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

**th 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**

*j*. 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

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