Non-coherent Integration For Signal Detection

N-Point FFT on a standard gaussian noise results in real and imaginary results that are gaussian distributed with variance N/2. This results in the magnitude of the FFT being Rayleigh distributed with mode parameter

. Assuming

 represents the magnitude of the FFT output at frequency i for the jth non-coherent integration time. Then after M non-coherent integrations (that is just adding the output of N-point FFTs on top of each other) the magnitude at the ith frequency is given by 

. If 

 are independent variables arising from gaussian noise then the variance is now 

. if 

 is a valid signal then

 resulting in the variance being 

 . So the ratio of the  variance of the valid signal to that of the gaussian noise after M non-coherent integration has increased by M. The video below is an example of this non-coherent integration method in action.

Binary-FSK signal that shifts between 1000KHz and 1200KHz is detected using non-coherent integration of FFT data

#detection #integration #noncoherent #statistics