## Chapter 9

## Continuous Spectra Estimation

The object of this chapter is mainly a discussion of the power spectral density’s (PSD) estimation. In this field, it is customary to separate two cases:

- When the statistical properties of the observation depend on a finite, and usually small number of parameters, the model is said to be
*parametric.* To be more precise, it means that knowing the few useful parameters is enough to find the exact probability distribution of the observation. We have already encountered an example of the parametric model: the AR-*P* model in the case of a white Gaussian input. Knowing the (*P* + 1) parameters *a*_{1} …, *a*_{P}, *σ*^{2} is enough to determine the probability distribution. Without the Gaussian hypothesis, and although it is no longer possible to write the probability distribution of the observation precisely, it is still possible to estimate some useful quantities, such as its spectrum, given a finite number of parameters: this is sometimes called a *semiparametric* model.
- Otherwise, the model is said to be
*non-parametric.* In the first paragraph of this chapter we will study a situation in which the only hypothesis is that the process is WSS. Knowing the spectrum requires the estimation of an infinity of parameters, that is the set of covariance coefficients.

### 9.1 Non-parametric estimation of the PSD

#### 9.1.1 Estimation from the autocovariance function

We have seen that expressions 8.32 can be used to estimate *R*(*k*) from a series of *N* observations *X*(l), …, *X*(