Any one of electrical products composed of microelectronic devices can be considered as an electronic product. Electronic products which can be seen everywhere play an essential role in our daily life and production. However, interrupting signals arising in circuits where electronic products are fitted can cause wave distortions of signals of these circuits, which have an extremely negative impact on the normal operation and life cycle of electronic products. Therefore, to eliminate the harmful effects of interrupting signals on the normal operation and life cycle of electronic products, efficient digital filter algorithm [1] are needed for the interrupting signals.
A number of digital filter algorithms have been used for the interrupting signals, such as short time Fourier transform (STFT) [2], fast Fourier transform (FFT), wavelet transform (WT) [3], mathematical morphology (MM) [4], etc. STFT and FFT have ideal filtering effects on interrupting signals except for signals interrupted by high frequency damped components. When processing signals interrupted by high frequency damped components, STFT and FFT need to linearize the attenuation factors of high frequency damped components in the interrupting signals, which seriously influences the filtering effects of STFT and FFT. WT needs not linearize the attenuation factors of high frequency damped components in the interrupting signals when it is used to filter the high frequency damped components in the interrupting signals [5]. Therefore, WT has ideal filtering effects on all types of interrupting signals. Nonetheless, WT requires considerable amount of calculation [6], and its performance is severely affected by the selection of mother wavelet function [7]. MM, a nonlinear analysis method based on signal processing in time domain, is excellent in noise removing and calculation speed [8]. Nonetheless, size and shape of structure element (SE) are often determined based on experience, which limits the applicability of MM [9].
To avoid the disadvantages of the above algorithms, in this paper, a novel digital filter algorithm based on morphological lifting scheme and median filter (MLSMF) is proposed. The proposed algorithm employs morphological lifting scheme for its ability of information preserving and median filter for its advantage in noise removing.
Ⅱ. METHODOLOGICAL BACKGROUND A. Morphological TransformMM was initially proposed to characterize physical or mechanical properties of certain materials, such as the permeability of porous media, by examining the geometrical structure of them. It was used as a quantitative description of shapes and sizes for binary images [10].
The original principle of MM stems from set theory. MM provides an algebraic formulation to apply neighbourhood operations on signals. The main notion of MM is the interaction between the signals under analysis and an SE, where the signal and SE are considered as sets of points. The SE, as a probe, slides through the signal as a moving window, inspects its interaction with the signal, and detects specific features in the neighbourhood of every point in the signal [10].
1) Dilation and Erosion Erosion: and dilation are two basic operations of MM. Let
$ \begin{align} f \oplus g(x) = \max\limits_t\{f(x+t)+g(t)x+t \in D(f), t\in D(g)\} \end{align} $  (1) 
and
$ \begin{align} f \ominus g(x) = \max\limits_t\{f(x+t)g(t)x+t \in D(f), t\in D(g)\} \end{align} $  (2) 
respectively, where
2) Opening and Closing: Opening is an operator that performs dilation on a signal eroded by the same SE. The definition is given as follows [12]:
$ \begin{align} f \circ g = (f \ominus g)\oplus g \end{align} $  (3) 
where
Closing, on the other hand, can be defined by the duality of opening as [12]
$ \begin{align} f \bullet g = (f \oplus g)\ominus g \end{align} $  (4) 
where
A useful and very general technique for constructing new wavelet decompositions from existing ones has been recently proposed by Sweldens [13] and is known as the lifting scheme. A typical case of lifting scheme consists of three stages: split, predict, and update [6].
1) Split Stage: Let
2) Predict Stage: The detail signal
$ \begin{align} y_1' = y_1  P(x_1) \end{align} $  (5) 
where
Actually, the prediction procedure is equivalent to applying a high pass filter to the source signal
3) Update Stage: The approximation signal
$ \begin{align} x_1' = x_1 + U(y') \end{align} $  (6) 
where
Like WT, the aforementioned three stages are carried out on
If prediction and update operators are given by
$ \begin{align} &P(x_1)(n) = x_1(n) \vee x_1(n+1)\notag\\ &U(y_1)(n)=(0\vee y_1(n1) \vee y_1(n)) \end{align} $  (7) 
respectively, then
$ y_1'(n) = y_1(n)  (x_1(n) \vee x_1(n+1))\ \ \ \ ({\text{prediction}}) $  (8) 
$ x_1'(n) = x_1(n) + (0\vee y_1'(n1) \vee y_1'(n))\ \ \ ({\text{update}}). $  (9) 
The above lifting scheme is the socalled maxlifting scheme. In the above maxlifting scheme, as a prediction for
If prediction and update operators are given by
$ \begin{align} &P(x_1)(n) = x_1(n) \wedge x_1(n+1)\notag\\ &U(y_1)(n)=(0\wedge y_1(n1) \wedge y_1(n)) \end{align} $  (10) 
respectively, then
$ y_1'(n) = y_1(n)  (x_1(n) \wedge x_1(n+1))\ \ \ \ ({\text{prediction}}) $  (11) 
$ x_1'(n) = x_1(n) + (0\wedge y_1'(n1) \wedge y_1'(n))\ \ \ ({\text{update}}). $  (12) 
The above lifting scheme is the socalled minlifting scheme.
Ⅲ. THE PROPOSED DIGITAL FILTER ALGORITHMMLSMF is developed as an extension of the morphological lifting scheme discussed in [13], which effectively combines the ability of MLS in information preserving and MF in noise removing. In the implementation process of MLSMF, MLS and MF are applied to process interrupted signals sequentially. Let
Step 1: Decompose
$ x'_1(n)=x_0(2n) $  (13) 
$ y'_1(n)=x_0(2n+1). $  (14) 
Step 2: Process
$ \begin{align} & x''_1 (n) = \max\left\{\frac{(Ci)\times (x'_1 \circ g)(n)}{C}+\frac{i\times (x'_1 \bullet g)(n)}{C}\right\} \end{align} $  (15) 
where
$ \begin{align} g= F \times A \times \sin\left(\frac{2\pi j}{B}\right) \end{align} $  (16) 
where
Step 3: Apply morphological minlifting scheme to
$ \begin{align} & y''_1 (n) = \min\left\{\frac{(Ci)\times (y'_1 \circ g)(n)}{C}+\frac{i\times (y'_1 \bullet g)(n)}{C}\right\}. \end{align} $  (17) 
Step 4: Predict
$ \begin{align} y_1(n) = \frac{y''_1(n) + x''_1(n)}{2}. \end{align} $  (18) 
Step 5: Update
$ \begin{align} x'(n) = x_0(n)y_1(n). \end{align} $  (19) 
Step 6: Apply median filter to
$ \begin{align} x(n)={\rm{median}}\{0, x'(n), x'(n+1)\}. \end{align} $  (20) 
Therefore, for the given input signal
$ \begin{align} x_0 &\rightarrow \{x'_1, y'_1\} \rightarrow \{x''_1, y'_1\} \rightarrow \{x''_1, y''_1\}\notag \\ & \rightarrow \{x''_1, y_1\} \rightarrow \{x''_1, x'\} \rightarrow \{x''_1, x\}. \end{align} $  (21) 
Voltage signals from circuits where electronic products are fitted may be interrupted by various interrupting signals. In this paper, high frequency continuous interference, random background noise and damped oscillatory transient interference are mainly considered.
1) High Frequency Continuous Interference: Fig. 1 (a) presents a typical voltage signal disturbed by high frequency continuous interference whose mathematical expression is
$ \begin{align} f(x) =&\ \sin\left(\frac{x1}{64}\pi\right) + \frac{1}{5}\sin\left(\frac{16\times (x1)}{25}\pi + \frac{\pi}{6}\right)\notag \\ & \ + \frac{1}{10}\sin\left(\frac{32\times(x1)}{25}\pi + \frac{\pi}{3}\right). \end{align} $  (22) 
Download:


Fig. 1 (a) Voltage signal disturbed by high frequency continuous interference, (b) the processed result with MLSMF, (c) the processed result with MMF. 
In order to evaluate the filtering performance of the proposed algorithm, an index of relative error of fundamental frequency signal (
$ \begin{align} \sigma = \frac{\sqrt{\sum[f(x)F(x)]^2}}{\sqrt{\sum{[F(x)]^2}}} \end{align} $  (23) 
where
$ \begin{align} F(x) =\sin\left(\frac{x1}{64}\pi\right) \end{align} $  (24) 
and
Fig. 1 (b) and Fig. 1 (c) give the filtering results of MLSMF and MMF, respectively. In Fig. 1 (b) and Fig. 1 (c), the
2) Random Background Noise: Fig. 2 (a) shows a typical voltage signal polluted by 10 dB white Gaussian noise (WGN). The output of MLSMF and MMF are described in Fig. 2 (b) and Fig. 2 (c), respectively. As depicted in Fig. 2 (b) and Fig. 2 (c), MLSMF can filter WGN contained in the voltage signal shown in Fig. 2 (a) effectively and efficiently, and the filter performance of MLSMF is better than that of MMF. The performance of the proposed scheme is influenced by the signaltonoise ratio (SNR) of WGN. When the SNR of WGN decreases, which means the ascent of the ratio of WGN in the interrupted signal, value of
Download:


Fig. 2 (a) Voltage signal disturbed by random background noise, (b) the processed result with MLSMF, (c) the processed result with MMF. 
3) Damped Oscillatory Transient Interference: A sinusoidal voltage signal, whose amplitude is 1 V, disturbed by damped transient broadband interference, whose amplitude is 0.1 V, is shown in Fig. 3 (a). The filtering results of MLSMF and MMF on the signal shown in Fig. 3 (a) are described in Fig. 3 (b) and Fig. 3 (c), respectively. In Fig. 3 (b) and Fig. 3 (c), the calculation values of
Download:


Fig. 3 (a) Voltage signal disturbed by damped oscillatory transient interference, (b) the processed result with MLSMF, (c) the processed result with MMF. 
The values of
To verify the superiority of MLSMF, the performance of MLSMF has also been compared with that of continuoustime finite impulse response SavitzkyGolay filter (CTFIRSGF) [16] and infiniteimpulseresponse digital filter (IIRDF) [17]. Table Ⅰ also presents a comparison of the index of relative error of fundamental frequency signal of each interrupting signal obtained by the three methods, respectively. For each type of interrupting signal, 200 tests were carried out. As shown in Table Ⅰ, MLSMF is more effective than the rest in interrupting signal components filtering.
B. Field Data ExperimentIn this part, voltage signals with interrupting signal components from IEEE PES database [18] are adopted to test the effectiveness of the proposed digital filter algorithm [19].
Fig. 4 (a) shows a voltage signal with interrupting signal components from IEEE PES database. The filtering effects of the signal shown in Fig. 4 (a) by MLSMF and MMF are depicted in Fig. 4 (b) and Fig. 4 (c), respectively. In Fig. 4 (b) and Fig. 4 (c),
Download:


Fig. 4 (a) Voltage signal with interrupting signal components from IEEE PES database, (b) the processed result with MLSMF, (c) the processed result with MMF. 
The above simulation results show that, in filtering various interrupting signals existing in the circuits in which electronic products are fitted, MLSMF is more accurate than MMF, CTFIRSGF and IIRDF, which effectively demonstrates the rationality of combination of MLS and MF.
Another advantage of MLSMF is that in the process of noise filtering, the scheme only involves addition and subtraction, which reduces calculation burden greatly. In this paper, the computation time of MLSMF is shorter than 0.005 s, which is very good for the negative influence of noise in the interrupted signals being obviated in time.
Ⅴ. ConclusionThe MLSMF, which effectively combines the ability of MLS in information preserving and that of MF in noise removing, has been proposed to filter various interrupting signals existing in the circuits in which electronic products are fitted. A interrupted signal is processed by morphological maxlifting and minlifting schemes with SEs selected by numerous tests, and the output of morphological minlifting scheme is processed by median filter. Compared with existing algorithms, the algorithm proposed in this paper is more capable of filtering interrupting signal components contained in the disturbed signals. The proposed algorithm has been evaluated on a batch of test signals interrupted by three types of interruptions under a variety of conditions, and voltage signals from IEEE PES database. Simulation results have shown that the proposed algorithm has good performance in various interrupting signals filtering.
[1]  C. S. Kim, J. W. Sun, D. Liu, Q. S. Wang, and S. G. Paek, "Removal of ocular artifacts using ICA and adaptive filter for motor imagerybased BCI". IEEE/CAA J. of Autom. Sinica , pp.1–8, 2017. DOI:10.1109/JAS.2017.7510370 
[2]  R. Song, H. D. Guo, G. Liu, Z. Perski, H. Y. Yue, C. M. Han, and J. H. Fan, "Improved goldstein SAR interferogram filter based on adaptiveneighborhood technique, " IEEE Geosci. Remote Sens. Lett., vol. 12, no. 1, pp. 140144, Jan. 2015. http://xueshu.baidu.com/s?wd=paperuri%3A%28181cf12b279fabe667e418107978b7b0%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F6840298%2F&ie=utf8&sc_us=13589508400916834095 
[3]  M. Shoaib, N. K. Jha, and N. Verma, "Signal processing with direct computations on compressively sensed data, " IEEE Trans. Very Large Integr. (VLSI) Syst., vol. 23, no. 1, pp. 3043, Jan. 2015. http://www.computer.org/csdl/trans/si/2015/01/06742615abs.html 
[4]  D. G. Jang, S. H. Park, and M. Haha, "Enhancing the pulse contour analysisbased arterial stiffness estimation using a novel photoplethysmographic parameter, " IEEE J. Biomed. Health Inform., vol. 19, no. 1, pp. 256262, Jan. 2015. http://xueshu.baidu.com/s?wd=paperuri%3A%282e1b44d1862e495b05b7527a7c6247b3%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpubmed%2F25561448&ie=utf8&sc_us=2504167975830101875 
[5]  X. Y. Zhang, Q. G. Huang, and Z. Ren, "Combined digital filters of microcomputer protection based on wavelet transforms, " Relay, vol. 31, no. 12, pp. 4850, Dec. 2003. http://xueshu.baidu.com/s?wd=paperuri%3A%2859d28c2cebff98ba684062558821c1e0%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fen.cnki.com.cn%2FArticle_en%2FCJFDTotalJDQW200312014.htm&ie=utf8&sc_us=16040927435721228114 
[6]  N. A. Khan and T. Hameed, "An implementation of Haar wavelet based method for numerical treatment of timefractional Schrödinger and coupled Schrödinger systems". IEEE/CAA J. of Autom. Sinica , pp.1–10, 2016. DOI:10.1109/JAS.2016.7510193 
[7]  M. ValtierraRodriguez, R. de Jesus RomeroTroncoso, R. A. OsornioRios, and A. GarciaPerez, "Detection and classification of single and combined power quality disturbances using neural networks, " IEEE Trans. Ind. Electron., vol. 61, no. 5, pp. 24732482, May 2014. http://www.mendeley.com/research/detectionclassificationsinglecombinedpowerqualitydisturbancesusingneuralnetworks2/ 
[8]  H. J. A. M. Heijmans and J. Goutsias, "Nonlinear multiresolution signal decomposition schemes. Ⅱ. Morphological wavelets, " IEEE Trans. Image Process., vol. 9, no. 11, pp. 18971913, Nov. 2000. http://xueshu.baidu.com/s?wd=paperuri%3A%287fd2311c9074a94f7373fcbf0d076904%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpubmed%2F18262923&ie=utf8&sc_us=12839071320723405610 
[9]  Z. W. Wen, S. Ouyang, and H. J. Pei, "An improved lifting morphological wavelet method and its application in power quality disturbances detection, " in Proc. 2012 AsiaPacific Conf. Power and Energy Engineering Conf., Shanghai, China, 2012, pp. 15. http://xueshu.baidu.com/s?wd=paperuri%3A%289d653677c3812dbbf0f0cca52a11e6d8%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Ficp.jsp%3Farnumber%3D6307620&ie=utf8&sc_us=2173735910372742907 
[10]  Z. Lu, J. S. Smith, Q. H. Wu, and J. Fitch, "Identification of power disturbances using the morphological transform, " Trans. Inst. Meas. Control, vol. 28, no. 5, pp. 441455, Dec. 2006. http://www.mendeley.com/catalog/identificationpowerdisturbancesusingmorphologicaltransform/ 
[11]  M. C. Rang, Y. Wei, J. F. Zhang, G. Hu, and Y. N. Qiu, "Power cable fault location based on mathematical morphology and wavelet theory, " in Proc. Int. Conf. Sustainable Power Generation and Supply, Hangzhou, China, 2012, pp. 16. http://xueshu.baidu.com/s?wd=paperuri%3A%2853f153043921a46a4105bc883e6c2c7a%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F6493169%2F&ie=utf8&sc_us=2669938570526759526 
[12]  Y. Zhang, T. Y. Ji, M. S. Li, and Q. H. Wu, "Identification of power disturbances using generalized morphological openclosing and closeopening undecimated wavelet, " IEEE Trans. Ind. Electron., vol. 63, no. 4, pp. 23302339, Apr. 2016. http://xueshu.baidu.com/s?wd=paperuri%3A%28f2569a2a966c7b3349adc840d25d0177%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fdx.doi.org%2F10.1109%2Ftie.2015.2499728&ie=utf8&sc_us=12674116691213020314 
[13]  W. Sweldens, "The lifting scheme: a construction of second generation wavelets, " SIAM J. Math. Anal., vol. 29, no. 2, pp. 511546, Jan. 1998. http://xueshu.baidu.com/s?wd=paperuri%3A%283597aaad0a0171fcef52070a4e25c6ef%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fci.nii.ac.jp%2Fnaid%2F30006811691&ie=utf8&sc_us=14127313371705278294 
[14]  G. Piella and H. J. A. M. Heijmans, "Adaptive lifting schemes with perfect reconstruction, " IEEE Trans. Signal Process., vol. 50, no. 7, pp. 16201630, Jul. 2002. http://xueshu.baidu.com/s?wd=paperuri%3A%282d360a8af89abb3f8a72dd232a3b2718%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2002ITSP...50.1620P&ie=utf8&sc_us=9255597152720734129 
[15]  P. Chen and Q. M. Li, "Design and analysis of mathematical morphologybased digital filters, " Proc. CSEE, vol. 25, no. 11, pp. 6065, Jun. 2005. http://xueshu.baidu.com/s?wd=paperuri%3A%288fcb780a37279fd3235fa034996f9d96%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fen.cnki.com.cn%2FArticle_en%2FCJFDTOTALZGDC200511011.htm&ie=utf8&sc_us=11820658975695154028 
[16]  Y. B. Hong and Y. Lian, "A memristorbased continuoustime digital FIR filter for biomedical signal processing, " IEEE Trans. Circuits Syst. I: Regul. Pap., vol. 62, no. 5, pp. 13921401, May 2015. http://xueshu.baidu.com/s?wd=paperuri%3A%28fec7261d3163206898b8345248cc38d5%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fdx.doi.org%2F10.1109%2FTCSI.2015.2403033&ie=utf8&sc_us=13802924266043735150 
[17]  F. Xiao, "Fast design of ⅡR digital filters with a general chebyshev characteristic, " IEEE Trans. Circuits Syst. Ⅱ: Express Briefs, vol. 61, no. 12, pp. 962966, Dec. 2014. http://xueshu.baidu.com/s?wd=paperuri%3A%28c005b79dc688cb81061a9a26c0394291%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fdx.doi.org%2F10.1109%2FTCSII.2014.2362638&ie=utf8&sc_us=4743674208910597396 
[18]  IEEE Power Engineering Society, IEEE PES working group P1433 power quality definitions[online]. Available: http://grouper.ieee.org/groups/1159/2/testwave.html 
[19]  S. F. He, K. C. Li, and M. Zhang, "A new transient power quality disturbances detection using strong trace filter, " IEEE Trans. Instrum. Meas., vol. 63, no. 12, pp. 28632871, Dec. 2014. http://xueshu.baidu.com/s?wd=paperuri%3A%287739d23633a3e62d02b184b6df9f74af%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fdx.doi.org%2F10.1109%2FTIM.2014.2383771&ie=utf8&sc_us=10982006847585394800 