I have a time domain data which when converted to frequency domain gives a Gaussian spectrum. If I have limited number of samples in time domain, my frequency domain resolution is bad. In that case, if we try to estimate the second moment of the spectrum, the estimation would be biased if the actual spectrum width (in reality) is less than the frequency resolution.
The way I estimate the second moment is given below,
$$ \sigma = \sqrt{ \int_{-f_m}^{f_m} \frac{1}{P_T} [f - \mu]^2 |S[f]|^2 df } $$
Here, $f$ is the frequency axis, $S[f]$ is the $DFT$ of the original signal in time domain, $df$ is the frequency resolution, $f_m$ is the maximum frequency allowed and $P_T$ is the total power contained in the signal
$$P_T = \int_{-f_m}^{f_m} |S[f]|^2 df $$
Is there a functional relationship I can write in the following form?
$$ \sigma = f( N, \frac{df}{\sigma_{True}} ) $$
Where, $\sigma_{True}$ is the true spectrum with and $N$ is the number of frequency points.