Adaptive Filters - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 257

The equation above can also be obtained using (7.14),

] [ ] = () ()

] [

]

[

()

[

()

() = () ( )

w 0 xn

yn xn xn - 1 w 0 w 1

()

( )

w 1

xn - 1

which reduces to

() = ()() +- ( ) ( ) yn xnw 0 xn 1 w 1

which can also be obtained by summing the signals at the node of the two-weight
diagram shown in Figure 7.8.
As can be seen in Figure 7.8, the linear combiner with a single input is just an
FIR filter with adjustable coefficients. Although this is a very simple configuration,
it can handle many of the adaptive applications.

7.4 PERFORMANCE FUNCTION

In the preceding section we provided a structure for the filter whose characteristics
may be changed by adjusting the weights. However, we still need a way to judge
how well the filter is operating—a performance measure is needed. The perfor-
mance function will be based on the error, which is obtained from the block diagram
in Figure 7.1, with the time index incorporated:

() = () - () en dn yn (7.15)

The square of this function is

() = () - ()() + ()

2 2 2

en dn 2 dnyn yn (7.16)

which is the instantaneous squared-error function. In terms of the weights, it
becomes

() = () - () () + () () en dn

2 2

2 dn XWWXXW

T

n

T

n

T

n (7.17)

where the time index on the W has been dropped. Equation (7.17) represents a qua-
dratic surface in W , which means that the highest power of the weights is the squared
power. The strategy will be to adjust the weights so that the squared-error function
will be a minimum.
To understand the performance surface equation (7.17), consider the case of one
weight. The error surface then becomes

() = () - ()()() + () ()

en dn

2 2

2 dnxnw 0 xnw

2 2

0 (7.18)

which is a second-order function in w (0). To find the minimum, set the derivative of