Digital Signal Processing Reference
In-Depth Information
entry, respectively. The integration of the difference between the ratio of the two
spectral envelopes and the optimal ratio
opt
¼
1
jA
i
,nb
(
e
jV
)
j
2
(7
:
50)
2
jA
nb
(
e
jV
,
n
)
j
can be computed very efficient by using only the predictor coefficients and the
autocorrelations matrix of the current frame
a
i
,nb
R
ss
(
n
)
a
i
,nb
a
nb
(
n
)
R
ss
(
n
)
a
nb
(
n
)
1
:
d
lhr
(
n
,
i
)
¼
(7
:
51)
Since only the index
i ¼ i
opt
(
n
) corresponding to the minimum of all distances (and
not the minimum distance itself ) is needed, it is sufficient to evaluate
i
opt
(
n
)
¼
argmin{
d
lhr
(
n
,
i
)}
¼
argmin{
a
i
,nb
R
ss
(
n
)
a
i
,nb
}
:
(7
:
52)
Note that (7.52) can be computed very efficiently since the autocorrelation matrix
R
ss
(
n
) has Toeplitz structure. Beside the cost function according to (7.49), which
weights the difference between the squared transfer function in a linear manner, a var-
iety of others can be applied [12]. Most of them apply a spectral weighting function
within the integration over the normalized frequency
V
. Furthermore, the difference
between the spectral envelopes is often weighted in a logarithmic manner (instead
of a linear or quadratic). The logarithmic approach takes the human loudness percep-
tion in a better way into account. In the approaches discussed in this chapter, we have
again used a cepstral distance measure according to (7.30).
For obtaining the codebook entries iterative procedures such as the method of
Linde, Buzo, and Gray (LBG algorithm [27]) can be applied. The LBG-algorithm
is an efficient and intuitive algorithm for vector quantizer design based on a long train-
ing sequence of data. Various modifications exit (see [25, 30] e.g.). In our approach the
LBG algorithm is used for the generation of a codebook containing the spectral envel-
opes that are most representative in the sense of a cepstral distortion measure for a
given set of training data. For the generation of this codebook the following iterative
procedure is applied to the training data.
1.
Initializing
:
Compute the centroid for the whole training data. The centroid is defined as the
vector with minimum distance in the sense of a distortion measure to the com-
plete training data.
2.
Splitting:
Each centroid is split into two near vectors by the application of a perturbance.
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