Discrete Stochastic Processes and Optimal Filtering

3.2. WSS processes and spectral measure

In this section it will be interesting to note the influence on the spectral density of the temporal spacing between the r.v. For this reason we are now about to consider momentarily a WSS process where θ is a constant and where jθ has the significance of duration.

3.2.1. Spectral density

DEFINITION.– We say that the process X_θ possesses a spectral density if its covariance can be written in the form: and S_XX (u) is then called the spectral density of the process X_θ.

PROPOSITION.–

Under the hypothesis :

1) the process X_θ admits a spectral density S_XX;

2) S_XX is continuous, periodic of period, real and even.

Figure 3.2. Covariance function and spectral ...