Understanding Communication Theory Jargon for Statistical Linear Models Coming from a statistical background, I understand the following formula:

\begin{equation}
y = X\beta + \epsilon
\end{equation}

as a linear regression model, with data input-output pair: $(X,y)$, noise term, $\epsilon$, and regression coefficient vector $\beta$. 

Now I am confused in how this system is described in communication theory. I see terms such as Rayleigh Fading, MIMO, Multi-access, Multi-antenna, Band-limited etc ... in relation to equations that look exactly like statistical linear regression. 

So far I suppose I can work out that for communication theory,

\begin{equation}
y = Hx + \epsilon
\end{equation}

$x$ is some input signal at a transmitter antenna, $\epsilon$ is noise again, $y$ is the signal at the receiver antenna ... but what is $H$? I don't understand how this term should be described, and how it is modelled with respect to all those previous terms I have listed? Reading journal papers I see the simple system as above described with lots of jargon I can't follow, and I'm unaware of the implications this has on the modeling.

If someone could clarify all this jargon and connections for me that would be suuuper helpful!
The equation $y =  hs + n$ is a discrete-time model of a certain kind of wireless communications system.


$s$ is a complex number, drawn from a finite set called a "constellation", that represents the information being transmitted.
$n$ is a complex, circularly-symmetric Gaussian random variable with zero mean and variance usually denoted by $N_0$.
$h$ represents the effect of the communications channel. In this simple model of a wireless channel, $h$ is a complex Gaussian random variable with zero mean and variance one. This model is called Rayleigh-fading because $|h|^2$ is a Rayleigh-distributed random variable.
$y$ is the received complex number. From it, the receiver must make an estimate of $s$. This estimate is usually calculated by first estimating $h$, and then finding $$ y' = \frac{h^* y}{|h|^2} = s + \frac{h^*n}{|h|^2} $$ Here you can see the importance of $|h|^2$ to enhance the noise, which is why the channel is called Rayleigh. The estimate of $s$ is calculated by finding the constellation element that is closest (in Euclidean-distance sense) to $y'$.