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Digital Signal Processing Using Matlab by André Quinquis

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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:

images

The quality of an estimate is evaluated using its bias and its variance defined below:

images

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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 ...

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