Digital Signal Processing Reference
In-Depth Information
random functions corresponding to the
L
-point Gaussian random sequences
contained in the codebook is searched. In MPLPC and RPELPC, the search
is in time, but through a set of delayed impulse response functions. For a
given technique, the criterion for finding the optimum excitation function is
the same. The objective is to determine the shape matrix
X
and the associated
gain
g
(assuming MPLPC and RPELPC have normalized shape vectors) so
that
gX
produces a synthetic signal that minimizes the weighted error
e(n)
shown in Figure 7.4, i.e.
=
−
ˆ
(7.7)
e
k
s
w
s
k
where
s
w
is the weighted original reference signal,
s
k
is the synthesized signal
(with pitch, LPC and perceptual-weighting filter contributions), and
k
denotes
the particular excitation.
Let
H
be an
L
ˆ
L
matrix whose
j
th
row contains the (truncated) combined
impulse response
h(n)
of the pitch, LPC and perceptual weighting filters
caused by a unit impulse
δ(n
×
−
j)
,i.e.
h(
0
) h(
1
)
···
h(L
−
1
)
0
h(
0
)
···
h(L
−
2
)
=
(7.8)
H
.
.
.
.
0
0
···
h(
0
)
If
s
m
denotes the output of the cascaded filters with zero input, i.e. thememory
hangover from previously synthesized frames, then the reference signal
s
to
˜
be matched can be described as,
=
−
(7.9)
s
˜
s
w
s
m
⇒
=
˜
−
(7.10)
e
k
s
g
k
X
k
H
=
˜
−
(7.11)
s
g
k
ˆ
s
k
where,
=
(7.12)
ˆ
s
k
X
k
H
and
X
k
and
g
k
are the
k
th
excitation shape and gain vectors. The criterion is
minimum-squared error, thus our objective is to minimize
E
k
where,
e
k
e
k
E
k
=
(7.13)
and
T
denotes transpose. The optimum amplitude vector
g
k
for the
k
th
candidate excitation can be computed from equations (7.11) and (7.13) by
requiring the error
e
k
to be orthogonal to our estimation
s
k
,i.e.
ˆ
s
T
=
0
(7.14)
e
k
ˆ
k
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