Let's break down the problem and provide a **step-by-step solution** along with a **MATLAB code** for plotting the **sum rate** of RSMA, NOMA, and OMA considering **imperfect SIC**. --- ## **Step 1: Understanding the Concepts** - **RSMA:** Rate-Splitting Multiple Access. Users decode a common message and private messages. - **NOMA:** Non-Orthogonal Multiple Access. Users share power domain, and SIC (Successive Interference Cancellation) is performed to separate signals. - **OMA:** Orthogonal Multiple Access. Users are allocated orthogonal resources (e.g., time or frequency). - **Imperfect SIC:** In NOMA/RSMA, SIC is not perfect; residual interference remains after cancellation. This is often modeled by a parameter `epsilon` (0: perfect, >0: imperfect). --- ## **Step 2: Define System Model** - **2 users** (User 1: weak, User 2: strong) - **Channel gains:** \( h_1 < h_2 \) - **Transmit SNR:** Vary SNR from 0 to 30 dB - **Imperfect SIC factor:** `epsilon` (e.g., 0.01) - **Power allocation:** \( a_1, a_2 \) for NOMA/RSMA --- ## **Step 3: Rate Formulas** ### **OMA** Each user gets half the resource (0.5): \[ R_{OMA,1} = 0.5 \cdot \log_2(1 + \text{SNR} \cdot |h_1|^2) \] \[ R_{OMA,2} = 0.5 \cdot \log_2(1 + \text{SNR} \cdot |h_2|^2) \] ### **NOMA (with imperfect SIC, two users)** Power allocation: \( a_1 + a_2 = 1,\, a_1 > a_2 \) - **User 1 (weak):** Decodes its own signal treating User 2 as interference. \[ R_{NOMA,1} = \log_2\left(1 + \frac{a_1 SNR |h_1|^2}{a_2 SNR |h_1|^2 + 1}\right) \] - **User 2 (strong):** Decodes User 1 first (imperfectly), then its own with residual interference \( \epsilon \). \[ R_{NOMA,2} = \log_2\left(1 + \frac{a_2 SNR |h_2|^2}{\epsilon a_1 SNR |h_2|^2 + 1}\right) \] ### **RSMA (simplified 2-user with imperfect SIC)** - Common part: Split a fraction of power for common message (\( a_c \)), rest for private (\( a_1, a_2 \)), \( a_c + a_1 + a_2 = 1 \). - Both users decode common message first, then private. Residual interference for imperfect SIC (private part). - To keep it simple, we assume \( a_c = 0.2,\, a_1 = 0.4,\, a_2 = 0.4 \). - **Common message rate:** \[ R_c = \min\left\{ \log_2\left(1 + \frac{a_c SNR |h_1|^2}{(a_1 + a_2) SNR |h_1|^2 + 1}\right), \log_2\left(1 + \frac{a_c SNR |h_2|^2}{(a_1 + a_2) SNR |h_2|^2 + 1}\right) \right\} \] - **Private rates (with imperfect SIC):** \[ R_{p,1} = \log_2\left(1 + \frac{a_1 SNR |h_1|^2}{\epsilon(a_2) SNR |h_1|^2 + 1}\right) \] \[ R_{p,2} = \log_2\left(1 + \frac{a_2 SNR |h_2|^2}{\epsilon(a_1) SNR |h_2|^2 + 1}\right) \] - **Total sum rate for RSMA:** \[ R_{RSMA} = R_c + R_{p,1} + R_{p,2} \] --- ## **Step 4: MATLAB Code** ```matlab % Parameters SNR_dB = 0:2:30; SNR = 10.^(SNR_dB/10); h1 = 0.8; % Weak user h2 = 2.0; % Strong user epsilon = 0.01; % Imperfect SIC factor % Power allocation a1 = 0.8; a2 = 0.2; % NOMA (User 1: more power) ac = 0.2; ap1 = 0.4; ap2 = 0.4; % RSMA sumrate_OMA = zeros(size(SNR)); sumrate_NOMA = zeros(size(SNR)); sumrate_RSMA = zeros(size(SNR)); for k = 1:length(SNR) snr = SNR(k); % OMA r1_oma = 0.5 * log2(1 + snr * h1^2); r2_oma = 0.5 * log2(1 + snr * h2^2); sumrate_OMA(k) = r1_oma + r2_oma; % NOMA with imperfect SIC r1_noma = log2(1 + (a1*snr*h1^2) / (a2*snr*h1^2 + 1)); r2_noma = log2(1 + (a2*snr*h2^2) / (epsilon*a1*snr*h2^2 + 1)); sumrate_NOMA(k) = r1_noma + r2_noma; % RSMA (simplified) rc1 = log2(1 + (ac*snr*h1^2) / ((ap1+ap2)*snr*h1^2 + 1)); rc2 = log2(1 + (ac*snr*h2^2) / ((ap1+ap2)*snr*h2^2 + 1)); rc = min(rc1, rc2); rp1 = log2(1 + (ap1*snr*h1^2) / (epsilon*ap2*snr*h1^2 + 1)); rp2 = log2(1 + (ap2*snr*h2^2) / (epsilon*ap1*snr*h2^2 + 1)); sumrate_RSMA(k) = rc + rp1 + rp2; end % Plot figure; plot(SNR_dB, sumrate_RSMA, 'r-o','LineWidth',2); hold on; plot(SNR_dB, sumrate_NOMA, 'b-s','LineWidth',2); plot(SNR_dB, sumrate_OMA, 'k--','LineWidth',2); grid on; xlabel('SNR (dB)'); ylabel('Sum Rate (bits/s/Hz)'); legend('RSMA','NOMA (imperfect SIC)','OMA','Location','NorthWest'); title('Sum Rate Comparison: RSMA, NOMA (imperfect SIC), OMA'); ``` --- ## **Step 5: Explanation and Final Answer** - The code **plots the sum rate** of RSMA, NOMA (imperfect SIC), and OMA for two users as a function of SNR. - **Imperfect SIC** is modeled by the parameter `epsilon` (set to 0.01). - **RSMA** generally outperforms NOMA and OMA, especially at high SNR and under imperfect SIC. - You can **adjust** `h1`, `h2`, `epsilon`, and power allocation to simulate different scenarios. --- ### **Final Answer:** **The above MATLAB code will plot the sum rate versus SNR for RSMA, NOMA (with imperfect SIC), and OMA. The consideration of imperfect SIC is reflected in the NOMA and RSMA formulas via the parameter `epsilon`.** If you run this code, you will see RSMA achieves higher sum rates than NOMA (with imperfect SIC), which in turn outperforms OMA. The gap increases as SNR increases and as SIC imperfection becomes more significant.