Currently I am interested in finding the equivalent real value model of the following system with beam forming vector

$y = h \times \mathop {{w_1} \times {x_1}}\limits_{{\text{WANTED}}\,\,{\text{SIGNAL}}} + h \times \mathop {{w_2} \times {x_2}}\limits_{{\text{INTERFERENCE}}} + n$

Where $h$ is the channel coefficient and $h \in {\mathbb{C}^{2 \times 1}}$.

${w_1},{w_2}$ is the beam forming vector and ${w_1} \in {\mathbb{C}^{2 \times 1}},{w_2} \in {\mathbb{C}^{2 \times 1}}$

$E\left\{ {{{\left| {{x_1}} \right|}^2}} \right\} = E\left\{ {{{\left| {{x_2}} \right|}^2}} \right\} = 1$.For simplicity sake, the symbol energy is assume to be unity

$n$ is the additive gaussian noise

The dimension is $2\times1$ because it is assume that the transmitter has $2$ transmit antenna and receiver has $1$ receive antenna

$\mathbb{C}$ means that $h,{w_1},{w_2}$ would take on complex values.

I have been looking for such transformation and found an existing result in the literature

The complex value MIMO system model could be presented as $y = Hs + n$

The real equivalent system model \begin{array}{l} \widetilde y = \widetilde H\widetilde s + \widetilde n\\ \widetilde y = \left[ {\begin{array}{*{20}{c}} {{\rm{Real}}\left( y \right)}\\ {{\rm{Imag}}\left( y \right)} \end{array}} \right],\widetilde s = \left[ {\begin{array}{*{20}{c}} {{\rm{Real}}\left( s \right)}\\ {{\rm{Imag}}\left( s \right)} \end{array}} \right],\widetilde n = \left[ {\begin{array}{*{20}{c}} {{\rm{Real}}\left( n \right)}\\ {{\rm{Imag}}\left( n \right)} \end{array}} \right]\\ \tilde H = \left[ {\begin{array}{*{20}{c}} {{\rm{Real}}\left( H \right)}&{{\rm{ - Imag}}\left( H \right)}\\ {{\rm{Imag}}\left( H \right)}&{{\rm{Real}}\left( H \right)} \end{array}} \right] \end{array} I do not know if this result have any implication toward my original problem and I do not know how to proof this rigorously but this has been my best effort so far.

Could you kindly help me with this problem.

Thank you for your enthusiast !


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