At least when using Gaussian white noise, the DFT of the noise is the same distribution with a smaller variance: https://dsp.stackexchange.com/questions/24170/w

At least when using Gaussian white noise, the DFT of the noise is the same distribution with a smaller variance: https://dsp.stackexchange.com/questions/24170/what-are-the-statistics-of-the-discrete-fourier-transform-of-white-gaussian-nois https://dsp.stackexchange.com/questions/24170/what-are-the-s... I don't think the same neat result holds when you use uniform white noise, but I haven't done the math.