Digital Signal Processing Reference
In-Depth Information
Therefore,
s
T
(
˜
−
s
k
).
ˆ
=
0
(7.15)
s
g
k
ˆ
k
s
k
[
s
k
]
−
1
⇒
g
k
=
˜
(7.16)
s
ˆ
s
k
ˆ
ˆ
By substituting equation (7.16) into equation (7.11), equation (7.13) can be
rewritten as
s
k
[
s
k
]
−
1
s
T
E
k
=
˜
s
[
I
−
ˆ
s
k
]
(7.17)
s
k
ˆ
ˆ
ˆ
˜
where
I
is the identity matrix. The vector
g
k
and matrix
X
k
that yield the
minimum value of
E
k
over all
k
are then selected as the optimum excitation.
The above expression for
E
k
is generalized for all the possible forms
of excitations and is, therefore, rather more complicated than required in
practical cases. The [
s
k
]
−
1
inversion, for instance, is unnecessary in most
cases, as illustrated below using codebook excitation.
ˆ
s
k
ˆ
s
k
=
σ
(scalar)
(7.18)
ˆ
s
k
ˆ
s
k
σ
s
˜
ˆ
=
=
g
k
(scalar)
(7.19)
g
k
s
T
s
T
⇒
E
k
=
˜
−
(7.20)
s
˜
g
k
ˆ
s
k
˜
s
T
=
˜
−
s
˜
Q
k
(7.21)
Rewriting equations (7.19) and (7.20) in time-domain samples form,
−
L
1
0
˜
s(i)
s
k
(i)
ˆ
i
=
g
k
=
(7.22)
L
−
1
s
k
(i)
0
ˆ
i
=
L
−
1
L
−
1
s
2
(i)
E
k
=
0
˜
−
g
k
0
ˆ
s
k
(i)
˜
s(i)
(7.23)
i
=
i
=
and, substituting
g
k
into equation (7.23), we can rewrite equation (7.21) as,
L
−
1
s(i)
2
0
ˆ
s
k
(i)
˜
−
L
1
=
i
s
2
(i)
E
k
=
0
˜
−
(7.24)
L
−
1
i
=
s
k
(i)
0
ˆ
i
=
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