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

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6.9. Example of the application of the LMS algorithm

Let us recall the modelization of an AR process.

image

Thus image

By multiplying the two members by XKl and by taking the expectations, it becomes:

images

If image then image

As BK is a white noise and unique BK is dependent on XK.

Thus, by stating:

images

images

and image

By noting a0 = 1 and by using the matrix expression, this becomes:

images

For an AR process of order 1, let the following AR process be XK = −a XK−1 + BK where BK is a centered white noise of variance image.

The problem consists ...

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