Detection in MIMO systems

Hi, pls I'll ask about the system model, in many papers the author is use the real-valued equivalent of the complex-valued model to make the derivation ease. Like the equations in attachment, or in matlab code like this: Y=HX+W
H_complex = (randn(Nr,Nt)+1j*randn(Nr,Nt))/sqrt(2); % Flat Rayleigh fading channel
w_complex = sigma(index1)*(randn(Nr,N)+1j*randn(Nr,N))./sqrt(2); % Noise signal
y_complex = H_complex*s,(index2-1)*N+1:index2*N)+w_complex; % The received signal
H = [[real(H_complex) -imag(H_complex)]; [imag(H_complex) real(H_complex)]]; % turn into real-valued model
y = [real(y_complex); imag(y_complex)];
w = [real(w_complex); imag(w_complex)];
now I need someone show me the equations, if I want to use the complex-valued model instead of real-valued model to change my code matlab.;-)