Analysis of Seismocardiographic Signals Using Polynomial Chirplet Transform and Smoothed Pseudo Wigner-Ville Distribution

Analysis of Seism ocardiographic Signals Using Polynomial Chirplet Transform and Smoothed Pseudo Wigner-Ville Distribution Amirtaha Taebi , Student Member , IEEE , Hansen A. Mansy Biomedical Acoustics Research Laboratory , University of Central Florida , Orlando , FL 32816 , U SA { taebi@knights. , hansen .mans y@ } ucf. edu Abstract —Seismocardiographic ( SCG ) signals are chest surface vibrati ons induced by car diac activi ty . These signals may offer a method for diagnosing and monitoring heart function. Successfu l classi fication of SCG signals i n health and disease depends on accurate signal characterization and feature extraction. One approach of determining signal features is to estimate its time- frequency characteristics . In this regard, four different time - frequency d istrib ution (TFD) approaches were used including shor t - time Fourier transform (STFT), polynomial chirplet transform (PCT), Wigner - Ville d istribution (WVD), and smoothed pseudo Wigner - Ville dist ribution (SPWVD) . Syntheti c SCG signals with know n time - f requen cy properties were generated and used to evaluate the a ccuracy of the differ ent TFD s in extracting SCG spectral characteristics . Usi ng diff erent TFD s , the instantane ous frequenc y ( IF) of each synthetic signal was determined an d the erro r (NRMSE) in estimat ing IF was calculated. STFT had lower NRMSE than WVD for synthetic signals considered . PCT and SPWVD w ere , however, more accurate IF estimators especially for the signal with time -varying frequenc ies . P CT and SPWVD also provided better dis crimination betw e en signal frequency compo nents. There fore, the result s of this study suggest that PCT and SPWVD would b e more reliable methods for estimating IF of SCG signals. Analysis of actual S CG signals showed that th ese signals had multiple spectral components with slightly time -varying fr equencies . More studies are needed to investigate SCG spec t ral properties f or hea lthy subjects as w ell as patients with different cardiac conditions. Keywords— Seismocardiographic signals ; Time - frequency analysis; Polynomial chirplet transform; Wigner - Ville distribution; Smoothed pseudo Wigner - Ville dis tribution . I. I NTRODUCTION ardiovascular diseas e is a lead ing cause of death in the United States t hat accounts for 24.2% of total mortal ity in 2010 [1] . Associated death rate may be decre ased by im proving current cardiac diagnostic and patient monitori ng methods. Therefore, a nalys is of blood fl ow dyna mics [2], [3] a nd the heart related signals [4] has become an active area of research. Auscultat ion of hea rt sounds i s a widel y used proce dure perform ed during phy sical e xaminati ons and can provi de useful diagnostic informa tion. Comput er analy sis of heart sound and vibrati on provides additiona l quantita tive inform ation that may be helpful for d etecting a variety of heart c ondition s . Card iac vibrations m easured non - invasively at the chest surface are called s eism ocardiographic ( SCG ) signals. These signals ar e believed t o be caused by mechanical activities of the heart (e.g. valves closure, cardiac co ntraction, blood moment um changes, etc.). The characteristics of these sig nals were found to c orrelate with d ifferen t cardiovascular patho logies [4] – [7] . These signals may also c ontain useful inform ation that is complementary to other diagnostic methods [8] – [10] . For e xa mple, due to the mechanical nature of SCG , it may cont ain inform ation that i s not manifested in electrocard iographic (ECG) signals. A typical SCG contains t wo main e vents during each hear t cycle . These events can be called the first SCG ( SCG 1) and the seco nd SCG ( SCG 2). The se SCG events contain relatively low - frequency component s. Since t he human auditory sensit ivity decreases at low freque ncies , examinatio n of SCG si gnals m ay not be optimall y done by t he unaided human ea r [11], [12]. T herefore, a computer assisted analysis may help find possible correlation s between the SCG signal characteristics and cardio vascular conditions. M any m ethods have been use d to invest igate the SCG features [13] – [15] incl uding tim e - frequency distributi on s (TFD) [16] . The current study aim s at compari ng the performance of some available TFD methods for esti mating the spectral content of SCG signals. TFDs have be en used to identify the tim e- frequency features of a wi de range of biom edical signals such as ECG [17] , EEG [18] , PCG [19] , and SCG [16] . The performance of each TFD technique depends o n its und erlying a ssumptions , which can consequently lead to different levels of accuracy in estimating spectral content of th e signals under cons iderat ion. One com mon approach f or TFD estim ation is the short - tim e Fourier transfo rm (STFT). STFT is re latively simple but can’t track ste ep signal temporal changes [19], [20]. In additi on, the Heise nbe rg uncertainty p rinciple leads to resolution limitat ion s . For instance , improvin g the STFT resolution in time doma in worsens the resoluti on in the fr equency dom ain and vice versa. The Wigner - Ville distrib ution (WVD) is another TFD method tha t wa s propose d for tim e - frequen cy analy sis of non - stationary sig nals [21] . The WVD provides a finer re solution i n both tim e and freque ncy directions com pared to S TFT at the cost of the introduction of artif act pea ks and increased aliasing in the time - frequen cy plane [22], [23] . For exam ple, ali asing in the WVD will t ake place for fre quencies above ¼ of t he sampli ng frequency compared to the ½ o f the sampli ng frequency i n STFT [24] . The resulting artifacts may overlap w ith th e actual frequency components, and theref ore they may result in a misleading interpretation of the TFD. The artifacts can be C reduced by sm oothing WVD in both tim e and frequency. When WVD is only smoothed i n the frequenc y direc tion, the new distributi on is called pseudo WVD and when it i s smoot hed in both tim e and frequenc y, it is known as smoot hed pseudo W VD (SPWVD). However, s moothing ca n negative ly affect resolution a nd a compromise needs to be reach ed between reduction of interferen ce terms and ref ining the temporal - spectral reso lution. The general pro perties of t he STFT a nd WVD have be en discuss ed in li terature [19] , [25], [ 26] . C hirplet transfo rm (CT) that might be considere d a generalization of STF T and wavelet transform (W T) [27] , involves a c omp lex function of time, frequency, scale an d chirp rate; where the chirp rate can be defined as the instantaneous rate of change in t he signal frequency [28] . Since the convent ional CT is base d on a kernel wi th linea r instanta neous freque ncy (IF), it may provide inaccur ate TFD est imations for signals t hat have nonlinear IF traj ectory. Polynomial chir plet transfor m (PCT) wa s proposed to solve this probl em [20] . It wa s developed b ased on a polynom ial nonline ar kernel t hat makes it more approp riate for the analysis of sig nals with either line ar or nonlinear continuous I F traject orie s. So me previ ous studies c ompared t he performanc e of different TF D methods for the anal ysis of cardi ac signals . For instance, Obaidat [19] utilized STFT, WVD, and W T for the analysi s of phonocar diogram si gnal s . Com paring the re soluti on among the different methods of interest , he conclude d that WT performed better than STFT a nd WVD in providing m ore details of the he art sound s . White [ 29] exploit ed the p seudo W igner - Ville distribu tion to d etect, analyze and c lassify heart murmurs in PCG signals . Taebi & Mansy [16] employed ST FT, PCT and continuous WT with di fferent m o ther functi ons to dete rmine t he most sui table TFD t echnique for ana lysis of SCG signal s . They concluded t hat PCT e stimate d the insta ntaneous fre quency of the SCG signal with higher accuracy than the o ther methods considered . Understandi ng differe nt chara cteris tics of SCG , including its TFD, ma y lead to a bett er unders tanding of heart fu nction. Furthermore, successful classificat ion of SCG sign als in health and disease d can provi de a possible method for diagnosing and monitori ng cardiac m ech anical activities. This paper is ai med at compari ng the perform ance of different m ethods of estim ating the TFD of SC G signals. He re, est imations from two methods (i.e., STFT and PCT) that were found accurate in an ear lier stu dy [16] were compared with two addi tional m ethods (WV D and SPWVD). Res ults of thi s study wi ll help guide the choic e of optimal TFD analysis methods of SCG signals. The theo ry o f different TF D techni ques and de finition o f the sy nthetic te st signals use d are describe d in secti on II. Results are presented and discus sed in sect ion III , followed by conc lusions in s ection IV . II. M ETHODOLOG Y The TFD of t he signals unde r consi derati on was estimated using f our diff erent m ethods: short - t ime Fourie r transf orm, p olynomial chirplet trans form, Wigne r - Ville d istribution , an d smoothe d p seudo Wi gner - Ville Distribution. This section provides the definition s and properti es of t he TFD techni ques of interest as well as descr iption s of the synthetic test signa ls used and the m ethods of SCG data acquisition . A. TFD M etho ds STFT : T he STFT of a signal  (  ) can be expressed as:    =   (  )  (    )       (1) where  ,  (  ) , t an d  are   1 , the window function, ti me and frequency, respect ively. In thi s TFD met hod, the signa l  (  ) is divided int o a number of sub - rec ords that are shorter tha n  (  ) . The purpose of using the window function,  , is to decrease spectral l eakage when the Fast Fourier Tra nsform is applie d to each sub - record. This a pproac h assume s that the si gnal in ea ch sub - record is stationa ry, i.e. t he signal i s assumed t o have non - varying spectral characteristi cs in each sub - record [30] . When steep signa l non - stationarity is ab sent in all sub - records , high quality TFD estimates are expected for the whole signal duration. When steep non - stationarity is present , the sub - records need to be shorte ned to red uce the non - stationarity in i ndividual sub - records, which woul d enhance tempora l resoluti on. This will, howe ver, cause deteriora tion in the frequency resolution. Hence, refi ning tem poral and f requency resolut ions are t wo competing effects and a co mpromise will need to be reached to ac curately estimate the instantane ous freque ncy . WVD : For a signa l  (  ) , with the analytic associate  (  ) , the WVD is define d as,    =   (  +   )   (     )       (2) where the su perscript * denotes t he comple x conjug ate, and t and  ar e time and frequency, respectively. Th e analytic associate of the signal  (  ) is defined as:  (  )   (  ) +  [  (  )] , where  [  (  )] is the Hilbert tra nsform of the sign al  (  ) , and defined as,  [  (  )] =  .  .   (  )  (    )    =    .  .   (  )     (3) where  .  . denotes the Cauchy principal value. Sinc e the WVD separates the signals i n both time and frequency dire ctions, it was suggested as a possible method of analysis fo r non - stationary sig nal s [31], [32] . PCT: For s ignals where IF is a nonli near function of time , CT will have li mited ability in tracking IF [33] . There fore, a new CT - based TFD technique with nonl inear fre quency rotati on and shift oper ators and a polyn omial ke rnel was pro posed t o effectively estimate the nonlinear IF traj ectory of non - stationary signals [20] . T his techni que is cal led PCT and can provide a TFD with finer re solutio n compared t o conventi onal CT for signals with e ither linear o r nonlinea r IF trajectory. More detailed definition of PCT is provi ded in [20] . B. Tes t S ignals Two synthetic test signals that are similar to SCG were generated and analyzed us ing the TF D techni ques unde r considerat ion. T he test s ignals were generat ed such tha t they have durati ons and fre quency cont ent simi lar to the a ctual SCG signals. The synthetic signals h ad known IF and hence can serve as the gold st andard for t esting the a ccuracy of the TFD unde r considerat ion. The analysis was aimed at comparing the accuracy of different TFD technique s for estim ating the frequency contents of SC G signals. The amplitude, instanta neous f requency and mathematical d escription of the tes t signals are listed in Table I. C. IF Error A nalysis The IF eq uations of s yntheti c signal s (i.e., th ose with known IFs) were estimated usin g different TFD techniqu es. To evaluate the accuracy of the d ifferent T FD methods, t he root - m ean - square error (RMSE) b etween actual and estimated IF values was calculated as:  =   (   ,     ,  )     (4) where   ,  and   ,  are the signal actual and estimated IF at time i , respecti vely , and n is the total num be r of data poin ts. RMSE value s are then no rmaliz ed by dividing RMSE by the mean actual instantaneou s frequency,      , of each signal as follows:  =       (5) Normalized root - mean - square error (NRM SE) was used i n t he current study as the cr iterion to quantify the accuracy of the different TF D technique s in est imating IF . Here, lower NRMSE values woul d indica t e higher accu rac y. The signal processi ng steps for eval uating t he most appr opriate TFD methods f or time - freq uency a nalysis of SCG signals are shown in Fig. 1. D. Data A cquisition of S CG Actual SCG signal s from 8 healthy subjects w ere also analyze d using the T FD techni ques under c onsiderat ion. After IRB approval, a lig ht - weight (2 gm) accelerometer (PCB P iezotronics , Dep ew, NY) was used to measure the SCG signal. The sensor wa s placed over the ches t of volunt eer s at the left sternal bor der and t he 4 th intercostal space. The signal was digitiz ed at a sam pling frequen cy of 320 0 Hz and down - sampled to 320 Hz. Ma tlab (R2 01 5b, The MathWorks , Inc, Natick, M A) was used t o both acquire and proc ess all signals. III. R ESULTS AND D ISCUSSIONS T he time series and TFD of the synthetic test signals under consideration are shown in F ig. 2 and Fig. 3 . The TFDs were estimated using STFT, PC T, WVD, and SPWVD and shown in subfigures b, c, d, and e, respectively. The P SD was also calculated from the TFDs, and norm alized with respect to the signal e nergy. P SDs are pre sented i n the le ft side of the Fig. 2 and Fig. 3 . Here, t he signal s did not hav e s ignifica nt energy above 70 Hz. Therefore, the spectral information is show n for frequenci es up to 70 Hz . Table II l ists the tem poral and spectral resoluti ons of differe nt TFD te chniques . The NR MSE bet ween the actual and calculated instantaneous frequenc y are reported in Table III for d ifferent TFDs. Temporal and frequenc y resolut ion: In g eneral, coa rser resolution i s not de sirable a s it m ay lead to higher err ors in estimating IF. Table II shows that STFT had coarser t emporal resolution c ompared to other methods (th e latter three methods had the sam e tempor al resolut ion ) . SPWVD a nd STFT techniques had the fine st and coa rsest spec tral resol utions, respectively (Table II ). The WVD had a finer resoluti on than PCT (about 0 .11 and 0.22 Hz, respect ively) a t the cost of increased artifacts that may result fr om cross - terms and aliasing. As expected, alias ing was seen in the curre nt study for the WVD and SPWVD TFD of a ll signal s with fol ding fre quencie s of   . /2 , where   . is the Nyqui st freque ncy. T ABLE I. L IST OF AMP LITUDE , INSTANTANEOUS FREQU ENCY AND EQUATION OF T HE SIMULATED SIGNALS USED IN THE CURRENT STUDY . Signal Amp li tude Instantaneous Frequency (Hz) Signal Equation     =      0 0 <   0. 25 0.5  0.5 cos ( 14  (   0 . 75 ) ) 0. 25 <   0 . 40 0 0. 40 <   0. 70 0. 45  0 . 45 cos ( 14  (   0 . 75 ) ) 0. 70 <   0. 83 0 0. 83 <   1. 00   = 20 and 40   =   sin ( 2  ( 20 )  + 94 ) + 0.9   sin ( 2  ( 40 )  + 188 )     =      0 0 <   0. 25 0.5  0.5 cos ( 14  (   0. 75 ) ) 0. 25 <   0 . 40 0 0. 40 <   0. 70 0. 25  0 . 25 cos ( 14  (   0. 75 ) ) 0. 70 <   0. 83 0 0. 83 <   1. 00   = 40 and 2610    430  + 20 (where 0    0. 15 )   =   +   where:   =  0.5   sin (2  ( 40 )  )   =      0 0 <   0. 25  0. 25 <   0. 40 0 0. 40 <   0. 70  0. 70 <   0. 83 0 0. 83 <   1. 00 where:  =   sin (2  ( 870    215  + 20 )  ) where 0    0. 15 Fig. 1. Block diagram describing the signal processing steps for evaluating the most appropriate TFD methods for time - frequency analys i s of the S CG signals . A. S ynthetic SCG with Time - independe nt F requencies Th is synthetic SCG consisted of two freque ncy com ponents at 20 and 40 Hz . Fig. 2 . d shows that WVD TFD was contami nated by relatively strong artifacts between the actual signal freque ncy c omponents. In ad diti on, WVD did not distinguis h between t he two com ponents and had notice able leakage, w hich led t o a PSD wi th 3 peaks a t 20, 3 0, and 40 Hz. The 30 Hz peak is clearly due additional inter ference terms . The PSD graphs of STFT , PCT, an d SPWVD c orrect ly showed t w o peaks corre sponding t o the a ctual si gnal com ponents. While STFT and PCT demonstrated less leakage than WVD, they showed possible more leakage than SPW VD where the sp ectral peaks appeared more co ncentrated. The WVD had the highest error among t he methods (NR MSE of 0.6970 ), which may be, a t least in part, due to the interference terms in W VD. STFT and PCT had lower N RMSE values compared to SPWVD (Table III ); however, the latter had the finest spec tral resolution and a relatively sma ll er leakage. B. Synthetic SCG with Var ying F requency C omponents Th is synthetic SCG consis t ed of two freq ue ncy compone nts . The fi rst c ompone nt had a varyin g freq uency ranging from 20 to 7 Hz while t he seco nd com ponent was a fixed fre quency at 40 Hz. In addition, the signal was cont aminated with white noise with a signal - to - noise ratio of 10. PCT and SPWVD estimated the s ignal IF with higher accuracy than ot her met hods (NRMSE of 0.281 3 and 0.2 672, respectively). STFT had a higher NRMSE of 0.4431, and WVD had the lowest accuracy with an NRMSE of 1.2202. WV D had finer temporal and spectral resolu tions than PCT and STFT at the cost of int roducti on of art ifact peaks in t he time - frequency plane (Fi g. 3 .d). These artifacts were significantly reduced after smoothi ng in tim e and frequency dom ains (Fi g . 3 .e). In additio n, WVD a ppeare d to ha ve the highest a rtifac ts. This leakage in WVD w as also reduc ed by employing S PWVD. It can also be see n in Fig. 3 that STFT and PCT could distinguish between t he two compone nts o f the si gnal; ho wever, t hey app eared to hav e slightly higher l eakag e bet ween the two frequenc y peaks compared to the SP WVD. T ABLE II . T EMPORAL AND SPECTRAL RESOLUTION FOR DIFFERENT SIGNALS AND TFD TECHNIQ UES FOR FREQU EN CIES UP TO 70 H Z . STF T TENDED TO HAVE COARSER TEM PORAL AND SPEC TRAL RESOLUT ION COMPARED TO PCT. SPWV D HAD THE FINES T TEMPORAL AND SPECTRAL RESOLUTION . STFT PCT WVD SPWVD Temporal R esolution (ms) 12.5 3.1 3.1 3.1 Spectral Resolution (Hz)   0.6250 0.2462 0.1231 0.0004   2.5000 0.2462 0.1231 0.0038 T ABLE III . NRMSE BETWEEN THE ACTUAL A ND CALCULATED IF FOR DIFFERENT TFD METHODS . T HE MOS T APPROPRIATE TFD TECHN IQUES WOULD BE THOSE WITH THE LOWE ST NRMSE VALUES . STFT PCT WVD SPWVD   0.0199 0.0214 0.6970 0.0325   0.4431 0.2813 1.2202 0.2672 Fig. 2. Synthetic S CG with ind ependent fr equencies: ( a) Time se ries. Ti me - frequency distribution using (b) STFT, (c) PCT, (d) WVD, and (e) SPWVD, respectively. Fig. 3 . Synth etic S CG with varying frequen cy compone nts : (a) Time series. Ti me - frequency distr ibution using (b) STFT, (c) PCT, ( d) WVD, and (e) SPWVD, res pectively. The STFT is relatively simple and mostly su itable for stationary signals [30] . However, SCG signals are usually rhythm ic, and may ha ve dif ferent amplitude, time duration , an d spectral properties f or each cardiac cycle [34], [ 35] . Thus, STFT m ay not be the be st opt ion for time - frequency analysi s of SCG signals. TFD of th e synthetic signals in the current stud y suggest that PCT and SPWVD consisten tly had low NRMSE, less artifacts and lower leakage. These two techniqu es also provide d bette r discrim ination betwee n signal f requenc y compone nts. Th e PCT and SPWVD performance was better than STF T for the synthetic signals, which may be due to the ir finer resolution. SPWVD had more accurate IF estimation s than WVD fo r the synthetic SCG s ignals. The pe rform ance of S TFT varied dependi ng on t he synt hetic si gnal un der cons iderati on, but it did not s eem t o provide notice able ad vantages over SPWVD or P CT , except for simplicity. These tr ends will be helpful in the interpret a tion of the results of the actual SCGs TFD. Since PCT and SPWVD provided m ore accurate IF values than STFT , we decided to utilize PCT and SPWVD for estimating the frequency c ontent of the ac tual SCG si gnals. C. Actual SCG S ignal s Fig. 4 shows the TFDs of an actual SCG f or two cardiac cycles. The figure sugg ests that there were two SCG even ts ( SCG 1 and SCG 2) for each cardiac cycle. U sing PCT and SPWVD, SCG 1 and SCG 2 appeared to be localized in the time - frequency dom ain at frequencies of 18.75 and 37 .50 Hz. Th is was also clearly seen in the PSD of the PCT and SPWVD (Fig. 4 .c and 4 .e, respec tively). The PSD of the WVD, inst ead, showed a noisy broadband rather than two separate peaks, which may be due to presence of the artifact peaks. Comparing Fig 4 .e with Fig. 2 .e and 3 .e, one could al so concl ude that the high - frequency c omponent of the SCG 1 behaved m ore as a fixed frequency , while the lower - frequency com ponent behave d mo re as a chirp (with a slig htly decreasing frequency with time). Also, the results su ggested that there a r e certain inter - cycle cardiac variability (F ig. 4 ). For example, t he lower - frequenc y component of SCG 1 and SCG 2 had more energy than t heir higher - frequency component during the first cardiac cycle. However, i n the sec ond cyc le, more e nergy of SCG 1 was concentrat ed in the higher - fre quency com ponent. The source of this varia bility is unk nown but m ay be rel ated to the known heart beat variability [16 ] . Fig. 5 .b shows the SCG IF estimated by PCT and SPW VD. This figur e shows that the domi nant IF of SCG 1 wa s time - dependent and signi ficantly varied among t he two cardiac cycles shown. For the first cycle, the dominant frequency f ell from 21 Hz to below 10 Hz, whil e in the s econ d cycle, t he dominant frequency start ed around 20 Hz then peaked to 37.5 Hz, a nd finall y dropped t o below 1 0 Hz. SCG 2 had a slight dow nward trend from 22 Hz to 19 Hz in both the first and second cardi ac cycl es before dr opping to below 10 Hz. While these tr ends may be of diagn ostic val ue, further i nvestigat ions would be needed to see if they are consistent for several cardiac cycles and patients. The TFD te chniques unde r considera tion were used to e stima te the dominant frequency of the SCG from 8 healthy s ub jects ( Table I V ). The data suggests that there is a broad range of IF in dif ferent s ubjects. T he agreem ent between STFT and PCT appeare d to be hi ghest foll owed by SP WVD while WVD ha d the most disagreem ent with other methods . IV. C ONCLUSIONS This study aimed a t c omparin g the pe rform ance of fo ur differe nt TFD a pproac hes in est ima ting the i nstanta neous frequenc ies of SCG signals. M ethods i ncluded S TFT, P CT, WVD and SPWVD. In the curr ent study, the temporal and spectral resoluti on of the SPWVD a nd STFT was finer a nd Fig. 4. Actu al SCG signal: (a) Time ser ies. Time - frequency distribution using (b) STFT, (c) PCT, (d) WVD, and (e) SPWVD, respectively. Fig. 5. (a) T i me representation of S CG signal, (b) E stimating signa l instantaneous frequency using P CT and SPWVD . T ABLE IV . D OMINANT FREQUENCY (H Z ) OF THE ACTUAL SCG SIGNALS CALCULATED USING STFT, PCT, WVD, AND SPWVD. Subject # STFT PCT WVD SPWVD 1 6.25 6.15 3.93 3.82 2 30.00 30.28 21.42 30.40 3 11.25 11.32 16.12 12.92 4 17.50 17.23 17.85 15.88 5 28.75 28.31 26.95 27.82 6 7.50 7.63 18.95 7.38 7 20.00 19.94 13.05 19.57 8 33.75 32.98 23.63 31.38 coarser t han other me thods, res pectivel y. The accuracy of differe nt methods in dete rmining t he IF was tested us ing synthetic test signals with known TFD , and the estimated IF was com pared t o actual IF val ues. The errors in estim ating IF were highes t for WVD. These results may be a ttributed to the limitations of W VD (e.g. negative energ y values and ar tifacts in the TF D) and su ggested t hat the m ethod woul d not be a good choice for estimating the TFD characteristics of SCG signals. The SPWVD and PCT had the lowest error in estimating I F followe d by STFT. These results may be a scrib ed to the fine temporal and spec tral re soluti on of P CT and SP WVD. Therefor e, this study s uggest ed that t he PCT a nd SPWVD would be be tter estimators of t he freque ncy cont ent of SCG signals. TFD of an actual SCG was also estimated using PCT and SPWV D. Result s showe d that this signal had two p rimary frequenc y compo nents tha t are slightl y time - va rying. F urther studies m ay be warranted t o inves tigate the freque ncy contents of SCG signals in health and disease. A CKNOWLEDGEMENTS This study was suppo rted by NIH R44HL099053 . R EFERENCES [1] S. L. Murphy, J. Xu, an d K. D. Kochanek , “Deaths: final data fo r 2010.,” Natl. Vital Stat. Rep. , vol. 61, no . 4, pp. 1 – 117, 2013. [2] F. Khalili and H. A. Mansy, “Blood Flow through a Dysfu nctional Mechanical Heart Valve,” in 38th Annu Int Conf IEEE E ng Med Biol Soc , 2016. [3] F. Khalili, P. P. T . Gamage, and H. A. Mansy, “He modynamics of a Bileaflet Mechanical Heart Valve with Diff erent Levels of Dysfunction,” J Appl Biotechnol Bioeng , vol. 2, no. 5, p. 00044, 2017. [4] A. Taebi and H. A. Mans y, “Time - frequency An alysis of Vibrocardiographic Signals,” in 2015 BMES Annual Meeting , 2015. [5] U. Morbiducci, L. Scal ise , M. De Melis, and M. Grigioni, “Optical vibrocardiography: A novel to ol for the optical monitoring of cardiac activity,” Ann. Biomed. E ng. , vol. 35, no. 1, pp. 45 – 58, 2007. [6] L. Scalise and U. Morbiducci, “Non - contact cardiac monitoring fro m carotid artery usin g optical vibrocardiog raphy.,” Med. Eng. Phy s. , vol. 30, no. 4, pp. 490 – 497, 2008. [7] G. Cosoli, L. Casacanditella, F. Pietr oni, A. Calvaresi, G. M. Revel, and L. Scalise, “ A novel approach fo r features extraction in physiological signals,” in Medica l Measurements and Applicatio ns (MeMeA), 2015 I EEE Internat ional Symposium on , 2015, pp. 380 – 385. [8] A. Taebi and H. A . Mansy, “Grouping Si milar Seismocardiographic Signals Using Respiratory In formation,” in Signal Processing in Medicine and Bi ology Symposi um (SPMB), 2017 I EEE , 2017, pp. 1 – 6. [9] A. Taebi, R. H. Sandler, B. Kakavan d, and H. A. Mansy, “Seismocardiographic Signal Ti ming with Myocardial Strain,” in Signal Processi ng in Medicine and Biology Symposium (SPMB), 2017 IEEE , 2017, pp. 1 – 2. [10 ] B. E. Solar, A. Taebi, and H . A. Mansy, “Classification of Seismocardiographic Cycles into Lung Vo lume Phases,” in Signal Processing in Me dicine and Biology Symposium ( SPMB), 2017 IEEE , 2017, pp. 1 – 2. [11] L. P. Feigen, “Physical characteristic s of sound and hearing,” Am. J. Cardiol. , vol. 28, no. 2, pp. 130 – 133, 1971. [12] A. A. Luisad a, The sounds of the normal heart . Warren H. Green , 1972. [13] A. Taebi and H. A. Mansy, “Time - frequen cy Description of Vibrocardiographic Signals,” in 38th Annual Inte rnat ional Conference of the IEEE En gineering in Medicine an d Biology Society , 2016. [14] A. Taebi and H. A. Mansy, “Eff ect of Noise on Time - frequency Analysis of Vibrocardiog raphic Signals,” J. Bioeng. Biomed. Sci. , vol. 6(202), p. 2, 2016. [15] A. Taebi and H . A. Mansy, “Nois e Cancellation fro m Vibrocardiographic Signals B ased on the Ensemble E mpirical Mode Decomposition,” J. Biotechnol. Bioeng. , vol. 2, no. 2, p. 0002 4, 2017. [16] A. Taebi and H. A. M ansy, “Time - Frequency Distribution of Seismocardiographic Sign als: A Comparative Study,” Bioengineering , vol. 4, no. 2, p. 32, 2017. [17] J. A. Crowe, N. M. Gibson, M. S. Woolfson, and M. G. S omekh, “Wavelet transform as a poten tial tool for ECG a nalysis and compression,” J. Biomed. Eng . , vol. 14, no. 3, pp. 268 –2 72, 1992. [18] P. Celka, B. Boashash, an d P. Colditz, “Preprocessin g and time - frequency analysis of n ewborn EEG seizures,” IEEE Engineering in Medicine and Bi ology Magazine , vol. 20, no. 5. pp. 30 – 39, 2001. [19] M. S. Obaidat, “Phonocardiogra m signal analysis: techniques and performance comparison,” J. M ed. Eng. Technol. , vol. 17, no. 6, pp. 221 – 227, 1993. [20] Z. K. Peng, G. Meng, F. L . Chu, Z. Q. L ang, W. M. Zhang, an d Y. Yang, “Polynomial chirplet tran sform with application to instantaneous frequency est imation,” IEEE Tr ans. Instru m. Meas. , vol. 60, no. 9, pp. 3222 – 3229, 2011. [21] J. Ville, “Theorie et app lications de la notio n de signal analytique, ” Cables Transm. , vol. 2A, no. 1, pp. 61 – 74, 1948. [22] W. Martin and P. Flandrin, “Wigner - Ville Sp ectral Analysis of Nonstationary Processes,” IE EE Trans . Acoust. , vol. 33, no. 6, pp. 1461 – 1470, 1985. [23] S. C. Bradford, J. Yang, and T . H. Heaton, “Variatio ns in the Dynamic Properties of Structures: Th e Wigner - Ville Distribution,” 8th U.S. Natl. Conf. Earthq. Eng. , no. 1439, 2006. [24] F. Auger, P. Flandrin, P. Gonçal vès, and O. Lemoine, “ Time - frequency tool box,” CNRS Fr. Univ. , 1996. [25] P. Flandrin, Temps- Fréquence (Traité des Nouvelles Technologies, série Traitement du S ignal) . Paris: Hermès, 1993. [26] B. Boashash, Time- frequency si gnal analysis and pr ocessing: a comprehensive reference . Ac ademic Press, 2015. [27] S. Mann and S. H aykin, “Adaptive ch irplet transform: an adaptive generalization of the wavelet transform,” Opt. Eng. , vol. 31, no. 6, pp. 1243 –1 256, 1992. [28] S. Mann and S. Haykin, “The Chirplet Transform: Physical Considerations,” IEEE Trans. Signal Proce ss. , vol. 43, no. 11, pp. 2745 – 2761, 1995. [29] P. White, W. Collis , and A. Salmon, “ Analysing heart murmurs using time - frequency m ethods,” Pr oc. IEEE - SP Int. Symp. Time - Frequency Time - Scale Anal. , vol. 2, no. 1, pp. 2 – 5, 1996. [30] S. M. Kay and S. L. Marple, “Sp ectrum Analysis — A Modern Perspective,” Proc. I EEE , vol. 69, no. 11, pp. 1380 – 1 419, 1981. [31] P. Mansier, C. Médigue, N. Charlotte, C. Vermeiren, E. Coraboeuf, E. Deroubai, E. Ratner, B. Chevalier, J. Claira mbault, F. Carré, T. Dahkli, B. Bertin, P. Briand, D. Stro sberg, and B. Swyngh edauw, “Decreased heart rate variability in tran sgenic mice overexpressing atrial β1 -adrenoceptors,” Am. J. Physiol. - Hear. Circ. Physiol. , vol. 271, no. 4 40 – 4, pp. H 1465 – H1472, 1996. [32] H. H. Huang, H . L. Chan, P . L. Lin , C. P. Wu, and C. H. Hua ng, “Ti me - frequen cy spectral analysis of heart rate variab ility during induction of g eneral anaesthesia.,” Br. J. Anaes th. , vol. 79, no. 6, pp. 754 – 758, 1997. [33] Y. Yang, Z. K. P eng, W. M. Zhang , and G. Meng, “ Frequency - varying group de lay estima tion using frequency domain polynom ial chirplet transform,” Mech. Syst. Signal Process. , vol. 46, no. 1, pp. 146 – 162, 2 014. [34] B. W. Hyndman, R. I. Kitney, and B. M. Sayers, “ Spontaneous rhythms in physiolo gical control systems.,” Nature , vol. 233, no. 5318, pp. 339 – 341 , 1971. [35] M. J. Campbell, “Spectral anal ysis of clinical signals: an interface between medical statisticians and medical engineers,” S tat Methods Med Res , vol. 5, no. 1, pp. 51 – 66, 19 96.