Modification of Burg's adaptive algorithm

In Burg's algorithm, the backward residuals don't form an orthogonal basis of the space of the regressors ($Z^y,\cdots,Z^ny$). Effectively, we minimize the energies of the forward and backward residuals not separately, but together. Thus, the recursions implied for the backward residuals do not correspond to a Gram-Schmidt orthogonalization of the space of the regressors.

Therefore, my idea is to use two reflection coefficients, K^r_k and $K^{\varepsilon}_k$, as in the LSL algorithm, so that the forward and backward residuals are minimized separately. This is simply done by writing:  

$$\left\{\begin{array} {ll} &\varepsilon_0(T)=y(T)\;, \; r_0(T... ...1}(T-1)-K^{\varepsilon}_k\varepsilon_{k-1}(T)\end{array}\right.$$ (12)

Now, to compute the reflection coefficients K^r_k and $K^{\varepsilon}_k$, we minimize separately the energies of the forward and backward residuals:

$$\left\{\begin{array} {lllll} {\cal E}^{\varepsilon}(K^r_k)&=... ...)-K^{\varepsilon}_{k}\varepsilon_{k-1}(t)]^2 \end{array}\right.$$

This yields to the following time-varying formulations of the reflection coefficients:

$$\begin{eqnarray} K^r_{k,T}&=&{\sum_{t=0}^{T_{max}-1}\lambda^{\vert T-t\vert}\var... ...t=0}^{T_{max}-1}\lambda^{\vert T-t\vert}\varepsilon^2_{k-1}(t)}\;.\end{eqnarray}$$ (13) (14)

Both coefficients have the same numerator; actually, Burg's coefficient is their geometric mean. Once again, these coefficients can be easily computed if we split their numerators and denominators into past and future summations. Indexing the denominators with r and $\varepsilon$ as we did with the reflection coefficients, we have now:  

$$\left\{\begin{array} {lllll} N^{-}_{k}(T+1)&=&\lambda(\varep... ...ambda D^{r+}_{k}(T)&;&D^{r+}_{k}(T_{max})=0. \end{array}\right.$$ (15)

These recursions, joined to the formulas (13) and (15), give my modified version of Burg's adaptive algorithm. The advantage of Burg's algorithm is that it is more stable, because the reflection coefficients are always smaller than one. However, this version allows me to construct an orthogonal basis for the space of regressors of the general prediction problem, and thus leads to the corresponding algorithm.