NPS-MA-93-001 NAVAL POSTGRADUATE SCHOOL Monterey, California INVERSE PROBLEMS FOR ORTHOGONAL MATRICES, TODA FLOWS, AND SIGNAL PROCESSING by L. Faybusovich G. S. Airunar W. B. Gragg Technical Report For Period July 1992 - September 1992 Approved for public release; distribution unlimited Prepared for: Naval Postgraduate School Monterey, CA 93943 FedDocs D 208.14/2 NPS-MA-93-001 NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943 Rear Admiral R. W. West, Jr. Harrison Shull Provost Superintendent This report was prepared in conjunction with research conducted for the Naval Postgraduate School and funded by the Naval Postgraduate School. Reproduction of all or part of this report is authorized. 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Gragg 13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year, Month, Day) 15 PAGE COUNT Technical from 7/92 to 9/92 921013 6 16 SUPPLEMENTARY NOTATION 17 COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number) 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. 20 DISTRIBUTION/AVAILABILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION SUNCLASSIFIED/UNLIMITED SAME AS RPT DTIC USERS UNCUSSTFTFJ) 22a NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE (Include Area Code) 22c OFFICE SYMBOL WILLIAM B. GRAGG (408) 646-2194 MA/Gr DD Form 1473, JUN 86 Previouseditionsareobsolete SECURITY CLASSIFICATION OF THIS PAGE S/N 0102-LF-014-6603 SECURITY CLASSIFICATION OF THIS PAGE DD Form 1473. JUN 86 (Reverse) security classification of this page 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 orthogonal 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 Â€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.anFderi,i•s•t•h,eenstiasnfd/aierdcasncoanlia-r fFourrtahneyr,A/G| qAM|2. d\Siin—ce diefgagnd, 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 = 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/==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,î•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-Â). (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=hat,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' for all i. Otherwise one can satisfied. But then 0\ = 02- In particular, take diag((i, • • •,tÔdiag^, • • •,en). Aî = 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 „ (2.4) b0t,ieol•no•n•g,s»»t—o1.OHCo+nsiadnedrrn,o=w,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 = 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, oî = -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 - = flow < exp(O(0)t)euexp(O(0)t)ei >= Let 0(t) =|| Oij(t) be the solution to the =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 <7, = CT,(o,+lit+i - 0,it), [1] GO.nS.thAemmEiagre,npWr.oBb.leGmrfaogrgOarntdhoLg.onRaeilchMeal-, i = l,--,n — 1. Recalling that o,,, = —vx-\V{, trices, Proceedings of the 25th Confer- i = l,---,n, un = ±1, and v} + a} = 1, we ence on Decision and Control, IEEE, New obtain V{V + b{Oi = or York, 1063-1966 (1986). X evolve under Luk, ed.), Proc. SPIE vol. 826, 143-145 the Toda flow. Consider the family of rational (1987). functions G.S. Ammar and W.B. Gragg, Schur [3] f (z)=. Flows, preprint. t 1 1 We clearly have [4] A. Bunse-Gerstner and L. Eisner, Schur parameter pencils for the solution of the unitary eigenproblem, Lin. Alg. Appl., vol. 154-156,741-778(1991). On the other hand, [5] M.Chu, On the global convergence of the Toda lattice for real normal matrices and ft(z) =< Q(t)eu [zE - 0(0)1-^(061 >= its application to the eigenvalue problem, SIAM J. Math. Anal., vol. 15, 98-104 < exp(O{0)t)eu [zE - O(Q)]-1fxp(O(0)Qe > (1984). 1 < ezp(O(Q)t)ei,exp(O{0)t)ei > P.G. Eberlein and C.P. Huang, Global [6] ^ hj{t) convergence of the QR-algorithm for unitary matrices with some resultsfor normal t=0 matrices, SIAM J. Numer. Anal., vol. 12, Here 97-104 (1975). bi{t) =< exp(O(0)0ei,O(0),exp(O(0)0ei > [7] L.E.Faybusovich, QR-algorithm and gen- . eralized Toda flows, Ukranian. Mat. J., Thus Ti(t) = T.i(t) = hi{t)/h (t),i > 0. We vol. 45, 944-952 (1989). clearly have [8] W.B. Gragg, Positive definite Toeplitz hi(t) = hi+1(t) + fet_i(t). matrices, the Arnoldi process for isomet- ric operators, and Gaussian quadrature Thus, on the unit circle (in Russian), in Nu- *i = T.+l + Tt-1 -2T.T!, merical Methods in Linear Algebra (E.S. — Nikolaev, ed.), Moscow Univ. Press, 1-8 = • 0,1 . (1986). [9] W.B.Gragg, The QR-algorithm for unitary Hessenberg matrices, J. Comput. Appl. Math., vol. 16, 1-8 (1986). [10] W.B. Gragg and L. Reichel, A divide and conquer method for unitary and orthogonal eigenproblems, Numer. Math., vol. 57, 695-718 (1990). [11] H.J. Landau, Maximum entropy and the moment problem, Bull. Amer. Math. Soc. vol. 16,47-77 (1987). [12] V.F. Pisarenko, The retrieval of harmon- ics from a covariance function, Geophys. J. R. Astr. Soc, vol. 33, 347-366 (1973).

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

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