In practice, one has only one trajectory of a random signal in order to perform any estimation. We focus here on the spectrum estimation problem. Let $\boldsymbol{X}$ a weak-sense stationary random signal, and $\boldsymbol{X_N} = \lbrace x_0,x_1,\ldots,x_{N-1} \rbrace$ one observed trajectory of $\boldsymbol{X}$. We present here non parametric estimation of the spectrum density of $\boldsymbol{X}$.
The periodogram of $\boldsymbol{X}$ is simply the power spectrum of the observed trajectory $\boldsymbol{X_N}$. That is, we estimate the spectrum density $S(\nu)$ of $\boldsymbol{X}$ by $$ S_N^{Per}(\nu) = \left\vert \sum_{n=0}^{N-1} x_n e^{i\frac{2\pi}{N} \nu n}\right\vert^2 $$
The periodogram is a biased estimator (but asymptotically unbiased). One can show that for a gaussian white noise, the periodogram is not consistent, ie. its variance does not goes to $0$ when $N$ tends to infinity.
Welch method consists of a windowing of the observed trajectory using overlapping windows, compute the modified (windowed) periodogram for each windows, and then take the average of all the spectrums. It can be shown that Welch’s method is asymptotically unbiased, with a significant improvement on the variance of the spectrum estimator.