Adaptive prediction error filtering

We still have a data sequence y(t) ($t=0,\cdots,T_{max}$). For each time T and order k, we define the forward and backward residuals $\varepsilon_{k}(T)$ and r_k(T); a single reflection coefficient K_k and Burg's recursions are used:  

$$\left\{\begin{array} {ll} &\varepsilon_{0}(T)=y(T) \;, r_{0}... ...{k}(T)=r_{k-1}(T-1)-K_k \varepsilon_{k-1}(T) \end{array}\right.$$ (9)

To compute the reflection coefficients K_k, we minimize the weighted energy of the forward and backward residuals of order k. The weights are centered on the particular time T:

It is important to notice that the residuals depend here on the whole set of data, since the energy ${\cal E}(K_k)$ is a summation on the whole time window. However, this expression of the energy induces an adaptive formalism, since more emphasis is put on the residuals near the time T of interest than of the residuals far from this time. Minimizing this expression with respect to K_k leads to a time-dependent reflection coefficient:  

$$K_{k,T}={2\sum_{t=0}^{T_{max}-1}\lambda^{\vert T-t\vert}\var... ...da^{\vert T-t\vert}(\varepsilon_{k-1}^2(t)+r^2_{k-1}(t-1))} \;.$$ (10)

Then Hale (1981) showed that the reflection coefficients K_k,T can be easily computed, if the numerator and denominator of expression (11) are split between past and future summations. For example, the numerator becomes: In the same way, we can split the denominator between D^-_k(T) (summation up to $T\!-\!1$), and D⁺_k(T) (summation from T to T_max). Then, it is straightforward to show that:

$$\left\{\begin{array} {lllll} N^{-}_{k}(T+1)&=&\lambda(2\vare... ...\lambda D^{+}_{k}(T)&;&D^{+}_{k}(T_{max})=0. \end{array}\right.$$ (11)

These recursions, together with the recursions (10) and the expression (11), form the adaptive version of Burg's algorithm.