Digital Signal Processing Reference
In-Depth Information
may be negative for all integers
k
K
for some
K.
So, in view of the ordering
convention (22.46), we can assume in general that the first
K
values of
α
k
are
nonzero, for some
K
≥
≤
N.
22.4.1 Final form of the solution
From Eq. (22.44) we have
K−
1
K−
1
K−
1
1
σ
s
α
k
=
D
σ
q
k
σ
q
k
−
k
=0
k
=0
k
=0
where
K
is the number of nonzero values of
α
k
.
Substituting the preceding
equation into the power constraint (22.31) we obtain
K−
1
p
σ
s
1
σ
s
σ
q
+
=0
D
=
(22
.
47)
K−
1
σ
q
=0
Thus
⎧
⎨
p
0
+
K−
1
=0
σ
q
k
σ
s
σ
q
−
σ
q
k
0
≤
k
≤
K
−
1
K−
1
=0
α
k
=
(22
.
48)
σ
q
⎩
0
otherwise.
The expression for the minimized MSE can be obtained by substituting this into
Eq. (22.37):
σ
s
K−
1
σ
q
2
p
0
+
K−
1
=0
K
)
σ
s
.
E
mmse
=
+(
N
−
(22
.
49)
σ
q
=0
K
)
σ
s
The second term (
N
−
arises because whenever
α
k
=0,the
k
th term in
Eq. (22.37) reduces to
σ
s
.
22.4.2 Choice of K
We have to figure out the value of the integer
K
in Eq. (22.48). For su
ciently
large power
p
0
we can assume
K
=
N
, but in general this is not so. To appreciate
this, recall that the ordering convention in Eq. (22.32) implies that the quantity
p
0
+
K−
1
σ
q
=0
−
σ
q
k
K−
1
=0
σ
q
is a decreasing function of
k.
Suppose we choose
K
=
N.
Then
α
2
N−
1
computed
from Eq. (22.48) might turn out to be negative (for a given power
p
0
). If this is
the case we take
K
=
N
−
1 and try it again. We proceed like this by successively
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