====== Superposition Encoding ======

===== Demonstrate interests of SupEnc =====


BS        -----------------> Far user (f)
            -----> Near user (u)

==== Signal transmit from the BS ====

$x = \sqrt{1-\alpha} s_n + \sqrt{\alpha} s_f$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_01.png?type=equation}}

==== Capacity of far user ====

$c_f(\alpha) = \log \left(
  1 + \frac{\alpha\mathrm{SNR}_f |h_f|^2}
               {1+(1-\alpha)\mathrm{SNR}_f |h_f|^2}
  \right)$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_02.png?type=equation}}

$\mathrm{SNR}_f =
  \frac{\mathrm{P}_{\mathrm{TX}}}{\sigma^2} \mathrm{L}_f$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_03.png?type=equation}}

==== Capacity of near user ====

$c_n(\alpha) = \log( 1 + (1-\alpha)\mathrm{SNR}_n |h_n|^2)$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_04.png?type=equation}}

==== Assumption of p-% for far user capacity ====

$\frac{c_f(1) - c_f(\alpha)}{c_f(1)} \leq \rho$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_05.png?type=equation}}

Get $\alpha$

compute $c_n(\alpha)$ versus $\mathrm{L}_n$

==== Remark if $\mathrm{SNR}_f$ is small compared to 1 ====

$c_f(\alpha) \approx
  \frac{\alpha\mathrm{SNR}_f |h_f|^2}
         {1+(1-\alpha)\mathrm{SNR}_f |h_f|^2}$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_06.png?type=equation}}

thus

$\alpha = \frac{(1+\mathrm{SNR}_f |h_f|^2)(1-\rho)}
                      {1+\mathrm{SNR}_f |h_f|^2(1-\rho)}$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_07.png?type=equation}}

If $\mathrm{SNR}_f |h_f|^2$ is negligible compared to 1, $\alpha = 1 - \rho$

{{/home/alm002/nobackup/notes/RLT/Superposition_Encoding/equation_08.png?type=equation}}