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Singular value analysis of predictor matrices
Bazán, Fermin S.V; Toint, Philippe
Published in:
Mechanical Systems and Signal Processing
DOI:
10.1006/mssp.2000.1370
Publication date:
2001
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Early version, also known as pre-print
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Citation for pulished version (HARVARD):
Bazán, FSV & Toint, P 2001, 'Singular value analysis of predictor matrices', Mechanical Systems and Signal
Processing, vol. 15, no. 4, pp. 667-683. https://doi.org/10.1006/mssp.2000.1370
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Federal University of Santa Catarina
Florianopolis - SC - Brasil
http://www.mtm.ufs .br
Singular Value Analysis of Predi tor Matri es
by F.S.V.Bazan
1
andPh. L. Toint
2
Revised September22,2000
To appearinMe hani alSystems andSignal Pro essing
1
Department of Mathemati s,
FederalUniversityof Santa Catarina,
Florianopolis, Santa Catarina88040-900, Brasil.
Email: ferminmtm.ufs .br
2
Department of Mathemati s,
Fa ultes UniversitairesNDde laPaix,
61,ruede Bruxelles,B-5000 Namur,Belgium.
F. S.V. Bazan
Ph. L. Toint
Revised September22,2000
Abstra t
Predi tor matri es arise in problems of s ien e and engineering where oneis interestedin
predi ting future information from previous ones using linear models. The solution of su h
problemsdependsonana urateestimateofapartofthespe trum(thesignal eigenvalues) of
thesematri es. Inthispaper,singularvaluesofpredi tormatri esareanalyzedandformulaefor
their omputation are derived. Byapplying awell-knowneigenvalue-singularvalueinequality
toourresults,wededu elowerandupperbounds onthemodulusofsignaleigenvalues. These
bounds depend on thedimensionof the problem and allowus to showthat themagnitude of
signaleigenvaluesisrelativelyinsensitivetosmallperturbationsinthedata,providedthesignal
isslightlydampedandthedimensionoftheproblemislargeenough. Thetheoryisillustratedby
numeri alexamplesin ludingtheanalysisofasignalarisingfromexperimentalmeasurements.
Key words. Singularvalues, eigenvalues,linear predi tion,timeseries,exponentialmodeling
1 Introdu tion
Predi tormatri esoftenariseinanumberofareassu hasmodalanalysis,spee hpro essing,system
identi ation, et , where predi tion of future from previous information, is of primary interest.
In many ases, this predi tion is omputed using linear models. The nature of the information
itself depends on the parti ular appli ation under study and often has the form of dis rete time
series, ommonly knowninengineering asdis rete time signals. Morepre isely,given a setof real
or omplex-valued observations h ` ;h `+1 ;:::;h `+N 1
, a linear predi tion model assumes that the
future value h
`+N
hastheform
f 1 h ` + f 2 h `+1 ++ f N h `N 1 =h `+N ; `0: (1.1)
DepartamentodeMatemati a,UniversidadeFederaldeSantaCatarina,Florianopolis,SantaCatarina88040-900,
In this formulation, N is the order of the model and the f's, are omplex oeÆ ients known as
predi tor parameters. These oeÆ ientsre e tintrinsi propertiesofthesignalsu hasfrequen ies,
planewaves,dampingfa tors, et ,whoseestimationfroma nitedatasetfh
k g
L
k=0
,isanimportant
problemin areas su has system identi ation and spe tral estimation, among others. The linear
predi tion model is also urrently used in other areas su h as e onomy and zoology; see [21 ℄ and
[6 ℄ for details. In this work, we fo uson those appli ations where the data are assumed to be of
theform h k = n X j=1 r j (e s j t ) k = n X j=1 r j z k j ; k =0;1;:::; where r j ;s j 2C,I s j 6=s k for j 6=k, s j = j +i! j ; { = p 1, and j 0. In these appli ations
theproblemisto estimatetheparametersr
j ,s
j
and thenumbern(thesignalparameters), froma
nitesamplingoftheobservedsignal. Wealwaysassume!
j
t<. The lassi alapproa hforthe
problemis relativelysimple in thenoiseless ase: the parameters s
j
are extra tedfrom theroots
of theso- alledforward predi tor polynomial
P f (t)= f 1 + f 2 t++ f N t N 1 t N ; (1.2) whose oeÆ ients f j
areestimated bysolvingtheset oflinear predi tionequations
2 6 6 6 4 h ` h `+1 h `+N 1 h `+1 h `+2 h `+N . . . . . . . . . h `+M 1 h `+M h `+M+N 2 3 7 7 7 5 2 6 6 6 4 f 1 f 2 . . . f N 3 7 7 7 5 = 2 6 6 6 4 h `+N h `+N+1 . . . h `+M+N 1 3 7 7 7 5 ; (1.3)
where we assume M N n, and n is the rank of the oeÆ ient matrix. The underlyingidea
behindthisisthatnzeros ofP
f
(t),knownassignalzeros,areoftheformz
j
=e
sjt
,aresultearly
proven by R.de Prony for the ase M = N = n. For details of Prony'smethod, see Se tion 9.4
in Hildebrand [11 ℄. Appli ations of Prony's method are en ountered in number of areas su h as
a ousti s, ele tromagneti s,and stru tural dynami s,among others,see, e.g., Magda, Straussand
Wei [20 ℄, Braun and Ram [7 , 8℄, Kumaresan [16 ℄, and Kurka [19 ℄. A new theoreti al approa h
for Prony's method is des ribed in Wei and Majda [25 ℄. On e the signal zeros are available, the
problemofestimatingtheparameters r
j
issimpleand we donot ommentanyfurtheraboutthis.
One ould alsouse thelinearpredi tion equationsinthereverse order,repla ing(1.1) by
b 1 h ` + b 2 h `+1 ++ b N h `N 1 =h ` 1 ; `1: (1.4)
Then theparameters s
j
are extra tedfrom thezeros of theba kward predi tor polynomials
P b (t)= b N + b N 1 t++ b 1 t N 1 t N ; (1.5)
whose oeÆ ients are estimated as above. In this ase, the signal zeros are z j
= e
j
[16 ℄.
However, asonlythesignal zeros areofinterest,one is fa edwiththeproblem ofseparatingthem
from the other N n zeros, alled extraneous zeros, whi h appear as a onsequen e of hoosing
N >nsin enisnotknowninadvan e. ThisseparationturnsouttobediÆ ultsin etheextraneous
zerosdependonhowone hoosesthe oeÆ ients
f
j
fromthein nitelymanysolutionsofthesystem
(1.3). Further detailsabout the zeros of predi tor polynomials an be found in Kumaresan [16 ℄,
and also inCybenko [9 ℄, wherethe problemis examined inthe framework of in nite dimensional
Hilbert spa es. Amore re ent approa h, wheresignal zeros areviewed aseigenvaluesof predi tor
matri es(thesignaleigenvalues), anbefoundinBazanandBezerra[2 ℄andBezerraandBazan[4 ℄.
The problem, however, be omesdiÆ ultwhen the dataare orrupted bynoise, sin eboththe
rank nandthe parameters
f
j
mustbeestimated froma linearpredi tionsystemwhose oeÆ ient
matrix is generally of full rank, though this an be often ir umvented by taking N n, see,
e.g., [17 ℄, [16℄, [22 ℄ and [23℄. But, as polynomial root- nding methods are time onsuming, new
approa hesbased on estimates of theso- alled signal spa es (the rowor olumn spa e ofthe data
matrix) are a tually preferred. In these te hniques, the signal zeros emerge as eigenvalues of a
smallnnmatrix. Methodsinthis ategory in ludeKung'smethod[18 ℄,EigensystemRealization
Algorithm (ERA) of Juang and Pappa [14℄, HTLS of Van Hu el, Chen, De anniere and Van
He ke [24 ℄,OPIAofBazanand Bavastri[1 ℄,andthematrixpen ilmethodofHuaand Sarkar[13 ℄,
among others. Many referen es about both polynomial and subspa e approa hes an be found
in [23 ℄. However, despite the bursting a tivity in new approa hes, little is known about signal
eigenvaluesensitivity,an intrinsi omponent of theproblem.
The goal ofthispaperis to perform asingularvalueanalysis of predi tormatri es,theresults
of whi h provide insight into the sensitivity of these eigenvalues. Our analysis relies on the fa t
that, sin etheeigenvalues
j
ofanymatrixA2CI
NN
satis es theinequalities
N j j j 1 j =1;2;;N; (1.6)
(seeGolubandVanLoan[10 ℄,page318),where
N
and
1
denotethesmallestandlargestsingular
value of A, then reliableinformationabout signal eigenvalue sensitivity an be drawn from (1.6),
provided these singular values are available. We provide analyti formulae for all singularvalues
of a lass of predi tor matri es and analyze the asymptoti behavior of the bounds (1.6) for N
large. Asa onsequen e, we showthat themagnitudeof thesignal eigenvaluesbe omes relatively
insensitiveto smallperturbationsonthedata, providedmild onditionsaresatis ed.
We rstanalyzeinSe tion2thelo alizationofsignaleigenvaluesextra tedfromthespe trumof
predi tormatri es. The mainresultsarepresentedinSe tion3wherewegivean exa tdes ription
andthatitasymptoti allybe omessmallprovidedthesignalisslightlydamped. Finally,Se tion4
presentssome numeri alresultswhi hillustrate ourtheoreti al analysis.
2 Predi tor matri es and basi results
Predi tor matri es are de ned as follows in the noiseless ase. Let H(`) be the M N Hankel
matrixof thesystem(1.3). We saythat theNN matrixF isa forward predi tor matrix if
H(`+1)=H(`)F; 8`0: (2.1)
Similarly,B is aba kward predi tor matrix iffor`1,itsatis es
H(` 1) =H(`)B: (2.2)
Noti ethat H(`) an be fa toredas
H(`)=VZ ` R W; (2.3) where Z =diag (z 1 ;:::;z n ); R =diag (r 1 ;:::;r n
), V is an M n Vandermonde matrix with z
j 1
k
asits(j;k) entryand W thetransposeofthematrixthat onsistsoftheN rstrows ofV. Hen e,
we have that (2.3) is a full-rank fa torization of H(`) and thatfor all `0, rank[H(`)℄=n. The
olumnspa eof H(`),denotedbyR(H(`)), isspannedbythe olumnsofmatrixV,whiletherow
spa e ofH(`),R(H(`)
),isspannedbythe olumnsofW
. Herethesymbol
standsfor omplex
onjugate transpose. From (2.3) also follows thatN(H(`)), the nullspa e of H(`),is equalto the
nullspa e of W,and that therefore
R(H(`)
)=[N(W)℄
?
: (2.4)
Furthermore, ifP denotestheorthogonal proje toronto R(H(`)
),then we havethat
P =H(l) y H(l)=W y W =W W y ; (2.5) where y
stands forthe Moore-Penrose pseudo-inverse. The reader is referred to [5℄ for detailson
proje tionsandpseudo-inverses. Wenowobservethat(2.1)and(2.2)havein nitelymanysolutions
be auseH(`) is rank de ient. The solutionsset of(2.1), S
F ,is S F =fF jF = b F +(I P)X; X2R NN g; (2.6) where b F = H(l) y H(l+1). Similarly, if we set b B = H(l) y
H(l 1) , then the set of ba kward
predi tormatri es is S B =fBjB = b B+(I P)X; X2R NN g: (2.7)
matrixfor, ifone substitutesF in(2.1), one hasthat,
WF =ZW; (2.8)
whi h showsthat therowsof W arelefteigenve tors ofF orrespondingto thesignaleigenvalues.
In thesame way,(2.2) impliesthat
WB =Z
1
W; (2.9)
and thustherows ofW areleft eigenve torsofB asso iatedwiththeeigenvaluesz
1
j
. However, if
signal eigenvaluesare independent of whi h predi tormatrixis hosen in the lass, thisis notthe
asefortheextraN neigenvalues(theextraneouseigenvalues). Itturnsoutthataanalysisofthe
lo alization ofthese extraneouseigenvaluesispossible,providedwe restri tourselvesto asuitable
lass of predi tor matri es. We fo us here on two interesting hoi es: the matrix
b
F (or
b
B) and
minimalnormpredi tormatri es with ompanion stru ture. This last lass overs,ifpredi tion is
arried outintheforwarddire tion,predi tormatri es oftheform
F =[e 2 e 3 e N f℄; where e j
isthe j-th ve tor of the anoni al basis and thelast olumnve tor, f =[f
1 f 2 f N ℄ T ,
isthe solutionofthelinear system
H(`)f =H(`+1)e
N
: (2.10)
of minimal2-norm. If predi tionis arried out inthe reverse order instead, ba kward ompanion
predi tormatri es havethe form
B =[be 1 e 2 e N 1 ℄;
wherethe rst olumn ve tor,b=[b
1 b 2 b N ℄ T
,isthe solutionofthesystem
H(`)b=H(` 1)e
1
; `1 (2.11)
of minimal 2-norm. The following result gives information about the eigenvalues of the above
predi tormatri es.
Theorem 1 Let
b
F and
b
B be as in (2.6) and (2.7), respe tively, and let F and B be the forward
and ba kward minimal norm ompanion predi tor matri es. Then,provided N >n,we havethat
(b) j k
()j<1; k =1;:::N n where() denotes an extraneous eigenvalue and () any of the
matri es b F,
b
B, F or B.
Proof. We note that to prove (a) for F, it suÆ es to prove that e
1
f 6= 0: Observe that f 2
R(H(l)
). Wethenverifythate
1
doesnotbelongtoeitherN(W)ortoitsorthogonal omplement.
The rst of these two laims follows from the fa t that We
1
= e, where e is the ve tor inCI
n of
all ones. To see the se ond, we onsider the system W
x = e
1
. If we sele t n equations of this
system startingfrom these ond one, we obtaina square nonsingularhomogeneous system whose
unique solution is x = 0. However, this is in ontradi tion with the rst equation, whi h shows
that the system is in ompatible. Thus e
1 =
2 R(H(l)
), whi h ensures that e
1
f 6= 0, as laimed.
One an similarly he k that B is nonsingular,and thuspart (a)of thetheorem holds. Theproof
of (b) involving ompanion matri es an be found in [2 ℄. We now prove that (b) for
b
F and
b B.
In order to see that the extraneous eigenvalues of
b
F fall inside the unit ir le, noti e that, as
b
F = H(l)
y
H(l+1) = W
y
ZW, it is immediate to see that (
b F) = fz 1 ;:::;z n g[f0g (see Horn
and Johnson[12℄,Theorem1.3.20). Asimilarargument anbeappliedfor
b
B,whi h on ludesthe
proof.
As thistheorem des ribes ompletely thelo ations ofall eigenvaluesof thepredi tor matri es
b F,
b
B, F and B, what remains to do is to determine their singular values. We rst start with a
te hni allemmathatallowsusto omputethesingularspe trumof ompanion predi tormatri es.
Thedeterminationof thesingularspe trumof
b
B and
b
F isslightlymore involvedand ispostponed
to thenext se tion.
Lemma 2 Let u 1 , u 2 , v 1 and v 2 be ve tors inCI N
, N > 2, su h that at least one of the inner
produ ts v 1 u 1 or v 2 u 2
is di erent of 1. Suppose that the rank-two modi ation of the identity
given by I+u 1 v 1 +u 2 v 2
isnonsingular. Then we have that,
det(I+u 1 v 1 +u 2 v 2 )=1+v 1 u 1 +v 2 u 2 +v 1 u 1 v 2 u 2 v 2 u 1 v 1 u 2 ; (2.12)
and that, the asso iated hara teristi polynomial is
p()=(1 ) N 2 [ 2 (2+v 1 u 1 +v 2 u 2 )+1+v 1 u 1 +v 2 u 2 +v 2 u 2 v 1 u 1 v 2 u 1 v 1 u 2 ℄: (2.13)
Proof. We assume, without loss of generality, that v
1 u
1
6= 1. It then follows that I +u
1 v
1 is
nonsingular sin e det(I +u
1 v 1 ) = 1+v 1 u 1
6= 0. Hen e, using properties of the determinant, we
have that det(I+u 1 v 1 +u 2 v 2 ) = det(I+u 1 v 1 )det(I+(I+u 1 v 1 ) 1 u 2 v 2 )) = (1+v 1 u 1 )(1+v 2 (I+u 1 v 1 ) 1 u 2 );
right-handside. Ontheother hand,given that p()=det (I +u 1 v 1 +u 2 v 2
I)=det((1 )I+u
1 v 1 +u 2 v 2 );
sin ep(1)=0 andN >2, wehave that =1 isan eigenvalueof I+u
1 v 1 +u 2 v 2 . If 6=1, p()=(1 ) N det (I +(1 ) 1 u 1 v 1 +(1 ) 1 u 2 v 2 );
and these ond partof thelemma isthen obtainedby applying(2.12) inthisequation.
Thusit suÆ es to extra t the eigenvalues asso iatedwith a quadrati polynomial in(2.13) to
obtain the eigenvalues of the perturbed matrix, sin e the remaining ones are equal to one. We
illustrate this by onsidering the problem of omputing the singular spe trum of the ba kward
ompanion matrix B introdu ed above. In fa t, sin e the singular values of B an be omputed
from theeigenvalues ofBB
,we observe that BB = [be 1 ::: e N 1 ℄ 2 6 6 6 4 b e 1 . . . e N 1 3 7 7 7 5 =bb +e 1 e 1 ++e N 1 e N 1 ;
whereb istheminimum2-normsolutionof 2.11, and hen ethat
BB =I+bb e N e N : ByapplyingLemma2toBB ,withu 1 =v 1 =bandu 2 = v 2 =e N
;we ndthatthe hara teristi
polynomial of BB is p() = (1 ) N 2 [ 2 (1+kbk 2 )+4jb e N j 2
℄. Hen e, we have that the
singularspe trumof B,(B),is ofthe form
(B)=f 1 (B);1;1;:::;1; N (B)g; where 2 1 (B); 2 N (B)= kbk 2 +1 p (kbk 2 +1) 2 4je N bj 2 ) 2 : (2.14)
This result is notnew, (see forinstan e [15 ℄), butthe authors are notaware of a proof along the
linesdevelopedhere. We nowuseittoobtainanimportanteigenvaluebound. Sin eforea hsignal
eigenvaluewehave that,
N (B)jj 1 (B) s kbk 2 +1+ p (kbk 2 +1) 2 4je N bj 2 ) 2 p 1+kbk 2 ; (2.15)
an interesting upperbound an be immediatelyderived providedkbk is smallenough(remember
inthis asejj1). Inour ontext,asshownin[3 ℄,itisfortunatethatkbk
2
0inmanypra ti al
appli ations, provided the dimension of the problem is suÆ iently large. If thisis true, then the
form of the upperboundindi ates that it ould be rather tight. Hen e, thispreliminaryanalysis
suggeststhatreliablebounds ouldbeobtainedprovidedtheyonlydependonquantitiessimilarto
theright-hand sideof (2.15). Unfortunately,no lowerboundof interest an beobtained from the
left inequalitybe ause
N
(B)0, whi hmotivatesour sear hfora better lowerbound.
3 Signal eigenvalue bounds
Inspite ofthepromisingqualityofthe above upperbound, thelinkof thesignaleigenvalueswith
the solution of a \large eigenvalue problem" seems to generate a new in onvenien e, in that it
appears to require that the predi tion matrix is suÆ iently large. In this se tion, we shall show
thatthis an be ir umventedprovidedsignaleigenvalue boundsare derived byusingthesingular
spe trum of predi tormatri esobtained via orthogonal proje tions. We saythat thenn matrix
F P
,is aforwardpredi tormatrix obtainedbyorthogonal proje tionif, itis oftheform
F P =V 1 FV 1 ; (3.16) whereV 1
denotesanyNnmatrixwithorthonormal olumnsthatspanR(H(`)
). Themotivation
for this de nition relies on the fa t that the spe trum of F
P
, (F
P
), only ontains the n signal
eigenvalues,sin e (F
P
)=(PF)=(
b
F) when zeroeigenvaluesare dis arded. Similarly,B
P isa
ba kwardpredi tormatrix obtainedbyorthogonalproje tionifit isof theform
B P =V 1 BV 1 : (3.17) The spe trum of B P
then onsists of the re ipro al of the signal eigenvalues. Thus, if eigenvalue
boundsarederivedfromthesingularspe trumof thesematri es,thesignaleigenvaluesarerelated
to asmallnneigenvalueproblem. The purposeof thisse tion istherefore to develop asingular
valueanalysis ofthese matri es and to analyzethe orresponding signaleigenvaluebounds.
Theorem 3 Suppose
b
B is the ba kward predi tor matrix introdu ed in (2.7). Then the singular
spe trum of b B, ( b B), satis es ( b B)=(B P )[f0g; (3.18)
2 1 (B P ) = 2+kbk 2 kp N k 2 + p (kbk 2 +kp N k 2 ) 2 4je N bj 2 2 ; i (B P ) = 1; (i=2;:::;n 1); 2 n (B P ) = 2+kbk 2 kp N k 2 p (kbk 2 +kp N k 2 ) 2 4je N bj 2 2 ; (3.19) whereb and p N
are the rst and the last olumn ve tors of B and the proje tor P, respe tively.
Proof. From 2.7 we havethat
b B =PB=V 1 V 1 B. Hen e, b B b B=B V 1 V 1 V 1 V 1 B =(V 1 B) (V 1 B); and therefore ( b B)=(V 1 B): (3.20)
Ontheother hand, ifwe introdu e A=V
1 B,thenB P =AV 1 and B P B P =AV 1 V 1 A =APA : (3.21) But,asP =V 1 V 1 =W y W =W W y by (2.5),then A =B V 1 =B PV 1 =B W W y V 1 =W Z W y V 1 ;
where the last equalityis be ause of (2.9). Hen e, sin e W
y W
= I, where I denotes the nn
identitymatrix, wehave, using(2.9) again,that
PA =W W y W Z W y V 1 =B W W y V 1 =B V 1 =A : (3.22)
Usingthispropertyin(3.21), wededu ethatthesingularvaluesofB
P
arethesingularvaluesofA,
and thus the rst part of the theorem follows from (3.20). To prove the se ond statement, noti e
that AA =[V 1 b;V 1 e 1 ;:::;V 1 e N 1 ℄ 2 6 6 6 4 b V 1 e 1 V 2 . . . e N 1 V 1 3 7 7 7 5 =V 1 bb V 1 +V 1 e 1 e 1 V 1 ++V 1 e N 1 e N 1 V 1 ; an berewrittenas AA =I+xx yy ;
wherex=V 1 b,andy=V 1 e N
. ByapplyingLemma2toAA withu
1 =v 1 =xandu 2 = v 2 =y,
we obtain that the spe trum of this matrix is formed by n 2 eigenvalues equal to 1, and the
remainingones, obtained fromtherootsof thepolynomialin (2.13),are givenby
1 ; 2 = 2+kxk 2 kyk 2 p (kxk 2 +kyk 2 ) 2 4(x y) 2 2 :
Now, observe that kp
N k =kV 1 V 1 e N k=kV 1 (V 1 e N )k =kV 1 yk=kyk; sin eV 1 isan isometry,and that kxk 2 =b V 1 V 1 b=b Pb=b b;=kbk 2 ; jx yj=jb V 1 V 1 e N j=j(b P)e N j=jb e N j sin e b 2 R[H(`)
℄. We now observe that the largest value of the above roots,
1
, say, is larger
than one be ause theeigenvaluesof B arelarger than one inmodulus. Assume nowthat
2 >1.
Then we obtainfromits de nitionthat
kxk 2 kyk 2 2 >(kxk 2 +kyk 2 ) 2 4(x y) 2 ;
whi h an be simpli ed to kxk
2 kyk 2 < jx yj 2
. This is impossible as it ontradi ts the
Cau hy-S hwarz inequality, and we therefore dedu e that
2
1. These roots are therefore
2 1 (B P ) and 2 n (B P
), respe tively,whi h on ludesthe proof.
Thus, we have obtained an exa t des ription of the singular spe trum of predi tor matri es
obtainedbyproje tionandtheannulus(1.6) whi hprovideslowerandupperboundsforthesignal
eigenvalues. However,noti ethattheseboundsarenotimmediatelyusefulbe ausetheyarederived
from the expressions of the singular values given by the last theorem, whi h depend themselves
on the proje tor P. In order to over ome this diÆ ultywe prove the following result, where we
reintrodu ethe minimumnorm solutionf of(2.10).
Theorem 4 Suppose that A
N
is the annulus de ned by
A N = ( z2CI j 1 p 1+kfk 2 jzj p 1+kbk 2 ) ; (3.23)
whereN is the dimensionof the predi tor matrix B, then the eigenvalues of B
P
belong toA
N .
Proof. We shallprovethatboth
1 (B P ) and n (B P )belongto A N
. Infa t,using(3.19), wehave
2 1 (B P )= 2+kbk 2 kp N k 2 + p (kbk 2 +kp N k 2 ) 2 4je N bj 2 2 1+kbk 2 ;
1 P N n P N we rst show that B P = F 1 P
: usingthe de nitions of both matri es, (2.5), (2.8), (2.9), and the
fa tthat WW y =I,we have that B P F P =V 1 BV 1 V 1 FV 1 =V 1 PBPFV 1 =V 1 W y WBW y WFV 1 =V 1 W y Z 1 WW y ZWV 1 =I;
as laimed. We nextobserve thatthisenablesus to ompute
n (B P ) as n (B P )=1= 1 (F P ); (3.24) and 1 (F P
) anbedeterminedinawaysimilartothatusedforthesingularvaluesoftheba kward
predi tormatrix. Thisyieldsthat
1 (F P ) 2 = 2+kfk 2 kp 1 k 2 + p (kfk 2 +kp 1 k 2 ) 2 4je 1 fj 2 2 1+kfk 2 ; where p 1
is the rst olumn of P and f the minimumnorm solutionof (2.10). This ensures that
theleft inequalityof(3.23) issatis edby
n (B
P
),whi h ompletes theproof.
Wenowmake the ru ialobservation that,dependingon thedimensionoftheproblem N,the
innerandouter radiiofA
N
be ome ex ellentapproximationsof
1 (B P ) and n (B P ), respe tively.
This anbeseenasfollows. Sin ee
N
bistheindependent oeÆ ientofthe hara teristi polynomial
asso iatedto the ompanion matrixB,whi hisnot zerobyTheorem1,then
je N bj= N n Y k=1 j b k j n Y j=1 j j j; (3.25) where b k
aretheso- alledextraneous eigenvaluesand
j =z 1 j . Hen e,asj b k j<1 byTheorem1,
and,sin efor N largeenough, je
N bj
2
0,from (3.19) we havethat
2 1 (B P )1+kbk 2 . A similar reasoningon je 1 fjgivesthat 2 1 (F P )1+kfk 2
,andthequalityof theseapproximationsimproves
when N in reases.
Beforestating our nal result,we introdu etwo te hni allemmas.
Lemma 5 De ne G Q = V 2 BV 2 , where V 2
is an N (N n) matrix whose olumns form an
orthonormal basis of N(H(`)). Then,
2 j (G Q ) = 1; j=1;2:::;N n 1; 2 N n (G Q ) = 1 (1+kbk 2 )kq 1 k 2 ; (3.26) whereq 1
Lemma 6 Letb and f be the rst and last olumn ve tor of B andF respe tively. Then 1+kbk 2 1+kfk 2 = n Y j=1 jz j j 2
for N >n. Moreover, both kbk and kfk de rease monotoni ally when N in reases.
Proof. The proof isinappendixA2.
We now returnto ourmainobje tive and ontinue analyzingthebehaviorof thewidth ofA
N
asfun tionof kbk andkfk, and, onsequently,of N. Note that, be ause ofLemma 6,thisredu es
to analyzingthe norm ofthe forward oeÆ ients kfk. But, sin e (2.8) isequivalentto the system
Wf =Z
N
e,where eistheve tor inCI
n
ofall ones, whi h follows from(2.3), and,as
kfk=kW y Z N ekkW y k p n N ; (3.27)
where we set =maxfjz
j
j; j=1;:::;ng,it suÆ es to analyze kW
y
k asfun tion of N. We now
hoose, fornotational onvenien e, N =pn, p>1. We also write
W =[W 0 DW 0 ::: D p 1 W 0 ℄; where W 0
is thenn Vandermonde matrixwhose (j;k) entry is z
k 1
j
, and D=Z
n
. Usingthese
de nitions,one an thenprove that thesmallestsingularvalue ofW,
n (W), satis es n (W) n (W o ) 0 p X j=1 2n(j 1) 1 A 1=2 ; (3.28) where =minfjz j
j; j=1;:::;ng(see [3 ℄,Theorem1, fordetails). Hen e,we have that
1) if
j
<0,i.e. thesignalis damped, from(3.27) we have thenthat
lim N!1 kfk=0; (3.29) be ause N ! 0 and kW y k=1= n
(W) isbounded as <1 ensures that thesum in (3.28)
is nite;
2) ifthesignalisundampedinstead,i.e.,
j
=0,then(3.28) impliesthatkW
y
k!0asN !1
lim N!1 (1+kbk 2 )= n Y j=1 jz j j 2 (1+ lim N!1 kfk 2 )= n Y j=1 jz j j 2 :
To on ludethisdis ussion,wenotethatthelastpartofLemma6,thislastlimitand(3.23),ensure
thefollowingresult.
Theorem 7 Theannulusthat ontainsthesignaleigenvaluesasso iated toB
P
hasamonotoni ally
de reasing width, i.e. A
N+1
A
N
: Moreover, it is asymptoti ally des ribed by
A 1 =fz2CI =1jzj n Y j=1 jz j j 1 g: (3.30)
Thus, we have shown that the quality of the signal eigenvalues bounds dependson the speed
at whi hkfk
2
approa hesto zero asthe dimensionN in reases. However, as illustratedin(3.27),
this speed depends on the behavior of kW
y
k as a fun tion of N, whi h ultimately depends on
the nature of the signal itself. The authors' experien e is that in most of pra ti al appli ations,
moderatevaluesofN aresuÆ ienttoensurevaluesofthenormsofthepredi tor oeÆ ientssmaller
thanone (see,for instan e,theexamples dis ussedin[3 ℄).
Now onsider the ase of slightly damped signals. For su h signals, we know that the signal
eigenvaluesarerelatively loseto one, whi h we have shown to imply,forlarge N,thatthe width
of the annulus (1.6) is small. Sin e these radii provide ex ellent approximations for
1 (B P ) and n (B P
),thesesingularvaluesmustbe losetoea hother. Furthermore,thestabilityofthesingular
values(seeGolubandVanloan[10 ℄,Se tion8.3.1)guaranteesthatasmallperturbationofthedata
willnotalterthisproperty. This,inturn,impliesthatthewidthoftheannulus(1.6)remainssmall,
evenafterasmallperturbation. Asa onsequen e,themagnitudeofsignaleigenvalues annotvary
byalargeamount. Further,sin etheseeigenvalues annotfalloutsideoftheannulus,thisproperty
suggests thatthe eigenvaluesthemselvesshouldbeinsensitive to smallperturbationson thedata.
3.1 Conne tion of Predi torMatri es with Certain Subspa e-Based Methods
Ourgoal hereisto showthatthere existsa loserelationshipbetweenpredi tormatri es obtained
byproje tionorthogonal and ertain matri es used by two known modal parameteridenti ation
methods. Spe i ally, we shall relate the forward predi tor matrix of (3.16) with those matri es
usedbyERA and OPIA,and showthatall these matri es sharethe same eigenvalues. Infa t, we
noti ethat OPIAuses an nn matrixof theform
S =V
whereV 2CI isamatrixwhose olumnsarerightsingularve torsrelatedtothelargestsingular
valuesof H(`). This shows that the matrix used by OPIA is indeed a predi tor matrix obtained
byproje tion. Ontheother hand, thematrixusedbyERA is
S E = 1=2 U H(`+1)V 1=2 ; (3.32)
where V is as before, U ontains the left singular values related to the largest singular valuesof
H(`)andadiagonalmatrix ontainingthesesingularvalues. Followingreferen e[1 ℄(seerelations
(17), (20), (22) and (27) therein), it an be proved that S
E
=
1=2 S
1=2
. Thisshows thatboth
S and S
E
share the same eigenvalues, and the result ontinues to hold regardless of whether the
signalisperturbedornot. Wethus on ludethatauni edsignaleigenvalueperturbationanalysis
overingERAand OPIAusinganappropriatepredi tormatrixisalwayspossible. Areportabout
thissubje tisinpreparation andshouldappearin afuture ontribution.
4 Numeri al Examples
Inthisse tionwe presenttheresultsofsome omputersimulationswhi hillustratethebehaviorof
thebounds(3.23) andtheroleofkW
y
k. We onsidertwonumeri alexamples. The rstisextra ted
from the spe ialized literatureof the signal analysis eld, and the se ond is a signal synthesized
from experimental measurements. For ea h example, we ompute kW
y
k and the bounds (3.23),
(1+kfk 2 ) 1=2 and (1+kbk 2 ) 1=2
,forseveral valuesofN.
4.1 Bounds for the Signal Eigenvalues of a Syntheti Signal
Thisexample illustratestheboundsasso iatedwiththe sampledsignal de nedby
h k =e ( 0:01+20:20{)k +e ( 0:02+20:22{)k ; k =0;1;:::;
whose signal eigenvalues are z
1 = e ( 0:01+20:20{) and z 2 = e ( 0:02+20:22{)
. In this ase, we have
that jz
1
j = 0:9900, jz
2
j = 0:9802 and the upper bound limit is
Q n j=1 jz j j 1 = 1:035 (note that
n=2). Thissignalisoften usedfortesting the apabilityofalgorithmsto extra t frequen iesand
de ay fa tors from noisy signals, be ause the two losely spa ed frequen ies are easily seen as a
single one when additive noise ispresent (see,e.g, [13 ℄). The behaviorof the boundsasfun tions
of N is displayedin Figure 1-(b). In Figure 1-(a) we show thebehaviorof kW
y
k on a logarithmi
s ale. From these gureswe see the marked monotoni de rease of both kW
y
k and the width of
the annulusA
N
for in reasing values of N. In this example, we also verify that for N 60, our
0
20
40
60
80
100
120
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
0
20
40
60
80
100
120
0
0.5
1
1.5
2
2.5
(a) (b)Figure 1: (a)log (kW
y
k) asfun tionof N. (b)Bounds3.23asfun tion ofN
4.2 Bounds for Signal Eigenvalues related to a Me hani al System
In this example we illustrate the behavior of the bounds (3.23) for a synthesized signal obtained
from experimental measurements of thefreeresponse ofa real vibratorysystem. Fullinformation
about the pro edure used to synthesize the signal an be found in [3 ℄. For this ase, the signal
eigenvalues ome in omplex onjugate pairsand n=10. Theseeigenvaluesareshown inTable 1.
The behaviorof theupper and lowerbounds (3.23) asfun tions of N, whi h we denote here by
j z j jz j j jz j j 1 1 0.9699 0.2248{ 0.9956 1.0044 2 0.9532 0.2931{ 0.9972 1.0028 3 0.9844 0.1619{ 0.9976 1.0024 4 0.9921 0.1055{ 0.9977 1.0023 5 0.9972 0.0585{ 0.9989 1.0011
Table1: SignalEigenvaluesofsynthesized signaland orrespondingmoduli.
L N
and U
N
respe tively,is displayed forN 30 inFigure 2-(b). The rapidde rease ofthe width
of theannulusA
N
isagain very apparent.
Noti ethat,be ausejz
j
j1,thesignalisslightlydamped(seeFigure2-(a)). Inordertobetter
illustratethebehaviorof theboundsasfun tionsof N,we have omputed theirdistan esto their
orrespondinglimits,L 1 =1;andU 1 = Q 10 j jz j j 1
=1:0262; respe tively. Thesedistan esaswell
as the norms kW
y
k are shown in Table 2 for ertain values of N. This table also illustrates the
de reaseof kW
y
kasthee e t ofin reasingN. We alsonotethattheboundsagreewellwiththeir
0
1
2
−5
−4
−3
−2
−1
0
1
2
3
4
5
Tempo (seg.)
0
0
50
100
150
200
250
300
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(a) (b)Figure 2: (a)Signalrelated to a me hani al system. (b) Bounds(3.23) asfun tionof N
N 10 20 30 40 50 60 U N 415.0088 11.7560 4.6708 2.9155 2.1564 1.7303 U N U 1 413.9826 10.7298 3.6445 1.8893 1.1302 0.7041 L N 0.0025 0.0873 0.2197 0.3520 0.4759 0.5931 L 1 L N 0.9975 0.9127 0.7803 0.6480 0.5241 0.4069 kW y k 5:454810 9 0:001210 9 1:822810 4 0:097410 4 0:009610 4 0:001410 4 N 200 220 240 260 280 300 U N 1.0411 1.0410 1.0379 1.0367 1.0352 1.0338 U N U 1 0.0149 0.0148 0.0117 0.0105 0.0090 0.0076 L N 0.9857 0.9858 0.9888 0.9899 0.9913 0.9926 L 1 L N 0.0143 0.0142 0.0112 0.0101 0.0087 0.0074 kW y k 0.1127 0.1104 0.1095 0.1055 0.1044 0.1033
Table2: Behaviorof bounds3.23 andkW
y
kasfun tions of N.
5 Con lusions and perspe tives
We havedevelopedasingularvalue analysisof ertainpredi tormatri esthatenabledusto derive
a losed form for their singular spe trum. Using these results, we have derived lower and upper
bounds forthe so- alled signal eigenvalues, bothdependingof thedimension of theproblem. By
analyzing the in uen e of the dimension on these bounds, we have shown that they an be ome
very tight provided the dimension of the problem is suÆ iently large and the signal is slightly
damped. Thiswasillustrated withnumeri al examples in ludingtheanalysis of the bounds fora
signalrelated to a vibratingstru ture.
We may anti ipate interesting appli ations of our results to ertain subspa e-approa hes for
these methods, providedthey an be shown to dependon spe i predi tor matri es obtainedby
proje tion. This hallengingdevelopment istheobje t ofongoing resear h.
Appendix
A Proof of Lemma 5
Noti ethat bis theminimumnorm solutionof(2.11) and thereforeb2R(H(`)
). From this G Q =V 2 BV 2 =[0;V 2 e 1 ;:::;V 2 e N 1 ℄V 2 ; an berewrittenas G Q =V 2 J V 2 uv ; (A.1)
whereJ isthepermutationmatrixJ =[e
N ;e 1 ;:::;e N 1 ℄,and u and v
arerespe tively,thelast
and rst rowofV
2
. Theproofof thetheoremis thenbasedon the omputation oftheeigenvalues
of G
Q G
Q
. We startthenbyobservingthat
G Q G Q =G G G uv vu G+(u u)vv ; (A.2) whereweset G=V 2 JV 2
. Analyzingthe rst termofthe right-handsidewe see that
G G=V 2 J V 2 V 2 JV 2 =V 2 J (I V 1 V 1 )JV 2 =I V 2 J V 1 V 1 JV 2 ; (A.3)
where thelast equalityfollows from the theorthogonality of J. Onthe other hand, observe that
J V 1 an be rewrittenas, J V 1 = w V " 1 = b V 1 V " 1 + w b V 1 0 =A + w b V 1 0 whereV " 1 isthematrixofV 1
onsistingof allrows ex ludingthelast,w
is thelastrowof V
1 ,and A =BV 1
:Hen e,wehave that
V 2 J V 1 =V 2 w b V 1 0 =V 2 e 1 (w b V 1 )=v(w b V 1 ); (A.4) sin eV 2 A
=0by(3.22). Substitutingthisrelationin(A.3) yields
G G=I (kwk 2 b e N e N b+kbk 2 )vv : (A.5)
We now analyze the se ond term ofthe right-hand sideof (A.2). Noti e that by usingu =V 2 e N and v =e 1 V 2 ,we have, G u=V 2 J V 2 V 2 e N =V 2 J (I V 1 V 1 )e N =V 2 J e N V 2 J V 1 V 1 e N :
But,ifone observesthatJ
e N =e 1 and V 1 e N
=w,we have,using(A.4), that
G u=v v(w b V 1 )w=v v(w w b V 1 V 1 e N )=v v(w w b e N ); sin eb2R(H(`) ),and hen e, G uv =(1 kwk 2 +b e N )vv : (A.6)
This,inturn,impliesthat thethirdtermof theright-hand sideof (A.2) is
vu G=(1 kwk 2 +e N b)vv : (A.7)
Repla ing now (A.7), (A.6) and (A.5) into (A.2) and taking into a ount that kwk
2 +kuk 2 = 1; be ause[w u
℄isthe lastrowof theorthogonalmatrix [V
1 V 2 ℄, we dedu e that G Q G Q =I (1+kbk 2 )vv :
From this relation, we see that N n 1 eigenvalues of G
Q G
Q
are equal to the unity, while
the remaining one is 1 (1+kbk
2 )kvk
2
. The proof on ludes by noting that kq
1 k = kQe 1 k = kV 2 V 2 e 1 k=kV 2 vk=kvk. B Proof of Lemma 6
We rst deriveauxiliaryresultsinvolvingthe termsof theratio
1+kbk
2
1+kfk
2
asfun tions of N. Forthis, we onsidertwo onse utive valuesof N, and usethe subs ript
[N℄ to
denote thedependen e ofthe onsidered quantitieson N. We startbyobservingthat
W y [N+1℄ W y [N+1℄ =(W [N+1℄ W [N+1℄ ) 1 : =A [N+1℄ (B.8)
and thattheW
[N+1℄ W [N+1℄ =A 1 [N+1℄ is arank-one modi ationof A 1 [N℄ =W [N℄ W [N℄ ,i.e. A 1 [N+1℄ =A 1 [N℄ +Z N ee Z N =ZA 1 [N℄ Z +ee ;
whereweusedthetworepresentationsW [N+1℄ =[W [N℄ Z e℄=[e ZW [N℄
℄:Applyingthe
Sherman-Morrison formula to ea h ofthese forms, we derivethat
A [N+1℄ =A [N℄ A [N℄ Z N ee Z N A [N℄ 1+e Z N A [N℄ Z N e =Z A [N℄ Z 1 Z A [N℄ Z 1 ee Z A [N℄ Z 1 1+e Z A [N℄ Z 1 e : (B.9)
Now, theproje tor asso iated withthevalueN is given by
P [N℄ =W y [N℄ W [N℄ =W y [N℄ [e ZW [N 1℄ ℄=[W y [N℄ e W y [N℄ ZW [N 1℄ ℄
be ause of (2.5) and the de nition of W
[N℄ . This yields kp 1;[N℄ k =kW y [N℄
ek: Combiningthis with
(B.8), we obtain kp 1;[N+1℄ k 2 = e A [N+1℄
e. Using this relation in the rst equality of (B.9), we
obtain kp 1;[N+1℄ k 2 = e A [N℄ e e A [N℄ Z N ee Z N A [N℄ e 1+e Z N A [N℄ Z N e = kp 1;[N℄ k 2 (e 1 W y [N℄ Z N e)(e Z N W y [N℄ e 1 ) 1+e Z N A [N℄ Z N e = kp 1;[N℄ k 2 je 1 f [N℄ j 2 1+kf [N℄ k 2 ; (B.10) where f [N℄ =W y [N℄ Z N
e is the minimum norm solution of the system (2.10). On the other hand,
usingtheequalitybetweenthe left-handsideand thelastright-handsideof (B.9), we have that
e A [N+1℄ e=e Z A [N℄ Z 1 e e Z A [N℄ Z 1 ee Z A [N℄ Z 1 e 1+e Z A [N℄ Z 1 e :
Thisis nothingbut
kp 1;[N+1℄ k 2 =kb [N℄ k 2 kb [N℄ k 4 1+kb [N℄ k 2 = kb [N℄ k 2 1+kb [N℄ k 2 ; whereb [N℄ =W y [N℄ Z 1
e. Thisimpliesthat
1+kb [N℄ k 2 = 1 1 kp 1;[N+1℄ k 2 : (B.11)
We nowobserve that, usingthe fa tthatp
1;[N℄ =e 1 q 1;[N℄ , (B.10) an berewrittenas 1 kp 1;[N+1℄ k 2 =kq 1;[N℄ k 2 + je 1 f [N℄ j 2 1+kf [N℄ k 2 ;
1=(1+kb [N℄ k 2 )kq 1;[N℄ k 2 + je 1 f [N℄ j 2 1+kf [N℄ k 2 (1+kb [N℄ k 2 ); orequivalently,byTheorem5, 2 N n;[N℄ (G Q )=je 1 f [N℄ j 2 1+kb [N℄ k 2 1+kf [N℄ k 2 : (B.12)
The nal partof ourproofdependson two important observations. The rst is that, ase
1 f
[N℄ , is
theindependenttermofthe hara teristi polynomialofF,thatisnonzerobe auseofTheorem2,
itis equalto theprodu toftheeigenvaluesof F. That is,
je 1 f [N℄ j= N n Y k=1 j b k j n Y j=1 jz j j; (B.13) where the b
's are the extraneous eigenvalues of F, and the z's are the signal eigenvalues. The
se ond is that the extraneous eigenvalues of B are the onjugate of those of F, as proved in [2 ℄,
Theorem3.2. Hen etheprodu t oftheirmodulusisequalto theprodu tof thesingularvaluesof
V 2 BV 2 =G Q
. Usingnow Theorem5,wededu e that
2 N r;[N℄ (G Q )= N n Y k=1 j b k j 2 : (B.14)
The rstpart ofthe theoremthen followsfrom (B.14), (B.13) and (B.12).
Now, observethat, usingthe se ondequalityof(B.9) and thede nitionsof f and b,
kf [N+1℄ k 2 = e Z N+1 A [N+1℄ Z N+1 e = e Z N A [N℄ Z N e e Z N A [N℄ Z 1 ee Z A [N℄ Z N e 1+e Z A [N℄ Z 1 e = kf [N℄ k 2 jf [N℄ b [N℄ j 2 1+kb [N℄ k 2 ;
whi h shows that kfk de reasesmonotoni allywith N. The same on lusion then follows forkbk
[1℄ F.S.V.BazanandC.Bavastri.Anoptimizedpseudo-inversealgorithm(OPIA)formulti-input
multi-output modal parameter identi ation. Me hani al Systems and Signal Pro essings,
10:365{380, 1996.
[2℄ F.S.V. Bazan and L.H.Bezerra. Onzerolo ationofpredi torpolynomials. Numer.Linear
Algebra With Appli ations, 4(6):459{468, 1997.
[3℄ F. S.V. Bazan, Ph. L. Toint, and M. C. Zambaldi. A onjugate-gradients based method for
harmoni retrievalproblemsthatdoesnotuseexpli itsignalsubspa e omputation.Te hni al
Report16,Department of Mathemati s,FUNDP,Namur, Belgium,November1997.
[4℄ L. H. Bezerra and F. S. V. Bazan. Eigenvalue lo ation of generalized ompanion predi tor
matri es. SIAM Journal on Matrix Analysis and Appli ations, 19(4):886{897, 1998.
[5℄
A.Bjork. Numeri al Methods for least squares problems. SIAM, Philadelphia,USA, 1996.
[6℄ G.E.BoxandG.M.Jenkins.TimeSeriesFore astingandControl.Holden-Day,SanFran is o,
1970.
[7℄ S.Braun andY.Ram. Determinationofstru turalmodesviathepronymodel: Systemorder
and noiseindu edpoles. A oust. So . Am., 81(5):1447{1495, 1987.
[8℄ S.BraunandY.Ram. Timeandfrequen yidenti ationmethodsinover-determinedsystems.
Me hani al Systemsand Signal Pro essings,1(3):245{257, 1987.
[9℄ G. Cybenko. Restri tions of normal operators, Pade approximation and autoregressive time
series. SIAM Math. Anal.,4:753{767, July1984.
[10℄ G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press,
Baltimore,thirdedition,1996.
[11℄ F.B. Hildebrand. Introdu tion to Numeri al Analysis. M Graw-Hill,New York, 1956.
[12℄ R.A.HornandC.R.Johnson. Matrixanalysis. CambridgeUniversityPress,Cambridge,UK,
1990.
[13℄ Y. Hua and T. P. Sarkar. Matrix pen il method for estimating parameters of exponentially
damped/undampedsinusoidsinnoise. IEEETrans.OnA oust.Spee handSignalPro essings,
identi ation and modal redu tion. Journal of Guidan e, Control and Dynami s, 8:620{627,
1985.
[15℄ Fuadd Kittaneh. Singularvalues of ompanion matri es and bouds on zeros of polynomials.
SIAM Journal on Matrix Analysis and Appli ations, 16(1):333{340, January 1995.
[16℄ R. Kumaresan. On the zeros of the linear predi tion-error lters for deterministi signals.
IEEETrans. OnA oust. Spee h and Signal Pro essing,ASSP-31(1):217{221, February 1983.
[17℄ R.KumaresanandD.W.Tufts.Estimatingtheparametersofexponentiallydampedsinudoids
andpolezero-modelinginnoise.IEEETrans.OnA oust.Spee handSignalPro essing,
ASSP-30:833{840, De ember1982.
[18℄ S.Kung. A new identi ationand model redu tion algorithm via singular value
de omposi-tions. In Pro eedings of the 12
th
Asilomar Conferen e on Cir uits Systems and Computers,
pages705{714, Pa i Grove California, 1978.
[19℄ P.R. G. Kurka. Modal parameter identi ation in the time domain from a random for e.
Me hani al Systemsand Signal Pro essing,4(5):393{404, 1990.
[20℄ G.Majda,W. A.Strauss,andM. Wei. Computationosexponentialsintransient data. IEEE
Trans. On Antennas and Propagation,37(10):1035{1043 , 1989.
[21℄ J. Makhoul. Linear predi tion: A tutorial review. Pro eedings of the IEEE, 63(4):561{580,
1975.
[22℄ M. A. Rahman and K. B. Yu. Total least squares approa h for frequen y estimation using
linearpredi tion. IEEETrans. A oust. Spee h Signal pro ess., ASSP-35:1440{1454, 1987.
[23℄ A.Van DerVeen, E.F. Deprettere, and A.LeeSwindlehurst. Subspa e-based signalanalysis
using singular value de omposition. Pro eedings of the IEEE, 81(9):1277{1309 , September
1993.
[24℄ S.Van Hu el, H. Chen,C. De anniere, and P. Van He ke. Algorithm fortime-domain NMR
data ttingbasedontotalleastsquares.JournalofMagneti Resonan eA,110:228{237,1994.
[25℄ M. Wei and G. Majda. A new theoreti al approa h for Prony's method. Linear Algebra and
