An Empirical Signal Separation Algorithm for Multicomponent Signals Based on Linear Time-Frequency Analysis

Transcription

1 An Empirical Signal Separation Algorithm for Multicomponent Signal Baed on Linear Time-Frequency Analyi Lin Li,, *, Haiyan Cai, Qingtang Jiang, Hongbing Ji School of Electronic Engineering, Xidian Univerity, Xi'an 77, China Department of Mathematic and Computer Science, Univerity of Miouri-St. Loui, St. Loui 63, United State {lilin, {haiyan_cai, Abtract: The empirical mode decompoition (EMD) i a powerful tool for non-tationary ignal analyi. It ha been ued uccefully for ound and vibration ignal eparation and time-frequency repreentation. Linear time-frequency analyi (TFA) i another powerful tool for non-tationary ignal. Linear TFA, e.g. hort-time Fourier tranform (STFT) and wavelet tranform (WT), depend linearly upon the ignal analyi. In the current paper, we utilize the advantage of EMD and linear TFA to propoe a new ignal recontruction method, called the empirical ignal eparation algorithm. Firt we repreent the ignal with STFT or WT, and then by uing an EMD-lie procedure, extract the component in the time-frequency (TF) plane one by one, adaptively and automatically. With the iteration carried out in the ifting proce, the propoed method can eparate non-tationary multicomponent ignal with fat varying frequency component which will be mixed together when EMD i ued. The experiment reult demontrate the efficiency of the propoed method compared to tandard EMD, enemble EMD and ynchroqueezing tranform. Index Term: time-frequency analyi; empirical mode decompoition; wavelet tranform; ignal eparation.. Introduction Separation of multicomponent non-tationary ignal ha broad application in many engineering field, uch a eimic ignal analyi, vibration fault diagnoi, biomedical ignal analyi, peech enhancement, onar ignal proceing, etc. The EMD algorithm along with the Hilbert pectrum analyi (HSA), introduced by Huang et al. in [], i an efficient method to decompoe and analyze non-tationary ignal. The EMD, conidered a an adaptive ignal analyi, eparate a given ignal into a number of component, called intrinic mode function (IMF), then calculate the intantaneou frequency (IF) of each IMF by the HSA. In recent year, effort to improve the performance of EMD focu on envelope fitting, including interpolation *Correponding author lilin@xidian.edu.cn

2 algorithm and boundary expanion. In [], the cubic pline function i ued to generate the upper envelope and lower envelope by interpolating the local maxima and local minima repectively. An alternative B-pline algorithm for the EMD i introduced in []. And the blending cubic pline interpolation operator and correponding error bound are alo derived in [3]. The direct contruction algorithm of envelope mean i propoed baed on the contrained optimization for narrow-band ignal [4]. In addition, [5] propoe an alternative EMD uing the filter to calculate the envelope mean. In the iteration proce, the boundary expanion error will gradually effect the accuracy of IMF. Uually the ymmetry rule i ued to predict the extrema beyond the two boundarie. A lope-baed method i introduced to retrain the boundary effect in [6]. The capability of EMD to decompoe two ub-ignal i dicued in [7], and the influence of ampling on the EMD i dicued in [8]. And in [9], the EMD i conidered a a filter ban, whoe ub-band number, band-width and cutoff frequency are adaptive and data-driven. An invere filter cheme of EMD i propoed in [], which aim to reduce the cale mixing. The enemble EMD (EEMD) propoed in [], i a noie-aited ignal analyi method. EEMD utilize the advantage of the tatitical characteritic of white noie to force the enemble to exhaut all poible olution in the ifting proce. On the other hand, time-frequency analyi ha been tudied for many year, including linear TFA and non-linear TFA, ee the overview in [], [3], [4], [5] and [6]. Compared with EMD, TFA i of olid theoretical bai. The reearch tendency of TFA i to enhance the TF reolution and energy concentration. The uncertainty principle [6] mae a tradeoff between temporal and pectral reolution unavoidable. To overcome thi important hortcoming, the time-frequency reaignment method i ued to produce a better localization of ignal component [7]. The ynchroqueezing tranform (SST) i introduced in [8] and further developed in [9]. SST i a pecial type of reaignment method on the continuou wavelet tranform (CWT) to enhance the energy concentration of the time-frequency plane. The ub-ignal are recontructed by extracting the IF curve in the SST plane one by one. SST wor well for ingle-tone ignal, but not for broadband time-varying frequency ignal. The generalized SST and intantaneou frequency-embedded SST (IFE-SST) are propoed in [] and [] repectively, with both of them changing broadband ignal to narrow-band ignal and IFE-SST preerving the IF of the original ignal. A multitapered SST i introduced in [] to enhance the concentration in the TF plane by averaging over random projection with ynchroqueezing. A hybrid EMD-SST cheme i introduced in [3] by applying the modified SST to the IMF obtained by the EMD. STFT-baed SST i dicued in [3] and [4], and STFT-baed econd-order SST i propoed in [5]. The matching demodulation tranform and it ynchroqueezing are dicued in [6]. The linear and ynchroqueezed time-frequency repreentation are reviewed in [7] in detail.

3 In thi paper, we utilize the advantage of EMD and linear TFA to propoe a new ignal eparation method, called the empirical ignal eparation algorithm (ESS). In the ifting procedure of EMD, only feature in time domain, e.g. maximum and minimum extrema are ued to decompoe the given ignal. So EMD i enitive to noie and uually reult in mode mixing and artifact. Baed on linear TFA (STFT or WT), we ue an EMD-lie algorithm to extract the ignal component in the TF plane one by one. Our algorithm i different from the empirical wavelet tranform in [8], which extract the component by deigning an appropriate wavelet filter ban. Compared with the ignal recontruction by SST, our algorithm i adaptive and can be ued for multicomponent ignal diatifying the adaptive harmonic model (AHM) in [3]. The remainder of thi paper i organized a follow. The linear TFA and SST are reviewed in Section. The ESS i introduced in Section 3. We alo dicu ome propertie of ESS in Section 3. The numerical reult and comparion are provided in Section 4.. Linear Time-Frequency Analyi and Synchroqueezing Tranform A multicomponent amplitude and frequency-modulated (AM-FM) ignal i given by [6], N N ( t) j ( t) Aj ( t)co j ( t) j j where t () conit of N monocomponent ignal. To allow for the extraction and eparation of component from a given ignal, the IF of the j-th component j ( t) Aj ( t)co j ( t) i () t, and the amplitude A () t and the IF j j () t are poitive, lowly varying compared to () t. j j.. The Linear Time-Frequency Analyi Let u tart with the hort-time Fourier tranform (STFT). For a real-valued window function u L with unit norm, and a ignal L, the STFT of t () i defined by, i V t, u te d for t,, where t and denote time and angular frequency, repectively. Note that, where i the frequency. When the Gauian function u t e t with 4 i ued a the window function, the STFT i called the Gabor tranform [4].

4 The ignal t () can be recontructed by the invere STFT, uing the following formula, it ( ) V, t u t e dtd And t () can alo be recontructed by ( t) V t, d u() Note that ut () i required to be non-zero and continuou at, ee [3]. Now we dicu the TF reolution of STFT. The time center of the window function ut () i defined by t u t u() t u() t dt dt and the equivalent duration of ut () i defined by u t t u u() t dt u() t dt The Fourier tranform of ut () i denoted by uˆ( ). Then the frequency center and equivalent bandwidth of ut () are denoted a û and û, obtained by Eq. (6) and Eq. (7) with u and t replaced by û and, repectively. From the Heienberg uncertainty principle, we have ee detail in [6]. u and uu ˆ, and the equality i obtained for Gauian function only, û are alo called temporal and pectral reolution, repectively. The uncertainty principle impoe an unavoidable tradeoff between temporal and pectral reolution. CWT i another method of linear TF analyi. Lie STFT, CWT ue a window to locate a ignal to it time and frequency content. However, the window hape of CWT change continuouly at different frequency intant. Let ( t) L ( ) be a fix function. ˆ( ) i the Fourier tranform of () t. If ˆ atifie the condition, ˆ( ) for then () t i called an analytic wavelet. ˆ d C ( ) Let u denote () (, ) t a the correponding family of wavelet () t, ab ( ab, ) t b () t a a where a i the cale, and b i the hift. The Fourier tranform of () (, ) t i written a ab

5 ˆ ˆ i b ( ab, )( ) ae The CWT of a ignal t ( ) L ( ) i defined by tb W a, b,, ( t) a dt ab a An analytic ignal t () can be recovered from W ( a, b ) by (ee e.g. [3], [3] and [33]) t W a, ba tdbda C ab, For an analytic CW () t, if C ˆ( ) d, another recontruction formula can be alternatively ued (ee [8] and [9]), ( b) W a, ba da C For real ignal, the CWT with integral involving b i invertible too. Let L be a real-valued ignal. Conider the analytic ignal with aociated, which i denoted by ( a t ) ( t ) i ( ) ( t ) where () i the Hilbert tranform of. The following recontruction formulae exit in L ( ) [9], ( t) e W a, ba ab, tdbda C ( b) e W a, ba da C With the definition in Eq. (6) and Eq. (7), we can alo calculate the temporal and pectral reolution of wavelet ( ab, ), which are given by t b t t dt ab, a a a ab, tb dt a a ˆ ib ˆ ae d ab, ˆ ˆ, ab ˆ ib ae d a

6 Frequency (Hz) Scale Figure. The STFT (left) and CWT (right) of a LFM ignal where and are the temporal and pectral reolution of the mother wavelet, repectively, ˆ t ab, and ˆ ab, are the center of ab, and ˆ ab,, repectively. Therefore, we obtain, ˆ ˆ ab ab, where the equality i obtained for the Gauian function only. From Eq. (8), Eq. (9) and Eq. (), we now the localization window of ( ab, ) i not fixed when CWT i ued. For larger a, viz. the low frequency band, it provide a narrower frequency window and hence a higher frequency reolution, but a lower time reolution. On the other hand, the oppoite i true. Figure how the difference between STFT and CWT. The imulation ignal i a linear frequency modulation (LFM) ignal, where the frequency change linearly from.hz to.hz... The Synchroqueezing Tranform Baed on the linear TFA above, the SST i a TF reaignment approach, which aim to harpen the TF repreentation while eeping the temporal localization. SST ha been proved to be well adapted to multicomponent ignal in [9]. However, it i difficult to eparate an unnown multicomponent ignal to pecific component without any prior nowledge and/or appropriate retriction. Eq. () i called the AHM [3], with the retriction decribed by Aj C ( ) L ( ); j C ( ) inf Aj( t) c; up Aj( t) c t t inf j( t) c; up j( t) c t t Aj ( t) j ( t); j ( t) j ( t) for all t, where and c c. In addition, the component are aid to be well-eparated if j( t) j ( t), j ( t) j ( t) d j ( t) j ( t),

7 for all t and ome d. A continuou function f ( t) A( t)co ( t) with At () and () t atifying condition () and () i called an intrinic mode type (IMT) function in [9]. The CWT-baed SST (WSST) wor through queezing the CWT, where the wavelet i required to be admiible (defined in Eq. (9)). The reference IF function (local IF) baed on CWT i defined by, i bwa, b ab, W a, b Then the WSST i defined by T, b W a, b a, b a da a: W a, b where, which i ued to reduction numerical error and noie influence (ee the dicuion in [8] and [9]). [3] Similar to the WSST, the STFT-baed SST (FSST) wor through queezing the STFT. The local IF i defined by i tvt, t, e V t, Then the FSST i defined by P t, V t, t, d u() : V t, The SST i ued not only to harpen the TF repreentation, but to eparate the multicomponent ignal into it contituent mode. For a multicomponent ignal in [], the j-th component can be recontructed by WSST and FSST, repectively: j( b) lim T, bd C j b ( t) lim P t, d j t where i the width of the zone to be ummed up around the ridge correponding to the IF of the j-th component. j 3. The Empirical Signal Separation Algorithm Let u tart with the queezing and recovery of a monocomponent ignal by SST. Conider x ( t ) co. t.8 co.. t where i the random initial phae. xt () i a inuoidal frequency modulation ignal which can be found in radar ytem. Figure how one of the waveform of ignal xt () and it IF. Note the ampling frequency i Hz.

8 Frequency (Hz) Frequency (Hz) Scale Frequency (Hz) Frequency (Hz) Figure. Waveform of ignal xt () (left) and it intantaneou frequency (right) Figure 3. CWT (top left) and STFT (top right) of xt (), WSST (middle left) and FSST (middle right) of xt (), the waveform recontructed from WSST (bottom left) and the waveform recontructed from FSST (bottom right). Figure 3 how the SST and the waveform recontruction reult by the Synchroqueezing Toolbox by E. Brevdo and G. Thaur [9, 4, 9]. The wavelet ued i Morlet wavelet with parameter and and the number of voice ued i 64. The window function for STFT i the tandard Gauian, with parameter mean and tandard

9 deviation.. The threhold in both WSST and FSST i equal to 3. The width of the zone to be ummed up around the ridge for waveform recontruction i 5 (the dicrete form, unitle), for both WSST and FSST. We find that the SST wor well when the IF change lowly, but not well when the IF change fat. See WSST and FSST in Figure 3. Thi can alo be found in the recovered waveform, ee the bottom row in Figure 3. Theoretically, for a monocomponent ignal, one can increae the width to improve the recovery performance. However it i not alway thi to proce multicomponent ignal ince increaing the width will reult in the mixture of component. 3.. Signal Recovering by Linear Time-Frequency Analyi Directly Conider the IMT function ( t) A( t)co ( t), the CWT of t () i approximated by i ( b W a, b A( b) e ) a( b) For a ingle-tone ignal ( t) Acoct, epecially, it CWT i given by ˆ i bc W, ˆ a b Ae ac If the center frequency of ˆ( ) i, W a, b will concentrate around a ( c). The IF function of t () i a, b c. Then by Eq. (4), all point ( ab, ) in CWT plane can be "queezed" to ( cb, ) in the WSST plane. However, if there i a ignificant change of frequency in the ignal, the efficiency of ynchroqueezing of IF depend on the intantaneou bandwidth of ignal in the range of the analyi wavelet. Looing bac at Figure 3, ynchroqueezing perform well near the extrema of IF of the original ignal, where the IF change lowly. Note that by increaing the width and uing econd-order SST, we can enhance the accuracy of the recontruction waveform continuouly in Figure 3. However, the important problem i how to deal with the multicomponent ignal. Next, let u conider another numerical example given by y( t) ( t) ( t) 3( t) n( t) co.t co(.6 t) co.t co(.6 t) co.38 t n( t), 3 where nt () i an additive noie. Figure 4 how the IF of the three component in yt () and alo the mixed waveform with an additive Gauian white noie. The ampling rate i Hz and the ignal duration i from to 5. Oberve that () 3 t tart at 65 and end at 384. Figure 5 how the CWT, STFT, WSST and FSST of the multicomponent ignal yt (), repectively. The correponding parameter of thee algorithm are the ame a thoe ued in Figure 3. Note that both WSST and FSST do not wor well for the two inuoidal frequency modulation ignal. It eem FSST i better than WSST in Figure 5. But

10 Frequency (Hz) Frequency (Hz) Scale Frequency (Hz) Frequency (Hz) when changing the wavelet parameter in WSST, we can obtain better repreentation for either the high frequency component () t or the low frequency component 3 () t. When uing SST to extract the component, we need to now exactly the number of component and the width of extraction window firt. So it i not adaptive. Actually, we can ue the CWT or STFT to recover a ignal too, ee Eq. (5) and Eq. (4). But for multicomponent ignal, the recovering algorithm baed on CWT or STFT i alo not adaptive. [7] indicate that the ynchroqueezed tranform cannot improve TF reolution of STFT and CWT in the ene if the component are not reliably repreented by STFT or CWT, then they will not be well eparated by SST. By comparing the reult in Figure 3 and Figure 4, and our other experiment, we find that the CWT and STFT are more table than WSST and FSST in the whole TF plane Figure 4. The real IF of yt () (left) and the waveform with ignal-to-noie ratio (SNR) 5dB (right) Figure 5. Analyi of multicomponent ignal yt (): real part of CWT (top left), real part of STFT (top right), WSST (bottom left) and FSST (Bottom right).

11 3.. Extraction of TF Ridge Conidering the AHM condition and noie effect, we ue the extrema of the real part and imaginary part of STFT or CWT to extract the TF ridge. We tae CWT a example in the following analyi. Suppoe f ( t) ( t) n( t), where nt () i a white noie with mean and variance, and ( t) A( t)co ( t) i an IMT function. The CWT of n f() t i given by W a, bw a, bw a, b f n i ( b) ˆ tb A( b) e a( b) n( t) dt. a a The expectation and variance of W a, b are given by f ˆ i ( b E W a, b W a, b A( b) e ) a( b) f and Var Wf a, b nca where Ca ( ) varie with cale a. See Appendix A for the proof of Eq. (35) and the following Eq. (36) and Eq. (37). Suppoe a, b i a point on which e W a, b gain it local extreme value. The ignal-to-noie ratio of e W f a, b i defined by e W a, b r ( a, b ) Var e W f a, b and we have A( b ) ˆ a ( b ) r ( a, b ) Ca n Analogouly, the ignal-to-noie ratio of W a, b i f W ( ) ˆ ( ) a, b A b a b r ( a, b ) r ( a, b ) Var W, nca f a b Note that for imaginary part Im W f a, b, the ignal-to-noie ratio i the ame a Eq. (36) with e replaced by Im, where a, b i a local extreme point of Im W a, b. Furthermore, the abolute operation i nonlinear, which mean that the extracted ridge i not alway conitent and unbiaed for the IF etimation. After we extract the extrema of the real part and imaginary part firt, we ue the cubic pline to interpolate the extema to obtain the ridge of IF.

12 3.3. Effect of the Window/Wavelet Parameter Unle otherwie noted, the wavelet ued in the remainder of thi paper i Morlet wavelet. Morlet wavelet i compoed of a complex exponential multiplied by a Gauian window, given by t it () t c, e e e where, c, i a normalizing contant ubject to ( t) dt Then we have c, e e The Fourier tranform of Morlet wavelet i given by ˆ( ), c e e Note that ˆ( ) d. The frequency center of () t i given by 3 ˆ( ) d 4 e ˆ 3 ˆ( ) d 4 e e The bandwidth of () t i given by 3 4 ˆ ˆ e d e 4 ˆ 3 ˆ d 4 e e Uually, we let and.5, then. Therefore, the center and bandwidth of ˆ( ˆ ) are dependent on and, repectively. In addition, by Eq. (8) and Eq. (9), we have, ˆ ˆ a, ab, a a a. ab, By Eq. (3) and Eq. (4), CWT, around a ( c), with cale upport zone W a b of a ingle-tone ignal t () with Morlet wavelet will concentrate a c c

13 K For a multicomponent ignal with pure harmonic ( t) A coct, where A and c c, to eparate each component with CWT, the wavelet hould atify c c for all, which i equivalent to c c Thu it eem that the bigger value of come better eparation of a multicomponent ignal in the CWT plane. c c However, thi concluion may only correct for pure harmonic. To thi regard, let u conider a linear frequency modulation ignal with phae zone i r () t ct t, intantaneou frequency () h ra, ee the proof in the Appendix B. where r ( t) Aco ct t t c rt and chirp rate () t r. Then it cale upport h h a c rt c rt Suppoe c c and c r t c r t. To eparate each component with CWT, the wavelet hould atify h h c r t c r t Conider and h h on the ridge a () t c r t and a () t c r t, repectively, we have r r c r t 4 c r t c r t c r t 4 4 Note that Eq. (49) i a fourth order inequality with repect to, where. So Eq. (49) may hold for ome pecific interval on of. Therefore, by electing different the parameter and, we can change the poition of the ridge in the CWT plane, and hence change the cale upport zone around the ridge. The width of cale upport zone depend on both the bandwidth of ab, and the intantaneou bandwidth of the ignal to be analyzed. For a monocomponent ignal, we

14 Scale Scale Scale Scale Figure 6. CWT of yt () with different parameter:, (top left), 4, (top right),,.5 (bottom left) and,.5 (bottom right). hould chooe uitable and to mae ure the CWT i well concentrated and it ridge eep away from the upper and lower bound of the CWT plane (finite dicrete). For a multicomponent ignal, except for the condition above, we need to eparate different component a well a poible. Tae the noie-free multicomponent ignal in Figure 4 ( yt () in Eq. (3) with nt ( ) ) for example. Figure 6 how the CWT with different parameter. When, increae from to 4, the ditribution move from the bottom to the top in the CWT plane. On the other hand, when, increae from.5 to.5, the ditribution hold at the ame poition but with increaing frequency reolution for the low frequency component. From the left column of Figure 6, we find that the component with lowet frequency i well-repreented, but the component with high frequency are not concentrated. On the other hand, from the right column of Figure 6, the two component with high frequency are well-repreented, but the component with the lowet frequency i not concentrated. That mean we cannot obtain a good repreentation for both high frequency component and low frequency component imultaneouly for CWT and SST CWT-baed and STFT-baed empirical ignal eparation Baed on the above dicuion, we utilize the advantage of EMD to extract the ignal component adaptively. The conventional EMD only ue the feature in time domain, e.g. maximum and minimum extrema to decompoe a ignal. Here, we preent a new EMD-lie method by ifting in the TF plane. We need no prior nowledge of the ignal to be decompoed. The new method i more efficient to eparate component cloe to each other than EMD and SST a

15 demontrated in the experiment reult in Section 4. The propoed algorithm are called CWT-baed empirical ignal eparation (CWT-ESS) algorithm and STFT-baed empirical ignal eparation (STFT-ESS) algorithm. In our method, different ignal component are extracted by changing the parameter of Morlet wavelet automatically. Becaue the higher frequency component alway ha the ridge with narrower bandwidth when uing a uitable window (ee figure 6), namely high energy concentration, o we firt extract the high frequency component with maximal amplitude. We ue the frequency pectrum to determine the value of parameter adaptively. Suppoe the effective frequency range of the ignal i [ pq, ] with p q, and the ampling rate i Hz. Actually, one can ue the principle of 3-dB bandwidth to find p and q, which are given by where ˆf i the Fourier tranform of f, t and ˆ ˆ p min : arg f ( ) f ( ), q max : arg fˆ ( ) fˆ ( ), arg max fˆ ( ). Suppoe the ignal length i N ( N i a m power of, namely N, m ) and the number of voice i n v. We chooe a proper to mae the Morlet wavelet tranform of the ignal ditribute in the center of the CWT plane. Becaue the domain of the cale i n { a v,,,, mn } with ample frequency Hz, we define the center area (half) of the CWT with the cale v { a n v : mn 4, mn 4,,3mn 4} for example. Then by a ( c) in Eq. (43), we calculate v v v 3m 4 p, m 4 q. Here to avoid unexpected error, we et the range of to be [,5 ], namely, max ( ),, if ( ) 5 ; 5, if ( ) 5. By Eq. (45) or Eq. (49), we could determine the range of theoretically. But ince we have no prior information about the input ignal, uch a number of component, intantaneou frequency etc. We chooe a in [9] for implification. Next we calculate the CWT. After that, baed on Section 3., we find an extreme point (with abolute value exceed a given threhold ) which i cloet to the line of cale a and time b N in the CWT plane. Then we find extrema belong to the ame component next to the current point one by one. Note that extrema of the real part and the imaginary part, the maximum and the minimum, are alway being alternately along the time line. And the interval

16 between two adjacent extrema varie lowly according to the AHM. With thee condition, we can to extract all the maximum and minimum extrema from one component. Now we preent the tep of the CWT-ESS for a given ignal f() t. Firt, initialize, r( t) f ( t), Step : Determine the window/wavelet parameter. Uing the Fourier tranform to calculate the tart and cut frequencie of the ignal rt (), namely p and q. Then determine the value of. Step : Extract the maximum and minimum extrema of the high frequency component in the CWT plane (both real part and imaginary part). Step 3: Interpolate the extrema by the cubic pline, and extract the lice ct () on the interpolation ridge d () t in the CWT plane. c( t) e W ( ), ˆ f d t t ( ) m t ( ) d ( t) where m () t i the IF of the -th IFM, and for Morlet. Then iterate (top when atify the top criterion), and obtain the -th IFM. Step 4: For the reidual r, let, f r, repeat Step ~3, obtain the other IFM. The top criterion for Step 3 i that when OS (. in thi paper), top iterating [5]. OS t t imf ( t) c ( t) l f ( t) imf ( t) c ( t) l where imf () t and c () l t are defined in Algorithm. Note that when the STFT i ued, Eq. (53) and Eq. (54) are replaced by c( t) Vf d( t), t u() and m ( t) d ( t) Furthermore, becaue of the equal TF reolution in the whole STFT plane, we imply ue a contant window for STFT-ESS, which mean we will ignore Step.

17 Algorithm Input f() t, let r( t) f ( t),. Iteration: l, imf t, r ( t) r( t). Calculate p, q, then While max W ( a, t), do r l l, W a, t. Calculate the interpolation ridge d () t, r l c ( ) ( ), ˆ l t e Wr d ( ) l t t, imf ( ) ( ) ( ) t imf t cl t, r ( ) ( ) ( ) l t rl t cl t c ˆ l ( t) e Wr d ( ), ( ) l t t. While OS, do ll, c ( ) ( ), ˆ l t e Wr d ( ) l t t, imf ( ) ( ) ( ) t imf t cl t, r ( ) ( ) ( ) l t rl t cl t, c ˆ l ( t) e Wr d ( ), ( ) l t t. r( t) r ( t),. l 4. Numerical reult In thi ection, variou numerical example are ued to validate the propoed method. Firt we deal with the two-component non-tationary ignal zt () in Figure 7. Component ha the low frequency, while Component the high frequency. Both of their amplitude are modulated ymmetrically by the Gauian function, ee the original waveform in Figure 7. And their frequency modulation are ymmetrical triangular LFM. The waveform are continuou. Figure 8 how the eparation reult by different method. Note that the parameter of WSST and FSST algorithm are the ame thoe in Figure 3 and Figure 5. In addition, the number of component and the width of extraction window are et in advance to be and 5, repectively. We jut diplay the firt IMF and econd IMF of the decompoition reult of EMD by uing the code upplied by T. Oberlin [4]. Becaue of the mode mixing (a explained in [7]), EMD cannot eparate thee two component. For CWT-ESS and STFT-ESS, alo we jut diplay the firt two mode. By comparing the waveform of the eparation reult to the original one, we can find that our method are uperior to the EMD, WSST and FSST.

18 Frequency (Hz) Figure 7. The two-component LFM ignal zt (): the real IF (left), the mixed ignal (right) Figure 8. The eparation reult by different method: two component of the original waveform (top left), two component obtained by EMD (top right), WSST (middle left), FSST (middle right), CWT-ESS (bottom left) and STFT-ESS (bottom right). In each ub-picture, the top and the bottom are correponding to Component and Component, repectively. We ue the relative root mean quare error (RMSE) to evaluate the eparation performance, which i defined by R n n T j j ( ) j ( ) j T n

19 Relative RMSE Relative RMSE where ( n) i the recontruction of ( n ), j j i the tandard deviation of ( n ), and ( n ) i the j-th component j j j of a multicomponent ignal. Figure 9 how the relative RMSE under different ignal-to-noie ratio (SNR) for the two-component triangular LFM ignal zt (). The additive Gauian white noie are added to the original ignal with SNR from to db. Under each SNR, we ue the Monte-Carlo experiment for run. Note that the parameter for recovering with WSST and FSST are the ame a Figure 8. Becaue of noie, here we ue the EEMD [] intead of EMD. And for the EEMD, it repeat time for average by adding independent noie with tandard deviation.. Obviouly, from the reult in Figure 9, the algorithm CWT-SST and STFT-SST introduced in thi paper are uperior to EEMD, WSST and FSST. Then, we move to the three-component ignal yt () in Eq. (3), whoe IF and waveform are hown in Figure 4, and WSST and FSST are hown in Figure 5. Becaue the SST baed recontruction method are upervied which require the input of the number of component of the ignal and the parameter, the eparation performance can be improved a lot by changing the window width manually. However, it i very difficult to get the bet window width and component number adaptively and automatically. So, in thi experiment, we jut compare the eparation performance of the unupervied method in thi paper, i.e. EEMD, CWT-ESS and STFT-ESS WSST FSST EEMD CWT-ESS STFT-ESS WSST FSST EEMD CWT-ESS STFT-ESS SNR (db) SNR (db) Figure 9. The eparation performance of the two-component triangular LFM ignal: Component (left) and Component (right).

20 Relative RMSE Relative RMSE Relative RMSE.8.6 EEMD CWT-ESS STFT-ESS SNR (db).8.6 EEMD CWT-ESS STFT-ESS SNR (db).8.6 EEMD CWT-ESS STFT-ESS SNR (db) Figure. The eparation performance of the three-component ignal yt () in Eq. (34): Component (top), Component (middle) and Component 3 (bottom) Figure. The waveform (left) and pectrum (right) of a electroencephalography ignal Frequency (Hz)

21 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) Figure. Separation reult of the electroencephalography ignal by different method: EMD (firt row), WSST (econd row), FSST (third row), CWT-ESS (fourth row) and STFT-ESS (lat row). Finally, we deal with the electroencephalography (EEG) ignal analyzed in [5]. We chooe a hort truncation of R-to-R pea interval (RRI) ignal, which i diplayed in Figure. Here we jut ue thi ignal to tet our method, and will not introduce the medical pathogeny and phenomenon of the ignal, which can be found in [5]. Without loe generality, we uppoe the ample rate i Hz. Although the pectrum tructure of thi ignal i very complicated, we can ee that there are three main component located at about.68 Hz (Component ),.996 Hz (Component ) and

22 .699 Hz (Component 3). By analyzing the TF ditribution of thi ignal, we find that the three component are eparated. However, becaue of bacground noie and interference, it i difficult to decompoe thi ignal in frequency domain or TF domain. We ue different method to eparate thi ignal, including EMD, WSST, FSST, CWT-ESS and STFT-ESS. Figure how the eparation reult with the left column howing the firt three mode and the right column howing the correponding pectrum. Note that we ue blac color for Component 3, red for Component and blue for Component. Becaue we have no prior information about the original ignal, hence the RMSE cannot be applied to evaluate the eparation performance. Here we analyze the pectrum of the eparation reult. We ay a method can eparate thi ignal well if the pectrum of the eparation reult are jut correponding to the three frequency component in Figure. EMD cannot eparate Component and Component 3 becaue of the mode mixing. All of WSST, FSST, CWT-ESS and STFT-ESS can decompoe thi ignal to it three main component. But by comparing the waveform and pectrum, we find that Component 3 by WSST and Component by FSST are not good. Both CWT-ESS and STFT-ESS enhance the energy of the three component ignificantly. STFT-ESS ha the bet eparation performance for Component and Component 3. CWT-ESS i the bet among all thee method for the average eparation performance of the three component. Concluion In thi paper, we have introduced a new EMD-lie method to eparate multicomponent ignal baed on linear TF analyi. Our method i adaptive and unupervied. We have hown that the propoed method eparate the model better than EMD and the method baed on tandard SST. In the ifting proce of our method, we adjut the parameter for CWT according to the pectrum of the ignal adaptively. For multicomponent ignal with IF of component interecting each other, the method of how to eparate thee ignal both by our method and the tandard SST will be the ubject for further tudy. Acnowledgement Thi wor wa upported in part by the National Natural Science Foundation of China (Grant No. 687) and Simon Foundation Collaboration Grant for Mathematician (Grant No ). Appendix A. Proof of Eq. (35)-(37). From W a, bw a, b W a, b, we have f n

23 Var W f a, b Var Wna, b t Var n( t b) dt a a N t Var lim lim n( t b) t N t N a a t lim lim t, a a N n N t N where we now n( t b) i independent with n( j t b), if j. Throughout thi paper, we jut let ampling rate a Hz, which mean t, and conider the duration of f t i finite, namely,, N. Then we have N Var Wf a, b n nca a a, (A.) where N Ca a a varie with cale a. Thi how (35). Note that Var W f a, b i irrelevant to the time tranlation b. By (36), for real and imaginary part of CWT, we have E e Wf a, b e Wa, b, E Im W f a, b Im Wa, b. (A.3) Analogouly, Conider the Morlet wavelet, where Var W f a, b Var e W f a, b Var Im W f a, b Var e W f a, b Var Im W f a, b nca. ˆ i real. When b or b, and a ( b ), where ˆ arg max ( ), then a, b i the point with extreme value of the real part e W a, b, with ignal to noie ratio e W ( ) ˆ ( ) a, b A b a b r ( a, b ) Var e W f a, b nc a The ignal-to-noie ratio i the ame for the extrema of the imaginary part. Analogouly, for W a, b, the ignal to noie ratio i defined by f

24 W a, b r ( a, b ) Var W f a, b n A( b ) ˆ a ( b ) Ca Thee how Eq. (36) and Eq. (37). B. Proof of Eq. (47). For t given in (38), when, ˆ for. Thu for t given by (46), when ( t) c rt, we have t bdt W a, b ( t) a a ( ax b) xdx r x A i caxcb a x abxb i x e e e e dx x x i carabxra x cbrb A i carabxra x cbrb A e dx e e I,, a, be I,, a, b, where c,. According to the Fourier tranform of a linearly-chirped Gauian pule [3], dx g iht i t 4g ih e dt e g ih we have Let ca rab, and define h( ) by A icbirb I,, a, b e e ira carab 4 ira Then, Note that e I,, a, b, Hence, ra h( ) e. (B.4) A icbirb ra I,, a, b e e h( ). (B.5) ira 4 ira

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