# **meeting 11/07**

**Advisor: Prof. Chih-Yu Wang \
Presenter: Shao-Heng Chen \
Date: Nov 07, 2023**
<!-- Chih-Yu Wang -->
<!-- Wei-Ho Chung -->

## **Steering vectors**
- The stacked received signal of all $N_k$ users
$$
\begin{align*}
\mathbf{y} &= (\mathbf{H}_{2} \mathbf{\Phi} \mathbf{H}_1 + \mathbf{H}_{3}) \mathbf{F} \mathbf{x} + \mathbf{n} = \tilde{\mathbf{H}}\mathbf{F} \mathbf{x} + \mathbf{n}
\end{align*}
$$
- $\mathbf{H}_1 \in \mathbb{C}^{N_s \times N_t}$ is the BS-RIS channel
- $\mathbf{H}_{2} \in \mathbb{C}^{N_k \times N_s}$ is the RIS-users channel
- $\mathbf{h}_{k, 2} \in \mathbb{C}^{N_s \times 1}, \forall k = 1, ..., N_k$ is the channel vector from RIS to the $k$-th user
$$
\begin{align*}
\mathbf{h}_{k, 2} =& \ \hat{\mathbf{h}}_{k, 2} + \Delta\mathbf{h}_{k, 2} \in \mathbb{C}^{N_s \times 1} \\
\Delta\mathbf{h}_{k, 2} &= \psi\frac{\mathbf{h}_{k, 2}^{N_1}}{\| \mathbf{h}_{k, 2}^{N_1} \|_2}, \;\; \mathbf{h}_{k, 2}^{N_1} \sim \mathcal{CN}(0, 1)
\end{align*}
$$
- $\mathbf{H}_{3} \in \mathbb{C}^{N_k \times N_t}$ is the BS-users channel
- $\mathbf{h}_{k, 3} \in \mathbb{C}^{1 \times N_t}, \forall k = 1, ..., N_k$ is the channel vector from BS to the $k$-th user
$$
\begin{align*}
\mathbf{h}_{k, 3} =& \ \hat{\mathbf{h}}_{k, 3} + \Delta\mathbf{h}_{k, 3} \in \mathbb{C}^{1 \times N_t} \\
\Delta\mathbf{h}_{k, 3} &= \psi\frac{\mathbf{h}_{k, 3}^{N_2}}{\| \mathbf{h}_{k, 3}^{N_2} \|_2}, \;\; \mathbf{h}_{k, 3}^{N_2} \sim \mathcal{CN}(0, 1)
\end{align*}
$$
- The $\mathbf{H}_1 \in \mathbb{C}^{N_s \times N_t}, \hat{\mathbf{h}}_{k, 2} \in \mathbb{C}^{N_s \times 1}$ and $\hat{\mathbf{h}}_{k, 3} \in \mathbb{C}^{1 \times N_t}$ are modeled as
$$
\begin{align*}
\mathbf{H}_1 &= \kappa_{LoS} \ \textbf{a}_{RIS}(\gamma^r, \eta^r)\textbf{a}_{BS}(\theta^t)^H, \\
\hat{\mathbf{h}}_{k, 2} &= \kappa_{LoS} \ \textbf{a}_{UE}(\theta^r)\textbf{a}_{RIS}(\gamma^t, \eta^t)^H, \\
\hat{\mathbf{h}}_{k, 3} &= \kappa_{NLoS} \ \textbf{a}_{UE}(\theta^r)\textbf{a}_{BS}(\theta^t)^H,
\end{align*}
$$
- where $\textbf{a}_{BS}, \textbf{a}_{RIS}$ and $\textbf{a}_{UE}$ are the steering vectors at the BS, RIS, and UE, respectively
- the values of $AoD$ and $AoA$ are randomly generated between $0$ and $2\pi$
- $\kappa_{LoS}$ and $\kappa_{LoS}$ are the complex channel gains modeled by $\mathcal{CN}(0, 1)$ and $\mathcal{CN}(0, 10^{-0.1\mu})$ with Ricain factor $\mu = 10 dB$
- K. Ying, Z. Gao, S. Lyu, Y. Wu, H. Wang and M. -S. Alouini, "[GMD-Based Hybrid Beamforming for Large Reconfigurable Intelligent Surface Assisted Millimeter-Wave Massive MIMO](https://ieeexplore.ieee.org/abstract/document/8964330)," in *IEEE Access*, vol. 8, pp. 19530-19539, 2020. (Cited by 91)
- $\theta^t$ and $\theta^r$ denote the angles of departure $(AoD)$ at the BS and angles of arrival $(AoA)$ at the UE
- $\gamma^t$ and $\gamma^r$ are the azimuth $AoD$ and $AoA$ at the RIS
- $\eta^t$ and $\eta^r$ are the elevation $AoD$ and $AoA$ at the RIS
- Considering the uniform linear array $(ULA)$ structure at the BS and UE, and the uniform planar array $(UPA)$ structure at the RIS, the steering vectors are expressed as
$$
\begin{align*}
\textbf{a}_{BS}(\theta) =& \; \textbf{a}_{UE}(\theta) = \sqrt{\frac{1}{N}} \cdot [ 1, \ e^{j2\pi\frac{d_{a}}{\lambda}\sin(\theta)}, \, \ldots, \ e^{j2\pi\frac{d_{a}}{\lambda}(N - 1)\sin(\theta)} ]^T \\
\textbf{a}_{RIS}(\gamma, \eta) =& \; \textbf{a}_{y}(\gamma, \eta) \ \otimes \ \textbf{a}_{z}(\eta) \\
=& \; \sqrt{\frac{1}{M}} \cdot [ 1, \ \ldots, \ e^{j2\pi\frac{d_{RIS}}{\lambda}((M_y - 1)\sin(\gamma)\cos(\eta) + (M_z - 1)\sin(\eta))} ]^T \\
\textbf{a}_y(\gamma, \eta) &= \sqrt{\frac{1}{M_y}} \cdot [ 1, \ e^{j2\pi\frac{d_{RIS}}{\lambda}\sin(\gamma)\sin(\eta)}, \ldots, \ e^{j2\pi\frac{d_{RIS}}{\lambda}(M_y - 1)\sin(\gamma)\sin(\eta)} ]^T \\
\textbf{a}_z(\eta) &= \sqrt{\frac{1}{M_z}} \cdot [ 1, \ e^{j2\pi\frac{d_{RIS}}{\lambda}\cos(\eta)}, \ldots, \ e^{j2\pi\frac{d_{RIS}}{\lambda}(M_z - 1)\cos(\eta)} ]^T
\end{align*}
$$
- $d_a$ and $d_{RIS}$ represent the antenna and RIS element spacing, which are set to half wavelength $\frac{\lambda}{2}$
- J. Yuan, Y. -C. Liang, J. Joung, G. Feng and E. G. Larsson, "[Intelligent Reflecting Surface-Assisted Cognitive Radio System](https://ieeexplore.ieee.org/document/9235486)," in *IEEE Transactions on Communications*, vol. 69, no. 1, pp. 675-687, Jan. 2021. (Cited by 130)
- $N$ represents $N_t$ and $N_r$ for the case of BS and UE, respectively
- $M = M_y \times M_z$, where $M_y$ and $M_z$ represent the numbers of horizontal and vertical elements of the RIS

### **Array response implementations**

### **Channel matrices**

## **Training**
### **Setting**

### **Mean episode reward**

## **Random action rewards**

<img src='https://hackmd.io/_uploads/r1DPhQRMa.png' width=50% height=50%>
<img src='https://hackmd.io/_uploads/rJ7gkN0M6.png' width=50% height=50%>

## **TODO list**
- Last week
- [x] Inference
- [ ] CDF Plot
- [x] more anttenas do help
- [x] more users don't affect the performance
- [x] lower bits do have similar performance
- [x] more bits don't actually help
- [x] lower the Tx power doesn't affect the performance
- This week
- [x] 模擬裡面要考慮 RIS element 跟 BS 還有 UE 的距離造成的相位差 (phase shift)
- [x] 每個 RIS element (跟 BS 還有 UE) 經過的通道要不一樣
- [x] 跟 Random action 的比較