# Bi-Objective Power Optimization of Radio Stripe Uplink Communications
## 2 Radio Stripes Network Model
### Payload Transmission and Reception  
Let \(s_{i}\in\mathbb{C}\) denote the signal transmitted by the \(i\)th UE, with power \(p_{i}\) and let \(\mathbf{s}=[s_{1},\cdots,s_{K}]^{T}\). The received signal at AP \(l\), \(\mathbf{y}_{l}\in\mathbb{C}^{N}\), includes the data transmitted by all \(K\) users, being written as  
\[\mathbf{y}_{l}=\mathbf{H}_{l}\mathbf{s}+\mathbf{n}_{l}, \tag{6}\]where \(\mathbf{H}_{l}=[\mathbf{h}_{1l},\cdots,\mathbf{h}_{Kl}]\in\mathbb{C}^{N\times K}\) is the channel matrix and \(\mathbf{s}=[s_{1},\cdots,s_{K}]^{T}\sim\mathcal{N}_{\mathbb{C}}(0,\mathbf{Q})\) is the signal vector, with \(\mathbf{Q}=\text{diag}(p_{1},\cdots,p_{k})\) and \(\mathbf{n}_{l}\sim\mathcal{N}_{\mathbb{C}}\left(0,\sigma^{2}\mathbf{I}_{N}\right)\) denotes the receiver noise. Equation (6) can be rewritten as  
\[\mathbf{y}_{l}=\mathbf{\hat{H}}_{l}\mathbf{s}+\mathbf{w}_{l}, \tag{7}\]  
where \(\mathbf{\hat{H}}_{l}=\mathbf{H}_{l}-\mathbf{\hat{H}}_{l}=\left[\mathbf{\hat{h }}_{1l},\cdots,\mathbf{\hat{h}}_{Kl}\right]\) is the matrix of CSI and \(\mathbf{\hat{H}}_{l}\) denotes the matrix of channel estimation errors. Additionally, \(\mathbf{w}_{l}=\mathbf{\hat{H}}_{l}\mathbf{s}+\mathbf{n}_{l}\) represents a vector containing both the estimation error and the noise term, and \(\mathbf{w}_{l}\sim\mathcal{N}_{\mathbb{C}}(0,\mathbf{\Sigma}_{l})\) where  
\[\mathbf{\Sigma}_{l}=\sum_{i=1}^{K}p_{i}\mathbf{\hat{R}}_{il}+\sigma^{2}\mathbf{ I}_{N}. \tag{8}\]  
The RS requires a UL sequential linear processing algorithm to obtain the signal estimation vector for each AP \(l\) along the sequential network, i.e., \(\mathbf{\hat{s}}_{l}=[s_{1l},\cdots,s_{Kl}]^{T}\), where \(\hat{s}_{kl}\) denote the individual estimates for each UE \(k\). The starting point of the sequential algorithm, after receiving \(\mathbf{y}_{1}\), is the computation of the signal's estimation vector for AP 1, i.e., \(\mathbf{\hat{s}}_{1}\). This information is then sent to AP 2 which performs its own estimation, \(\mathbf{\hat{s}}_{2}\), which is based, not only on \(\mathbf{y}_{2}\), but also relies on the information embedded in \(\mathbf{\hat{s}}_{1}\). Thus, each AP \(l\) along the network, after receiving the signal estimation vector from AP \(l-1\), \(\mathbf{\hat{s}}_{l-1}\), performs their estimation, \(\mathbf{\hat{s}}_{l}\), and forwards this information to AP \(l+1\). This process is iterative until AP \(L\) is reached, forwarding the final signal vector estimate, \(\mathbf{\hat{s}}_{L}\), to the CPU. The sequential algorithm proceeds as follows: AP \(l\), upon receiving the signal estimation vector from AP \(l-1\) computes its own soft estimate following  
\[\mathbf{\hat{s}}_{l}=\mathbf{A}_{l}\mathbf{\hat{s}}_{l-1}+\mathbf{B}_{l} \mathbf{y}_{l}, \tag{9}\]  
where \(\mathbf{A}_{l}\in\mathbb{C}^{K\times K}\) is a receiver combining matrix that allows the local detection to leverage on the estimate computed from previous AP, \(\mathbf{\hat{s}}_{l-1}\), while \(\mathbf{B}_{l}\in\mathbb{C}^{K\times N}\) depends on the local received signal, \(\mathbf{y}_{l}\). Assuming that \(\mathbf{\hat{s}}_{0}=\mathbf{0}_{K}\), Equation (9) can be generalized for \(l\in\{1,\cdots,L\}\) as  
\[\mathbf{\hat{s}}_{l}=\mathbf{B}_{l}\mathbf{z}_{l}, \tag{10}\]  
where \(\mathbf{z}_{l}=\left[\mathbf{y}_{1}^{H},\cdots,\mathbf{y}_{l}^{H}\right]^{H}\) is the augmented received signal at AP \(l\) and the augmented receiver matrix combining is defined as [10; 11]  
\[\mathbf{\hat{B}}_{l}=\left\{ \begin{array}{ll}[\mathbf{A}_{l}\mathbf{B}_{l- 1}\quad\mathbf{B}_{l}],&\text{if }l>1\\ \mathbf{B}_{1},&\text{if }l=1.\end{array} \right. \tag{11}\]  
Equation (11) can be rewritten as \(\mathbf{\hat{B}}_{l}=\left[\mathbf{\hat{B}}_{l1},\mathbf{\hat{B}}_{l2},\cdots,\mathbf{\hat{B}}_{ll}\right]\in\mathbb{C}^{K\times NI}\) where  
\[\left\{ \begin{array}{ll}\mathbf{\hat{B}}_{l_{K}^{\infty}}=\mathbf{A}_{l} \mathbf{A}_{l-1}\cdots\mathbf{A}_{\xi+1}\mathbf{B}_{\xi},&1\leq\xi<l\\ \mathbf{\hat{B}}_{ll}=\mathbf{B}_{l}.\end{array} \right.