Probability of False Alarm versus SNR I'm currently testing the performance of an energy detector using two tests. One test involves finding the optimal threshold to achieve a probability of detection(Pd) of 0.9 and a probability of false Alarm (Pfa) of 0.1.

Having completed the first test, the next test I want to do is to see how varying SNR affects both Pd and Pfa, given a constant threshold. If Pd is the probability of the detector detecting a signal given the signal of interest is being transmitted and pfa is the probability of the detector detecting a signal given the signal of interest is not being transmitted how exactly can I vary SNR to get Pfa. I'm sure its something that I'm not quite understanding but if the signal of interest is not being transmitted and all that the receiver is picking up is noise, how is there a SNR in the case of Pfa?
Hi: In statistics we call that the probability of type I error ( rejecting when true ) and type II error ( accepting when false ). The way it's done there is that, once you make an assumption about the distribution of the data ( i.e: normal, t, whatever ), you decide on the null and the alternative, along with what you want the P(type I error ) to be ( say 0.05 ) and then, given the assumption about the distribution, your type II error probability is then known or can be calculated ( sometimes with difficulty ). Your case may be slightly different but I would think that a statistical signal processing book would have a "hypothesis testing" section which would explain what I am describing in more detail.  I hope this helps a little. I put it in the answer section even though it's not really an answer because it was too long for a comment.