A Signal Period Detection Algorithm Based on Morphological Self-Complementary Top-Hat Transform and AMDF - MDPI
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Article
A Signal Period Detection Algorithm Based on
Morphological Self-Complementary Top-Hat
Transform and AMDF
Zhao Han and Xiaoli Wang *
School of Mechanical, Electrical and Information Engineering, Shandong University, Weihai 264209, China;
hanzhao@mail.sdu.edu.cn
* Correspondence: wxl@sdu.edu.cn; Tel.: +86-138-6302-6640
Received: 14 November 2018; Accepted: 7 January 2019; Published: 11 January 2019
Abstract: Period detection technology for weak characteristic signals is very important in the
fields of speech signal processing, mechanical engineering, etc. Average magnitude difference
function (AMDF) is a widely used method to extract the period of periodic signal for its low
computational complexity and high accuracy. However, this method has low detection accuracy
when the background noise is strong. In order to improve this method, this paper proposes a
new method of period detection of the signal with single period based on the morphological
self-complementary Top-Hat (STH) transform and AMDF. Firstly, the signal is de-noised by the
morphological self-complementary Top-Hat transform. Secondly, the average magnitude difference
function of the noise reduction sequence is calculated, and the falling trend is suppressed. Finally,
a calculating adaptive threshold is used to extract the peaks at the position equal to the period of
periodic signal. The experimental results show that the accuracy of periodic extraction of AMDF after
Top-Hat filtering is better than that of AMDF directly. In summary, the proposed method is reliable
and stable for detecting the periodic signal with weak characteristics.
Keywords: weak characteristic signal; period detection; single period; self-complementary Top-Hat
transform; average magnitude difference function; adaptive threshold
1. Introduction
The weak characteristics signal period detection in noise background has a very important
position in many engineering applications, such as pitch detection in speech processing [1], vibration
period of rolling bearing impact fault extraction in mechanical engineering [2], the vibration condition
of motor stator and rotor fault detection in the power system [3], etc.
At present, there are many more classic period extraction algorithms such as the autocorrelation
function (ACF) [4], average magnitude difference function (AMDF) [5–7], cepstrum [8], wavelet
transform [9], etc. Among all period detection techniques, those based on AMDF and ACF are more
widely used than other methods. AMDF has the advantages of low computation complexity and
high precision, but with the increase of delay, the decrease of frame overlap for its calculating will
lead to the phenomenon of a falling trend [10,11]. In addition, the traditional AMDF will appear to
mistake showing the location of the lowest notch besides zero point rather than showing the real
period. The result of the ACF method is similar with the AMDF’s, which is the extreme points located
at integer multiple of the detection signal period. However, the choice of these points will be interfered
with by many factors, such as the size of signal frame, the type of window function and the reduction of
the average. Therefore, the errors of double frequency and half frequency resulted from using the ACF
will always appear in practical applications. For the problems that appeared in the above methods,
Information 2019, 10, 24; doi:10.3390/info10010024 www.mdpi.com/journal/informationInformation 2019, 10, 24 2 of 12
researchers have proposed some improved algorithms. Shimamura proposed weighted autocorrelation
(WAC) that using AMDF weight ACF makes the peaks of pitch period more prominent in the speech
pitch period extraction process [12]. But the results of that are sometimes inaccurate, for the falling
trend of the mean value of AMDF will lead to an unreasonable weighting factor. The extended average
magnitude difference function (EAMDF) proposed by Muhammad spreads over the second half
of the previous frame, the current frame, and the first half of the next frame [13]. While that is a
good correction of the falling trend in AMDF, it doesn’t solve the problem of misjudgment caused
by false peaks in period extraction fundamentally, so this method always generates errors in doing
so. In contrast, this paper chooses AMDF with lower computational complexity as the main period
detection method.
The AMDF method is susceptible to random noise [14,15], resulting in the notch not being obvious
enough, so it needs a filtering method to denoise the signal in advance. Among the many filtering
methods, mathematical morphology filtering is a kind of time-domain nonlinear filtering method with
clear physical meaning, practicality and high efficiency, which is widely employed in some fields, such
as power system signal processing [16], mechanical fault diagnosis [17–19], electrocardiogram (ECG)
measurement [20] and image processing [21–23]. Self-complementation Top-Hat (STH) transform is a
denoising method based on some operations of that, which is good at suppressing the background
noise to a great extent and retaining the details of the original signal [24]. Therefore, in order to improve
the shortcomings of traditional AMDF method, this paper proposes a period detection scheme applied
to single period signals which combines AMDF with morphology self-complementation Top-Hat
transform. In this paper, this method is applied to detect the period of weak pressure signal in the gas
outlet of the mechanical diaphragm gas meter. This signal has periodicity due to the cyclical variation
of the internal working state of the diaphragm gas meter. Also, when the gas flow rate is stable, its
period is fixed. The results show that the method can extract the period accurately, and verify its
accuracy and low computational complexity.
The rest of this paper is organized as follows. In Section 2, some related principles, and the
proposed signal processing process combining AMDF with STH transform are presented. Section 3
applies the improved method to experiments. Section 4 analyzes the experimental results. Conclusions
and remarks on possible further work are given finally in Section 5.
2. Related Theory and Improved Scheme
2.1. Principle of AMDF Algorithm
The conventional AMDF is defined as [25]:
N − τ −1
D (τ ) = ∑ | x (n) − x (n + τ )|, (1)
n =0
where x (n) are the frames of input signal whose length is N, τ is the lag number.
D (τ ) has the same characteristic of period with the original signal, and there are notches located
in the position of Tp and its multiple where Tp is the period of the periodic or quasi periodic original
signal. Therefore, the period of the useful signal can be determined by calculating D (τ ) to find the
position of the highest notch (expect zero position).
2.2. Mathematical Morphology Filtering
The principle of mathematical morphology is designing a structure operator which could
move constantly in the signal and match the signal exactly, suppress the noise and keep the detail.
The morphology transformation includes several kinds of basic operations, the eroding operation,
dilating operation, open operation, close operation, and the combination of them [26]. This method was
mainly used in image processing at the initial stage. In recent years, it has been used in one-dimensional
signal filtering and has achieved good results [27–30]. Suppose that x (n)(n = 0, 1, . . . , N − 1) isInformation 2019, 10, 24 3 of 12
one-dimensional signal, g(m)(m = 0, 1, . . . , M − 1) is the structure operator, and N is greater than M.
Then the dilating operation ⊕ and the eroding operation are expressed as [28]:
( x ⊕ g)(n) = max{ x (n + m) − g(m)},
(2)
( x g)(n) = min{ x (n + m) − g(m)}.
Based on the dilating operation and eroding operation, open operation ◦ and closed operation •
are formed. The open operation can suppress and eliminate the positive impulse noise in the signal,
while the closed operation can filter the negative impulse noise. They are expressed as [29]:
( x ◦ g)(n) = ( x g ⊕ g)(n),
(3)
( x • g)(n) = ( x ⊕ g g)(n).
2.3. Self-Complementation Top-Hat (STH) Transform
The morphology Top-Hat transformation includes the white Top-Hat (WTH) operation and the
black Top-Hat (BTH) operation. WTH and BTH are expressed as:
WTH(n) = x (n) − ( x ◦ g)(n),
(4)
BTH(n) = ( x • g)(n) − x (n).
STH is defined as the sum of WTH and BTH, which is [29]:
STH(n) = WTH(n) + BTH(n) = ( x • g)(n) − ( x ◦ g)(n). (5)
In one-dimensional signal processing, STH can increase the peak value of the signal, and then
enhance the impact characteristics of the signal, which is conducive to peak extraction. In addition,
the selection of the size and shape of structure operator has a great influence on the signal processing
results in morphology operations. There are many kinds of structure operators with different shapes,
such as triangles, circles, and other polygons and their combinations. These shapes should approximate
the graphical characteristics of the detection signal, and the computation complexity should also be
taken into account when making a selection [30].
2.4. An Improved Method Based on AMDF and STH
In order to solve the problems that the falling trend of AMDF is serious and the amplitude of
AMDF is susceptible to noise, this paper combines the traditional AMDF method with the mathematical
morphological filtering method STH to form a new periodic detection method, which can be called
STH-AMDF. This method is mainly divided into three steps as follows.
(1) Collect the periodic signal to be detected, and filter that by STH transform.
(2) Use AMDF transform the filtered signal, and suppress the falling trend.
(3) Set the dynamic threshold to extract peaks.
In step (1), a structure operator needs to be selected before morphology filtering. Since the noise
in the collected signal is mainly from the rotating structure inside the diaphragm gas meter, a simple
flat zero structure (linear structure) element is selected for filtering [17,24]. In step (2), to suppress the
falling trend, this paper divides the AMDF value by the number of overlapping points. And then, the
notches are converted to peaks by taking the reciprocal of the non-zero amplitude. This improved
AMDF based on Formula (1) is defined as
τ
D (τ ) = , (6)
N − τ −1
∑ | x (n) − x (n + τ )|
n =0Information 2019, 10, x 4 of 12
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where all variables have the same meaning as Formula(1). D(τ ) has the same characteristic of
period all
where with the original
variables have signal,
the sameand there are
meaning as peaks
Formula located
(1). Din(τthe position
) has the same Tp and its multiple.
of characteristic of period
with The
the original signal, and there are peaks located in the position of T and its multiple.
periodic signal used in this paper is the pressure signal ofpthe outlet of the traditional
The periodic
diaphragm signal
gas meter. used
Since thatinisthis paper
related is the
to the modelpressure signal ofofthe
and structure theoutlet of thegas
gas meter, traditional
network
pressure and background noise, the signal has no fixed shape, which also results in having network
diaphragm gas meter. Since that is related to the model and structure of the gas meter, gas no fixed
pressure
range of and
AMDFbackground
spectrumnoise,
values.theTherefore,
signal has the
no fixed
step shape,
(3) useswhich also results
the dynamic in having
threshold no fixed
method to
range ofthe
extract AMDF
peaksspectrum values.
in the AMDF Therefore,
spectrum. Thethe stepchart
flow (3) uses thepeak
of the dynamic threshold
extraction method
algorithm to extract
is shown in
the peaks
Figure 1. in the AMDF spectrum. The flow chart of the peak extraction algorithm is shown in Figure 1.
Start
Calculate AMDF spectral sequence
Calculate the minimum value Vmin of the
first half points in the AMDF spectrum
Calculate the mean value Vmean of the first
half points in the AMDF spectrum
Determine the threshold V:
V = (Vmean -Vmin )*k+Vmean
Search for peak position in AMDF
sequences using threshold V:
L(L1,L2,…,Ln)
Computing L sequence difference results：
W(W1,W2,…，Wn-1)
Select the median Wm in the W
sequence as the extraction period
Calculate the period numbers
End
Figure 1. The
The flow
flow chart of the peak extraction algorithm.
Figure 11 indicates
indicatesthat
thatsupport
supportfor forthe
thelength
lengthofofAMDF
AMDF spectral sequence
spectral sequence is N.
is ToN extract the peak
. To extract the
position
peak and peak
position andinterval in that sequence,
peak interval the minimum
in that sequence, value Vmin
the minimum value Vmin and
and mean value Vmean
mean of theVfirst
value m ean
of the N/2
to the first points
to theinNthe2 sequence
points inare thecalculated
sequencefirst, areand then calculate
calculated first, andthe peak
then extraction
calculate threshold
the peak
V. The expression for V is
extraction threshold V. The expression for V is
V = (Vmean − Vmin ) × k + Vmean , (7)
V = (Vmean − Vmin ) × k + Vmean , (7)
where k is the coefficient of threshold calculating. The principle of Formula (7) is to screen out the peaks
that arek above
where a certain range
is the coefficient of the mean
of threshold of the spectral
calculating. sequence,
The principle and the method
of Formula (7) is toof determining
screen out the
this range
peaks thatrefers to the difference
are above between
a certain range of the
the mean
meanand of the minimum in the spectrum.
spectral sequence, and the method of
Setting anthis
determining appropriate
range refersthreshold
to the anddifference
performing peak identification
between the mean and and extraction
the minimum of the AMDF
in the
sequence
spectrum. play an important role in the calculation of the signal period. If the threshold is too small, the
aperiodic
Settingpeak
an value in the AMDF
appropriate thresholdsequence will be introduced,
and performing so that the calculated
peak identification value of
and extraction of the
spectralsequence
AMDF peak interval is too
play an small, resulting
important role in the incalculation
the computational flowperiod.
of the signal higher Ifthan
thethe actual value.
threshold is too
If the threshold is too large, some effective peaks in the AMDF sequence
small, the aperiodic peak value in the AMDF sequence will be introduced, so that the calculated will be omitted, so that the
calculated
value of thevalue of the
spectral peakspectral
intervalpeak interval
is too small,isresulting
too large,inresulting in a computational
the computational flow higher flowthanlower
the
than the
actual actualIfvalue.
value. The effective
the threshold is toopeak appearing
large, in the AMDF
some effective peaksspectrum
in the AMDFof the gas cycle pressure
sequence will be
omitted, so that the calculated value of the spectral peak interval is too large, resulting in aInformation 2019,
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computational flow lower than the actual value. The effective peak appearing in the AMDF
cycle signal
spectrum of of
thethe
gasmembrane gas meter
cycle pressure is usually
cycle signal a membrane
of the narrow peak. gasThe peak
meter height is
is usually increased
a narrow by
peak.
reducing
The peakthe signal
height noise, andby
is increased thereducing
increase theof the peaknoise,
signal heightand
doesthenot have an
increase ofexcessive
the peak influence
height doeson
the
not calculation of the Vinfluence
have an excessive mean value.onTherefore,
the calculation of the Vmof
the robustness eanthe signal
value. cycle extraction
Therefore, algorithm
the robustness of
can be improved by increasing the threshold coefficient k appropriately. Through
the signal cycle extraction algorithm can be improved by increasing the threshold coefficient k a large number of
calculation tests,
appropriately. the k ofathis
Through paper
large is 1.4.of calculation tests, the k of this paper is 1.4.
number
Next,
Next, the
theposition
positionsequence
sequence L isLobtained by selecting
is obtained the peak
by selecting thevalue,
peakand the interval
value, and thesequence
interval
W between the peaks is obtained by L. Then
sequence W between the peaks is obtained by L . Then the the median W of the W sequence is
m median Wm of the W sequence selected as the
is
period
selectedlength
as theofperiod
the signal.
lengthFinally,
of the the number
signal. of periods
Finally, is extracted
the number by that
of periods length. by that length.
is extracted
Summarizing
Summarizing the the above
above steps,
steps, the
the period
period detection
detection method proposed in
method proposed in this
this paper
paper isis shown
shown inin
Figure 2.
Figure 2.
Start
Input data
Use the STH transform to filter the
background noise
Use the AMDF transform and
suppress the falling trend
Calculate the dynamic threshold to extract
the period
Calculate the period
numbers
End
Figure 2. The flow chart of the peak detection method based on AMDF and STH algorithm.
3. Experimental Process
3. Experimental Process and
and Results
Results
In
In this
thispaper, thethe
paper, measured periodic
measured pressure
periodic signalssignals
pressure are collected from the from
are collected outletthe
of the mechanical
outlet of the
diaphragm gas meter with the stable gas source, and the fluctuation of the pipeline
mechanical diaphragm gas meter with the stable gas source, and the fluctuation of the pipeline network is
eliminated. The sampling frequency is 6 times/second, and the known period range is
network is eliminated. The sampling frequency is 6 times/second, and the known period range is 5 5 seconds to
90 s.
seconds to 90 s.
3.1. Simulation of Filtering with STH Transform
3.1. Simulation of Filtering with STH Transform
In the actual signal collecting process, there are many sources of noise and they are random.
In the actual signal collecting process, there are many sources of noise and they are random. At
At the same time, for the collected signal, it’s difficult to get the signal without noise. As such, the
the same time, for the collected signal, it’s difficult to get the signal without noise. As such, the
random noise is added to the collected signal, and then the morphological transform is used to filter
random noise is added to the collected signal, and then the morphological transform is used to filter
the signal to verify its effect. Additive white Gaussian noise is a good way to simulate a variety of
the signal to verify its effect. Additive white Gaussian noise is a good way to simulate a variety of
noise sources, so it’s used in this simulation. In addition, the mean value of the collected signal is near
noise sources, so it’s used in this simulation. In addition, the mean value of the collected signal is
0, and the energy is low. The specific details of the experiment are as follows.
near 0, and the energy is low. The specific details of the experiment are as follows.
The signal with a length of 200 to be measured and the addition of Gaussian white noise (0, 2) are
The signal with a length of 200 to be measured and the addition of Gaussian white noise (0, 2)
shown in Figure 3.
are shown in Figure 3.Information 2019, 10,
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Information 2019, 10, x 6 of 12
(a) (b)
(a) (b)
Figure 3. Pressure signal before filtering: (a) The original signal; (b) The signal after adding Gaussian
Figure 3. Pressure signal before filtering: (a) The original signal; (b) The signal after adding Gaussian
white noise.
white noise.
Figure 3 demonstrates that the original signal has periodicity and the peak value is obvious,
Figurethe
but when 3 demonstrates
Gauss noise is that the original
added, the peak signal
signal
value has periodicity
is periodicity
submergedand and thenoise.
the
in the peak value
peak value is obvious,
is
According obvious,
to the
reference [30], the length of the structural element is related to the period of the signal. Since the
but when the Gauss noise
noise is
is added,
added, the
the peak
peak value
value is
is submerged
submerged in
in the
the noise.
noise. According to the
reference
reference
period length[30],
[30],the thelength
range length
is 30 ofof
thethe
points structural
structural element
to 540 points,elementis related
this is
paper to
related the
toperiod
selects the of the
period
a linear signal.
of Since
the signal.
structural elementthe period
Since
withthea
length
length range
period length isrange
of 11 points 30 points 30topoints
is perform
to 540 points,
to 540this
morphological paper
points, thisselects
paper
filtering a selects
on linear structural
a linear
the signal, element
structural
which with a the
element
can remove length
with of
noisea
11 points
length of to
11 perform
points to morphological
perform filtering
morphological on the signal,
filtering on which
the
component contained in the signal and retain the characteristic pulse component. Firstly, the can
signal, remove
which the
can noise
remove component
the noise
contained
component
morphological in contained
theopen
signal and
in the
operation retainis the
signal characteristic
and retain
performed on the pulse
the component.
characteristic
signal with GaussianFirstly,
pulse the morphological
component.
white noise, and the open
Firstly, the
peak
operation
morphological is performed
open on
operation the signal
is with
performed Gaussian
on the white
signal noise,
with and the
Gaussian
in the original signal can be eliminated, and the fluctuation component with the width larger than peak
white in the
noise, original
and thesignal
peak
can
in
thethebe original
eliminated,
structural andcan
signal
element the
is befluctuation
eliminated,
filtered out, and component
and thethe
then with
fluctuation
signal theiswidth largerwith
component
compared than
withthethe
the structural
width larger
calculation element
than
result of
is
thefiltered
the structuralout,
open operation and then
element the signal
is filtered
to extract is compared
out, and
positive then
peaks with
inthe
thesignalthe calculation
signal;isthe compared result of the
with theclose
morphological open operation
calculation
operation result to
of
of the
extract
the
noisyopen positive
operation
signal can peakstoinextract
obtain the
thesignal;
negative the morphological
positive peaks in
peaks, and closethe
thefinally
signal; operation
theSTH of the noisy
morphological
sequence signal
close
containing can
theobtain
operation of the
positive
negative
noisy peaks,
signal can and finally
obtain the the STH
negative sequence
peaks, containing
and finally the
the positive
STH
and negative peaks information of the original signal can be calculated, as shown in Figure 4. and
sequencenegative peaks
containing information
the positive
of the
and originalpeaks
negative signalinformation
can be calculated, as shown
of the original in Figure
signal can be 4.calculated, as shown in Figure 4.
Figure 4.
Figure Results of
4. Results of self-complementary
self-complementary Top-Hat
Top-Hat transform
transform for
for the
the noisy
noisy signal.
signal.
Figure 4. Results of self-complementary Top-Hat transform for the noisy signal.
It
It can
can be
be seen
seen from
from Figure
Figure 44 that
that the
the peaks
peaks of
of the
the noisy
noisy signal
signal after
after STH
STH transformation
transformation become
become
very obvious,
It can be which
seen is
from very beneficial
Figure 4 that for
the the peak
peaks of extraction.
the
very obvious, which is very beneficial for the peak extraction. noisy signal after STH transformation become
very The
The mathematical
obvious, which is morphology
mathematical very filtering
beneficial
morphology for themethod
filteringpeak is used
extraction.
method to denoise
is used the gas
to denoise thepressure signal, signal,
gas pressure which
has the
whichThe characteristics
hasmathematical of low computational
morphology
the characteristics complexity,
filtering method
of low computational good denoising
is used togood
complexity, denoisecharacteristics
the gas
denoising and low-pass
pressure signal,
characteristics and
characteristics.
which has the Its core is to
characteristics detect
of the
low position of
computational the target in the
complexity, signal
good by structural
denoising elements, obtain
characteristics
low-pass characteristics. Its core is to detect the position of the target in the signal by structural and
low-pass
elements, characteristics.
obtain the geometricIts core is toinformation
shape detect the position of the target
and its relationship in in
thethe signal
signal, andbythen
structural
realize
elements, obtain the geometric shape information and its relationship in the signal, and then realizeInformation 2019, 10, 24 7 of 12
Information 2019, 10, x 7 of 12
the geometric shape information and its relationship in the signal, and then realize the judgment of
signal
the characteristics.
judgment Incharacteristics.
of signal addition, selecting the most selecting
In addition, commonly theused flatcommonly
most structure elements
used flatasstructure
filtering
windows as
elements canfiltering
effectively extract useful
windows features from
can effectively theuseful
extract signal features
and further
fromreduce the computational
the signal and further
complexity
reduce of the algorithm.
the computational complexity of the algorithm.
3.2. Simulation
3.2. Simulation of
of Improved
Improved AMDF
AMDF and
and Peak
Peak Extraction
Extraction Algorithm
Algorithm
A gas
A gas pressure
pressuresignal
signalsegment with
segment a length
with of 300
a length of and
300 its
andtraditional AMDFAMDF
its traditional calculation results
calculation
are presented in Figure 5.
results are presented in Figure 5.
(a)
(b)
Figure 5. Simulation
Simulation results
results of
of the traditional AMDF algorithm:
algorithm: (a) The original signal; (b) The
traditional AMDF result.
According to
According to Figure
Figure 5,
5, with
with the
the increase
increase of
of lag
lag time, the peak
time, the amplitude of
peak amplitude of the
the conventional
conventional
AMDF will decrease. Therefore, in this paper, the improved AMDF method (Formula (6))
AMDF will decrease. Therefore, in this paper, the improved AMDF method (Formula (6)) is
is applied,
applied,
and the
and the result
result of
of that
that is
is shown
shown in
in Figure
Figure 6.
6.Information 2019, 10, 24 8 of 12
Information2019,
Information 2019,10,
10,xx 88of
of12
12
Figure6.6.Simulation
Figure Simulationresults
Simulation resultsof
results ofthe
theimproved
improvedAMDF
AMDFalgorithm.
algorithm.
Figure66indicates
Figure indicatesthat thatthetheimproved
improvedAMDF AMDFmethod methodcan canmake
makethe theperiodic
periodiccharacteristic
characteristicof ofthe
the
signalbe
signal beexpressed
be expressedinin
expressed in
the the
theform form
form of peaks
of peaks
of peaks
andand and eliminate
eliminate
eliminate the downward
the downward
the downward trend
trend trend
of of the
the traditional
the traditional
of traditional
AMDF
AMDF
calculation
AMDF calculation
results. In
calculation results.
addition,
results. In addition, as
as can beasseen
In addition, can
canfrombe seen
be seen from
Figure
from Figure
6, Figure
peak values6, peak
6, peak values
arevalues are
different
areand different anditit
it is difficult
different and
is difficult
to
is difficultthe
extract to extract
to extractby
peaks theusing
the peaksfixed
peaks by using
by using fixed threshold,
threshold,
fixed threshold,
so so the
the dynamic
so thethreshold
dynamic is
dynamic threshold
calculated
threshold isis calculated
calculated
by using the by
by
using
algorithm the algorithm
shown in shown
Figure
using the algorithm shown in Figure 1. in
1. Figure 1.
Figure777is
Figure isthe
is thesimulation
the simulationresult
simulation result diagram
resultdiagram
diagramof of the
ofthe AMDF
theAMDF
AMDFpeaks peaks extracting
peaksextracting method.
extractingmethod.
method. The
The The red
redredline isis
line
line
the
is
the thethreshold
threshold
threshold line.
line. ItItcan
line. can beseen
It can
be seen fromfrom
be seen
from thefigure
the figure thatthis
the figure
that this method
thatmethod isiseffective
this method effective inextracting
extracting
is effective
in thepeaks.
in extracting
the peaks.
the
However,
peaks. However,
However, the peak
the peak amplitudes
theamplitudes
peak amplitudes are still not
arenot
are still obvious,
stillobvious,
not obvious,so the
so the determined
sodetermined
the determined threshold
threshold
threshold cannot
cannotcannot extract
extract
extract all
all
peaks.
all peaks.
peaks.
Figure 7.
Figure7. Simulation
7.Simulation results
Simulationresults of
resultsof the
ofthe peak
thepeak extraction
peakextraction algorithm.
extractionalgorithm.
algorithm.
Figure
3.3. Simulation of STH-AMDF Algorithm
3.3.Simulation
3.3. SimulationofofSTH-AMDF
STH-AMDFAlgorithm
Algorithm
In order to make the experimental results more obvious, this paper selects a sampling signal
Inorder
In ordertotomake
makethe
the experimentalresultsresultsmore
moreobvious,
obvious, thispaper
paper selectsaasampling
samplingsignal
signalof
of
of length 700 (Figure 8a). experimental
In the first group, AMDF operationthis
and peak selects
extraction were performed
length
length 700 (Figure 8a). In the first group, AMDF operation and peak extraction were performed
directly700 (Figure
on the 8a). In
sampled the (Figure
signal first group, AMDF
8b). In operation
the second and
group, peak
STH extraction
filtering were performed
was performed on the
directlyon
directly onthe
thesampled
sampledsignal
signal(Figure
(Figure8b).
8b).InInthe
thesecond
secondgroup,
group,STH
STHfiltering
filteringwas
wasperformed
performedononthe
the
sampled signal (Figure 8c), followed by AMDF operation and peak extraction (Figure 8d).
sampled signal (Figure 8c), followed by AMDF operation and peak extraction
sampled signal (Figure 8c), followed by AMDF operation and peak extraction (Figure 8d). (Figure 8d).Information 2019, 10, 24 9 of 12
Information 2019, 10, x 9 of 12
(a) (b)
(c) (d)
Figure
Figure8.8. Simulation
Simulationresults
resultsof
of the
the STH-AMDF
STH-AMDF algorithm:
algorithm: (a)
(a)The
Theoriginal
originalsignal;
signal;(b)
(b)The
Theresult
resultof
of
AMDFof
AMDF oforiginal
originalsignal
signaland
andpeak
peakextraction
extractionalgorithm;
algorithm;(c)
(c)The
Thesignal
signalafter
afterSTH
STHtransform;
transform;(d)
(d)The
The
resultof
result ofSTH-AMDF
STH-AMDFalgorithm.
algorithm.
Asshown
As shownin inFigure
Figure8b, 8b,the
thesecond,
second,sixth,
sixth,eleventh,
eleventh,fifteenth
fifteenthand andsixteenth
sixteenthpeaks
peaksin inthe
theAMDF
AMDF
spectrum are
spectrum are lower
lower thanthan the
the extraction
extraction threshold,
threshold, and and the
the peaks
peaks after
after the
the nineteenth
nineteenth peak peak areare
pseudo-peaks and their peak intervals are significantly smaller than the
pseudo-peaks and their peak intervals are significantly smaller than the real periodic values. real periodic values. Although
the error of
Although this
the calculation
error of this can be avoided
calculation canby beusing
avoidedthe median
by using of the
the median
peak interval
of theaspeak
the period value,
interval as
when
the the signal
period data
value, is shortened,
when the signal the data
probability of occurrence
is shortened, of the cycle calculation
the probability of occurrence errorofwill
theincrease
cycle
greatly. Therefore,
calculation the signal
error will needsgreatly.
increase morphological
Therefore,filtering
the tosignal
enhance the useful
needs peak information
morphological filteringof the
to
original weak
enhance periodic
the useful peaksignal and filter of
information outthe
part of the noise.
original weakAccording to Figure
periodic signal and8d, afterout
filter morphological
part of the
filtering,
noise. the AMDF
According spectral
to Figure 8d,shape
after of the signal is clearly
morphological filtering,improved
the AMDF for spectral
extracting periodic
shape of thefeatures,
signal
is clearly improved for extracting periodic features, which is manifested by the distinctreduction
which is manifested by the distinct increase of periodic peaks, uniform distribution, increase of of
pseudo-peaks
periodic peaks,and low height.
uniform The threshold
distribution, reduction set of
bypseudo-peaks
Formula (7) is and approximately
low height.inThe thethreshold
middle of set the
effective peak, higher than the pseudo-peaks and lower than all the effective
by Formula (7) is approximately in the middle of the effective peak, higher than the pseudo-peaks peaks. This method greatly
improves
and lower the
thanaccuracy and robustness
all the effective of the
peaks. This weak signal
method greatlyperiodic
improves extraction algorithm.
the accuracy and robustness of
the weak signal periodic extraction algorithm.
4. Discussion
4. Discussion
In order to verify the optimization effect of self-complementary Top-Hat filter on AMDF peak
extraction,
In orderit is
tonecessary
verify thetooptimization
calculate theeffect
meanofof self-complementary mp in AMDF
effective peak heightTop-Hat filterspectrum
on AMDFand the
peak
extraction, it is necessary to calculate the mean of effective peak height mp in AMDF spectrum and
the mean value mn in spectrum sequence with the effective peak removed. The definition of the
peaks recognition accuracy parameter P is as follows:Information 2019, 10, 24 10 of 12
mean value
Information 2019,m10,
n in
spectrum sequence with the effective peak removed. The definition of the 10
x peaks
of 12
recognition accuracy parameter P is as follows:
mp m
P =P = . p . (8)
(8)
mn m n
The larger
The the PP value
larger the valueis,is,the
thehigher
higherthe thepseudo-peak
pseudo-peakvalue
value is,
is, and
and the
the easier it is to filter the
the
pseudo-peak and
pseudo-peak and reduce
reduce thethe pseudo-peak
pseudo-peakinterference.
interference.The
Thesmaller
smallerthethePP value is, the closer the the
effective peak
effective peakheight
heightis is
to the pseudo
to the peak,peak,
pseudo whichwhich
increases the probability
increases of periodic
the probability of error extraction.
periodic error
When the PWhen
extraction. valuethe P value
is small, it usually leads
is small, it to the extraction
usually leads to period value being
the extraction smaller
period valuethan
beingthe actual
smaller
value,the
than which
actualleads to the
value, flowleads
which calculation
to the value being larger
flow calculation thanbeing
value the actual value.
larger than The
the experimental
actual value.
The
data experimental
were collecteddatafrom were collected
the pressure fromofthe
signals the pressure signals
outlet of five of the
different outlet of five
specifications different
of household
specifications
diaphragm gas ofmeters
household
underdiaphragm gasconditions.
different fire meters underEachdifferent firecontains
test group conditions.
five Each test group
complete signal
contains five
cycles. The complete
accuracy parameter P from The
signal cycles. usingaccuracy
the AMDF parameter
method and P the
from using themethod
STH-AMDF AMDF ismethod
shown
and the STH-AMDF
in Figure 9. method is shown in Figure 9.
The
Figure 9.9. The
Figure peaks
peaks recognition
recognition accuracy
accuracy parameter
parameter distribution
distribution usingusing the AMDF
the AMDF and STH-
and STH-AMDF
AMDF methods.
methods.
The Figure
The Figure99shows
shows that thethe
that effective peaks
effective in STH-AMDF
peaks in STH-AMDFsequences with morphological
sequences filtering
with morphological
are easier to extract than those in AMDF sequences. The STH-AMDF period extraction
filtering are easier to extract than those in AMDF sequences. The STH-AMDF period extraction algorithm
can better avoid
algorithm the calculation
can better avoid theerrors caused errors
calculation by the interference
caused by the peaks, and it is convenient
interference peaks, and to set
it isa
reasonable extraction
convenient threshold to
to set a reasonable ensure the
extraction accuratetoidentification
threshold of the periodic
ensure the accurate peaks, soofasthe
identification to
achieve accurate detection of the signal period.
periodic peaks, so as to achieve accurate detection of the signal period.
5. Conclusions and Future Work
5. Conclusions and Future Work
In order to extract the periodic number of noisy weak characteristic periodic signals more
In order to extract the periodic number of noisy weak characteristic periodic signals more
accurately, a STH-AMDF method of extracting period based on morphological self-complementary
accurately, a STH-AMDF method of extracting period based on morphological self-complementary
Top-Hat transform and average magnitude difference function is proposed in this paper. Traditional
Top-Hat transform and average magnitude difference function is proposed in this paper.
AMDF can reflect the periodicity of the signal, but when there is noise, its peaks are not obvious
Traditional AMDF can reflect the periodicity of the signal, but when there is noise, its peaks are not
enough, which is not conducive to peak extraction. Therefore, STH is applied to filter the noise and
obvious enough, which is not conducive to peak extraction. Therefore, STH is applied to filter the
improve the impact characteristics of the signal. Experiments show that the improved algorithm
noise and improve the impact characteristics of the signal. Experiments show that the improved
can increase the peak amplitude and set dynamic threshold to extract peak value, which verifies the
algorithm can increase the peak amplitude and set dynamic threshold to extract peak value, which
accuracy and effectiveness of this method.
verifies the accuracy and effectiveness of this method.
In future work, we will do some significant researches on the following aspects. First, we will
In future work, we will do some significant researches on the following aspects. First, we will
make a meaningful study about the real-time period detection for the signal of changing period.
make a meaningful study about the real-time period detection for the signal of changing period.
Secondly, we will collect more pressure data of the exhaust port of domestic gas meters and
optimize the algorithm to make it suitable for family life.Information 2019, 10, 24 11 of 12
Secondly, we will collect more pressure data of the exhaust port of domestic gas meters and optimize
the algorithm to make it suitable for family life.
Author Contributions: Z.H. proposed innovative idea; Z.H. and X.W. conceived the algorithm and wrote the first
draft; X.W. improved the algorithm; Z.H. performed the experiments; Z.H. and X.W. analyzed the results; Z.H.
drafted the manuscript; X.W. provided writing advice; Z.H. and X.W. read and approved the final manuscript.
Funding: This research received no external funding.
Acknowledgments: We gratefully acknowledge the technical assistance of DL850E ScopeCorder.
Conflicts of Interest: The authors declare no conflict of interest.
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