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estimate instead of gradient vector, and has been widely used in many electronic sys-
tems because of simple structure, stable performance, small calculated amount and easy
realization, etc [4]. The basic idea of LMS algorithm is that it makes mean-square error
be minimum between the output signal of filter and desired signal by adjusting the
weights parameters of filter.
Adaptive filtering usually includes two basic processes: filtering process and the
parameters adjustment process of filter that are composed of a feedback loop, as shown
in Fig. 1 [5], where
X
(
n
)
are
N
filtered input signal and can be expressed as follows:
T
. (1)
X
(
n
)
=
[
x
(
n
),
x
(
n
),
"
,
x
N
n
(
)]
1
2
Wn
are
N
filter weights and can be expressed as follows:
()
. (2)
()
T
Wn wnwn w n
() [ (),
=
(), ,
"
1
2
N
y
(
n
)
is filtered output signal,
d
(
n
)
is reference signal, and
e
(
n
)
is error signal be-
tween
y
(
n
)
and
d
(
n
)
. Error signal
e
(
n
)
controls adaptive filter coefficients.
y
(
n
)
X
(
n
)
−
e
(
n
)
+
d
(
n
)
∑
Fig. 1.
The principle diagram of adaptive filter based on LMS algorithm
Then filtered output signal can be written as follows:
N
=
∑
=
T
y
(
n
)
w
(
n
)
x
(
n
)
=
X
(
n
)
W
(
n
)
. (3)
i
i
i
1
and estimate error signal can be written as follows:
T
e
(
n
)
=
d
(
n
)
−
y
(
n
)
=
d
(
n
)
−
X
(
n
)
W
(
n
)
. (4)
According to the rule of least mean-square error, the parameters of optimal filter
should make performance function—mean-square error
2
n
be minimum. Be-
E
[
e
(
)]
cause the following equation hold that
2
2
T
2
E
[
e
(
n
)]
=
E
[(
d
(
n
)
−
y
(
n
))
]
=
E
[(
d
(
n
)
−
X
(
n
)
W
(
n
))
]
(5)
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