This paper presents a novel approach to the extrac- tion of weaker periodic signals in the presence of noise and a stronger periodic waveforin The signals.
EXTR.ACTION OF MULTIPLE PERIODIC WAVEFOR.MS FROM NOISY DATA
Depart.ment of Electronics k Electrical Cornmiinication Engg., Indian 1nst.it.ut.eof Technology, Kharagpur-721302, INDIA
ABSTRACT This paper presents a novel approach to the extraction of weaker periodic signals in the presence of noise and a stronger periodic waveforin The signals may have any shape, not necessarily sinusoidal in nature. The approach is based on the Singular Value Decomposition. The extraction procedure consists of two steps, identifying the strongest periodic component and consequent configuring the data matrix followed by extraction of this component. This process is repeated till all the components have been recovered The approach is both simple and robust and requires no additional reference inputs. The simplicity of the scheme is however, at the expense of certain inherent limitations, which are also invcstigated.
Such problems are typically solved using additional inp1it.s which may consist of the interfering component, only as in (31 or otherwise as in [4]. Apart from the ext,ra cost. incurred in such approaches due to multiple sensors, t h e quality of the additiona.1 inputs is crucial to the satisfact,ory performance of the ext,raction sclieme. This paper discusses an approach which does not require any training data. The simplicity of t,he scheme is however, at. the expense of certain iiihm=nt, limit.at.ions, which are also investiga t.ed. 2. SIGNAL CONFIGURATION USING SVD
The SVD of a n decomposit ion
777
x
7)
matrix
A
is defined as [5] t'he
A = UCV' 1. INTRODUCTION
(1)
U = [ul,.. . ,U,,] E RmXm , V = [ V I , . . . ,v,] E and U'U = I , V'V= I a.nd E E L V X * . The mat,rix E is equal to [ d i a g { a l , . . . ,aP}: Opxlm-,,l] or it,s t,ranspose, depending on whether m < n or where
Extraction of weaker signals i n tlie presence of noise and a stronger periodic waveform is an important problem in various applications. Extraction of sinusoidal waveforms in the presence of noise has been widely studied in the past. However, in many reallife situations, the signals generated are periodic, but rarely sinusoidal. The problem of recovery of multiple sinusoids in a noisy environment has hcen addressed by several researchers i n the past [1,2]. In many practical situations, however, one encounters noisy signals with multiple periodic components, which are not sinusoidal. Indeed, tlie amplitude and shape of one or more components may be unknown. For example, Widrow considers [3] the extraction of the fetal cardiac waveform from an additive mixture of the mat.ernal and fetal cardiac signals with noise. In an underwater environment, sonar signals emitted from several ships, or other objects of interest may be recorded simultaneously. I n practice. the sound emitted from a ship is periodic. but rarely sinusoidal
R V ' X ~
m 2 n ; where p = tnin.(m,n). The real numbers ( T I , . . . , a,, are called t,he singular values of A and are convent~ionallyarranged as 01 2 . . . 2 up 2 0. The column vectors U ; and w,,which correspond to t,he singular value U , , are called the ith left singular vector aiid t.he it.11 right singular vector respectively. Consider a strictly periodic signal {x(.)} = {z(l)?x(2) . . . . ? } , withaperiodoflengthn, t,liat is, z( b n ) = x( k), k >_ 1. Observations from 777 periods are nssumed to be available. Then, the corresponding data matrix E Rmxn is formed by part,it.ioning t,Iie signal into periods a.nd placing each period ( i n phase) as a row of as shown below:
+
x
x,
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uset1 iiist eat1 of t.lie sample mean i n ea.ch case because
x
The matrix has m repeated rows and is of rank 1. Consequently it has only 1 non-zero singular value and 771 - 1 zerq singular values. Now consider the case of a periodic waveform with time-varying amplitude plus noise. This includes as a special case, a periodic signal plus noise. inay now have full rank but u1 would The matrix he very large compared to the rest of the singular Values, assuming that the additive noise is small. The ratio will he high.
x
2
3. PERIODICITY DETECTION AND P ER10 D LEN GTH DE T ER.MIN AT10 N
Consider the signal
it is a better indicator of t,he cent,ral t,endency for skewed tl ist,ri ti it t ions. The plot, of 6 w i t h row length not only detects periot1icit.y but, also reveals the period length. This procedure for period length determinat,ion has, however some 1iniit.ations. The plot of 6 will show peaks not, only at, t,he t,rrie period length but also a t its higher muIt.iples. Hence, choosing the correct period length implies a judicious choice of the peak and verification of t he esist.ence of harmonics. Further, since small row lengt,hs iinply t,hat the samples considered fall ivit,Iiin t.he linear range of t,lie data, the curve st,art,sfrom higlt initial values wl~iclican he misleading. T h e procediire for period determination hence consist,s of ( I ) discarding t,he initial monot,onically decreasing range of t.he 6 curve, and (2)choosing the first, peak tlrt~c~rminetl hy the height as well as separat.ioti ancl wat.cliing for hartnonics at. higher multiples of t.liis length. 4. PATTER,N ESTIMATION AND EXT R.ACT I 0 N
where r(k) is a perfect,ly periodic signal wit.li period I a.nd e ( k ) is whit,e noise. Since the inlierent periodicity of the signal is buried iinder a.tltlitive noise,it. is necessary to provide a. measitre of periodicity tliat would indicat,e t,he st,rength or weakness of it,s periodic content.. For t.liis purpose, it. is assumed t,liat, t.he period lengt,li is known and tlie dat.a matris is formed as indicated in t,lie previous section by aligning the rows corresponding to the periods. It follows from the discussion in Sect.ion 2 t.hat. t,he rat.io for will he large for d a h matrices of signals with a strong repeat,ing pat,tern. Hence the following ’measure of periodicit,y’ is proposed
2
6=
($
(4)
Not,e t.hat. 6 will be very liigli when the row length of the corresponding data niatris equals t.he period 1engt.h of the periodic part, of t,lie signal or its Iiiglicr mult.iples. Hence it,s profile of variation wit.li row length ca.n he esamined detect.ion of periodicity in a signal. The comput,at,ion of 6 for a. part.iciilar row lengt,li n is actually performed as t h e sample mediaii of 6 values oht,ained from ( N - n + I ) overlapping d a t a windows, where t,he kt,h d a h window is formed from signal samples a.t. k - m i ? , to k. Here 7)) refers t,o the number of rows in each t1iit.a window. N reprcscnt.~ the 1engt.h of t,lie ent.ire dat,a srqitence while n rcfrrs to the column lengt,h. The sample median valiie is
Consider a signal of t.lie form as in (3) wit.11 a data n i a t r i s Y For a sufficient.ly high SNR, when the row Iengt,li equals I , the SVD of Y will yield only one significant, siiigitlar value. Since the smaller singular values arise on account of t.he noise. a reconstruction of ? ( A - ) tmyd on t.he largest. singular value can be matle by estract,ing t,lie p n n c z p a l p n f t e n i a s follows: If t,he noise e ( k ) is zero. t,he d a t a mat,rix (with row lengt,Ii rqual t.o I ) is of ra.nk 1. Let u1,~1 and V I he the prime singular value and t.he corresponding singular vectors when t,he noise is present. It can he sltorvn t h a t o I u 1 is ~ blie ~ hest rank 1 a.pproximation of t,lie dat.a mat.rix in the sense that it minimises IIYZll~ under t,he const,raint rank(Z) = 1. Thus the priiicipal pat.tera reconst,ruction of the signal would he t.he seqitriice obt~ainecl by lining u p the rows of U l U ,VT.
Now consider n consecut,ive ohservat.ions from the niotlel nP
!AL.) =
C-I.l,(k) +4k)
(5)
2=1
wlic~rru l , r c y r c w n t s t lie nuniber of
periodic compo-
nents. Each signal component in the above model is a>\itnictl to he at least comparable in power to r ( k ) . The s>htematir part of thc signal is perfectly periodic w i t Ii t lip period equal to the lowest common niiiltiple (IA’11) of tlie constituent signals
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Assuming that the signal component,s have well separa.t,ed frequencies, the period lengt,lis can he determined from t,he varia.t,ion of 6 wit,h row length, in a successive manner. For example, consider the case when n,,, = 2 in the model (5) and t,he signal component,s have unknown period lengt,hs I 1 and 12 and corresponding fundament.a.1 frequencies f1 and f2. The value of I producing the largest. 6 is chosen as 11. N0t.e that often in a. periodic signal overt,ones have a higher power tha.n t,he fundamenhl frequency. In t8he6 plot, however, overt.ones will correspond to a smaller period length as compared t,o that of t,lie fundamental frequency and t.he correct peak must. he verified from t,he presence of 1ia.rmonics. The principal patt.ern is now used to form an estimate of the component, (21,(.)}. This is subtracted from {y(.)} and t.he ent.ire procedure repeat,ed t.o first. est.imat.e 12 and then form the corresponding principal pa.tt,ern to estimate {z,?(.)}. The algorit,hm for general n P can be summarizetl int.0 t.he following st,eps:
1. Det.ermine the period lengt,li 1, for tlte tlominant, period from the plot. of 6 versus row length. 2 . Configure the mat.ris wit,h row Icngt,li lj a n d masimuin numher of rows. The principal pat,tern is used t.0 reconstruct t.he shape a.nd amplit,ude of the dominant. signal coinpotlent,. 3. Suht.ra.ct t,he est.imat,etl prime component. ohtained in St,ep 2 from t,he entire d a h lengt.11. Repeat steps 1-2 until n p coniponent.s have been identified.
If nr, is not known apriori, the procediire may be repeated until the 61 plot of the residuals does not reveal any special feature. 5. LIMITATIONS OF THE APPR.OACH
The present. approach of pat,t.ern ext,ract,ion suffers from the inlierent, problem of identifiability. M‘lten one of t,he signal constit,iient.s has a.n unknown period length which is a suh-mult,iple of the coml~ined periods of the rest, t,his t.echnique will fail t.o itlent.ify it, or extract, it suhsequently. Even when all i.lie period lengt,lis are known, a. signal component, may not he complet,ely separable from t,he rest.. Rtrt.hrr, if the power of the weakest component happens t,o b e less t,han t,liat of the hackground noise. any at.t.emt t,o ident.ify it,s period lengt.lt may yield inconcliisive result,s. 6. R.ESULTS
0.012
hl
s
L
L maternal fetal
0.008
-
E
0.004 0 -6
-4
-2
0
2
4
6
8
10
SNR Figure 1: MSE vs. SNR for the extracted components.
I n order to verify the performance of t.he proposed technique t.wo experimentasusing ECG data were carried out.. Experiment 1:Sbircf.y of the effect ofnoise on the algorithm. In t.liis experiment, 25 replica of a typical ECG cycle is added to an adequate numher of replicae of a t,ypical fet,al ECG cycle. To this is added a 50 IIz. siniisoitlal signal with power 10 d B less than that of the fetal component, t o simulate t,he interference due t,o line frequency and noise (additive white Gaussian). The resukant signal is used to simulate t.he ECG clat,a obt,ained from the abdominal lead of an expecting mother. The mean square error per sample of t,he maternal and fetal ECG are plotted versus SNR i n Figure 1. In both cases, t,he MSE is found to monotonically decrea.se w i t h increase in SNR. Experiment 2 : A real-life application. Real data (oht.ainer1 from Widrow) is analyzed wit,li the above t,ecliniqite. Figure 3 shows the plot of 6 for the original signal from which t.he period of the maternal ECG is det.ermined t.o he 75. A spurious peak is ohserved at. row 1engt.h 37 which may he explained a s follows. The mat,ernal ECG component has a region of small fluct.uat.ions within each period spa.nning more than half of its length. If the data is aligned wit.11 a row 1engt.h of 37 such that the region of high act.ivity falls wit.hin a period, the data inat,ris will have alt,ernat,erows of ’high a.ct.ivity’ and ‘small fluct.itat.ion’ regions. This will lead to a reconst,riict,ion cliiit,e close to t,hat with row length 75 t.hough t,he MSE is likely to be higher. The 6 plot cannot, be est,entled for a row length of 150 owing to t,lie short.age of data, hence t,he so-determined period lcngt,li is verified by visual inspection. The est.ract.ed maternal component is shown in
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4
M
3
A G
N I T
2 1
D E
10 20
30 40 50 GO SNR
70 80
50
100 150 200 250 300 Sainpli ng i nst.ant
Figure 4: T h e composite ECG signal and the extracted fetal component.
Figure 2: Plot of 6 for t,he composite signal.
0
0
50 100 150 200 250 300 350 400 Sampling instant
Figure 3: T h e extracted maternal component.
Figure 3. The residual dat,a series, free of t,lie mat.erna1 coniponent appears t.0 be extremely noisy. A periodic40 is observed. The plot. of 6 ity of appr~ximat~ely shows a peak a t 40 but shows many spurious peaks on account of the noise. The period lengt,li of 40 is retained since t,his also happens to he t,he most. frequently ocurring peak-tGpeak lengt,li. Since the period lengths change slightJy from one period t o mother, the periods are first aligned using the peaks (or troughs). Linear interpola.tion is t.hen performed so that ea.ch cycle has the dpt.ermined period lengt,h. After extra.ction, each cycle is converted to tthe original scale. The extract.ed fet,al ECG wa.veform is plotted in Fig.4 along with the comp0sit.e ma.ternal ecg. This is comparable to t,he reconstructed fetal waveforms obtained by Widrow mit.11 the help of additional sensors for adaptive noise cancelling i n [3, Pg.336,337].
this approach suffers from certain limitheranalysis.
References
7. CONCLUDING REMARKS
Thus, it is found t,hat a simple SVD-based t.echnique can produce a reasona.ble reconst.ruct,ionof two periodic waveforms from noisy data. The advantages of t.his procedure are a ) elimina.t,ion of t,he need for training data, h) availability of computat~ionally stable algorithms for implement,ation [GI. Though
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[I] R. 0. Schmidt, “A Signal Subspace Approach to hlult,iple Emitter Location and Spectral Esttimat.ion,” Ph .D. disserlalion, Stanford University, 1981.
[a] R.
Roy and T.I
