variance $ s $. AWGN $ z $ gets added to the received signal at the destination, $z$ $\sim \mathcal{CN}(0,\sigma^2)$. User-$i$ transmits a symbol $ x_i $ from a complex finite constellation $ \mathcal{S}_i $ (like $M$-QAM or $M$-PSK) of unit average energy, *i.e*, $\mathbb{E} [\mid x_i \mid ^2]=1$. Let $ P $ be the average power constraint for each user. The received signal at the destination is thus represented by $$\begin{aligned}
\nonumber
y=\sqrt{P}h_1x_1+\sqrt{P}h_2x_2+z.\end{aligned}$$ We assume that perfect CSI *i.e.* the tuple $ (h_1,h_2) $ is available only at the destination.

![Two-user fading MAC with Gaussian noise[]{data-label="fig:mac"}](mac.eps){width="3in"}

At the destination the system can be viewed as a single user AWGN channel with the symbols drawn from a sum constellation $$\begin{aligned}
\nonumber
\mathcal{S}_{\text{sum}} &= \sqrt{P} h_1\mathcal{S}_1+ \sqrt{P} h_2\mathcal{S}_2\\
\nonumber
&= \sqrt{P} h_1(\mathcal{S}_1+\dfrac{h_2}{h_1}\mathcal{S}_2)\\
\label{eqneff}
&= \sqrt{P} h_1\underbrace{(\mathcal{S}_1+\gamma e^{j\theta}\mathcal{S}_2)}_{\mbox{$\mathcal{S}_{\text{eff}}$}},\end{aligned}$$ where $\gamma= | \frac{h_2}{h_1}|$, $\theta = \angle \frac{h_2}{h_1}$ and $\mathcal{S}_{\text{eff}}$ denotes the effective constellation.

Without loss of generality it can be assumed that $ \gamma \geq 1 $, as destination has knowledge of both $h_1$ and $h_2$ separately. If $\vert \frac{h_2}{h_1}\vert < 1$, then at the destination the ratio can be simply reversed to compute $\frac{h_1}{h_2}$. Which one among the two ratios is calculated 