Determining Power of White Noise Component of an Incomplete Time Signal I want to determine the power of the noise component of a time dependend signal x(t) (similar to picture below)



With constant sampling rate I use Parceval's Therorem and calculate the standard deviation with

$$
\sigma^2 = \frac{\text{abs}(\text{FFT}(x))^2}{N}
$$

where N is number of data points and fft matlab dft implementation. For the standard deviation according to white noise I remove the discrete frequencies of the spectrum and use above formula with the mean value of the remaining spectrum.

In some signals there are quite a few NaN data points (~ 10%), so said method can not be applied anymore.

Is there a mathematically correct method to determine the power of the noise component of the signal? Interpolation does not seem to be a good solution, because this would lead to an underestimation of the noise.
I assume you have information about the signal itself (Not the noise).

What you can do is use LS to estimate the signal (Easily compensate for the missing samples in the Model Matrix).
Once you have it estimated you can do 2 things (Which are the same):


Recreate the samples using the model. Subtract the measurements. You'd be left with IID Samples of the noise -> Use the empirical STD / Variance estimator.
If it is Linear Least Squares use its internal estimation of the noise Variance.


Good Luck!