Introduction
Here, we want to estimate the theoretical PSD of the signal x(t):
\(\gamma_x\left(\omega\right)=TF\left[R_x\left(\tau\right)\right]\lim\limits_{T \to \infty} {\frac{E\left\{\left|X_T(\omega)\right|^2\right\}}{T}}\) (29)
\(R_{xc}\left(k\right)=TF\left[x(t)f(t)\right]=\int_{0}^{T}{x(t)f(t)e^{-j\omega t}dt}\) (30)
A window f(t) of width T weighs the signal's FT.
The function \(X_T(\omega,\ T)\frac{\left|X_T(\omega)\right|^2}{T}\) is R. V. called a periodogram, which provides information on the frequency behaviour of the stationary process x(t) inside the interval \(\left[0,\ T\right]\) . Hence the first idea of taking for \(\gamma_x\left(\omega\right)\) the limit of\( X_T(\omega,\ T)\) when T tends towards infinity, so be it:
\(\gamma_x\left(\omega\right)= \lim\limits_{T \to \infty}{\frac{\left|X_T(\omega)\right|^2}{T}} \)(31)
Due to its poor performance, particularly in terms of the estimation variance, people rarely use this estimator. Indeed, although asymptotically unbiased, this estimator is not consistent.
Instead, and for reasons of environment and/or precision, analogue techniques based on filtering, quadrature, and integration based on the intuitive notion of PSD already encountered, as well as digital techniques such as the periodogram, correlogram, or even parametric methods, are used for reasons of environment and/or precision [15].