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Figure 3. Autocorrelation differences for a phase shift depending on speeds ranging from 90 to 170
beats per minute
The goal of
Fourier analysis
is to write the
series (
x
i
)
i
∈
{1
,…,n
}
as a (possibly infinite) sum
of multiples of the given base functions, which are
e
iνx
for all frequencies
ν
. A Fast Fourier Transfor-
mation (Cooley & Tukey, 1965) maps the given
time space into this frequency space and is valid
for audio data (Figure 2). The frequency space
is a special case of a function space. Therefore,
the transformation uses the infinite number of
complex valued dimensions of a Hilbert space.
Complex numbers are necessary because Fourier
transformations actually deliver two values: the
intensity of occurring frequencies and the phase
shifts.
The frequency space expresses a sort of cor-
relation between values in terms of frequencies.
For some features it would be more appropriate
to express the correlation in terms of time de-
pendencies. Therefore, the transformation into
another space is used.
sampling rate. If we shift the original time series
by
shift
=
T
⋅
SR
⋅
60
∕ Z
for several values of
Z
we can determine the correlation between the
original and the shifted time series. Maximal
correlation corresponds to minimal difference
between the shifted and the original series. Figure
3 shows the differences of original values with the
shifted ones. Clearly, the difference at 97 beats
per minute is minimal.
Nonlinear dynamic systems can be described
with the aid of nonlinear differential equations.
The number of variables, which must be known
to completely describe the behavior of such a
system, corresponds to the number of dimen-
sions of this system. These variables are called
state variables
.
Definition 5
The basis of the
state
space
of a
dynamic system is given by
the
state
variables
of the system, that is, the variables which must
be known to
describe the system. The elements
of a state space represent the values of the
state
variables at the examined (time) points.
Definition 4
The calculation of correlations of
values between two points in
time, i and i
+
k,
produce the
correlation
space
, where for each
lag k their
correlation coefficient in
(-1
,
+1)
is
indicated.
The
state space
emphasizes characteristics
which can hardly be seen in the original space.
Since the state variables are often unknown, a
topologically equivalent space is constructed
(Takens, 1980). This is known as
reconstruction
of state space
.
Transforming audio data into the correlation
space eases the recognition of the speed of the
music, measured in beats per minute. Assuming
T
is the number of beats per measure and
SR
the
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