The Two-User Interference Channel

The two-user discrete memoryless interference channel consists of:

• Two input alphabets $\mathcal{X}_1, \mathcal{X}_2$
• Two output alphabets $\mathcal{Y}_1, \mathcal{Y}_2$
• A channel transition probability $P_{Y_1, Y_2 | X_1, X_2}(y_1, y_2 | x_1, x_2)$

Encoder $k$ maps message $M_k \in [1:2^{nR_{k}}]$ to a codeword $x_k^n(M_k)$. Decoder $k$ observes $Y_k^n$ and produces an estimate $\hat{M}_k$. A rate pair $(R_{1}, R_{2})$ is achievable if $P_e^{(n)} = \Pr(\hat{M}_1 \neq M_1 \text{ or } \hat{M}_2 \neq M_2) \to 0$.

The capacity region $C_{\text{IC}}$ is the closure of all achievable rate pairs.

The standard Gaussian IC model is: $$Y_1 = X_1 + a X_2 + Z_1$$ $$Y_2 = b X_1 + X_2 + Z_2$$ where:

• $X_k$ is the signal from transmitter $k$, with power constraint $\mathbb{E}[X_k^2] \leq P_k$
• $a$ is the interference channel gain from Tx 2 to Rx 1
• $b$ is the interference channel gain from Tx 1 to Rx 2
• $Z_k \sim \mathcal{N}(0, \sigma^2)$ is i.i.d. Gaussian noise

The key parameters are:

• $\text{SNR}_{1} = P_1 / \sigma^2$ and $\text{SNR}_{2} = P_2 / \sigma^2$ (signal-to-noise ratios)
• $\text{INR}_{12} = a^2 P_2 / \sigma^2$ (interference from Tx 2 at Rx 1)
• $\text{INR}_{21} = b^2 P_1 / \sigma^2$ (interference from Tx 1 at Rx 2)