SINGULAR 11 C d

-

- - - - -- -

-- - -

---...

--'f~'t MC-LoJ\.

f'

-,.'"1Q . .-vaM vJC()I.MI\.-J f(oA-tJc..

tA

)LjMai... 1~0G2.t1

d

I

/f~_

/f'> ~e <-. A eH g 8

~

~

11C

~

SINGULARVALUE DECOMPOSITION

: A

POWERFUL

CONCEPT AND TOOL IN SIGNAL

PROCESSING

Joos Vandewalle and Dirk Callaerts1 ESAT-laboratory

Department of Electric.al Engineering Katholieke Universiteit Leuven

K. Mereierlaan 94 3030 Heverlee, BELGIUM

tel 016/22 09 31 telex 25941 elekul fax 32/16/22 18 55

---

---

- - -

-Abstract

The increased computational capabilities of modern computers, workstations and VLSI technology on the one hand and the low cost of sensors and data acquisition equipment on the other hand, make it more and more att.ractive and feasible to use the Singular Value Decomposition (SVD) in signal processing. In this context it is argued that the SVD may become a widespread tooI in signal processing, similar to the role played by FFT in digital signal processing in the seventies and eighties.

At t.he conceptuallevel the Singular Value Decomposition (SVD) and t.he Generalized SVD (GSVD) provide a unifying framework and a numerically robust approach for the formulation and computation of new concepts, sueb as oriented energy and oriented signal-to-signal ratio's, angles between spaces and canonical correlation analysis. In our research group these concepts have been very instrumental in order to devise new algorithms for a wide variety of problems in signal processing and identification.

Keywords

SVD, GSVD, oriented energy, signal-to-signal ratio; signal processing, signal separation

Running title

- - -

---

----

--

-

---1

Introduction

Digit.al signal processing of mu1t.ichannel, mult.isensor or 2D images, oft.en includes fi1t.ering, signal separat.ion, error correction, compensat.ion, int.erpolat.ion, decimation, transformation, monitoring, data compression, feature extract ion, . . . Many of these tasks require least squares approximation, low rank approximation, norm and condit.ion number calculation ([12]). AI-though SVD was already well-known in numerical analysis as a reliable concept for these comput.ations ([11], [13]), it has only recently been rediscovered in signal processing ([2], [3], [5], [10], [20]). In most signal processing applicat.ions, the SVD provides a unifying framework, in which one can describe at once the conceptual foru1Ulation of the problem, the practical application and an explicit solution that is guaranteed to be numerically robust. In this way, the SVD has become a fundament al vehicle for the formulation and derivation of new concepts such as angles between subspaces, oriented energy, oriented signal-to-signal ratio, canonical correlation analysis, ... ([6], [7], [15], [16], [20]) and for t.he reliable computation of

the solut.ions to problems such as totallinear least squares, realization and identification of linear st.ate space modeis, source separation by subspace methods, etc. It is expected that SVD will become a standard tooI on t.he workbench of a designer of signal processing systems, like FFT was since the mid seventies.

The benefit.s of using the (generalized) singular value decomposition are most pronounced in t.hose signal processing applications :

.

where essentially rank decisions and t.he comput.ation of the corresponding subspaces determine the complexity and parameters of the model

.

where numeri cal reliability is of crucial importance and the potentialloss of nwnerical accuracy (caused e.g. by the squaring of matrices) cannot be tolerated.

.

where a conceptual framework, such as the geometric notion of oriented signal-to-signal ratio, may provide unrevealed addit.ional imight., $uch as in fact.or-analysis-like prob-1ems.

.

where robustness analysis, condit.ioning é\nd sensit.ivit.y opt.imizat.ion are crucial, and are linked t.oget.her with geomet.rical insight. and int.erpret.ation, for which t.he (G )SVD may provide meaningful quant.ificat.i(.u (roudifjplI m.tml",,!>:,.principle angles. .. .).

Reliable routines for the SVD have been worked out. by GOLUB ([8], [9]) in the sixties and are now available in many libraries like NAG and packages like MATLAB. These avoid the dangerous squaring of matrices. Moderately sized (order of magnitude 50 - 70) SVD's are

- -

---- --- ---

----now feasible on workstat.ions and PC's. Moreover the computat.ional burden may be reduced, since in many digital signal processing applications only part of t.he SVD informat.ion is needed or a priori information on the SVD can be incorporated. Moreover parallelization and vectorization research on systolic, hypercube, multiprocessor, or dedicated architectures may lead to efficient SVD sol vers for arbit.rary or structured matrices.

In t.his paper we first describe in section 2 the role of SVD in the design of digital signal processing systems in order to situate the overall picture to the reader. In section 3 and 4 the (generalized) singular value decomposition and its basic proper ties are described along with concepts like oriented energy and oriented signal-to-signal ratios. The usefulness of t.hese concepts in the design of digital signal processing algorithms is shown by describing signal separation principles, based on strength or relat.ive strength of signals in section 5. Some practical applications will further motivate the reader. It is concluded that. SVD is a useful concept and should be an effective toolOn the workbench of a designer of a digital signal processing system.

2

The

SVD

as a useful

tooI

in the

design

of digital

signal

processing

systems

Designers of digital signal processing systems for medical, teleconllllunicat.ion, aut.omat.ion, automotive, conSWller or fabrication applications, usually are given a variety of signals mea-sured with a certain precision (say 10%) and are requested to extract in real time useful information from these data. Because of the limited accuracy of the measurements, the data should be handled with care, while on the other hand the speed is rat her crucial because of the real time nature of the application. In the design process a number of tools, like FFT, are offered from applied mathematics, with which the designer can experiment in order to find a signal processing algorithm which meets the specifications. These tools can be used and evaluated at a high (software) level in the top-down design. Moreover , depending on t.he needs, these tools can be t.uned or adapted t.o t.he specific constraint.s of t.he applicat.ion, in order to obtain a cost-effective and reliable signal processing syst.em. FFT has been such a tooI in signal processing since t.he mid sevent.ies and SVD is expeded to become an additional one in the near future.

In recent years one can witness an enormous expansion of computat.ional po\\'er at, allievels: supercomputers, workstations, minicomputers and microcomputers, VLSI, et.c. In addit.ion the measurement power is increasing by cheap sensors, transducers and data acquisition equipment. However in practice it is much more difficult to increase the accuracy of the

- - - -- - - -

-

---measurements than to increase the volume of the measurements. Hence the need for methods and software which can extract more accurate information from the measured data is more acute even at the expense of Knops of computations. In this situation we believe that the singular value decomposition (SVD) is a very important tooI which allows to take profit of the expansion of the comput.ational power in order to improve the accuracy. SVD is weIl known in linear algebra for its solid numeri cal qualities. However it is an important concept in digital signal processing. On the other hand its widespread use is still hampered by the computational burden. Hence for SVD, like for FFT, special efficient algorithms (software), coprocessors and application spec.ific integrated circuits (ASIC's) (hardware) are needed.

3

SVD

and

oriented

energy

of a vector

sequence

In this section we will introduce the concept of oriented energy of a vector sequence and show that the SVD of the corresponding matrix constitutes the main tooI to compute extremal values for this concept.

Definition 1 : _{Oriented energy : ([6], [ï], [15])}

Consider a sequence of p-vectors {ai}, i = 1,..., q and arrange them as the

columns of a p Xq ma.trix A. Then Be [A] is called the energy of the vectorset in the direction of unit vector e E ~p and is defined as :

q

Be [A]

=

L

(et aï)2

=

liet AII2

i=1 (1)

with 11. 11 the Euclidean norm.

In words, the oriented energy of a vector sequence {ai} in some direction e is a sum of squared projections of the vectors ai onto direct ion e. More general, one can define the oriented energy of a vectorset in a subspace Q C ~p as

q

EQ [A]

=

L

IIFQ (a;)112

;=1 (2)

with FQ (ad the orthogonal projection of t.he vectorset onto subspace Q.

Figure 1 shows the polar plot of th~' '.'rient.'~r.l"I1-':'r:;""r" "..t. (Ir \ rl.or$ in t \\"0 dimensions. Such a polen plot is found hy drawing a vector with length Ee [A] in the direct.ion of a variabie unit vector e, and this for all possible directions e in the vectorspace ~2. The figure then shows in general a p-dimensional surface with p directions of extremal energy : one direction of maximal oriented energy, one of minimal energy and p - 2 directions with a saddle point.

-

---We will now present the basic SVD-theorem, for our purpose restricted to real matrices. Furt.hermore we preferred the "economical" factorization, where t.he dimensions are reduced as much as possible and therefore t.he pseudo-diagonal mat.rix is replaced by a square diagonal one.

Theorem 1 _{A l!.!.~nne-E~khart~Y O~~!L~he~~n~ ([9))} FOl' a.ny real p X q matrix A, there exists a rea.lfactorization

Ap,q

=

Up,p ~P.p V:,q (for p

<

q)

in which the matrix U is orthogonal (UUt

=

lp = UtU), the matrix V contains

p orthon01'mal columns (VtV

=

lp) and ~p.p is a real diagonal matrix with nonnegatit/e diagonal dements, called the singular values <Tiof the matrix A.

Proofs of the existence and uniqueness of the SVD of a matrix are found in [9] and references therein. Some properties, useful in the further outline of this contribution, are mentioned here without pro of.

.

The number of singular values, different from zero, equals the algebraic rank of the matrix A.

.

Any matrix A can be written as the sum of r decomposition)

rank( A) rank one matrices (dyadic

r

A

=

L

Ui <Tiv~ i=l

where Ui (v;) is the i-th column of U (V) in the SVD of A and <Tiis the i-th singular value.

(3)

. Norms of the matrix A can be computed knowing its singular values, since

p q

IIAII} =

LLalj

i=lj=l 2 2 <Tl+...+<Tr (4) IIAlb <Tl (5)

.

The best. rank I..~approximatioll of a mat.rix ..1p.q with known SVD is given by t.he part.ial dyadic decomposition k .1 - ~\1'<T'vt ..t"'-1/"- .: , J I , :=1 wifh /.- r ₍₆₎

then V Bp,q with ranktE}

=

I.~

min IIA - BII2 min IIA - Bil}

IIA - Akll2

=

<Tk+l

IIA - Akll}

=

<T~+l

+ ... + <T;

(7)

(8)

---

---

-

---

-.

Each Uj, coluJ1m of the matrix U in the SVD of A has the following property 'r:/x E !RP, with XtUi

IIxt AII2

IIxll2

0, (i=1,...,j-1)

< lIuj AW

=

(7] (9)

with (7j t.he j-th singular value of A.

From this last property we establish the link between the oriented energy concept and the SVD in the following theorem

Theorem 2 : _{!he relation between oriented energy and SVD :} Consider a. p X q rna.trix A with SVD as defined in Theorern 1. Then

(10) (11)

In words, the SVD of a p X q matrix A looks for directions in the vectorspace iRP, for which the oriented energy of A is extremal. For a more detailed analysis and more properties of the concept of oriented energy, we refer to [6], [i], [15], [17], [18], [20].

4

Oriented

signal-to-signal

ratio

and GSVD.

We can now define the oriented signal-to-signal ratio (5-5 ratio) of two vector sequences as the ratio of oriented energies, as follows

Definition 2 : _{Oriented signal-to-signal ratio: ([6], [i], [20])}

The oriented signal-to-signal ratio Re [.4.,B] f)f tUlf) H:ts of p-1Jectors {8i} a.nd {bj} , (i= 1,...,q and j = 1,...,k). in the diredif)lI ofl/.nit tIedOl' eE 3~P,is defined as :

Re[A,B] Ee[A]

Ee [BI

liet .-1112

ii~/.-BIP (12) Eu; [A] = lIuAlI2 = (7.

P P

'r:/ e =

L

7i Ui, Ee [A] =

L

71 (71

--

--A similar plot as Figure 1 can now be const.ructeu by urawing a vector with length Re[A, B] In the uirection of a variabie unit vector e, anu t.his for all possible directions e in the vect.orspace 1RP. In Figure 2 this is done in two dimcnsions (p

=

2). Figure 2.a shows the oriented energy polar plot of two vect.orsets {ad and {bd, while Figure 2.b is the polar plot for t.he orient.ed S-S ratio in aU directions. No\.e t.hat. t.here are again 2 directions (in general p) of extremal S-S ratio, but. in contrast wit.h the oriented energy, these directions are not orthonormal in this case. Figure 3 shows the S-S ratio polar plots that arise for different configurations of the oriented energies of matrices A and B.

A very important property of the S-S ratio is its invariance through a basis transformation in 1RP,characterized by the non-singular p X P matrix T ([6], [7]). Indeed, one can verify that

Re [A,B] = Re' [TA, TB]

T-te

,

-for e = IIT-tell

(13)

At. this point we present the basic Generalized SVD theorem, again in the most "economical" notations, as foUows

Theorem 3 : The Generalized SVD : _([9])

Let A be a p x q and B a p X k mal1'ix (p < q and p < k) , then there exist matrices UA (q xp) and UB (k xp), both with p o1,thono1'nwl columns, and a. non-singular

p XP matrix X such that

A B

X-t DA U~

X-t DB u1

(14)

where DA

= diag(al,...,ap)

and DB = diag(pl>"",Bp),

square diagonal p X P matrices and

~

2':

~

2':... 2':

~

'

(ai,,Bi 2': 0) , are r = rank(B).

Proof see [9]. The elements ofthe set <T(A,B) = (~,..., ~) are referred to as the generalized singular values of A and B. Each Xi, colunm of t.hp matrix X in the GSVD of (A, B) has the foUowing interesting property

v y E ~P,

lIyt AII2 Ily BII2 <: IIx~ AII2 . IIx~ Bllz

.,

a'"

_!... 1321 (15)

-

---

-

---In analogy with thc relation between the oriented energy concept and the SVD, a link bet.ween t.he orienl.ed signal-t.o-signal ral.io and t.he Gencralized SVD (GSVD) exists : TheorelTI 4 : Thc relati<>.n b.etw:e.cn 9.ri~nted_S-S r_~ti9.<!-mLGSy'Q :

Considel' a p X q mat1'ix A and a. p X k matrix B with GSVD as defined in Theo-rem 3. Then

Re [A,B] =

(;:r

(16)

Xi

for e

=

IIxill

where Xi is the i-th column of ma.t1'ix X in the GSVD of (A, B).

Proof: _{Define t.he linear transformation T = Di/ Xt , then, due to the invarianee}

property of the S-S ratio

Re [A,B] Re' [TA, TB] Re' [DËlDAU~,U1] Eet [DËl DAU~] (17) for e' ___U.__hT-t e___ IIT-t el! DB X-I_{--.---.---..-.}e

IIDB X-I ell The matrix G _{DËl DA U~ can be faetorized as follows}

sueh that

)

2 a. 2 !. Eue; [G]

=

(J"ei =

(

t3i (18) due to the relation between oriented energy and SVD of G, In other words, Eet [G] is extremal for e' = uei, such that Re [A, B] is extremal for e = libl = ni:lI'

o

The GSVD thus provides the extrcma of the oriented S-S rat.io aud also t.he directions in which those extrcma occur : tllI.'se are t.h(? r()hIl1111~of th" I1l<J.t.rÎxX. \,"hirh need not. to Iw ort.honorIuûL

C = Ue e vet with Ue

=

lp e

=

DËl DA Ve

=

UA

---

---5

Signal

separation

principles

Of ten, in multicharUlel signal processing a set. of measured, recorded, acquired, capt.ured, . . . signals is given as a linear combinat.ioll of some original signals called "source" signais, and corrupted by additive noise. The model of such a measured signal nti(t) is then

r

nti(t)

=

L

i.ij sAt)

+

ni(t)

j=l

with p the number of measured signals and r the number of source signals involved. The coefficients tij in the model are called transfer coefficients, since they essentially depend on the signal transfer from the source, that generates the signal, to measurement point., that picks up the signa!.

for i

= 1,...,p

(19)

For digital signal processing applications it is required that aU the signals are sampled, a process by wmch a continuous-time signal is transformed into a series of digital numbers. Ir we consider p signals over a certain time-interval with q samples, the model can be written in matrix notation as follows

Mp,q = Tp,r Sr,q + Np,q with r

<

p ~ q (20) where the p rows of the datamatrix !vI contain the q samples of the p recorded signaIs. In this equation, only M is known and it is the goal of signal separation to obtain an estimate of the original "souree" signals in matrix S.

5.1 SVD of the datamatrix ]v!

Suppose that we partition the r souree signals into two groups : ra desired souree signals and ru undesired source signals (r = ru + ra). We can then write for matrix Sr,q

q Sr,q =

(

Su

)

ru Sa _I rd (21)

and partition the r COIUlIUlSof the transfer mat.rix T siruilarly )." rd

Tp.r = (Til Ta) P (22)

These grouped colullms of T form s11hspafPs OIr f 11" I'.d.ill1"I1Si0l1<\1 folu11111 span' of A-Ip,q.

Suppose that the SVD of Mp,q

ru

M = (UI

can he partitioned as follo\\'s (Pt>= P - "u - I'a)

rd p.

C

0

n

(m

ru U2 _U3

)

_u L:d rd (23) 0 _Po

---Recent work ((17], [18], (19], (21], [22]) showed that :

.

t.hc st.rongcr t.he undesired signals are present. in the recorded signals and

.

t.he more orthogonal t.he subspaces Tu and Td of the transfer matrix T are,

the better the subspaces described by UI and U2 match with Tu and Td respectively. These

t.wo conditions can be met by recording t.he signals mi(t) as follows (PI + P2

=

p)

.

PI signals are recorded as a mixture of desired and undesired signals (pi

~

Td)

.

P2 signals are recorded such thaI. they piek up a combination of undesired signals only (P2

~

Tu)

Example 1 : Fetal ECG extract ion _{([1], [4], [17], [18], [19])}

We will illustrate this signal separation principle for the separation of the cutaneously recorded maternal and fetal electrocardiogram. It has been verified (and there exists a phys-ieal explanation (14]) that sufficiently far from the adult heart, its electrical activity can be described by three independent signals only. In the case of a pregnant woman, the obstet.ri-cian is interest.ed in the fetal electrocardiogram (FECG), recorded by only placing electrodes on the maternal skin. The main problem is however the very strong and undesired maternal electrocardiogram (MECG) that is omnipresent in these recordings. Suppose that we apply our signal separation principle to this biomedical application and we assume that the number of undesired signa! Tu = 3 (MECG), the number of desired signals Td

=

3 (FECG) (some experiments show t.hat it can be less than three, but this is a safe upper bound) and we suppose for simplicity that there are no other source signa!s involved.

We now record PI

=

3 signals on the maternal abdomen, close to the feta! heart, such that. they contain a mixture of maternal and fetal ECG (see Figure 4, the last three signais).

Furthermore we record P2

=

3 signa!s that piek up MECG only by placing electrodes on

the arms and thorax of the mother , far from the fetal heart (see Figure 4, the first three signa!s). By construct.ing the matrix 1\1 in this wa)', the SVD finds 6 orthonormal directions of extremal oriented energy. Thc three directions, associated with the t.hree largest singular values, correspond 1.0a subspace UM thaI. is a good estimate of t.he subspace Tu (the maternal subspace). The two following ones correspond 1..0a subspaCt~ [TF, thaI.. is a good est.imate of the fetal subspace and the last direcf.ioll ("('rrp~p'-)n"" lp ;\ ~lIh"l'a("f'>,-,t"dilllel1~il)J\ J fnr ot.her much weaker source signais. Project.ion of t.he dat.amat.rix M 'Jllto t.hese ort.honormal directions t.hen resuIts in the signals of Figurc 5. These signals are orthogonal estimates fOf the original source signals in S.

- -

---5.2 GSVD of seleded data intervals

In some signal processing applications it. occurs that, over a specific time interval, the unde-sired signals contribute considerably more in the measured signals than the desired ones (e.g. for t.he pulse-like ECG signals the maternal QRS-peaks often appear in between t.wo fetal QRS-peaks (see Figure 6)). In t.hat. case we can construct from t.he datamat.rix !vI a matrix Ap,k as a sequence of such intervals. This matrix contains in the p-dimensional colunm space of M mainly contributions from the undesired signals only.

From section 4 we know that the GSVD of the matrix pair (M, A) finds non-orthogonal directions Xi of extremal oriented signal (in M)-to-signal (in A) ratio. Let

~

he the gener-alized singula,r values of M alld A, arranged in non-increasing order. Then the colUlIUl Xi of X in the GSVD of (M, A) is a vector for which the oriented energy of matrix M is ~; times larger than the oriented energy of matrix A. The project ion of the datamatrix M onto the directions Xi, that correspond with the generalized singular values ~ 1, then results in the desired signais.

Example 2 : Fetal ECG extract ion [4]

Figure 6 shows 5 abdominally recorded potential signais, containing both maternal and fetal EeG. After arranging the signals in a p Xq datamatrix M, a matrix MM is composed

as a sequence of several maternal QRS-intervals, not coinciding with fetal complexes (some of those intervals are indicated on Figure 6). The two.largest generalized singular values are

al

=

9.986

{31

and a2

=

5.342 {32

and project ion onto the corresponding columns Xl and X2 of X results in the two MECG-free fetal heart signals of Figure ï.

Exalnple 3 : Speech enhancenl.ent

At the moment we are doing tests on speech signaIs, recorded using multiple microphone systems, and corrupted by stationary noise (e.g. a ventilator. a motor running at. constant rotation speed, et.c.). From the rerorded speech ~ignah in dat.amat.rix 1U. somt' int.ervals are selected, where speech is absent anel only I:hl.' 11'"i:...' i:, pft::;'~lIf. These inl.en'als arl' arranged in a special matrix Band again t.he GSVD is comput.ed of t.he mat.rix pair (M, B). Up till now we found an enhancement of t.he speech-to-noise ratio of about a factor 2.

- ---

- - - -

-6

ConclusÎon

In this contribution the use of the (Gencralized) Singular Value Decomposition for signal processing and specifically for signal separation purposes is advocatedo The SVD and GSVD are not only excellent tools in forulUlating and describing new geOluetrical concepts (oriented encrgy, oriented signal-to-signal ratio), they are also extremely useful in efficiently computing reliahle and elegant solutions for many signal processing problemso Due to the increased computational capabilities of modern computers and the development of dedicated algoritlulls with incorporation of on-line, adaptive ([2], [3]) and parallel techniques, the computational burden can be reduced substantiallyo Therefore the SVD should be an effective tooI on the workhench of a designer of digital signal processing systemso

References

[1] CALLAERTS, Do, VANDERSCHOOT, Jo, VANDEWALLE, Jo, SANSEN, Wo, VANTRAPPEN, G., JANSSENS, Jo, "An adaptive on-line method for the extraction of the complete fetal electrocardiogram from cutaneous multilead recordings", Jov.rnal

of Perinatal Medicine, vol. 14, no. 6, pp. 421-433, 1986.

[2] CALLAERTS, Do, VANDERSCHOOT, Jo, VANDEWALLE, J., SANSEN, Wo, "An on-line adaptive algorithm for signal processing using SVD", in Signal Processing lIl, Am-sterdam, The Netherlands : Elsevier Science Pub't (North Holland), EURASIP 86, The Hague, pp. 953-956, Sept. 19860

[3] CALLAERTS, Do, VANDEWALLE, Jo, SANSEN, Wo, MOONEN, Mo, " On-line AI-gorithm for Signal Separation based on SVD", in SVD and Signal Processing: Algo-rithms, Applications and Architectures, Deprettere, E., Editor, North-Holland, ppo 269-2ï6, Septo 198ïo

[4] CALLAERTS, D., DE MOOR, B., VANDEWALLE, .J., SANSEN, W., "Comparison of SVD-methods to extract. the fetal clectrocardiogram from cutaneous electrode record-ings", Submitted to Medical & Bioogical Eng. & Comp., 1989.

[5] DAMEN, A.A.. VAN DER KAlvL .1.. "Th<' 'I~" nf fit.. ~inJ;'ll"r \Ïall.le Df>("IlI\lposition in Electrocardiography", 11ff:d. Biol. Eng. Comput., \'01. 20, pp. 4;3-482, July 19820 [6] DE MOOR, Bo,VANDEWALLE, J., STAAR, Jo, "Oriented Energy and Oriented

- - ---- - -

--

-and Signa! Processing: A!g01'ithms, App!ications and Architectures, Deprettere, E., Ed-itor, North-Hollancl, Pl'. 209-232, Sept. 1987.

[71 DE MOOR, B., "Mathematical Concepts and Techniques for Modelling of Statie and Dynamic Systems", PhD. thesis, Dept. of Elec. Eng., Katholieke Universiteit Leuven, Belgium, June 1988.

[81 GOLUB, G.H., REINSCH, C., "Singular Value Decomposition and Least Squares Solu-tions", Numel'. 1v1ath., Vol. 14, pp. 403-430, 1970.

[91 GOLUB, G.H., VAN LOAN, C.F., "Matrix Computations", North Oxford Aeademy, 1983.

[10] HANSEN, P.C., NIELSEN, H.B., "Singular Value Deeomposition of images", Proc. of

3-th Scand. Conf. on Image Analysis, pp. 301-307, Copenhagen, 1983.

[lIl KLEMA, V.C., LAUB, A.J., "The singular value decompostion : its eomputation and some. applications", IEEE Trans. Automatic Control, Vol. AC-25, no. 2, Pl'. 164-167, 1980.

[12] LAWSON, C.L., HANSON, R.J., "Solving least squares problems", Englewood Cliffs : Prentice Hall Series, 1974.

[13] PARLETT, B.N., "The SYlmnetric Eigenvalue Problem", Englewood Cliffs : Prentiee Hall Series, 1980.

[14] PLONSEY, R., "Bioelectric Phenomena", New York : Me Graw-Hill, 1969.

[15] STAAR, J., "Coneepts for reliable modelling of linear systems with applieations to on-line identifieation of multivariable state spaee deseriptions", PhD. thesis, Dept. of Elee. Eng., Katholieke Universiteit Leuven, Belgium, 1982.

[16] STAAR, J., VANDEWALLE, J., "Singular Value Deeomposition : A reliable tooI in the algorithmie analysis of systems", JQurnal .4, Vol. 2:3, Pl'. 69-74, 1982.

[17] VANDERSCHOOT, J., VANDEWALLE, J., JANSSENS, J., SANSEN, W., VANTRAP-PEN , G., "Extraction of weak bioelectrieal signals by means of Singular Value Decompo-sition", in Analysis a.nd Oplimi:;alion of Systcms. Lecture Not.es in Control and Informa-tion Sciences 6:3, Bt'nS0ussan. A.. LiollS. ..r..L. ~(l:'. :~I'l."i!l=;'!r\\'rlag, Bedilt. Pr>. 434-448, 1984.

[18] VANDERSCHOOT, J., VANTRAPPEN, G., JANSSENS, J., VANDEWALLE, J., SANSEN, W., "A reliable method for fetal ECG extract.ion from abdominal reeordings",

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-

--in Jvledicallnfo1'matics Eumpe 84, Lecture Notes in Medical Informaties 24, Roger, F.H. et al, Eds., Springer Verlag, Berlin, pp. 249-254, 1984.

119] VANDERSCHOOT, J., CALLAERTS, D., SANSEN, W., VANDEWALLE, J., VANTR.APPEN, G., JANSSENS, J., "Two Methods for Optima] MECG Elimination and FECG Detection from Skin Electrode Signals ", IEEE Trans. Biomed. Eng., vol. BME-34, No. 3, pp. 233-243, March 1987.

120] VANDEWALLE, J., DE MOOR, B., "A variety of applications of Singular Value Decomposition in Identification and Signal Processing", in SVD and Signal Process-ing: Algorithms, Applications and Architectu1'es, Deprettere, E., Editor, North-Holland, Sept. 1987.

121] VANDEWALLE, J., VANDERSCHOOT, J., DE MOOR, B., "Source separation by adaptive singular value decomposition" , P~oc. IEEE ISCAS Conf., Kyoto 5-7 June 1985, pp. 1351-1354, 1985.

[22] VANDEWALLE, J., VANDERSCHOOT, J., DE MOOR, B., "Filteri~g of vector signals based on the singular value decomposition", nh Eu.ropean Conf. on Circuit Theory and

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-Figure captions

Figure 1: Polar plot. of t.hc orient.ed cnergy of a vector sequencc in t.wo dimensions.

Figure 2: (a). Polar plot of the oriented energy of two vector sequences in two dimensions. (b). Polar plot of the corresponding oriented signal-t.o-signal ratio in two dimensions.

Figure 3: Polar plot of the oriented signal-to-signal ratio for several configurations of the oriented energies of two vectorsets.

Figure 4: Six-channel rncasurernent set of 6 seconds long: the first three signals rnainly contain signals from the rnaternal heart only, while the last. three signals contain a mixture of maternal and fet.al heart signals.

Figure 5: The six signals t.hat result. from a projection of the data of Figure 4 onto the U-matrix in t.he SVD of M. The fourth and fifth signals arc fet.al souree signais.

Figure 6: A set of 5 recorded abdominal signals (interval of 6 seconds) cont.aining a nûxture of MECG and FECG.

Figurc ï: The two MECG-free signals obt.ained wit.h t.he GSVD-based signal separation met.hod as a project ion of the data signals ont.o the directions Xl and X2 respectively.

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