how a tetra repeater works

I want to extend the radio coverage and I need to install a repeater to pick up the signal and broadcast it except I’m confused about how it works:
the repeater has a gain = 90 dbm and an output power = 37 dbm. for example the repeater receives a signal equal to -50dbm in this case how its output power will be calculated:
case 1: output power = received signal (-60dbm) + gain (90dbm)
-50+90=40 dbm = 10 watt
case 2: it suffices to capture a signal (greater than the minimum reception limit) so that the repeater to amplify it to its maximum power which is for this repeater equal to 37 dbm

You’ve got yourself tied up with gain figures - if you have a 37dBm output - that’s 5 Watts in understandable units. Your input sensitivity - as for the weak signals, means a likely sensitivity figure around -75-90dBm? so in voltage terms, very little. What are you confused about in practice? You will lose a little going through the filters on in and out, and perhaps if you have a combiner for single antenna operation, that’s another loss - but the point for practical use is that a repeater with decent antenna in the best place beats a semi-duplex radio in performance because of where the antenna is - the filter/combiner losses are known, but so much better performance, the losses are manageable. You pick the output power to be a balance between the input and output path. If all your users are in vehicles with decent antennas an 10-25W output, receive signal strength will be higher, and the path length a little greater. That might need a higher power output to make the in and out paths the same. If your paths are shorter because of low power, hand-held radios, there could be no point in using a high power ouput - they could hear the repeater, but the repeater wouldn’t be reciprocal, it wouldn’t hear them. The lower power output would also make the filtering more efficient.

What are you struggling with?