How to derived an equivalent real-valued system model of the complex value MIMO system?

Question

I have been reading this paper

"Fifty Years of MIMO Detection: The Road to Large-Scale MIMOs" and see an equivalent real-values system model of the complex value MIMO system without any proof or at least any intuition

The complex value system model

$y = Hs + n$

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 really interested in the detail of this presentation.

Answer 1

Try to decompose into the real and imaginary components, i.e., \begin{align} y := y_{\rm R} + i y_{\rm I}, \end{align} where $y_{\rm R} := \operatorname{Real}\{y\} $ and $y_{\rm I} := \operatorname{Imag}\{y\} $.

Similarly, $H := H_{\rm R} + i H_{\rm I}$, $s = s_{\rm R} + i s_{\rm I}$, and $n = n_{\rm R} + i n_{\rm I}$.

Now, take your linear complex-valued system model and replace with the appropriated decomposed signal components, i.e., \begin{align} y &= Hs + n\\ \Longleftrightarrow &\\ y_{\rm R} + i y_{\rm I} &= \left( y_{\rm R} + i y_{\rm I} \right) \left( s_{\rm R} + i s_{\rm I} \right) + \left( n_{\rm R} + i n_{\rm I} \right) \\ &= \left( H_{\rm R} s_{\rm R} - H_{\rm I} s_{\rm I} \right) + i \left( H_{\rm I} s_{\rm R} + H_{\rm R} s_{\rm I} \right) \\ &= \left[1 \quad i \right] \begin{bmatrix} H_{\rm R} & -H_{\rm I} \\ H_{\rm I} & H_{\rm R} \end{bmatrix} \begin{bmatrix} s_{\rm R} \\ s_{\rm I} \end{bmatrix} + \left[1 \quad i \right] \begin{bmatrix} n_{\rm R} \\ n_{\rm I} \end{bmatrix}. \end{align}

In the real-valued system model, just treat the real and imaginary components as different coordinates per se. I hope it helps now.