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number value space allows a convenient way
to use basis transformations like the Fourier
transformation. Finally, the introduction of the
index component allows both equidistant and
nonequidistant value series.
The value series cannot be directly used as
representation for classification learners since
the number of values is usually too large and the
interaction of values is not adequately represented.
Therefore, only a small set of features is derived
from the raw series data, which are then used by
the classification algorithm. In this section 2 , we
propose a principled view of the building blocks
from which audio features are constructed from
series data. Later on, this systematization will
allow us to flexibly construct sequences in an
automatic way.
Definition 2 Let H be a vector space with an in-
ner product
f , . H is called H ilbert space if
the norm defined by
g
f = turns H into a
complete metric space, i.e. any Cauchy sequence
of elements of the space converges to an element
in the space.
f
f
The assumption of Hilbert spaces is no con-
straint, because all finite-dimensional spaces
with a scalar product (such as Euclidean space
with ordinary scalar product) are Hilbert spaces.
However, we use Hilbert spaces with an infinite
number of dimensions to introduce the concept
of Fourier transformations. Therefore, we need an
infinite-dimensional Hilbert space of functions.
Definition 3 Let P be a Hilbert space. If the ele-
ments f P are functions, P is called a function
space .
Basis transformations
Example 1 The set of all functions f :
with a finite integral
Basis transformations map the data from the given
vector space into another space. Audio data— like
all univariate time series— are originally elements
of the vector space 2 . The basis B of a vector
space V is a set of vectors which can represent
all vectors in V by their linear combination. The
only required operation on vector spaces as the
domain of transformations is the scalar product.
Since the most common basis transformation
performed on audio data is the transformation
into the infinite space of harmonic oscillations
we assume Hilbert spaces .
f 2 ( x ) dx
together with the inner product
f
,
g
=
f ( x ) g ( x ) dx
form a well known function space: L 2 .
Figure 2.Overlay of two curves, ν 1 = 2Hz, a 1 = 3 and ν 2 = 8Hz, a 2 = 1, shown left in time space, right in frequency
space after a Fourier transformation.
frequency
time
(a) Time space
(b) Frequency space
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