Signal Processing - May 2017 - 91
s [·] Was
Sampled at
2T Samples/s
Input Signal s [·] of Size M Containing Frequencies from 0 to T Hz
Fluctuation of Size M /2,
Containing Frequencies
from T/2 to T Hz
Trend of Size M /2,
Containing Frequencies
from 0 to T/2 Hz
Trend of Size M /4,
Containing Frequencies
from 0 to T/4 Hz
Trend of Size M /8,
Containing Frequencies
from 0 to T/8 Hz
. . .
.. .. ..
Fluctuation of Size M /4,
Containing Frequencies
from T/4 to T/2 Hz
DWT
Level
j=2
Fluctuation of Size M /8,
Containing Frequencies
from T/8 to T/4 Hz
DWT
Level
j=3
(a)
s [·] Was
Sampled at
2T Samples/s
Input Signal s [·] of Size M Containing Frequencies from 0 to T Hz
Subband of
Size M /2,
Containing
Frequencies
from
0 to T/2 Hz
Subband of
Size M /4,
Containing
Frequencies
from
0 to T/4 Hz
Subband of
Size M /8,
Containing
Frequencies
from 0
to T/8 Hz
Subband of
Size M /8,
Containing
Frequencies
from T/8
to T/4 Hz
DWT
Level
j=1
Subband of
Size M /2,
Containing
Frequencies
from
T/2 to T Hz
Subband of
Size M /4,
Containing
Frequencies
from
T/4 to T/2 Hz
Subband of
Size M /4,
Containing
Frequencies
from
T/2 to 3T/4 Hz
DWT
Packet
Level
j=1
Subband of
Size M /4,
Containing
Frequencies
from
3T/4 to T Hz
Subband of Subband of Subband of Subband of
Size M /8,
Size M /8,
Size M /8,
Size M /8,
Containing
Containing
Containing
Containing
Frequencies Frequencies Frequencies Frequencies
from T/4
from 3T/8
from T/2
from 5T/8
to 3T/8 Hz
to T/2 Hz
to 5T/8 Hz
to 3T/4 Hz
... ... ...
Subband of
Size M /8,
Containing
Frequencies
from 3T/4
to 7T/8 Hz
DWT
Packet
Level
j=2
Subband of
Size M /8,
Containing
Frequencies
from 7T/8
to T Hz
DWT
Packet
Level
j=3
(b)
FIGURE 1. (a) The traditional DWT decomposition tree, exemplified for j = 3, where the ^M/8h + ^M/8h + ^M/4h + ^M/2h = M -sample long signal y [·]
corresponds to the concatenation of all the leafs, considering a left-to-right and bottom-up procedure. (b) The corresponding DWT-packet tree, built
based on the NFO.
Apply individually, for each leaf
subband contained in the wavelet
decomposition tree at the level j,
regardless of coming from the regular DWT map or the DWT-packet
map, a circular right-sided shift of
(N - 2) /2 samples.
COR comes from the fact that the
convolution between s [·] and a wavelet filter produces a signal for which
the length corresponds to M + N - 1,
primarily. Then, once the downsampling
by two is applied [2], this length is reduced
to (M + N - 1) /2 = (M/2) + (N - 1) /2.
Finally, the wraparound procedure [2]
vanishes the effect of the term (N - 1) /2,
remaining only a (M/2) -sample-long
resulting signal. Thus, even though N
has no inf luence over the length of
the transformed signal, it does cause
shifts. Particularly, due to the fact that
N is usually even and half-shifts are
not defined in discrete wavelet systems, (N - 1) /2 can be treated as being
IEEE Signal Processing Magazine
|
May 2017
|
(N - 2) /2, implying that all the filters
but Haar's cause a shifting. Therefore,
COR consists of a circular right-sided
shift of (N - 2) /2 samples, regardless
of j. Notably, COR does not correct
delays originated from nonlinear phase
responses of the filters, which can be
attenuated with the use of specific wavelet families, as aforementioned.
Finally, a refined time-frequency
description based on the previously
defined strategies requires the use of
91
Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
Signal Processing - May 2017 - 5
Signal Processing - May 2017 - 6
Signal Processing - May 2017 - 7
Signal Processing - May 2017 - 8
Signal Processing - May 2017 - 9
Signal Processing - May 2017 - 10
Signal Processing - May 2017 - 11
Signal Processing - May 2017 - 12
Signal Processing - May 2017 - 13
Signal Processing - May 2017 - 14
Signal Processing - May 2017 - 15
Signal Processing - May 2017 - 16
Signal Processing - May 2017 - 17
Signal Processing - May 2017 - 18
Signal Processing - May 2017 - 19
Signal Processing - May 2017 - 20
Signal Processing - May 2017 - 21
Signal Processing - May 2017 - 22
Signal Processing - May 2017 - 23
Signal Processing - May 2017 - 24
Signal Processing - May 2017 - 25
Signal Processing - May 2017 - 26
Signal Processing - May 2017 - 27
Signal Processing - May 2017 - 28
Signal Processing - May 2017 - 29
Signal Processing - May 2017 - 30
Signal Processing - May 2017 - 31
Signal Processing - May 2017 - 32
Signal Processing - May 2017 - 33
Signal Processing - May 2017 - 34
Signal Processing - May 2017 - 35
Signal Processing - May 2017 - 36
Signal Processing - May 2017 - 37
Signal Processing - May 2017 - 38
Signal Processing - May 2017 - 39
Signal Processing - May 2017 - 40
Signal Processing - May 2017 - 41
Signal Processing - May 2017 - 42
Signal Processing - May 2017 - 43
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Signal Processing - May 2017 - 45
Signal Processing - May 2017 - 46
Signal Processing - May 2017 - 47
Signal Processing - May 2017 - 48
Signal Processing - May 2017 - 49
Signal Processing - May 2017 - 50
Signal Processing - May 2017 - 51
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Signal Processing - May 2017 - 53
Signal Processing - May 2017 - 54
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Signal Processing - May 2017 - 58
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Signal Processing - May 2017 - 60
Signal Processing - May 2017 - 61
Signal Processing - May 2017 - 62
Signal Processing - May 2017 - 63
Signal Processing - May 2017 - 64
Signal Processing - May 2017 - 65
Signal Processing - May 2017 - 66
Signal Processing - May 2017 - 67
Signal Processing - May 2017 - 68
Signal Processing - May 2017 - 69
Signal Processing - May 2017 - 70
Signal Processing - May 2017 - 71
Signal Processing - May 2017 - 72
Signal Processing - May 2017 - 73
Signal Processing - May 2017 - 74
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Signal Processing - May 2017 - 76
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Signal Processing - May 2017 - 85
Signal Processing - May 2017 - 86
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Signal Processing - May 2017 - 88
Signal Processing - May 2017 - 89
Signal Processing - May 2017 - 90
Signal Processing - May 2017 - 91
Signal Processing - May 2017 - 92
Signal Processing - May 2017 - 93
Signal Processing - May 2017 - 94
Signal Processing - May 2017 - 95
Signal Processing - May 2017 - 96
Signal Processing - May 2017 - 97
Signal Processing - May 2017 - 98
Signal Processing - May 2017 - 99
Signal Processing - May 2017 - 100
Signal Processing - May 2017 - 101
Signal Processing - May 2017 - 102
Signal Processing - May 2017 - 103
Signal Processing - May 2017 - 104
Signal Processing - May 2017 - 105
Signal Processing - May 2017 - 106
Signal Processing - May 2017 - 107
Signal Processing - May 2017 - 108
Signal Processing - May 2017 - 109
Signal Processing - May 2017 - 110
Signal Processing - May 2017 - 111
Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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