Digital Signal Processing Reference
In-Depth Information
This section presents the most common linear feature transformation. In
the next section, a well-known non linear transformation is given based on
neural networks.
As it was mentioned before, a feature space transformation implements a
mapping of all parameters from an original feature space to a transformed
feature space. In addition, a
linear
feature transformation can be imple-
mented by matrix multiplication with a matrix transform
A
. For training
the matrix transform, supervised methods can be used which require that
each feature vector
x
of the training data is labeled with a particular class.
In some cases, the labels are already available, where the labels can represent
small speech units such as syllables or phonemes. In the case where there is
no labels available, unsupervised clustering techniques can be used such as
K-means [Lloyd 82] or LBG [Linde 80] algorithms.
It is common to model the original feature space with Gaussian distri-
butions. Then, a class
j
can be characterized by a mean vector
μ
j
and a
covariance matrix
Σ
j
estimated as follows
2
:
1
N
j
Σ
j
=
μ
j
)
T
(
x
i
−
μ
j
)(
x
i
−
(2.11)
{
x
i
}εj
1
N
j
μ
j
=
x
i
(2.12)
{
x
i
}εj
where
x
is a
n
-dimensional feature vector of the original feature space, and
{
εj
is the set of feature vectors belonging to the class
j
.
Figure 2.6 shows the procedure of the space transformation, which is im-
plemented as follows:
x
i
}
y
(
p×
1)
=
A
(
p×n
)
x
(
n×
1)
μ
j
(
p×
1)
=
A
(
p×n
)
μ
j
(
n×
1)
Σ
j
(
p×p
)
=
A
(
p×n
)
Σ
j
(
n×n
)
A
p
(
n×p
)
(2.13)
The Gaussian parameters in the transformed space are then used by the
statistical decoder for performing classification. In this case, the statistical de-
coder is mainly based on HMM/GMM, as it will be explained in Section 2.3.3.
Next, several linear transformation are briefly introduced.
Maximum likelihood linear transformation (MLLT) can be seen as a model-
space transformation since it is acting on the model parameters, rather
than implementing a dimensionality reduction [Gopinath 98]. MLLT aims
to find a feature space transformation where the features are decorrelated. In
2
Notation: those parameters belonging to the original feature space are repre-
sented with a check notation (ˇ), while those parameters in the transformed
feature space do not have a particular notation. Additionally, all vectors are
organized as column vectors.
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