A Signal Period Detection Algorithm Based on Morphological Self-Complementary Top-Hat Transform and AMDF - MDPI

 
information
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,

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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) is
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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 =0
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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 a
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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.
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                                 (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 realize
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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.
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                   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).
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                                (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:
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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.
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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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