Digital Signal Processing Reference
In-Depth Information
3.2.1 The MSE Surface
A relevant aspect of the cost function obtained in (3.11) is that it is a quadratic
function of the parameters of the filter and describes an elliptic paraboloid
with a single minimum. This means that there is a single parameter vec-
tor that minimizes the MSE, the so-called Wiener solution. This is in fact an
important reason why the MSE is the dominant metric in linear supervised
filtering. In Figure 3.5, we present the MSE cost function and its contours in
a typical two-dimensional filtering problem.
The two plots confirm our initial comments and reveal the elliptical char-
acter of the cost function contours. Two properties of these elliptical contours
deserve attention:
1. Their eccentricity is related to the eigenvalues of the correlation
matrix
R
(see Appendix A). The larger the eigenvalues spread, the
most significant is the discrepancy between axes. Naturally, if the
eigenvalue spread is equal to 1, the contours become circular.
2. The directions of the eigenvectors of the correlation matrix deter-
mine the orientation of the axes of the contours.
In order to find the minimum of the MSE cost function, we follow a clas-
sical procedure: setting to zero the gradient of
J
MSE
. From (3.11), it comes
that
∇
J
MSE
(
w
)
=
2
Rw
−
2
p
(3.12)
By forcing it to be equal to the null vector, we obtain
∇
J
MSE
(
w
)
=
2
Rw
−
2
p
=
0
→
Rw
=
p
(3.13)
3
80
2
60
1
0
40
−1
20
−2
0
−3
5
10
−4
0
5
−4
−2
0
2
4
6
w
1
0
w
0
(a)
(b)
−5
w
0
−5
FIGURE 3.5
(a) The MSE cost function and (b) its contours.
