Signal Processing - May 2017 - 90
relevant information the latter contains
about the former, particularly allowing
the time support of frequencies to be
accurately found.
Solution
Differently from the DFT, which converts a signal from the time to the frequency domain, the DWT filters it. Thus,
significant amplitudes in y [·] come
from frequencies that originally exist in
s [·] and were not removed by the filters.
Based on this process, our solution consists of a DWT-based filtering applied
to s [·] followed by an inspection of the
dominant amplitudes contained in the
transformed signal y [·].
The practical procedure used to
obtain y [·] from s [·] has already been
described in detail in [2]; thus, it is not
repeated at this time. Nevertheless, there
are two main issues to be addressed to
solve the stated problem: the ideal level of
transformation, i.e., j, and the appropriate analysis filter pair, i.e., the low-pass
filter h [·] and the high-pass mirrored
finite impulse response half-band filter
g [·] [4]. Once j is defined, the generalized and well-known DWT decomposition tree [4] will be more or less
accurate depending on the order N - 1
of those filters, N being their length,
and on the family they belong to, such as
Haar, Daubechies, Coiflets, Symmlets,
and so on [4].
Notably, spectral description and temporal localization hold distinct requirements for ideality. On one hand, the
former offers the finest resolution, whenever j and N are maximized and the
associated filters exhibit linear-phase and
maximally flat responses at their pass and
stop bands. On the other hand, the latter
is foolproof whenever j and N are minimized because each sample of y [·] refers
just to a small part of s [·] . Furthermore,
according to Heisenberg's uncertainty
principle [4], time and frequency information are antagonistic. The better that is,
the worse this is, and vice versa. Considering the limits imposed by the real conditions, we may either balance time and
frequency information or prioritize one of
them based on our specific needs.
Disregarding temporal information,
the finest spectral resolution is guaran90
teed whenever the deepest decomposition level is adopted, i.e., j = log 2 (M) .
Additionally, the maximum possible
length for N, for which there is no upper
bound, is required. In practice, it usually
implies filters for which N $ 40. Maximally flat responses at their pass and
stop bands, such as those provided by
Daubechies' filters, may be used to avoid
improper floatations in y [·] due to a reduced or an excessive gain at some subbands [4], causing inaccurate frequency
magnitudes. Furthermore, distinct delays
for different frequency bands are also
present, whenever the phase responses of
the filters are not linear; thus, Symmlets
and Coiflets [4], which exhibit almost
linear responses, are also interesting
choices. Contrary to this, the best temporal resolution, which causes the poorest
spectral description, requires the lowest decomposition level, i.e., j = 1, and
the smallest support size of filters, i.e.,
N = 2, directly implying in the adoption
of Haar's filters [4].
Opposed to an extreme time or frequency resolution, equilibrium is reached
whenever the intermediary decomposition
level is chosen, i.e., the situation in which
j is the mean between the minimum,
one, and the maximum, log 2 (M) . This
results in j = 6(1 + log 2 (M)) /2@, where
6 · @ is the floor operator, and implies that
the corresponding frequency resolution
is r = T/2 j = ^T/26(1 +log 2 (M ))/2@h Hz, where
2T is the sampling rate at which s [·]
was digitalized and, according to Nyquist's theorem [4], T is its maximum
frequency content. Complementarily,
a balanced time and frequency accuracy also requires an intermediary
value for N, for which two is the lower
bound and there exists no upper bound.
To circumvent the missing bound, a
careful inspection of the procedure used
to calculate DWTs, as described in [2],
is useful. It allows the statement that at
the jth-level transformation based on a
filter of support size N, information
from (N - 1) (2 j - 1) + 1 subsequent
samples of s [·] is grouped together,
directly influencing the accurate time
support of frequencies.
The final issue to be solved, before formulating a solution to the stated problem,
involves the specific map to be adopted:
IEEE Signal Processing Magazine
|
May 2017
|
the regular DWT map or the DWT-packet
map. On one hand, the former only provides the finest resolution for the lowest
frequencies, which are usually those that
carry most of the useful information.
On the other hand, the latter provides a
uniform and equally distributed time-frequency resolution for all subbands, once
j, N, and the wavelet family are defined.
At each decomposition level of the former
map, the transformation produces two
half-size and half-band signals-trend
and fluctuation, respectively-containing the low and high frequencies of the
input signal that are subsequently concatenated to establish the transformed signal.
Contrastingly, the latter map contains
2 j subband signals of size (M/2 j) at the
jth decomposition level, as detailed in [5].
Figure 1 helps to recall these schemes.
Based both on the previous notes
and Mallat's algorithm [2], the use of a
regular DWT map, as exemplified in
Figure 1(a), requires strategy STG_A-(i)
and (ii) to solve our problem, assuming an
ideal situation:
i) The energy of the ith sample of the
jth -level trend contains the amplitude of frequencies between 0 and
(T/2 j) Hz, which are located within
the range {s i2 j, s (i +1) 2 j -1} , for i =
{0, 1, 2, ..., (M/2 j) - 1} .
ii) The energy of the ith sample of the
jth -level fluctuation contains the
amplitude of frequencies between
(T/2 j) and ^T/2 j -1 h Hz, which are
located within the range {s i2 j,
s (i +1) 2 j -1}.
Accordingly, the DWT-packet map built
considering the natural frequency ordering (NFO) [5, p. 111], which is exemplified in Figure 1(b), is associated with the
strategy STG_B:
The energy of the ith sample of the
bth subband at the jth-level contains the amplitude of frequencies
between (bT/2 j) and ((b + 1) T/2 j)
Hz, which are located within
the range {s i2 j, s (i +1) 2 j -1}, for
b = {0, 1, 2, ..., 2 j - 1} a n d i =
{0, 1, 2, ..., ^ M/2 j h - 1}.
To lessen the imprecise temporal
localization caused by the influence of
N, correction COR is required right after
calculating the DWT but prior to using
those strategies:
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
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Signal Processing - May 2017 - 18
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Signal Processing - May 2017 - 20
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Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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