Digital Signal Processing Reference
In-Depth Information
takes the form of a forgetting factor, i.e., that it tends
to attenuate the relevance of older samples, which is justifiable, for instance,
whenever one deals with a time-varying environment:
We assume that λ
(
n
)
λ
N
samples
−
n
λ
(
n
)
=
(3.68)
where
0
<
1 is a parameter that controls the degree of penalization of older
samples (if λ
λ
≤
1, they are not penalized at all)
N
samples
is the number of available samples
=
The cost function given by (3.67) can be rewritten as
N
samples
e
2
J
LS
(
w
)
=
λ
(
k
)
(
k
)
k
=
1
N
samples
e
T
=
λ
(
k
)
e
(
k
)
(
n
)
k
=
1
N
samples
d
d
T
w
w
T
x
x
T
=
λ
(
k
)
(
k
)
−
(
k
)
(
k
)
−
(
k
)
(3.69)
k
=
1
Further manipulation leads to
N
samples
λ
d
2
x
T
w
T
x
J
LS
(
w
)
=
(
k
)
(
k
)
−
λ
(
k
)
d
(
k
)
(
k
)
w
−
λ
(
k
)
(
k
)
d
(
k
)
k
=
1
w
w
T
x
x
T
+
λ
(
k
)
(
k
)
(
k
)
(3.70)
which finally yields
N
samples
d
2
T
w
w
T
π
w
T
J
LS
(
w
)
=
λ
(
k
)
(
k
)
−
π
(
N
samples
)
−
(
N
samples
)
+
(
N
samples
)
w
k
=
1
(3.71)
where
N
samples
x
T
(
N
samples
)
=
λ
(
k
)
x
(
k
)
(
k
)
(3.72)
k
=
1
