(16) |

As I already said, this expression of the energy forces the residuals
to depend on past *and* future data, because the summation symbol
extends to the whole time window. But the formalism is still adaptive,
because the weighting is non-uniformly centered around the time of interest
*T*. The minimization with respect to *K*^{x}_{k} leads to the expression:

(17) |

The denominator is indeed the one involved in the computation of the
reflection coefficient *K*^{r}_{k,T}, and we already saw how we can compute it
recursively. The numerator can also be split into past and future summations
*N*^{x-}_{k}(*t*) and *N*^{x+}_{k}(*t*) verifying:

(18) |

In conclusion, the idea of the algorithm is to perform at the same time
an adaptive self-prediction of the data *y*(*t*) with my modified version
of Burg's adaptive algorithm, and the adaptive prediction of *x* from *y* with
the recursions derived in this section. The self-prediction of *y*
is processed with the recursion formulas (13),
(15), and (16). The prediction of *x* is
performed with the recursions (17),
(18), and (19).

1/13/1998