Adaptive Filter Learing Notes 自适应滤波学习笔记08 Levinson-Durbin算法_levinson-durbin算法优缺点

Levinson-Durbin算法是用来解线性预测误差滤波的系数(prediction-error filter coefficients)和预测误差功率(prediction-error power)的递归算法，优点是减少计算量和存储空间

Levinson-Durbin算法的推导

先参考上一篇笔记Adaptive Filter Learing Notes 自适应滤波学习笔记07 线性预测的记号， R m + 1 a m = [ P m 0 ] , R m + 1 a m B ∗ = [ 0 P m ] . \boldsymbol{R}_{m+1}\boldsymbol{a}_{m}=\left[

$$\begin{matrix} P_m\\ \boldsymbol{0} \end{matrix}$$ \right],\quad \boldsymbol{R}_{m+1}\boldsymbol{a}_{m}^{B*}=\left[ $$\begin{matrix} \boldsymbol{0}\\ P_m \end{matrix}$$ \right]. Rm+1​am​=[Pm​0​],Rm+1​amB∗​=[0Pm​​]. $$\begin{array}{rl}{\mathbf{R}}_{m+1}\left[\begin{array}{c}{\mathbf{a}}_{m-1}\\ 0\end{array}\right]=& \left[\begin{array}{cc}{\mathbf{R}}_m& {\mathbf{r}}_m^{B\ast }\\ {\mathbf{r}}_m^{BT}& r(0)\end{array}\right]\left[\begin{array}{c}{\mathbf{a}}_{m-1}\\ 0\end{array}\right]\\ =& \left[\begin{array}{c}{\mathbf{R}}_m{\mathbf{a}}_{m-1}\\ {\mathbf{r}}_m^{BT}{\mathbf{a}}_{m-1}\end{array}\right]\\ =:& \left[\begin{array}{c}{P}_{m-1}\\ 0\\ {\Delta }_{m-1}\end{array}\right].\end{array}$$ ${\Delta }_{m-1}={\mathbf{r}}_m^{BT}{\mathbf{a}}_{m-1}={\mathbf{a}}_{m-1}^{BT}{\mathbf{r}}_m$. $$\begin{array}{rl}{\mathbf{R}}_{m+1}\left[\begin{array}{c}0\\ {\mathbf{a}}_{m-1}^{B\ast }\end{array}\right]=& \left[\begin{array}{cc}r(0)& {\mathbf{r}}_m^H\\ {\mathbf{r}}_m& {\mathbf{R}}_m\end{array}\right]\left[\begin{array}{c}0\\ {\mathbf{a}}_{m-1}^{B\ast }\end{array}\right]\\ =& [\begin{array}{c}{\mathbf{r}}_m^H{\mathbf{a}}_{m-1}^{B\ast }\\ \end{array}\\ =& \end{array}$$