Adaptive Filtering

5.4    THE WIENER SOLUTION

In Figure 5.10, we observe that there exists a plane touching the parabolic surface at its minimum point, and is parallel to the w-plane. Furthermore, we observe that the surface is concave upward, and therefore, calculus tell us that the first derivative of the MSE with respect to w₀ and w₁ must be zero at the minimum point and the second derivative must be positive. Hence, we write

$\frac{\partial J\left(w_0,w_1\right)}{\partial w_0}=0\frac{\partial J\left(w_0,w_1\right)}{\partial w_1}=0$

(5.90a)

$\frac{{\partial }^2\left(w_0,w_1\right)}{{\partial }^2{w}_0}>0\frac{{\partial }^{2}J\left(w_0,w_1\right)}{{\partial }^2\left(w_1\right)}>0$

(5.90b)

For a two-coefficient filter, (5.87) becomes

$J\left(w_0,w_1\right)=w_0^2{r}_x\left(0\right)+2w_0{w}_1{r}_x\left(1\right)w_1^2{r}_x\left(0\right)-2w_0{r}_{dx}\left(0\right)-2w_1{r}_{dx}\left(1\right)+{\sigma }_d^2$

(5.91)

Introducing next the above equation into (5.90a) produces the following set of equations:

$2w_0^o{r}_x\left(0\right)+2w_1^o{r}_x\left(1\right)-2$