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

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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, Information 2019, 10, 24; doi:10.3390/info10010024 www.mdpi.com/journal/information

Information 2019, 10, 24 2 of 12 researchers have proposed some improved algorithms. Shimamura proposed weighted autocorrelation (WAC) that using AMDF weight ACF makes the peaks of pitch period more prominent in the speech pitch period extraction process [12]. But the results of that are sometimes inaccurate, for the falling trend of the mean value of AMDF will lead to an unreasonable weighting factor. The extended average magnitude difference function (EAMDF) proposed by Muhammad spreads over the second half of the previous frame, the current frame, and the first half of the next frame [13]. While that is a good correction of the falling trend in AMDF, it doesn’t solve the problem of misjudgment caused by false peaks in period extraction fundamentally, so this method always generates errors in doing so. In contrast, this paper chooses AMDF with lower computational complexity as the main period detection method. The AMDF method is susceptible to random noise [14,15], resulting in the notch not being obvious enough, so it needs a filtering method to denoise the signal in advance. Among the many filtering methods, mathematical morphology filtering is a kind of time-domain nonlinear filtering method with clear physical meaning, practicality and high efficiency, which is widely employed in some fields, such as power system signal processing [16], mechanical fault diagnosis [17–19], electrocardiogram (ECG) measurement [20] and image processing [21–23]. Self-complementation Top-Hat (STH) transform is a denoising method based on some operations of that, which is good at suppressing the background noise to a great extent and retaining the details of the original signal [24]. Therefore, in order to improve the shortcomings of traditional AMDF method, this paper proposes a period detection scheme applied to single period signals which combines AMDF with morphology self-complementation Top-Hat transform. In this paper, this method is applied to detect the period of weak pressure signal in the gas outlet of the mechanical diaphragm gas meter. This signal has periodicity due to the cyclical variation of the internal working state of the diaphragm gas meter. Also, when the gas flow rate is stable, its period is fixed. The results show that the method can extract the period accurately, and verify its accuracy and low computational complexity. The rest of this paper is organized as follows. In Section 2, some related principles, and the proposed signal processing process combining AMDF with STH transform are presented. Section 3 applies the improved method to experiments. Section 4 analyzes the experimental results. Conclusions and remarks on possible further work are given finally in Section 5. 2. Related Theory and Improved Scheme 2.1. Principle of AMDF Algorithm The conventional AMDF is defined as [25]: N − τ −1 D (τ ) = ∑ | x (n) − x (n + τ )|, (1) n =0 where x (n) are the frames of input signal whose length is N, τ is the lag number. D (τ ) has the same characteristic of period with the original signal, and there are notches located in the position of Tp and its multiple where Tp is the period of the periodic or quasi periodic original signal. Therefore, the period of the useful signal can be determined by calculating D (τ ) to find the position of the highest notch (expect zero position). 2.2. Mathematical Morphology Filtering The principle of mathematical morphology is designing a structure operator which could move constantly in the signal and match the signal exactly, suppress the noise and keep the detail. The morphology transformation includes several kinds of basic operations, the eroding operation, dilating operation, open operation, close operation, and the combination of them [26]. This method was mainly used in image processing at the initial stage. In recent years, it has been used in one-dimensional signal filtering and has achieved good results [27–30]. Suppose that x (n)(n = 0, 1, . . . , N − 1) is

Information 2019, 10, 24 3 of 12 one-dimensional signal, g(m)(m = 0, 1, . . . , M − 1) is the structure operator, and N is greater than M. Then the dilating operation ⊕ and the eroding operation are expressed as [28]: ( x ⊕ g)(n) = max{ x (n + m) − g(m)}, (2) ( x g)(n) = min{ x (n + m) − g(m)}. Based on the dilating operation and eroding operation, open operation ◦ and closed operation • are formed. The open operation can suppress and eliminate the positive impulse noise in the signal, while the closed operation can filter the negative impulse noise. They are expressed as [29]: ( x ◦ g)(n) = ( x g ⊕ g)(n), (3) ( x • g)(n) = ( x ⊕ g g)(n). 2.3. Self-Complementation Top-Hat (STH) Transform The morphology Top-Hat transformation includes the white Top-Hat (WTH) operation and the black Top-Hat (BTH) operation. WTH and BTH are expressed as: WTH(n) = x (n) − ( x ◦ g)(n), (4) BTH(n) = ( x • g)(n) − x (n). STH is defined as the sum of WTH and BTH, which is [29]: STH(n) = WTH(n) + BTH(n) = ( x • g)(n) − ( x ◦ g)(n). (5) In one-dimensional signal processing, STH can increase the peak value of the signal, and then enhance the impact characteristics of the signal, which is conducive to peak extraction. In addition, the selection of the size and shape of structure operator has a great influence on the signal processing results in morphology operations. There are many kinds of structure operators with different shapes, such as triangles, circles, and other polygons and their combinations. These shapes should approximate the graphical characteristics of the detection signal, and the computation complexity should also be taken into account when making a selection [30]. 2.4. An Improved Method Based on AMDF and STH In order to solve the problems that the falling trend of AMDF is serious and the amplitude of AMDF is susceptible to noise, this paper combines the traditional AMDF method with the mathematical morphological filtering method STH to form a new periodic detection method, which can be called STH-AMDF. This method is mainly divided into three steps as follows. (1) Collect the periodic signal to be detected, and filter that by STH transform. (2) Use AMDF transform the filtered signal, and suppress the falling trend. (3) Set the dynamic threshold to extract peaks. In step (1), a structure operator needs to be selected before morphology filtering. Since the noise in the collected signal is mainly from the rotating structure inside the diaphragm gas meter, a simple flat zero structure (linear structure) element is selected for filtering [17,24]. In step (2), to suppress the falling trend, this paper divides the AMDF value by the number of overlapping points. And then, the notches are converted to peaks by taking the reciprocal of the non-zero amplitude. This improved AMDF based on Formula (1) is defined as τ D (τ ) = , (6) N − τ −1 ∑ | x (n) − x (n + τ )| n =0

Information 2019, 10, x 4 of 12 Information 2019, 10, 24 4 of 12 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

Information 2019, Information 2019, 10, 10, 24 x 55 of of 12 12 computational flow lower than the actual value. The effective peak appearing in the AMDF cycle signal spectrum of of thethe gasmembrane gas meter cycle pressure is usually cycle signal a membrane of the narrow peak. gasThe peak meter height is is usually increased a narrow by peak. reducing The peakthe signal height noise, andby is increased thereducing increase theof the peaknoise, signal heightand doesthenot have an increase ofexcessive the peak influence height doeson the not calculation of the Vinfluence have an excessive mean value.onTherefore, the calculation of the Vmof the robustness eanthe signal value. cycle extraction Therefore, algorithm the robustness of can be improved by increasing the threshold coefficient k appropriately. Through the signal cycle extraction algorithm can be improved by increasing the threshold coefficient k a large number of calculation tests, appropriately. the k ofathis Through paper large is 1.4.of calculation tests, the k of this paper is 1.4. number Next, Next, the theposition positionsequence sequence L isLobtained by selecting is obtained the peak by selecting thevalue, peakand the interval value, and thesequence interval W between the peaks is obtained by L. Then sequence W between the peaks is obtained by L . Then the the median W of the W sequence is m median Wm of the W sequence selected as the is period selectedlength as theofperiod the signal. lengthFinally, of the the number signal. of periods Finally, is extracted the number by that of periods length. by that length. is extracted Summarizing Summarizing the the above above steps, steps, the the period period detection detection method proposed in method proposed in this this paper paper isis shown shown inin Figure 2. Figure 2. Start Input data Use the STH transform to filter the background noise Use the AMDF transform and suppress the falling trend Calculate the dynamic threshold to extract the period Calculate the period numbers End Figure 2. The flow chart of the peak detection method based on AMDF and STH algorithm. 3. Experimental Process 3. Experimental Process and and Results Results In In this thispaper, thethe paper, measured periodic measured pressure periodic signalssignals pressure are collected from the from are collected outletthe of the mechanical outlet of the diaphragm gas meter with the stable gas source, and the fluctuation of the pipeline mechanical diaphragm gas meter with the stable gas source, and the fluctuation of the pipeline network is eliminated. The sampling frequency is 6 times/second, and the known period range is network is eliminated. The sampling frequency is 6 times/second, and the known period range is 5 5 seconds to 90 s. seconds to 90 s. 3.1. Simulation of Filtering with STH Transform 3.1. Simulation of Filtering with STH Transform In the actual signal collecting process, there are many sources of noise and they are random. In the actual signal collecting process, there are many sources of noise and they are random. At At the same time, for the collected signal, it’s difficult to get the signal without noise. As such, the the same time, for the collected signal, it’s difficult to get the signal without noise. As such, the random noise is added to the collected signal, and then the morphological transform is used to filter random noise is added to the collected signal, and then the morphological transform is used to filter the signal to verify its effect. Additive white Gaussian noise is a good way to simulate a variety of the signal to verify its effect. Additive white Gaussian noise is a good way to simulate a variety of noise sources, so it’s used in this simulation. In addition, the mean value of the collected signal is near noise sources, so it’s used in this simulation. In addition, the mean value of the collected signal is 0, and the energy is low. The specific details of the experiment are as follows. near 0, and the energy is low. The specific details of the experiment are as follows. The signal with a length of 200 to be measured and the addition of Gaussian white noise (0, 2) are The signal with a length of 200 to be measured and the addition of Gaussian white noise (0, 2) shown in Figure 3. are shown in Figure 3.

Information 2019, 10, Information 2019, 10, 24 x 66 of of 12 12 Information 2019, 10, x 6 of 12 (a) (b) (a) (b) Figure 3. Pressure signal before filtering: (a) The original signal; (b) The signal after adding Gaussian Figure 3. Pressure signal before filtering: (a) The original signal; (b) The signal after adding Gaussian white noise. white noise. Figure 3 demonstrates that the original signal has periodicity and the peak value is obvious, Figurethe but when 3 demonstrates Gauss noise is that the original added, the peak signal signal value has periodicity is periodicity submergedand and thenoise. the in the peak value peak value is obvious, is According obvious, to the reference [30], the length of the structural element is related to the period of the signal. Since the but when the Gauss noise noise is is added, added, the the peak peak value value is is submerged submerged in in the the noise. noise. According to the reference reference period length[30], [30],the thelength range length is 30 ofof thethe points structural structural element to 540 points,elementis related this is paper to related the toperiod selects the of the period a linear signal. of Since the signal. structural elementthe period Since withthea length length range period length isrange of 11 points 30 points 30topoints is perform to 540 points, to 540this morphological paper points, thisselects paper filtering a selects on linear structural a linear the signal, element structural which with a the element can remove length with of noisea 11 points length of to 11 perform points to morphological perform filtering morphological on the signal, filtering on which the component contained in the signal and retain the characteristic pulse component. Firstly, the can signal, remove which the can noise remove component the noise contained component morphological in contained theopen signal and in the operation retainis the signal characteristic and retain performed on the pulse the component. characteristic signal with GaussianFirstly, pulse the morphological component. white noise, and the open Firstly, the peak operation morphological is performed open on operation the signal is with performed Gaussian on the white signal noise, with and the Gaussian in the original signal can be eliminated, and the fluctuation component with the width larger than peak white in the noise, original and thesignal peak can in thethebe original eliminated, structural andcan signal element the is befluctuation eliminated, filtered out, and component and thethe then with fluctuation signal theiswidth largerwith component compared than withthethe the structural width larger calculation element than result of is thefiltered the structuralout, open operation and then element the signal is filtered to extract is compared out, and positive then peaks with inthe thesignalthe calculation signal;isthe compared result of the with theclose morphological open operation calculation operation result to of of the extract the noisyopen positive operation signal can peakstoinextract obtain the thesignal; negative the morphological positive peaks in peaks, and closethe thefinally signal; operation theSTH of the noisy morphological sequence signal close containing can theobtain operation of the positive negative noisy peaks, signal can and finally obtain the the STH negative sequence peaks, containing and finally the the positive STH and negative peaks information of the original signal can be calculated, as shown in Figure 4. and sequencenegative peaks containing information the positive of the and originalpeaks negative signalinformation can be calculated, as shown of the original in Figure signal can be 4.calculated, as shown in Figure 4. Figure 4. Figure Results of 4. Results of self-complementary self-complementary Top-Hat Top-Hat transform transform for for the the noisy noisy signal. signal. Figure 4. Results of self-complementary Top-Hat transform for the noisy signal. It It can can be be seen seen from from Figure Figure 44 that that the the peaks peaks of of the the noisy noisy signal signal after after STH STH transformation transformation become become very obvious, It can be which seen is from very beneficial Figure 4 that for the the peak peaks of extraction. the very obvious, which is very beneficial for the peak extraction. noisy signal after STH transformation become very The The mathematical obvious, which is morphology mathematical very filtering beneficial morphology for themethod filteringpeak is used extraction. method to denoise is used the gas to denoise thepressure signal, signal, gas pressure which has the whichThe characteristics hasmathematical of low computational morphology the characteristics complexity, filtering method of low computational good denoising is used togood complexity, denoisecharacteristics the gas denoising and low-pass pressure signal, characteristics and characteristics. which has the Its core is to characteristics detect of the low position of computational the target in the complexity, signal good by structural denoising elements, obtain characteristics low-pass characteristics. Its core is to detect the position of the target in the signal by structural and low-pass elements, characteristics. obtain the geometricIts core is toinformation shape detect the position of the target and its relationship in in thethe signal signal, andbythen structural realize elements, obtain the geometric shape information and its relationship in the signal, and then realize

Information 2019, 10, 24 7 of 12 Information 2019, 10, x 7 of 12 the geometric shape information and its relationship in the signal, and then realize the judgment of signal the characteristics. judgment Incharacteristics. of signal addition, selecting the most selecting In addition, commonly theused flatcommonly most structure elements used flatasstructure filtering windows as elements canfiltering effectively extract useful windows features from can effectively theuseful extract signal features and further fromreduce the computational the signal and further complexity reduce of the algorithm. the computational complexity of the algorithm. 3.2. Simulation 3.2. Simulation of of Improved Improved AMDF AMDF and and Peak Peak Extraction Extraction Algorithm Algorithm A gas A gas pressure pressuresignal signalsegment with segment a length with of 300 a length of and 300 its andtraditional AMDFAMDF its traditional calculation results calculation are presented in Figure 5. results are presented in Figure 5. (a) (b) Figure 5. Simulation Simulation results results of of the traditional AMDF algorithm: algorithm: (a) The original signal; (b) The traditional AMDF result. According to According to Figure Figure 5, 5, with with the the increase increase of of lag lag time, the peak time, the amplitude of peak amplitude of the the conventional conventional AMDF will decrease. Therefore, in this paper, the improved AMDF method (Formula (6)) AMDF will decrease. Therefore, in this paper, the improved AMDF method (Formula (6)) is is applied, applied, and the and the result result of of that that is is shown shown in in Figure Figure 6. 6.

Information 2019, 10, 24 8 of 12 Information2019, Information 2019,10, 10,xx 88of of12 12 Figure6.6.Simulation Figure Simulationresults Simulation resultsof results ofthe theimproved improvedAMDF AMDFalgorithm. algorithm. Figure66indicates Figure indicatesthat thatthetheimproved improvedAMDF AMDFmethod methodcan canmake makethe theperiodic periodiccharacteristic characteristicof ofthe the signalbe signal beexpressed be expressedinin expressed in the the theform form form of peaks of peaks of peaks andand and eliminate eliminate eliminate the downward the downward the downward trend trend trend of of the the traditional the traditional of traditional AMDF AMDF calculation AMDF calculation results. In calculation results. addition, results. In addition, as as can beasseen In addition, can canfrombe seen be seen from Figure from Figure 6, Figure peak values6, peak 6, peak values arevalues are different areand different anditit it is difficult different and is difficult to is difficultthe extract to extract to extractby peaks theusing the peaksfixed peaks by using by using fixed threshold, threshold, fixed threshold, so so the the dynamic so thethreshold dynamic is dynamic threshold calculated threshold isis calculated calculated by using the by by using algorithm the algorithm shown in shown Figure using the algorithm shown in Figure 1. in 1. Figure 1. Figure777is Figure isthe is thesimulation the simulationresult simulation result diagram resultdiagram diagramof of the ofthe AMDF theAMDF AMDFpeaks peaks extracting peaksextracting method. extractingmethod. method. The The The red redredline isis line line the is the thethreshold threshold threshold line. line. ItItcan line. can beseen It can be seen fromfrom be seen from thefigure the figure thatthis the figure that this method thatmethod isiseffective this method effective inextracting extracting is effective in thepeaks. in extracting the peaks. the However, peaks. However, However, the peak the peak amplitudes theamplitudes peak amplitudes are still not arenot are still obvious, stillobvious, not obvious,so the so the determined sodetermined the determined threshold threshold threshold cannot cannotcannot extract extract extract all all peaks. all peaks. peaks. Figure 7. Figure7. Simulation 7.Simulation results Simulationresults of resultsof the ofthe peak thepeak extraction peakextraction algorithm. extractionalgorithm. algorithm. Figure 3.3. Simulation of STH-AMDF Algorithm 3.3.Simulation 3.3. SimulationofofSTH-AMDF STH-AMDFAlgorithm Algorithm In order to make the experimental results more obvious, this paper selects a sampling signal Inorder In ordertotomake makethe the experimentalresultsresultsmore moreobvious, obvious, thispaper paper selectsaasampling samplingsignal signalof of of length 700 (Figure 8a). experimental In the first group, AMDF operationthis and peak selects extraction were performed length length 700 (Figure 8a). In the first group, AMDF operation and peak extraction were performed directly700 (Figure on the 8a). In sampled the (Figure signal first group, AMDF 8b). In operation the second and group, peak STH extraction filtering were performed was performed on the directlyon directly onthe thesampled sampledsignal signal(Figure (Figure8b). 8b).InInthe thesecond secondgroup, group,STH STHfiltering filteringwas wasperformed performedononthe the sampled signal (Figure 8c), followed by AMDF operation and peak extraction (Figure 8d). sampled signal (Figure 8c), followed by AMDF operation and peak extraction sampled signal (Figure 8c), followed by AMDF operation and peak extraction (Figure 8d). (Figure 8d).

Information 2019, 10, 24 9 of 12 Information 2019, 10, x 9 of 12 (a) (b) (c) (d) Figure Figure8.8. Simulation Simulationresults resultsof of the the STH-AMDF STH-AMDF algorithm: algorithm: (a) (a)The Theoriginal originalsignal; signal;(b) (b)The Theresult resultof of AMDFof AMDF oforiginal originalsignal signaland andpeak peakextraction extractionalgorithm; algorithm;(c) (c)The Thesignal signalafter afterSTH STHtransform; transform;(d) (d)The The resultof result ofSTH-AMDF STH-AMDFalgorithm. algorithm. Asshown As shownin inFigure Figure8b, 8b,the thesecond, second,sixth, sixth,eleventh, eleventh,fifteenth fifteenthand andsixteenth sixteenthpeaks peaksin inthe theAMDF AMDF spectrum are spectrum are lower lower thanthan the the extraction extraction threshold, threshold, and and the the peaks peaks after after the the nineteenth nineteenth peak peak areare pseudo-peaks and their peak intervals are significantly smaller than the pseudo-peaks and their peak intervals are significantly smaller than the real periodic values. real periodic values. Although the error of Although this the calculation error of this can be avoided calculation canby beusing avoidedthe median by using of the the median peak interval of theaspeak the period value, interval as when the the signal period data value, is shortened, when the signal the data probability of occurrence is shortened, of the cycle calculation the probability of occurrence errorofwill theincrease cycle greatly. Therefore, calculation the signal error will needsgreatly. increase morphological Therefore,filtering the tosignal enhance the useful needs peak information morphological filteringof the to original weak enhance periodic the useful peaksignal and filter of information outthe part of the noise. original weakAccording to Figure periodic signal and8d, afterout filter morphological part of the filtering, noise. the AMDF According spectral to Figure 8d,shape after of the signal is clearly morphological filtering,improved the AMDF for spectral extracting periodic shape of thefeatures, signal is clearly improved for extracting periodic features, which is manifested by the distinctreduction which is manifested by the distinct increase of periodic peaks, uniform distribution, increase of of pseudo-peaks periodic peaks,and low height. uniform The threshold distribution, reduction set of bypseudo-peaks Formula (7) is and approximately low height.inThe thethreshold middle of set the effective peak, higher than the pseudo-peaks and lower than all the effective by Formula (7) is approximately in the middle of the effective peak, higher than the pseudo-peaks peaks. This method greatly improves and lower the thanaccuracy and robustness all the effective of the peaks. This weak signal method greatlyperiodic improves extraction algorithm. the accuracy and robustness of the weak signal periodic extraction algorithm. 4. Discussion 4. Discussion In order to verify the optimization effect of self-complementary Top-Hat filter on AMDF peak extraction, In orderit is tonecessary verify thetooptimization calculate theeffect meanofof self-complementary mp in AMDF effective peak heightTop-Hat filterspectrum on AMDFand the peak extraction, it is necessary to calculate the mean of effective peak height mp in AMDF spectrum and the mean value mn in spectrum sequence with the effective peak removed. The definition of the peaks recognition accuracy parameter P is as follows:

Information 2019, 10, 24 10 of 12 mean value Information 2019,m10, n in spectrum sequence with the effective peak removed. The definition of the 10 x peaks of 12 recognition accuracy parameter P is as follows: mp m P =P = . p . (8) (8) mn m n The larger The the PP value larger the valueis,is,the thehigher higherthe thepseudo-peak pseudo-peakvalue value is, is, and and the the easier it is to filter the the pseudo-peak and pseudo-peak and reduce reduce thethe pseudo-peak pseudo-peakinterference. interference.The Thesmaller smallerthethePP value is, the closer the the effective peak effective peakheight heightis is to the pseudo to the peak,peak, pseudo whichwhich increases the probability increases of periodic the probability of error extraction. periodic error When the PWhen extraction. valuethe P value is small, it usually leads is small, it to the extraction usually leads to period value being the extraction smaller period valuethan beingthe actual smaller value,the than which actualleads to the value, flowleads which calculation to the value being larger flow calculation thanbeing value the actual value. larger than The the experimental actual value. The data experimental were collecteddatafrom were collected the pressure fromofthe signals the pressure signals outlet of five of the different outlet of five specifications different of household specifications diaphragm gas ofmeters household underdiaphragm gasconditions. different fire meters underEachdifferent firecontains test group conditions. five Each test group complete signal contains five cycles. The complete accuracy parameter P from The signal cycles. usingaccuracy the AMDF parameter method and P the from using themethod STH-AMDF AMDF ismethod shown and the STH-AMDF in Figure 9. method is shown in Figure 9. The Figure 9.9. The Figure peaks peaks recognition recognition accuracy accuracy parameter parameter distribution distribution usingusing the AMDF the AMDF and STH- and STH-AMDF AMDF methods. methods. The Figure The Figure99shows shows that thethe that effective peaks effective in STH-AMDF peaks in STH-AMDFsequences with morphological sequences filtering with morphological are easier to extract than those in AMDF sequences. The STH-AMDF period extraction filtering are easier to extract than those in AMDF sequences. The STH-AMDF period extraction algorithm can better avoid algorithm the calculation can better avoid theerrors caused errors calculation by the interference caused by the peaks, and it is convenient interference peaks, and to set it isa reasonable extraction convenient threshold to to set a reasonable ensure the extraction accuratetoidentification threshold of the periodic ensure the accurate peaks, soofasthe identification to achieve accurate detection of the signal period. periodic peaks, so as to achieve accurate detection of the signal period. 5. Conclusions and Future Work 5. Conclusions and Future Work In order to extract the periodic number of noisy weak characteristic periodic signals more In order to extract the periodic number of noisy weak characteristic periodic signals more accurately, a STH-AMDF method of extracting period based on morphological self-complementary accurately, a STH-AMDF method of extracting period based on morphological self-complementary Top-Hat transform and average magnitude difference function is proposed in this paper. Traditional Top-Hat transform and average magnitude difference function is proposed in this paper. AMDF can reflect the periodicity of the signal, but when there is noise, its peaks are not obvious Traditional AMDF can reflect the periodicity of the signal, but when there is noise, its peaks are not enough, which is not conducive to peak extraction. Therefore, STH is applied to filter the noise and obvious enough, which is not conducive to peak extraction. Therefore, STH is applied to filter the improve the impact characteristics of the signal. Experiments show that the improved algorithm noise and improve the impact characteristics of the signal. Experiments show that the improved can increase the peak amplitude and set dynamic threshold to extract peak value, which verifies the algorithm can increase the peak amplitude and set dynamic threshold to extract peak value, which accuracy and effectiveness of this method. verifies the accuracy and effectiveness of this method. In future work, we will do some significant researches on the following aspects. First, we will In future work, we will do some significant researches on the following aspects. First, we will make a meaningful study about the real-time period detection for the signal of changing period. make a meaningful study about the real-time period detection for the signal of changing period. Secondly, we will collect more pressure data of the exhaust port of domestic gas meters and optimize the algorithm to make it suitable for family life.

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