Inverse problems for orthogonal matrices, Toda flows, and signal processing

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by  Faybusovich, L.;Ammar, Gregory S.;Gragg, William B.

NPS-MA-93-001
NAVAL POSTGRADUATE SCHOOL
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INVERSE PROBLEMS FOR ORTHOGONAL
MATRICES, TODA FLOWS,
AND SIGNAL PROCESSING
by
L. Faybusovich
G. S. Airunar
W. B. Gragg
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11 TITLE (Include Security Classification) Inverse problems for orthogonal matrices, Toda flows, and
signal processing
12 PERSONAL AUTHOR(S)
L. Faybusovich, G. S. Ammar, W. B. Gragg
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FIELD GROUP SUB-GROUP orthogonal matrices, Toda flows, signal processing
19 ABSTRACT (Continue on reverse if necessary and identify by block number)
We consider Toda flows induced on the set of orthogonal upper Hessenberg matrices. The
explicit formulas for the evolution of Schur parameters are given.
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Inverse problems for orthogonal matrices, Toda flows, and signal
processing
L. Faybusovich
Department of Mathematics, University of Notre Dame
Ammar
G.S.
Department of Math. Sciences, Northern Illinois University
W.B. Gragg
Department of Mathematics, Naval Postgraduate School
Abstract
the amplitudes q/ can be recovered from the
first components of the normalized eigenvec-
We consider Toda flows induced on the set of tors of O. One can then use any of several al-
orthogonal upper Hessenberg matrices. The gorithms designed for unitary and orthogonal
explicit formulas for the evolution of Schur pa- Hessenberg eigenproblems [9, 6, 1, 10, 4]
rameters are given. to calculate the frequencies and amplitudes.
In the present paper we investigate another
1 Introduction aspect of orthogonal Hessenberg matrices.
Namely, we consider Toda flows on these ma-
Any symmetric nonnegative definite Toeplitz trices (referred to as Schur flows in [3]) and
matrix Tn+i of order n + 1 can be modeled obtain explicit formulas for the evolution of
as an autocorrelation matrix of a stationary the so-called Schur parameters under the Toda
signal [12] flow. Since Schur parameters determine or-
thogonal Hessenberg matrices uniquely, we ac-
v
xm = ]Pcr/cos(mw/ + 0/) + ym tually obtain an explicit description ofthe evo-
, lution ofa given orthogonal Hessenberg matrix
/=i
under the Toda flow.
where 0/ are arbitrary phase shifts and ym is a
zero mean white noise process whose variance
equals the smallest eigenvalue Anijn of Tn+\. 2 Inverse problem for orthog-
Assume that the eigenvalue \„un is simple, and onal Hessenberg matrices
let (r)o, • • •,T)n) be a corresponding eigenvector.
Then [12] the polynomial Let M be the set of positive Borel measures
V>n(A)=7/ + ...+ 7/nAn on C whiMch have the following properties. For
any /x £ the support of /x, which we denote
has n distinct roots Ai, • • •, An on the unit cir- by AM, consists of exactly n points which lie
cle, and the frequencies of xm are given by on the unit circle U and is such that: i) if
{exp(±iu;/}|'_1 = {Aj}^=1, where i denotes the A G A^, then the complex conjugate A is also
imaginary unit. One can construct [2] an or- in AM and /x{A} = //{A}; it) //{A} > for any A
thogonal Hessenberg matrix with character- in AM; Hi) n(C) = 1; iv) —1 AM. We further
OH+
istic polynomial proportional to %j)n. Moreover, introduce a class of orthogonal matrices
6
.
O =|| Oij || such that otJ = if i — j > 1 , Indeed, f\Wdp = £AeA/i A«A^{A} =
o,+i,, > for all i and det = 1. Finally, given ^A€A>j A'-J/z{A}, since A = A"1 for A G U.
a vector r = (r , • • •, Tn-\)T G i?n introduce a Further, since n{X} = fi{X} we have
corresponding Toeplitz matrix T(t) =|| Uj
||, £ £
where /tJ = T)»-j|- y-^{\}= v-v{A} = /vAi^.
Theorem 2.1 Given a positive definite sym- AGAM A6AM
emxeitsrticexaTcotelpylitoznemamteraisxuTr(er\)i GwitMh Toan=d 1exatchterley Let q = a + . . . + an_iAn_1 G Pn- We have
0H+
one G such that
L A»'d/i(A) = n =< ei,0*ci >, (2.1) < Q,Q >= m^=2oaiaj J/ A'A-'rf/x = J/ | q |2 dfj, > 0.
tca=l b0a,s•i•s•i,nnR—n1.anFder<e c>i,i•s•t•h,eenstiasnfd/aierdcasncoanlia-r fFourrtahneyr,A/G| qAM|2. d\Siin—ce diefgagnd<onnly=ifcaq(r\^)A^=),
proGduc0t/./+Cotnhveermsaetlryi,ce,fsorT(atn)y, y.T(Gt'M) araendPoasniy- pthoilsynisompioaslsi£b(lAe)o=nlyUti€Af„q( =X ~0. C=onbsoid+er• •t•h+e
tive definite Toeplitz matrices. Here 6n_iAn_1 + An. Since all roots of £ lie on
the unit circle we clearly have An£(l/A) =
r = I AV/x(A),r;=<e ,0«e >, 6o£(A),6o = il- Further, all coefficients of
t Jc 1 1 £ are real because A^ = A^. Consider the
i = 0, • • -,n — 1. linear operator : Pn —» Pn defined as fol-
kRneomwanrtko 2t.h2e eTxhpeeorrtsem(se2e.1e.gi.s [m8o],re[11]o)r. lWeses l—obwioss:—obrO\tAXh'o—g=o.naA.l—,+r1en,l_iait=Aivne0_1,t-o.-Wt-he,enn-socaw2l,aprOroApvr"eo"dt1uhca=tt
nevertheless give an independent proof to clar- <, > We should prove that
ify relationships between introduced objects. .
Remark 2.3 There is nothing mysterious <OA\AJ >=<XiiO~1Xj >
about the number —1 which we have eMxcluded for any i,j = 0,---,n - 1. The only non-
from the support of each measure in . This trivial case is i = n — l,j = 0. We have
simplifies notations a little bit. < OAn-\l >= -b - 6in - ...- frn-iTn-i,
WLeemnmeead the following elementary lemma. wCnh-eirAe"r-,1=. TfhceAn*d/<z(AA).^LOet^l_11>==cc +Tn._i. . ++
macatllysoynsete2om.r4tohfoLgevotencvat\ol,rms•a•it•n,rivRxn-n\. TsbehuecarhnethoearxtitshOtoesn{oerx=-- Chtiha7av"nte_c21,+==. c•-.o6+Oncl_ni+__i,..,-.Tih+=uCs0,n,^i•tO•i•As,"snu"f-f1ic1=i.eWncteAto+clpe.ra.or.vl+ey
t?,-,» = 1, • •• , n — 1 and detO = 1. cn_2An-1 +cn_1(-6 -6iA-...-6n_iAn-1)or
We can now outline a proof of Theorem 2.1. 1 = -Cn-l&o, c - Cn-l&l = 0, C\ — Cn_i&2 =
Proof: Denote by Pn the vector space of real 0, • •• , cn_2 - cn_i&n_i = 0. This yields 61 =
polynomials of degree less or equal n — 1. Set -co/60, h = -C1/&0, •••, 6n_i = -c„_2/6 .
We now use the relation An£(l/A) = 6 £(A). It
< A\AJ >= / A-^A). (2.2) follows that 6n_, = 6 6,, i = 0,---,n. Thus
Jc
We prove that (2.2) defines a positive definite barne-i/ebxoact=ly -tch,e_1r/e6qu,iried=con1,di•t•i*,o!n»s.. TThheusse
scalar product on Pn. Observe that we have constructed an orthogonal operator
f\iXjdfi= I' \{-jdn= fyXjdfi. O0,--s-u,cnh -th1a.t OfbcseA'rdv/ex t=ha<t th1e,0c'h1arac>t,eriisti=c
polynomial of O coincides with £. Thus the where all T{ > 0. We then can define the mea-
M
spectrum of is A^. In particular, detO = sure /x 6 by the conditions /x{A,} = r, and
1 (here we use the assumption that —1 £ equal to zero otherwise. We immediately see
A^). Let po = l, --, ,Pn-i be an orthonor- that equations (2.1) are satisfied. It remains to
mal basis in Pn obtained by the orthonor- prove that the measure fi is defined uniquely
malization of the basis 1, A,- • •, An_1. It is by conditions (2.1). Let /xjt € M,k = 1,2 be
clear that the matrix O of the operator O such that
is upper Hessenberg in this basis. Moreover,
stphean(epntor,ie•s• -,6;p+i_iiit) ar=e salplann(oln,z-e•r•o, A(1o-t1h)eriwsisaen, J/c AM//] = JIc A'<f/z2,
invariant subspace of O which is not true). i = 0,---,n - 1. Then we can construct
Without loss of generality one can suppose Ok,k = 1,2 such that conditions (2.1) are
that o.+i,, > for all i. Otherwise one can satisfied. But then 0\ = 02- In particular,
take diag((i, • • •,t^Odiag^, • • •,en). A^i = A^2, i.e., p,\ = p.2 because we have
Suppose we are given a positive definite for /x{A} the following system ofVandermonde
Toeplitz matrix T(t) and an orthogonal ma- equations:
trix O £ OH+ such that r, =< ei,0'ei >,i =
:
0, • • -,n — 1. Then
A6A
T(t) = VTV, (2.3) M
where V is the upper triangular matrix i- 0,•••,!»- 1.
[ci,Oci, • •• ,On-1ei] with positive entries on Let T(t) be a positive definite nxn Toeplitz
the main diagonal. But (2.3) is the Cholesky matrix and <,> be the corresponding scalar
decomposition of T(t). Hence it is uniquely product on Pn. Let
defined by T(t). In other words, the vectors
Oe\, • • •,On~1e\ are uniquely defined by T(r). Fj(A) = 6,-A' + . . .,4 >0,* = 0,---,n-l,
Since these vectors form a basis, the vectors
be the basis obtained by the orthonormaliza-
Oci, • • -,Oen_i are uniquely defined by our n_1
Toeplitz matrix. Thus by Lemma 2.4 the ma- tion procedure from the basis 1, A, •• • , A .
trix is uniquely defined by T(t). Given a Since pi
positive definite Toeplitz matrix T(t) we can is orthogonal to span(l, A, •• •, A'-1), we have:
endow Pn with a scalar product <,> and the Ap,(A) is orthogonal to span(A, • • •, A'). Fur-
sahsifwteopdeirdatboefrordee.finTehdenonussipnagn(Ll,eAm,m•a• • ,2.A4n-w2e) Vt?hier€, rPi+=i \bpet(s\u)/c6hit-hpat,+<1/£q,t+p1i e>=Pi+q1(0.) Lfeotr
can extend this operator to the orthogonal op- any q G P,+i. Since p, is orthogonal to Pt and
erator 0, defined on Pn such that detO = 1. both t and <f{ are orthogonal to AP,, we obtain
Then the matrix of O in the basis obtained by
orthonormalization of the basis l,A,---,An_1 Xpt(X)/St = pl+1(A)/<5,+1 + 7tY>„ (2.4)
b0t,ieol•no•n•g,s»»t—o1.OHCo+nsiadnedrrn,o=w<theeir,aOtileoinal>,fzunc=- fcourlastoimonesrheoalws7,,thia=t <0p,{•=••£,tnA*-p,2(.l/AAn).eHaesyncceal-
f(z) =< l,(zl- 0)-l l>. 1 = St/6{+1 + 7?2t«4 (2.5)
As is easily seen
In other words, if we know 70, ••-,7n-2 we
5
/(*) = £ can find Si, • • •,6n-i• Then using (2.4), one can
<=?*-*«' determine pi, • • -,pn-i and consequently using
again (2.4) the corresponding upper Hessen- uniquely defined by its auxiliary Schur param-
berg orthogonal matrix 0. We have by (2.4) eters <Ti(t) = Oi+i i{i). We now describe ex-
t
< Ap (A),p,(A) >= 7Ap,(0),t = 0,--,n-2. plicitly how these parameters evolve under the
t
Evaluating (2.4) at 0, we obtain pt+i(0) = Toda flow.
= ••,n- Thus =<
-7,£?£,+i, * 0, 2. o,+i,t+i
Ap,A,pt(A) >= -7t7t-i*?_i^?> * = l,--,n-2. Theorem 3.1
Further, o^i = -7oPo(°) = -70. Let us set
~ a
at(t) (0)
A~(7) ' '
= = =
<7t o,+i,t Si-i/h, Vi 7,_1£?_1,
i = 1,• -,n — 1. We obviously have tthe=i—1t,h••p•r,in!c»i-pa1l,miAnor=of1t.he HmearterixA,T((Jt)) =is
T
exp((O(0) + O(0) )0.
We
i = l,---,n- 1, f = 1. Further, on>n = kPrnooowf: that 0{t) = R(t)Q(Q)R(t)~l where
±\/l - <?n-i- The sign is defined by the condi- exp(O([70])0 = Q(t)R{t), Q(t) is orth,ogonal,
tion detO = 1. The quantities i/,-,a, are called and R(t) is an upper triangular matrix with
Schur parameters and auxiliary Schur param- positive entries on the main diagonal. We then
eters, respectively. As we saw above the Schur clearly have
parameters i/,-, i = 1, • •• , n — 1, determine *W
uniquely. = r<?^(
On the other hand, if we know the entries ffi(t) rt,«v) (3.1)
°i+i,i — ^7^»+i) t = 1 • •• ,n — 1 of the matrix
we can determine 7, by (2.5) up to a sign. In i = l,---,n- 1. Here R(t) =\\ rtf(*) ||. The
other words, the entries o;+i)t- (auxiliary Schur operator A/ R(t) naturally acts on the i— th
parameters) determine almost uniquely. exterior power A,' Rn by the following rule:
A* R{t){vi A ...A Vi) = R(t)vi A . . .R(t)vi for
3 Explicit formulas for the any v\, • • •, u, 6 Rn- We have, further, the fol-
lowing relations:
evolution of auxiliary Schur
parameters under the Toda r2u(t)=< R{t)eu R(t)ei >=
flow < exp(O(0)t)euexp(O(0)t)ei >=
Let 0(t) =|| Oij(t) be the solution to the <euT(t)ei >=Ai(t).
||
Toda flow
And more generally
d = [o,*o),
such that O(0) is upper Hessenberg orthogonal r2n(t)...rUt) =
and irreducible. Here itO = O- - OZ and 0-
i
is strictly lower triangular part of 0. Then < e A A e„/\ Y{t){e A ...A e,) >= A,(f.),
x . . . x
0{i) possesses the same properties and O(t)
(3.2)
—
converges when t 00 to a block diagonal i = 1, • •• ,n. By (3.2) we easily obtain
matrix. Each two by two block corresponds to
a pair of complex conjugate eigenvalues. The r,-+i,,-+i(Q _ y/A,-+i(t)A,--i(Q
blocks are arranged in the decreasing order of
»m(<) A,-(<)
real parts of eigenvalues [7, 5]. From the pre-
vious discussion we know that O(t) is almost The result now follows by (3.1).
We have the following differential equations for References
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[3]
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< ezp(O(Q)t)ei,exp(O{0)t)ei >
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