Real and complex two-user Gaussian interference channels

The two-user Gaussian interference channel has independent messages, separate encoders, and outputs $Y_1=h_{11}{X}_1+h_{21}{X}_2+Z_1,Y_2=h_{12}{X}_1+h_{22}{X}_2+Z_2\mathrm{.}$ For the real model, $X_i,h_{ji},Z_j$ are real and $Z_j\sim N(0,1)$. For the complex model, the inputs and gains are complex and $Z_j\sim \mathcal{C}\mathcal{N}(0,1)$ with density ${\pi }^{-1}{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2}$. The receiver-noise variables are independent across receivers and channel uses and independent of the messages, so the channel is memoryless. Each power budget satisfies $P_i\ge 0$. A length-$n$ code has independent uniform messages $M_i$ on $[m_i]$ for integers $m_i\ge 1$, deterministic separate encoders $X_i^n=f_i(M_i)$, and decoders ${\widehat{M}}_i=g_i(Y_i^n)$; no feedback is present. The expected average-power constraints are $n^{-1}{\sum }_{t=1}^n\mathbb{E}[X_{i,t}^2]\le P_i$ in the real model and $n^{-1}{\sum }_{t=1}^n\mathbb{E}\mathrm{\mid }{X}_{i,t}{\mathrm{\mid }}^2\le P_i$ in the complex model, with expectations under the uniform message law. The average joint error is $\mathrm{Pr}\{({\widehat{M}}_1,{\widehat{M}}_2)\mathrm{\ne }(M_1,M_2)\}$. For either model define ${\mathrm{S}\mathrm{N}\mathrm{R}}_1=\mathrm{\mid }{h}_{11}{\mathrm{\mid }}^2{P}_1,{\mathrm{S}\mathrm{N}\mathrm{R}}_2=\mathrm{\mid }{h}_{22}{\mathrm{\mid }}^2{P}_2,{\mathrm{I}\mathrm{N}\mathrm{R}}_1=\mathrm{\mid }{h}_{21}{\mathrm{\mid }}^2{P}_2,{\mathrm{I}\mathrm{N}\mathrm{R}}_2=\mathrm{\mid }{h}_{12}{\mathrm{\mid }}^2{P}_1\mathrm{.}$ Rates are bits per use, $R_i=n^{-1}{\log }_2{m}_i$. For a scalar unit-noise AWGN link $Y=aX+Z$ with power constraint $P$, the capacity is $\kappa {\log }_2(1+\mathrm{\mid }a{\mathrm{\mid }}^{2}P)$, attained by a centered real Gaussian input or a circular complex Gaussian input; $\kappa =1\mathrm{/}2$ in the real model and $\kappa =1$ in the complex model.