I am developing a system which coherently integrates $L$ sets of complex samples to allow detection of a sub noise floor signal. For one set of samples, I believe the CFAR threshold factor for a probability of false alarm $P_{FA}$ for a filter with $N$ noise cells is given by
$\alpha = N \left( P_{FA}^{-1/N}- 1 \right)$
Though I am unsure how to calculate the threshold factor when applying the filter to the result of the coherent integration.
The received signals are subject only to AWGN, though a general explanation would be of interest.
