Digital Signal Processing Reference
In-Depth Information
where the equality holds if and only if
g
has at most one nonzero element.
Moreover, from (4.42), we have
l
i
2
|
g
(
i
)
|
≤
l
(
1
)
(4.44)
being the equality valid if and only if
i
|
2
1. Hence, from (4.43)
and (4.44), we know that only for ideal solutions the equalities are satisfied.
Now, for the definition of a cost function, it is necessary to specify the
l
g
(
i
)
|
=
(
·
)
function. Let us consider the choice
α
x
2
, α
l
(
x
)
=
2α
x
−
>
0
(4.45)
which, as required, monotonically increases in 0
≤
x
<
1 and decreases for
x
1. By a direct substitution of (4.45) into (4.42), and performing some
algebraic manipulations, we obtain
>
E
2
|
2
E
|
2
c
4
(
y
(
n
))
y
(
n
)
|
y
(
n
)
|
J
=
))
−
(
1
+
α
)
E
2
|
2
+
2α
E
|
2
c
4
(
s
(
n
s
(
n
)
|
s
(
n
)
|
E
2
E
2
2
4
E
2
1
2
+
(
1
+
α
)
c
4
(
s
(
n
))
E
2
|
2
=
|
y
(
n
)
|
−
|
y
(
n
)
|
−
|
y
(
n
)
|
c
4
(
s
(
n
))
s
(
n
)
|
2
E
c
4
(
s
(
n
))
+
E
|
2
|
y
(
n
)
|
2α
(4.46)
s
(
n
)
|
Equation 4.46 can still be rewritten as
]
E
2
4
E
y
2
J
SW
c
=
sgn [
c
4
(
s
(
n
))
|
y
(
n
)
|
−
(
n
)
γ
1
E
2
2
2γ
2
E
2
+
|
y
(
n
)
|
+
|
y
(
n
)
|
(4.47)
where
2
+
(
1
+
α
)
c
4
(
s
(
n
))
γ
1
=−
E
2
|
2
(4.48)
s
(
n
)
|
and
c
4
(
s
(
n
))
γ
2
=
α
E
|
2
(4.49)
s
(
n
)
|
