# MIMO System Model in Matlab

I have this lame doubt which has ended up confusing me which is probably trivial. I am modelling a MU-MIMO system with a Rayleigh fading channel. I have created a wanted signal as an OFDM signal, and generated a 3D Rayleigh Channel matrix over all sub carriers. I want to now multiply the transmit vector(OFDM signal) to the 3D channel matrix. Should i convert both the channel matrix and wanted signal in frequency domain and add noise

$y(f) = H(f)x(f)+n(f)$, and then convert it back to time domain $\textrm{ifft}(y(f))$ and provide an estimate using MRC, ZF or MMSE methods.

$\hat{y}(t) = H(t)’ y(t)$

Or is it just a direct multiplication of

$y(t)=H(t)x(t)+n(t)$

as i understand from most of the literature (y = H x+n)is the usual system model in time domain. If so why isn't it convolved when its combined in time domain, similar to any communication system, where the input vector is convolved with the channel matrix to obtain the output vector in time.

## 1 Answer

In a wireless system, most of the noise is added by the receiver itself. The received signal is very weak (-100 dB or less is typical), and its power is comparable to the thermal noise of the receiver's amplifiers. That is the reason why noise is modeled as being added to the signal present in the antenna.

The noise still affects the estimated signal, of course; if the received vector is $r$, a naive ZF receiver would calculate

\begin{align} \hat{y} &= H^{-1}r \\ &= H^{-1}[Hy+n] \\ &= y + Hn. \end{align}

Since the noise $Hn$ is correlated, this receiver has terrible performance.