Information Technology Reference
In-Depth Information
•
Initialize the interval counter with
t
= 1, start
a new interval
I
t
and add the first point.
The function
F
summarizes values within a
window and thus prevents general windowing
from enlarging the data set too much. Since the
size of audio data is already rather large, it is
necessary to consider carefully the number of data
points which is handled more than once.
•
For the remaining points of the series, do:
1.
If
f
e
(
I
t,xi
) = 1
,
then add
x
i
to the current
interval
I
t
.
2.
Else close
I
t
, increase
t
by 1, and add
x
i
to the new
I
t
.
Definition 12
The
overlap
of a general window-
ing with step size s and width
w is defined as g
=
w ∕ s.
Typical examples of the decision function
f
e
refer to the gradient, delivering characteristics
such as, for example,
increase, decrease
. Signal
to symbol processing is applied to the index
dimension (time). If intervals have already been
found in the value dimension, these can be used
to induce intervals in the index dimension. For
instance, whenever an interval change in the value
dimension has been found, the current interval in
the index dimension is closed and a new one is
started. Figure 6 illustrates this combination.
Only for windowings with overlap
g
= 1 the
function can be omitted. Such a windowing per-
forms transformations for each window and is
called
piecewise
filtering.
.
Combining general windowing with the
mark-up of intervals allows consideration at
each interval being a window. This results in an
adaptive window width
w
and no overlap, that
is,
g
= 1. Of course, this speeds up processing
considerably.
functions
generalized Windowing
Many known operators on times series involve
windowing. Separating the notion of windows over
the index dimension from the functions applied
to the values within the window segment allows
it to construct many operators of the kind.
Transformations convert a series into another se-
ries. In contrast, functions calculate single values
from a series. The group of functions includes all
kinds of statistics like different averages, variance
and standard deviation. They refer to the value
dimension. We may also consider the index di-
mension, for instance, the point with the largest
value or highest amplitude. Often used functions
are those indicating peaks.
Definition 10
Given the series
(
x
i
)
i
∈ {1
,…,n
}
,
a trans-
formation is called
windowing
, if it shifts a window
of width w over
(
x
i
)
i
∈{1
,…,n
}
,
using a step size s and
evaluates in each window the function F:
y
j
=
F
(
x
i
)
i
∈{
j
•
s
+1,...,
j
•
s
+
w
}
Definition 13 (
k
-Peaks function)
The k-peak
function delivers for a series
(
x
i
)
i
∈{1
,…,n
}
the posi-
tion (index dimension), the height, and the width
of the k
largest peaks.
All y
j
together form again a series:
(
y
j
)
j
∈{
0,…,
⌊
(n-w) ∕ s
⌋
}
Definition 11
A windowing which performs an
arbitrary number of
transformations in addition
to the function F is called
general
windowing
.
It is an instance of finding extremes (minimum,
maximum). Similarly, the gradient of a regression
line can be formulated. For audio data, the spectral
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