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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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3.3. Spectral representation of a WSS process

In this section we explain the steps enabling us to arrive at the spectral representation of a process. In order not to obscure these steps, the demonstrations of the results which are quite long without being difficult are not given.

3.3.1. Problem

The object of spectral representation is:

1) To study the integrals (called Wiener integrals) of the images type obtained as limits, in a meaning to clarify the expressions with the form:

images

where S is a restricted interval of images, φ is a mapping with complex values (and other conditions), ZS = {Zu | uS} is a 2nd order process with orthogonal increments (abbreviated as p.o.i.) whose definition will be given in what follows.

2) The construction of the Wiener integral being carried out, to show that reciprocally, if we allow ourselves a WSS process Ximagesθ, we can find a p.o.i. images such that ∀jimages X may be written ...

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