I need mathematical expressions to compare two systems, one transmit and one receive antenna versus one transmit and 2 receive antenna in diversity. How much gain effectively will i get in the second case from 2 antennas in diversity not array, as in first case the antenna gain is already known?
Let me take an example, if I have 2 antennas with 3dBi gain each in receive diversity. In best case i can have no loss on both receive channels and receive twice the power on my receiver, so it means i have additional gain of 3dB. Am I correct in this or missing something??
You need to learn the difference between coherent and incoherent addition of signals. If you receive the signals on both antennas, and convert the power in each to a voltage - that's what a detector does - then add them together, the voltages add and it looks like twice the power of one antenna. Each antenna will also receive noise, and the noise power will also add, to make twice the noise power, so the SNR will not improve.
If you add the two antennas before putting it into a detector then you can get four times the power or 6 dB improvement in gain. The noise will increase by 3 dB so the SNR goes up by 3 dB. Unfortunately, it is also possible to get the two signals in this case to add up to zero so you get no signal. This is what happens when you add up the signals from two antennas coherently. In some directions (and frequencies) you get improvement and in some you get nothing. This is described by the array antenna pattern of the two antennas.
Gain isn't everything - SNR or something equivalent is what matters, so you need to know if noise matters and what happens to the noise as well as the gain.
On reviewing my answer there are some mistakes in it, although the conclusions are not misleading. Rather than me explaining it in detail it would be better if you read about the subject so that you can assess the system you are looking at yourself and select the best solution. Adding powers helps get rid of multipath and I believe is a common solution, but it wasn't what I mainly did, because I worked in coherent radar processing. There should be lots of papers on using channel or antenna diversity.
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