Digital Signal Processing Reference
In-Depth Information
where
Q
[
.
]
denotes quantization operation. The shape codebook is then searched and
the codebook vector which minimizes the expression,
N
σ
x
y
ki
)
2
D
k
=
(x
i
−ˆ
k
=
1
,
2
, ... ,L
(3.64)
i
=
1
is chosen for transmission. This search scheme, called open loop, is not
optimum. Better performance can be achieved with a closed loop scheme
where the shape is first found and then the corresponding gain is quantized
before computing the final error. Here, we assume an optimum gain
σ
k
to be
used for each of the
L
shape codebook entries and compute the corresponding
distortion
D
k
as:
N
σ
k
y
ki
)
2
D
k
=
(x
i
−
k
=
1
,
2
, ... ,L
(3.65)
i
=
1
We wish to find a vector
y
k
from the shape codebook with a gain value of
σ
k
such that the corresponding distortion
D
k
is minimized. However, we have
two unknowns, namely,
y
k
and
σ
k
.Tofind
σ
k
in terms of
y
k
we differentiate
(3.65) with respect to
σ
k
and set it to zero for minimum error gain. This gives
the following
σ
k
for the codebook vector
y
k
in relation to an input vector
x
,
N
(x
i
y
ki
)
i
=
1
σ
k
=
(3.66)
N
y
ki
i
=
1
If we substitute (3.66) into (3.65) we can write the distortion
D
k
independently
of
σ
k
as,
N
2
x
i
y
ki
N
i
=
1
(x
i
)
2
D
k
=
−
k
=
1
,
2
, ... ,L
(3.67)
N
i
=
1
y
ki
i
=
1
The first term of
D
k
in equation (3.67) does not change with
k
,and
hence it is not computed during the search of the shape. The shape is
found by maximizing only the second term in (3.67). During the codebook
search process, the most likely shape values are found by maximizing the
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