One oversimplified method to look for a valid signal in a frequency band is to first compute the FFT of the signal in the desired frequency band and then ascertain if any point in FFT is crossing a detection threshold. This can be achieved by computing the ratio of the maximum of the FFT output to that of the sample mean value. There are host of other applications that use detection procedure which is very similar to this max-mean ratio test.
My task is to find a closed form solution for the probability of false detection in this case. Typically when there is no signal the output of the detection process is a normally distributed data with certain mean and variance. Once in a while this normally distributed data can generate such high signal level that false detection are triggered. So the criterion for false detection is if the ratio of maximum to the sample mean of a random variable is exceeding a certain threshold (m).
Probability of false detection is
If I was able to derive the distribution of Y/Z then the above probability is just the CDF of this distribution computed at m. However, to derive this distribution, the joint distribution of Y and Z has to be first derived. A monte-carlo simulation of 10000 sample runs showed that Y and Z are weakly correlated with a correlation coefficient of 0.08.
Solution 1
The simplest approximation would be to assume N to be large enough so that Z has very low variance and becoming a constant. Hence
where
However for N=5, the plots starts to diverge at the extremes
Solution 2
To make the situation complicated (which I am really good at) you can approximate the joint distributions between Y and Z using copula. So I am going to pick the Gaussian copula and the correlation coefficient between Y and Z is
where
The apply a transformation of variables as X = V and Y = W + mV to derive the joint distribution of V and W. Finally integrate out V and you will get the distribution of W which is Y-mZ. Now compute the CDF of W at 0 and that will be the solution.
When I get time to implement that I will use a numerical solution to compute that and see how close this result will match with the CDF plot when N=5.