Digital Signal Processing Reference
In-Depth Information
Now defining,
L
−
1
φ(k, i)
=
h(n
−
k)h(n
−
i)
(7.57)
n
=
0
and,
L
−
1
0
˜
(k)
=
s(n)h(n
−
k)
(7.58)
n
=
simplifies equation (7.56) to a form,
−
M
1
(m
k
)
=
g
i
φ(m
i
,m
k
)
k
=
0
,
1
, ... ,M
−
1
(7.59)
i
=
0
which can be written in the form of a correlation matrix as,
=
(m
0
)
(m
1
)
.
(m
M
−
1
)
g
0
g
1
.
g
M
−
1
φ(m
0
m
0
)
(m
0
m
1
)
...
φ(m
0
m
M
−
1
)
φ(m
1
m
0
)
(m
1
m
1
)
...
φ(m
1
m
M
−
1
)
. . .
φ(m
M
−
1
m
0
)φ m
M
−
1
m
1
) ... φ(m
M
−
1
m
M
−
1
)
.
(7.60)
The optimum amplitudes
g
i
can now be solved utilizing the Cholesky decom-
position of the correlation matrix.
Using the above analysis, two forms of pulse amplitude re-optimization
procedure can be used [12]. One can re-optimize the amplitudes after all
of the
M
pulses have been located within a subframe, or after each new
pulse is located. Of course the latter method has the greater computational
burden of the matrix inversion, but the overall quality compared to the
former method is superior. If amplitude re-optimization takes place once at
the end of each subframe, the required matrix inversion size is
(M
M)
.Ifit
takes place after each new pulse is located, amplitude re-optimization occurs
M
times with matrix sizes of
(
1
×
M)
.Figure7.19showsthe
variation of number of pulses versus
segSNR
for three different algorithms.
Curve
(c)
is for a basic sequential MPLPC coder. It shows increasing
segSNR
as the number of pulses increases but, after 30 pulses per 160 samples, its
performance tends to saturate. Curves
(b)
and
(a)
arefortheimprovedMPLPC
algorithms, i.e. amplitude re-optimization after all pulses have been located
and amplitude re-optimization after location of each new pulse respectively.
Objective results show curve
(b)
giving lower
segSNR
than
(a)
,asexpected.
×
1
)
up to
(M
×
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