90
phase shift
w
0
(n)
w
1
(n)
+
-
e(n)
LMS
y(n)
x
1
(n)
d(n)
x
1
(n)
x
0
(n)
x
0
(n)
Figure 8.10 Single-frequencyadaptive notch filter
Digital Hilbert transform filters can be employed for this purpose. Instead of using
cosine generator and a phase shifter, the recursive quadrature oscillator given in Figure
6.23 can be used to generate both sine and cosine signals simultaneously. For a reference
sinusoidal signal, two filter coefficients are needed.
The LMS algorithm employed in Figure 8.10 is summarized as follows:
yn w
0
nx
0
n w
1
nx
1
n,
8:5:8
en dn À yn,
8:5:9
w
l
n 1 w
l
n mx
l
nen, l 0, 1:
8:5:10
Note that the two-weight adaptive filter W(z) shown in Figure 8.10 can be replaced with
a general L-weight adaptive FIR filter for a multiple sinusoid reference input x(n). The
reference input supplies a correlated version of the sinusoidal interference that is used to
estimate the composite sinusoidal interfering signal contained in the primaryinput d(n).
The single-frequencyadaptive notch filter has the propertyof a tunable notch filter.
The center frequencyof the notch filter depends on the sinusoidal reference signal,
whose frequencyis equal to the frequencyof the primarysinusoidal noise. Therefore the
noise at that frequencyis attenuated. This adaptive notch filter provides a simple
method for tracking and eliminating sinusoidal interference.
Example 8.8: For a stationaryinput and sufficientlysmall m, the convergence
speed of the LMS algorithm is dependent on the eigenvalue spread of the input
autocorrelation matrix. For L 2 and the reference input is given in (8.5.6),
the autocorrelation matrix can be expressed as
R E
x
0
nx
0
n x
0
nx
1
n
x
1
nx
0
n x
1
nx
1
n
!
E
A
2
cos
2
!
0
n
A
2
cos!
0
n sin!
0
n
A
2
sin!
0
n cos!
0
n
A
2
sin
2
!
0
n
4
5
A
2
2
0
0
A
2
2
P
T
R
Q
U
S:
378
ADAPTIVE FILTERING