# **meeting 09/12**
**Advisor: Prof. Chih-Yu Wang \
Presenter: Shao-Heng Chen \
Date: Sep 12, 2023**
<!-- Chih-Yu Wang -->
<!-- Wei-Ho Chung -->
### **System model**
- Downlink RIS-aided MU-MIMO system
- $N_k$ users with $N_r$ antenna
- one BS with $N_t$ antennas
- one RIS with $N_s$ elements
- each symbol has a length of $L$
- Channel model
- BS-RIS channel matrix $\mathbf{H}_1 \in \mathbb{C}^{N_s \times N_t}$
- RIS-$k$-th-user channel matrix $\mathbf{H}_{2, k} = \hat{\mathbf{H}}_{2, k} + \Delta\mathbf{H}_{2, k} \in \mathbb{C}^{N_r \times N_s}$
- BS-$k$-th-user channel matrix $\mathbf{H}_{3, k} = \hat{\mathbf{H}}_{3, k} + \Delta\mathbf{H}_{3, k} \in \mathbb{C}^{N_r \times N_t}$
- AWGN vector at the $k$-th user $\mathbf{n}_k \in \mathbb{C}^{N_r \times 1}$, $\mathbf{n}_u \sim \mathcal{CN}(0, \sigma_n^2 I_{N_r})$
- Beamforming matrices
- Precoding matrix for $k$-th user $\mathbf{F}_k \in \mathbb{C}^{N_t \times L}$
- Combining matrix for $k$-th user $\mathbf{W}_k \in \mathbb{C}^{N_r \times L}$
- The decoded signal at the $k$-th user
$$
\begin{align*}
\mathbf{y}_k &= \mathbf{W}_k^H (\mathbf{H}_{2, k} \mathbf{\Phi} \mathbf{H}_1 + \mathbf{H}_{3, k}) \mathbf{F}_k \mathbf{x}_k + \mathbf{W}_k^H \mathbf{n}_k \\
&= \mathbf{W}_k^H \ \tilde{\mathbf{H}} \ \mathbf{F}_k \mathbf{x}_k + \mathbf{W}_k^H \mathbf{n}_k
\end{align*}
$$
- $\mathbf{\Phi} \triangleq diag(\phi_1, ..., \phi_{N_s}) \in \mathbb{C}^{N_s \times N_s}$ is the diagonal reflection matrix of the RIS
- $\phi_i = \beta(\varphi_i) \cdot e^{j\varphi_{i}}, \; \forall i = 1, ..., N_s$
- in the simulation, the value of $N_s$ is set to $25$ (quite large?)
- $\beta(\varphi_i) = (1 - \beta_{min}) \cdot (\frac{\sin(\varphi_i - \mu) + 1}{2})^\kappa + \beta_{min}$
- $\varphi_i = \hat{\varphi_i} + \varphi_i'$
- actual phase shift = desired phase shift + phase error
- $\hat{\varphi_i} \in \mathcal{A} = \{ e^{(j\frac{ \; 2\pi n \;}{2^{bits \;}})} \}_{n = 0}^{ 2^{bits - 1}}, \; \forall i = 1, ..., N_s$
- in the simulation, the value of $bits$ is set to $8$, which give us $256$ choices
- $\varphi_i'$ folows the von Mises distribution with PDF $f(\varphi_i'(\mu, \kappa)) = \frac{\;e^{\kappa \cos(\varphi_i' - \mu)}\;\;\;}{\;2\pi I_0(\kappa)\;}$
- $I_0(\kappa)$ is the modified Bessel function of the first kind of order $0$
### **MSE derivation**
- I was wondering if the MSE of the Weighted Minimum Mean Squared Error (WMMSE) is the same as the MSE of the Min-Max MSE?
- The MSE of WMMSE
$$
\begin{align*}
\mathbf{E}_k = E\left\{ \| \mathbf{x}_k - \xi_k^{-1} \mathbf{y}_k \|^2 \right\}
= E\left\{ (\mathbf{x}_k - \xi_k^{-1} \mathbf{y}_k)(\mathbf{x}_k - \xi_k^{-1} \mathbf{y}_k)^H \right\}
\end{align*}
$$
- The MSE of Min-Max MSE
$$
\begin{align*}
\text{MSE} &= E\left\{ tr\{(\mathbf{x}_k - \mathbf{y}_k)(\mathbf{x}_k - \mathbf{y}_k)^H \}\right\}
= tr\left\{ E\{(\mathbf{x}_k - \mathbf{y}_k)(\mathbf{x}_k - \mathbf{y}_k)^H \} \right\} \\
&= tr\left\{ (\mathbf{I}_L - \mathbf{W}_k^H \tilde{\mathbf{H}} \mathbf{F}_k)(\mathbf{I}_L - \mathbf{W}_k^H \tilde{\mathbf{H}} \mathbf{F}_k)^H + \sigma_n^2\mathbf{W}_k^H\mathbf{W}_k \right\}
\end{align*}
$$
- The derivation process
$$
\begin{align*}
&E\left\{ (\mathbf{x}_k - \mathbf{y}_k)(\mathbf{x}_k - \mathbf{y}_k)^H \right\} = \ E\{ (\mathbf{x}_k - (\mathbf{W}_k^H \tilde{\mathbf{H}} \mathbf{F}_k\mathbf{x}_k + \mathbf{W}_k^H\mathbf{n}_k)) ((\mathbf{x}_k - (\mathbf{W}_k^H \tilde{\mathbf{H}} \mathbf{F}_k\mathbf{x}_k + \mathbf{W}_k^H\mathbf{n}_k))^H \} \\ \\
&= \ E\{ (\mathbf{x}_k - \mathbf{W}_k^H \tilde{\mathbf{H}} \mathbf{F}_k\mathbf{x}_k - \mathbf{W}_k^H\mathbf{n}_k) (\mathbf{x}_k - \mathbf{W}_k^H \tilde{\mathbf{H}} \mathbf{F}_k\mathbf{x}_k - \mathbf{W}_k^H\mathbf{n}_k)^H \} \\ \\
&= E\{ (\mathbf{x}_k - \mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k \mathbf{x}_k - \mathbf{W}_k^H\mathbf{n}_k) (\mathbf{x}_k^H - \mathbf{x}_k^H(\mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k)^H - \mathbf{n}_k^H\mathbf{W}_k) \} \;\;\; (\because (\mathbf{AB})^H = \mathbf{B}^H\mathbf{A}^H) \\ \\
&= E\{ \mathbf{x}_k\mathbf{x}_k^H - \mathbf{x}_k\mathbf{x}_k^H(\mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k)^H - \mathbf{x}_k\mathbf{n}_k^H\mathbf{W}_k \\
& \;\;\;\;\;\;\;\;\; - (\mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k)\mathbf{x}_k\mathbf{x}_k^H + \mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k\mathbf{x}_k\mathbf{x}_k^H(\mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k)^H + \mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k\mathbf{x}_k\mathbf{n}_k^H\mathbf{W}_k \\
& \;\;\;\;\;\;\;\;\; - \mathbf{W}^H\mathbf{n}_k\mathbf{x}_k^H + \mathbf{W}^H\mathbf{n}_k\mathbf{x}_k^H(\mathbf{W}_k^H\tilde{\mathbf{H}} \mathbf{F}_k)^H + \mathbf{W}^H\mathbf{n}_k\mathbf{n}_k^H\mathbf{W} \} \\ \\
& \;\;\;\;\;\; (\because E\{ \mathbf{x}_k\mathbf{x}_k^H \} = \mathbf{I}_L, \ E\{ \mathbf{x}_k\mathbf{n}_k^H \} = E\{ \mathbf{x}_k^H\mathbf{n}_k \} = 0, \ E\{ \mathbf{n}_k\mathbf{n}_k^H \} = \sigma_n^2 ) \\
&= E\{ \mathbf{I}_L - (\mathbf{W}_k^H\tilde{\mathbf{H}}\mathbf{F}_k)^H - 0 - \mathbf{W}_k^H\tilde{\mathbf{H}}\mathbf{F}_k + \mathbf{I}_L + 0 - 0 + 0 + \sigma_n^2\mathbf{W}_k^H\mathbf{W}_k \} \\ \\
&= (\mathbf{I}_L - \mathbf{W}_k^H\tilde{\mathbf{H}}\mathbf{F}_k)(\mathbf{I}_L - \mathbf{W}_k^H\tilde{\mathbf{H}}\mathbf{F}_k)^H + \sigma_n^2\mathbf{W}_k^H\mathbf{W}_k
\;\;\; (\because (\mathbf{I} - \mathbf{A}^H)(\mathbf{I} - \mathbf{A}) = \mathbf{I} - \mathbf{A} - \mathbf{A}^H + \mathbf{I})
\end{align*}
$$
- References
- K. -Y. Chen, H. -Y. Chang, R. Y. Chang and W. -H. Chung, "[Hybrid Beamforming in mmWave MIMO-OFDM Systems via Deep Unfolding](https://ieeexplore.ieee.org/document/9860467)," *2022 IEEE 95th Vehicular Technology Conference: (VTC2022-Spring)*, Helsinki, Finland, 2022, pp. 1-7.
<img src='https://hackmd.io/_uploads/ByyKUbs02.png' width=70% height=70%>
- X. Zhao, T. Lin, Y. Zhu and J. Zhang, "[Partially-Connected Hybrid Beamforming for Spectral Efficiency Maximization via a Weighted MMSE Equivalence](https://ieeexplore.ieee.org/document/9467491)," in *IEEE Transactions on Wireless Communications*, vol. 20, no. 12, pp. 8218-8232, Dec. 2021.
<img src='https://hackmd.io/_uploads/By2JUZsCh.png' width=70% height=70%>
<img src='https://hackmd.io/_uploads/HyabLWi0h.png' width=70% height=70%>