mathbf{H}_{l}\Big{)}\mathbf{w}_{l}^{*} \Big{\}}\] \[+d_{l}^{2}\Big{[}\Big{|}\Big{(}\mathbf{h}_{Bl}^{H}+\boldsymbol{ \theta}_{l}^{H}\mathbf{H}_{l}\Big{)}\mathbf{w}_{l^{\prime}}\Big{|}^{2}+\Big{|} \Big{(}\mathbf{h}_{Bl}^{H}+\boldsymbol{\theta}_{l}^{H}\mathbf{H}_{l}\Big{)} \mathbf{w}_{l}\Big{|}^{2}\Big{]} \tag{8}\] \[+\frac{1}{1+d_{l}^{3}}\frac{\Big{|}\Big{(}\mathbf{h}_{l}^{H}+ \boldsymbol{\theta}_{l}^{H}\mathbf{G}_{l}\Big{)}\mathbf{w}_{l^{\prime}}\Big{|} ^{2}}{2\mathfrak{R}\Big{\{}\mathbf{w}_{l^{\prime}}^{H}\Big{(}\mathbf{h}_{l}+ \boldsymbol{\mathbf{G}}_{l}^{H}\boldsymbol{\theta}_{l}\Big{)}\Big{(}\mathbf{h}_ {l}^{H}+\boldsymbol{\theta}_{l}^{{}^{H}H}\mathbf{G}_{l}\Big{)}\mathbf{w}_{l^{ \prime}}^{*}\Big{\}}+d_{l}^{4}}\]  
It is obvious that (8) is convex in the interest of maintaining conciseness and clarity, refer to [25] for comprehensive details.