meeting 09/26 - HackMD

# **meeting 09/26** **Advisor: Prof. Chih-Yu Wang \ Presenter: Shao-Heng Chen \ Date: Sep 26, 2023**   ### **System model** - Downlink RIS-aided MU-MISO system - $N_k$ single-antenna users ($N_r = 1$) - one BS with $N_t$ antennas - one RIS with $N_s$ elements - Channel model - BS-RIS channel matrix $\mathbf{H}_1 \in \mathbb{C}^{N_s \times N_t}$ - RIS-user channel matrix $\mathbf{H}_{2} = \hat{\mathbf{H}}_{2} + \Delta\mathbf{H}_{2} \in \mathbb{C}^{N_k \times N_s}$ - BS-user channel matrix $\mathbf{H}_{3} = \hat{\mathbf{H}}_{3} + \Delta\mathbf{H}_{3} \in \mathbb{C}^{N_k \times N_t}$ - AWGN vector $\mathbf{n} = [n_1, n_2, \ldots, n_{N_k}]^T \in \mathbb{C}^{N_k \times 1}$, $n_k \sim \mathcal{CN}(0, \sigma_n^2) \; \forall k$ - Transmit beamformer - Precoding matrix $\mathbf{F} \in \mathbb{C}^{N_t \times N_k}$  - The received signal at the $k$-th users $$ \begin{align*} y_k &= (\mathbf{h}_{k, 2}^T \mathbf{\Phi} \mathbf{H}_1 + \mathbf{h}_{k, 3}^T) \mathbf{F} \mathbf{x} + n_k \end{align*} $$ - $y_k$ is the received signal (complex scalar) - $n_k$ is the additive receiver noise at the $k$-th user, $n_k \sim \mathcal{CN}(0, \sigma_n^2)$ for all $k$ - $\mathbf{x} \in \mathbb{C}^{N_k \times 1}$ is the data streams for $N_k$ users - $\mathbf{F} \in \mathbb{C}^{N_t \times N_k}$ is the transmission beamforming matrix - $\mathbf{H}_1 \in \mathbb{C}^{N_s \times N_t}$ is the BS-RIS 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 $$ \begin{align*} \mathbf{h}_{k, 2} =& \ \hat{\mathbf{h}}_{k, 2} + \Delta\mathbf{h}_{k, 2} \in \mathbb{C}^{N_s \times 1} \\ \Delta\mathbf{h}_{k, 2} &= \psi\frac{\mathbf{h}_{k, 2}^{N_1}}{\| \mathbf{h}_{k, 2}^{N_1} \|_2}, \;\; \mathbf{h}_{k, 2}^{N_1} \sim \mathcal{CN}(0, 1) \end{align*} $$ - $\mathbf{h}_{k, 3} \in \mathbb{C}^{N_t \times 1}, \forall k = 1, ..., N_k$ is the channel vector from BS to the $k$-th user $$ \begin{align*} \mathbf{h}_{k, 3} =& \ \hat{\mathbf{h}}_{k, 3} + \Delta\mathbf{h}_{k, 3} \in \mathbb{C}^{N_t \times 1} \\ \Delta\mathbf{h}_{k, 3} &= \psi\frac{\mathbf{h}_{k, 3}^{N_2}}{\| \mathbf{h}_{k, 3}^{N_2} \|_2}, \;\; \mathbf{h}_{k, 3}^{N_2} \sim \mathcal{CN}(0, 1) \end{align*} $$ - The 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 (LOS **Rician** fading) - $\mathbf{H}_{2} \in \mathbb{C}^{N_k \times N_s}, \forall k = 1, ..., N_k$ is the RIS-$k$-th user channel (LOS **Rician** fading) - $\mathbf{H}_{3} \in \mathbb{C}^{N_k \times N_t}$ is the BS-user channel (direct path, NLOS **Rayleigh** fadding channel) - $\mathbf{\Phi} \triangleq diag(\phi_1, ..., \phi_{N_s}) \in \mathbb{C}^{N_s \times N_s}$ 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 $0$ ### **Problem formulation** - The objective is to minimize the worst case MSE (min-max MSE) $$ \begin{align*} \min\limits_{\boldsymbol{\Phi}} \;\; &\max\limits_{\Delta\mathbf{h}_{k, 2}, \; \Delta\mathbf{h}_{k, 3} \\ \; \forall k = 1, \ ..., \ N_k } \;\;\;\;\; \alpha \\ \textrm {s.t.} \;\; & \alpha \geq 0, \\ & E\left\{ (x_k - y_k)(x_k - y_k)^H \right\} \leq \alpha, \forall k = 1, \ldots, N_k, \\ & tr\{ \mathbf{F}\mathbf{F}^{H} \} \leq P_{t}, \\ & \| \phi_i \|_2^2 = \beta(\varphi_i) = (1 - \beta_{min}) \cdot (\frac{\sin(\varphi_i - \mu) + 1}{2})^\kappa + \beta_{min}, \; \beta_{min} \geq 0, \; \mu\geq 0, \;\; \forall i = 1, \ldots, N_s, \\ & \varphi_i = \hat{\varphi_i} + \varphi_i' \ , \; \varphi_i \in [0, 2\pi), \;\; \forall i = 1, \ldots, N_s, \\ & \hat{\varphi_i} \in \mathcal{A} = \{ e^{(j\frac{ \; 2\pi n \;}{2^{bits \;}})} \}_{n = 0}^{ 2^{bits - 1}}, \; f(\varphi_i'(\mu, \kappa)) = \frac{\;e^{\kappa \cos(\varphi_i' - \mu)}\;\;\;}{\;2\pi I_0(\kappa)\;}, \;\forall i = 1, \ldots, N_s, \\ & \| \Delta\mathbf{h}_{k, 2} \|_2 \leq \psi, \; \forall k = 1, \ldots, N_k, \\ & \| \Delta\mathbf{h}_{k, 3} \|_2 \leq \psi, \; \forall k = 1, \ldots, N_k. \end{align*} $$ ### **MSE derivation** - The resulted MSE of our Min-Max MSE problem $$ \begin{align*} \text{MSE} = E\left\{ (\mathbf{x} - \mathbf{y})(\mathbf{x} - \mathbf{y})^H \right\} = (\mathbf{I}_{N_k} - \tilde{\mathbf{H}} \mathbf{F})(\mathbf{I}_{N_k} - \tilde{\mathbf{H}} \mathbf{F})^H + \sigma_n^2\mathbf{I}_{N_k} \end{align*} $$ - The stacked received signal of all $N_k$ users $$ \begin{align*} \mathbf{y} &= \tilde{\mathbf{H}}\mathbf{F} \mathbf{x} + \mathbf{n} = (\mathbf{H}_{2} \mathbf{\Phi} \mathbf{H}_1 + \mathbf{H}_{3}) \mathbf{F} \mathbf{x} + \mathbf{n} \end{align*} $$ - The derivation process $$ \begin{align*} &E\left\{ (\mathbf{x} - \mathbf{y})(\mathbf{x} - \mathbf{y})^H \right\} = E\{ (\mathbf{x} - ( \tilde{\mathbf{H}} \mathbf{F}\mathbf{x} + \mathbf{n})) ((\mathbf{x} - (\tilde{\mathbf{H}} \mathbf{F}\mathbf{x} + \mathbf{n}))^H \} \\ \\ &= \ E\{ (\mathbf{x} - \tilde{\mathbf{H}} \mathbf{F}\mathbf{x} - \mathbf{n}) (\mathbf{x} - \tilde{\mathbf{H}} \mathbf{F}\mathbf{x} - \mathbf{n})^H \} \\ \\ &= E\{ (\mathbf{x} - \tilde{\mathbf{H}} \mathbf{F} \mathbf{x} - \mathbf{n}) (\mathbf{x}^H - \mathbf{x}^H(\tilde{\mathbf{H}} \mathbf{F})^H - \mathbf{n}^H) \} \;\;\; (\because (\mathbf{AB})^H = \mathbf{B}^H\mathbf{A}^H) \\ \\ &= E\{ \mathbf{x}\mathbf{x}^H - \mathbf{x}\mathbf{x}^H(\tilde{\mathbf{H}} \mathbf{F})^H - \mathbf{x}\mathbf{n}^H - (\tilde{\mathbf{H}} \mathbf{F})\mathbf{x}\mathbf{x}^H + \tilde{\mathbf{H}} \mathbf{F}\mathbf{x}\mathbf{x}^H(\tilde{\mathbf{H}} \mathbf{F})^H + \tilde{\mathbf{H}} \mathbf{F}\mathbf{x}\mathbf{n}^H - \mathbf{n}\mathbf{x}^H + \mathbf{n}\mathbf{x}^H(\tilde{\mathbf{H}} \mathbf{F})^H + \mathbf{n}\mathbf{n}^H \} \\ \\ & \;\;\;\;\;\; (\because E\{ \mathbf{x}\mathbf{x}^H \} = \mathbf{I}_{N_k}, \ E\{ \mathbf{x}\mathbf{n}^H \} = E\{ \mathbf{x}^H\mathbf{n} \} = 0, \ E\{ \mathbf{n}\mathbf{n}^H \} = \sigma_n^2 ) \\ &= E\{ \mathbf{I}_{N_k} - (\tilde{\mathbf{H}}\mathbf{F})^H - 0 - \tilde{\mathbf{H}}\mathbf{F} + (\tilde{\mathbf{H}}\mathbf{F})\mathbf{I}_{N_k}(\tilde{\mathbf{H}}\mathbf{F})^H + 0 - 0 + 0 + \sigma_n^2 \} \\ \\ &= \mathbf{I}_{N_k} - (\tilde{\mathbf{H}}\mathbf{F})^H - \tilde{\mathbf{H}}\mathbf{F} + (\tilde{\mathbf{H}}\mathbf{F})(\tilde{\mathbf{H}}\mathbf{F})^H + \sigma_n^2\mathbf{I}_{N_k} \\ \\ &= (\mathbf{I}_{N_k} - \tilde{\mathbf{H}}\mathbf{F})(\mathbf{I}_{N_k} - \tilde{\mathbf{H}}\mathbf{F})^H + \sigma_n^2\mathbf{I}_{N_k} \;\;\; (\because (\mathbf{I} - \mathbf{A})(\mathbf{I} - \mathbf{A})^H = (\mathbf{I} - \mathbf{A})(\mathbf{I} - \mathbf{A}^H) = \mathbf{I} - \mathbf{A}^H - \mathbf{A} + \mathbf{A}\mathbf{A}^H) \\ \\ &= (\mathbf{I}_{N_k} - (\mathbf{H}_{2} \mathbf{\Phi} \mathbf{H}_1 + \mathbf{H}_{3})\mathbf{F})(\mathbf{I}_{N_k} - (\mathbf{H}_{2} \mathbf{\Phi} \mathbf{H}_1 + \mathbf{H}_{3})\mathbf{F})^H + \sigma_n^2\mathbf{I}_{N_k} \end{align*} $$ - MSE simulation - With 10000 random samples ![](https://hackmd.io/_uploads/ryoo6Mygp.png) ```shell min MSE: 4.1847501204568065 max MSE: 889.880246708903 average: 112.24177325877862 median: 91.77701858761456 ``` - Phase Error distribution with 100000 samples ![](https://hackmd.io/_uploads/SJZwfmkxp.png) ```python location_mu = 0.43*np.pi, concentration_kappa = 1.5 ```