A Rayleigh-faded channel is often written as
which is short-hand for saying that
where is a normal distribution with mean and variance .
We have in the case of .
The magnitude of the channel follows a Rayleigh distribution where whose PDF is
or, taking , is equivalent to
Note that when , .
Sometimes the Rayleigh distribution is parameterized by . Other times it is parameterized by .
The distribution of follows an exponential distribution with PDF
where (when ) and whose mean is , implying that .
In other words, the average channel power is 1. This is convenient for abstracting the large-scale path loss (e.g., Friis) and large-scale fading (e.g., log-normal shadowing) from the small-scale fading distribution (e.g., Rayleigh fading).
One could also define use to write the PDF as
Note that can also be written as having distribution
where now it’s clear that is the average power of (i.e., its variance is its average power since is zero mean).