10.1. Theoretical background
Spectral analysis is one of the most important signal processing techniques and consists of identifying the spectral content of a time-varying quantity. The usual spectral analysis methods can be divided into two main classes:
– direct methods (selective filtering, periodogram);
– indirect methods (correlogram, parametric methods, etc.).
In an experimental framework, a signal spectral analysis can be performed using spectral analyzers, which represent an essential investigation tool in many applications. In real world applications, the observation period (continuous-time signals) or the number of samples (discrete-time signals) is finite. Thus, the spectral content will be estimated using the limited available information.
10.1.1. Estimate properties
Let us consider a stochastic process x(n) depending on a parameter a, and an estimate of this parameter obtained from N outcomes of x:
The quality of an estimate is evaluated using its bias and its variance defined below:
A low variance indicates a low dispersion of the estimate values around its mean E[â]. An estimate converges if its bias and variance will vanish when the number of observations N becomes ...