# How to derived an equivalent real-valued system model of the complex value MISO system with beam forming?

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 !