I have a Rayleigh channel that transmits discrete values according to the following: $$y(n) = h(n) * s_n + \omega_n$$ Where $y$ is the received message, $s_n$ is the symbol sent (constellation 2-PSM), $\omega_n$ is a white noise and $h(n)$ is a Gaussian circular random variable of mean 0 and variance 1. The first thing I am asked to do is determine an $A(n)$ such that, by sending $s_n/A(n)$, I am able to keep the instantaneous power constant. To find this value, I did the following:

  • Did not consider the noise, because it is impossible to know its value each time.
  • Then: $P = |y(n)|^2$. This means that P = $|h(n)|^2 |s_n|^2/P $, where $P$ is the instantaneous power. From there I obtained $A$.

I don't know if this the correct approach but it seems logical to me. Now the part I am stuck at is calculating the average power of the signal. I tried going via the correlation formula: $$P_a = R(0) = \mathbb{E}[|y(n)|^2] $$ But I don't know where to follow from here. I could try integration but I'm not sure on how to do that in a discrete signal. Also, should I take the noise into consideration when calculating the average power?

Reference: This is an exercise from Chapter 9 of Principles of Digital Communications, by R. Gallegher.


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