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

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6.4. Minimization of the cost function

If our system (filter) is linear and non-recursive, we will have a quadratic cost function and this can be represented by an elliptical paraboloid (dim 2) (or a hyperparaboloid if the dimension is superior).

We will call the graphs or same level cost surfaces, i.e. the graphs or surfaces defined vector image:

images

Let us give an example, the equation of the isocosts in the case of a second order filter:

images

using the stationarity of XK, we obtain after development the equation of the isocosts image

images

NOTE.– By identification, we easily find the coefficients of the ellipse function of the traditional form:

images

NOTE.– Still because of the stationarity of XK, we see that the coefficients arising in the expression of image are independent of “K” and this finding is valid for a filter ...

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