JACKSON-SZEGÖ SYSTEMS OF ORTHOGONAL … Bound... · JACKSON-SZEGÖ SYSTEMS OF ORTHOGONAL...

Philips J. Res. 44, 417-432, 1989 R 1218

JACKSON-SZEGÖ SYSTEMS OF ORTHOGONALTRIGONOMETRIC POLYNOMIALS REVISITED, AND

THE SPLIT LEVINSON ALGORITHM

by P. DELSARTE and Y. GENIN

Philips Research Laboratory Brussels, Av. Van Becelaere 2, Box 8, B-1170 Brussels, Belgium

AbstractThis paper is devoted to the theory of systems of real trigonometriepolynomials that are orthogonal with respect to a given positive measure.It gives a self-contained overview of the main algebraic results obtainedby Jackson and by Szegöconcerning that subject; they include recurrencerelations and explicit constructions in terms of unit circle orthogonalpolynomial systems. In addition, the paper deals with the split Levinsonalgorithm, which is a new efficient method for solving the linear predictionproblem in the stationary (Toeplitz) case. It explains how this algorithmfits into the Jackson-Szegö theory, and how it can be used to orthogonalizethe graded linear space of real trigonometrie polynomials with respect toa positive measure (determined by its Fourier coefficients).Keywords: Jackson-Szegö spaces, orthogonal trigonometrie polynomials,

positive definite Toeplitz matrices, split Levinson algorithm,unit circle orthogonal polynomials.

1. Introduetion

The classical theory of orthogonal polynomials on the real line started inthe second half of the last century, with significant contributions by Chebyshev,Christoffel, Markov, Stieltjes, and others (see Szegö's treatise 1)). Theimportance of the role played by this theory in pure and applied mathematicsis so obvious that there is no need to emphasize it here. The analogous theoryof orthogonal polynomials on the unit circle was introduced by Szegö in 1921,in relation with positive definite Toeplitz matrices 2). It progressively foundinteresting applications in various areas such as statistics, circuit and systemtheory, inverse scattering, and digital signal processing (DSP). A closeconnection between some well-defined subclasses of orthogonal polynomialsystems of both types (unit circle and real line) was discovered at once by

Philips Journalof Research Vol.44 No.4 1989 417

P. Delsarte and Y. Genin

Szegö himself"). It is based on the simple mapping eiB --+ cos () between theunit circle and the real interval [ -1, 1].About 20 years later, Jackson developed a theory analogous to that of

orthogonal polynomials for the case of real trigonometrie sums 4); his resultsinclude recursion formulas, a Christoffel-Darboux formula, and convergencetheorems (relative to the expansion of a function in some orthogonaltrigonometrie systems). Jackson's approach does not refer to Szegö's theoryof orthogonal polynomials on the unit circle. More recently, the same subjectof 'orthogonal trigonometrie sums' was treated by Szegö, under the name ofbi-orthogonal systems oftrigonometric polynomials S). The Szegöcontributionin question, which does not refer to Jackson's paper, provides explicitconstructions in terms of unit circle orthogonal polynomial systems, recurrencerelations, zero location properties, mechanical quadrature formulas, aChristoffel-Darboux type of formula, and convergence theorems.

Positive definite Toeplitz matrices are frequently met in applied mathematicsand, more specifically, in DSP applications, where they occur typically asautocorrelation matrices of sampled signal records. A central theme in thisarea is the linear prediction problem, which amounts to computing the firstcolumn of the inverse of a given positive definite Toeplitz matrix 6). Theintimate connection that exists between the theory of unit circle orthogonalpolynomial systems and the theory of positive definite Toeplitz matrices ismost clearly revealed by the following fact. The recurrence relation underlyingthe Levinson algorithm 7), which is the standard method for efficiently solvingthe linear prediction problem with Toeplitz structure, can be viewed as animplementation of Szegö's recurrence formula for orthogonal polynomials onthe unit circle.A few years ago, the present authors presented a novel numerical procedure,

termed the split Levinson algorithm, to solve the linear prediction problemfor real Toeplitz matrices 8,9). This method is closely related to the standardLevinson algorithm, but its computational complexity is smaller (by a factor of2 in terms of the number of multiplications). It involves some well-defined'symmetric predietor polynomials', and computes them by way of a simplerecurrence relation. This relation can be identified to a standard three-termformula of orthogonal polynomial theory by the change of variablez = eiB --+ x = cos((}j2). The one-to-one correspondence between unit circleorthogonal polynomial systems with 'symmetric measure' and [-1,1]orthogonal polynomial systems with 'even measure' thus obtained 8,10)

constitutes a refinement of Szegö's result 1,3) alluded to above. The splitLevinson algorithm can be extended to the case of complex Toeplitzmatrices 11-13). In this general setting, the natural mathematical framework

418 Philip. Journalof Research Vol.44 No. 4 1989


Jackson-Szegö polynomial systems

of the algorithm is the theory of bi-orthogonal trigonometrie polynomials,instead of the theory of orthogonal polynomials on the real line (seeMorgera 14)).

The purpose of this contribution is twofold. First, to revisit the algebraicpart of what could be called the Jackson-Szegö theory of orthogonaltrigonometrie polynomials so as to draw the attention of the DSP researchcommunity to a useful mathematical subject (Sec. 2), and to give new insightinto technical results such as recurrence relations and specific orthogonal bases(Sec. 3). Next, to show that the general split Levinson algorithm (for complexdata) fits nicely into the framework of the Jackson-Szegö theory, and that itprovides an efficient new method to compute an orthogonal basis (with respectto a given measure) for the graded space of real trigonometrie polynomials(Sec. 4). This aims at demonstrating the rich interplay between DSP researchand mathematical subjects such as orthogonal trigonometrie polynomialtheory.

2. Jackson-Szegö spaces

A complex Laurent polynomial x(z) is any finite sum of the formx(z) = Lk XkZ\ with complex coefficients Xk, where k ranges over a finite setof integers. The degree of a non-zero Laurent polynomial x(z) is the largestnon-negative integer n satisfying XII =F 0 or x ,; =F 0; it will be denoted by deg x.By convention, we set deg 0 = O. Replacing z by eiOin a Laurent polynomialx(z), we obtain a complex trigonometrie polynomial x(eiO), i.e. a linearcombination of the functions cos ke and sin ke with k ~ O. Note that deg X

equals the largest n such that either cos ne or sin ne occurs with a non-zerocoefficient in the expansion of x(eiO), for x =F o. When there is no risk ofconfusion we often use the term trigonometrie polynomial instead of Laurentpolynomial. This paper is mainly concerned with real trigonometriepolynomials x(eiO); in terms of Laurent coefficients, they are characterized bythe property x-k = xk for all k.

Consider a positive measure dil on the real interval [0,2n). From dil wecan define a non-negative definite inner product over the space of complextrigonometrie polynomials as follows. The inner product <x, y) of any twoLaurent polynomials x(z) and y(z) is given by

(2.1)

In the sequel, we assume that the underlying function Il has an infinite number


of points of increase. This means that the inner product of eq: (2.1) is strictlypositive definite, in the sense that the norm [x] = (x, X)1/2 of a non-zerotrigonometrie polynomial x cannot vanish. In particular, the requirement isfulfilled when the measure dIL is absolutely continuous, i.e. when it is of theform dIL(O) = w(O) dO where w denotes any Lebesgue-integrable function thatis positive almost everywhere. (Note that the theory is still applicable whenthis condition is not satisfied, provided one considers only Laurent polynomialswhose degree is less than the number of points of increase of IL.)Let:!ï denote the vector space of real trigonometrie polynomials. We shall

examine the question of orthogonalizing the degree-graded space :!ï withrespect to the given measure dIL, which essentially is the subject investigatedby Jackson 4) and by Szegö 5). More precisely, the problem can be introducedas follows. For any non-negative integer n, define :!ï" as the (2n + 1)-dimensional subspace of:!ï that consists of the real trigonometrie polynomialsof degree less than or equal to n. Thus, the functions 1; cos 0, sin 0; ... ; cos ni),sin nO constitute the 'standard basis' for :!ï" (they are orthogonal with respectto the standard measure dO). We are interested in characterizing (andcomputing) a du-orthogonal basis

(2.2)

of the space :!ï", where g;; is a member of 5k for r = 0 and 1. (We use g;; asan abbreviation for gk(z), with z = eiO.) The orthogonality requirements are

(2.3)

for 0::::;r, s s; 1 and 0::::;k, 1::::; n (except that gó is defined only with r = 0). Theparameter v;;in eq. (2.3) is the squared norm of g;;. If vi; = 1 for all k and r,then the basis (2.2) is said to be orthonormal.For a positive integer k let us define Yk as the two-dimensional subspace

of 5k that is the orthogonal complement of 5k _ 1. It is clear that the spacesYk (with 1::::;k < 00) are pairwise orthogonal and produce the orthogonaldecomposition

(2.4)

for all n ~ 0, where the EBsymbol denotes the orthogonal direct sum. Thespaces Yk will be referred to as the Jackson-Szegö spaces (subspaces of :!ï).By definition, the pair (g~, gf) is an orthogonal basis of Yk.As we shall see, the Jackson-Szegö spaces can be described explicitly in

420 Philips Journalor Research Vol.44 No.4 1989



terms of Szegö polynomials 1,2), which are defined as being orthogonal on theunit circle with respect to the measure dil. Recall that the system [bk(z)]k'=oof Szegö polynomials relative to dil consists of the (ordinary) polynomialsbk(z) = IJ= 0 bk,jzj, with complex coefficients bk,j' where bk,k =f. 0, that satisfythe orthogonality relation

(2.5)

for 0 ~ k, I< 00. Here, we assume bk(z) to be monic, i.e. to satisfy bk,k = 1, forall k;;::: O.The reciprocal of bk(z), denoted 6k(z), is defined by

~ _ k _

bk(z) = zkbk(ljz) = L bk,jZk- j.

j=O(2.6)

In the context of linear prediction theory, bk(z) is usually referred to as the(forward) predietor polynomial of order k (and bk(z) itself is called thebackward predietor polynomial of order k).Let us briefly recall the origin ofthis terminology and introduce some further

material. Define the trigonometrie moments (or Fourier coefficients) Ck of themeasure dil by

(2.7)

for all integers k. For any non-negative integer n, construct the Toeplitz matrixCn>of order n + 1, whose (k + 1, I+ l)th entry equals Ck-I, that is,

Co C-1 C-n

c= Cl Co C1-nn

Cn Cn-1 Co

(2.8)

Note that C, is Hermitian, since we have C-k = ëk for all k. Furthermore, C,is known to be positive definite, owing to the properties of the measure dil.In fact, C, is the Grammian matrix of the monomials 1, z, ... , z" with respectto the inner product (2.1). Let us emphasize that any positive definite Toeplitzmatrix C, can be described as above, in terms of a suitable positivemeasure 15,16).

The linear prediction problem 6) relative to the Toeplitz matrix Cn>viewed


as the covariance matrix of a complex stationary stochastic process, or as theautocorrelation matrix of a discrete-time complex-valued signal, requires oneto determine the unique vector an = [an,o, an,I' ... , an.nJT,with an,o = 1, thatsatisfies the system of linear equations

(2.9)

The solution polynomial an(z) = Lj=O an,jzj is called the predietor polynomialof order n. It can be written in the form

(2.10)

where bn(z) is the monie Szegö polynomial of degree n. This classical resultis easily proved by comparing eqs (2.5) and (2.9). Note the property(Jn = det Cn/det Cn-I' (The (J parameter in eq. (2.9) is the same as that in eq.(2.5); it can be interpreted as the prediction error squared norm.) ~For future use, let us recall that the predietor polynomials ak(z) = bk(z) are

related by the Szegö recurrence formula

(2.11)

with Pk = ak,k = Dk(O). The Szegö parameters Pk' also called reflectioncoefficients, satisfy Ipkl< 1 as a consequence ofthe identity (Jk = (Jk-I (1 -lpkI2).By the Schur-Cohn criterion, this implies that ak(z) is stable, in the sense thatit has no zeros in the closed unit disc [z]~ 1. The recurrence formula (2.11)constitutes the basis ofthe Levinson algorithm for solving the linear predictionproblem (2.9).

After this digression, we go back to the description of the Jackson-Szegöspace ~. For any complex parameter Uk' define the Laurent polynomial

(2.12)

As discovered by Szegö, the space ~ consists exactly of the trigonometriepolynomials gk(ei6) given by eq. (2.12), where Uk varies over the complex field 5).More precisely, the correspondence Uk - gk defines a real vector spaceisomorphism between the complex field and ~. The argument is the following.First, the functions (2.12) are seen to constitute a two-dimensional subspaceof !!ik. (For example, choosing Uk = 1 and Uk'= i yields a basis of the space.Let us stress the fact that the gk functions in eq. (2.12) are real.) Next, thissubspace is seen to be orthogonal to the space !!ik-i' since both Z-ka2k_I (z)

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Jaekson-Szegö pol ynomial systems

and zl-kIÎ2k_1 (z) are orthogonal to all Laurent monomials Zl witho ~ III ~ k - 1, as a consequence of eq. (2.5).

In view of eqs (2.9) and (2.12), the Jackson-Szegö space 9k is completelydetermined from the trigonometrie moments ej with 0 ~j ~ 2k - 1. It shouldhowever be noted that the computation of inner products (and norms) within9k requires the additional knowledge of e2k or, equivalently, of the reflectioncoefficient P2k' This will be made more explicit in the sequel.

By use ofthe Szegö recurrence (2.11)we can obtain an alternative descriptionof 9k. Consider the complex number transformation Uk __. Vk defined by

(2.13)

which yields (1 -lp2kI2)vk = Uk - P2kÜk' Note that eq. (2.13) induces areal-linear one-to-one mapping of the complex field onto itself. It followsfrom eqs. (2.11) and (2.13) that eq. (2.12) can be rewritten as

(2.14)

When Vk varies over the field of complex numbers, the trigonometriepolynomial (2.14) varies over the Jackson-Szegö space 9k. The correspondencebetween Vk and gk is real-linear and one-to-one. For an obvious reason, eqs(2.12) and (2.14) will be referred to as the odd-degree and even-degree Szegörepresentations of gk(Z).

Let us comment on the zeros of the members gk of 9k (see Szegö 5». In viewofthe stability property ofthe predietor polynomials, it follows from eq. (2.12),or from eq. (2.14), that gk(ei6) has 2k distinct zeros in the interval 0 ~ ()< 2n,for any non-zero gk in 9k. Furthermore, it is seen that two linearly independentelements of 9k cannot have a zero in common.

3. Recurrence relations and orthogonal bases

Consider any two trigonometrie polynomials gZ(z) and g~(z) in theJackson-Szegö space 9k. For r = 0 and r = 1, let uI; and vI; denote the complexnumbers that correspond to ,gHz) in the Szegö representations (2.12) and(2.14), respectively. By use of the orthogonality relation (2.5) we' obtain

(3.1)



for r, s = 0, 1. This allows us to express the Grammian matrix

(3.2)

of the pair (gr, gt) in terms of the parameter matrices

(3.3)

as the product Gk = (J 2k Uk V r. where the asterisk denotes the conjugatetranspose. From the reflection coefficient Pk construct the matrix

R = [1 PkJ.k Pk 1

(3.4)

By use of eq. (2.13) we obtain Uk = VkR2k, which implies that the Grammianmatrix (3.2) can be written in the remarkable form

(3.5)

This shows that (gr, g~) is a basis of 9k if and only if Vk is non-singular, andthat (gr, g~) is an orthonormal basis if and only ifV k satisfies (J 2k VkR2k V t = I.

From the results of Sec. 2 we can easily prove the following vector spaceidentity:

(3.6)

for 1=0, 1, ... , k - 1, where the left-hand side denotes the subspace of g-spanned by all products f g with f eSi and gE 9k. In particular, eq. (3.6) tellsus that 9k+l is a subspace of g-19k, with 9k-l E99k as its orthogonalcomplement. This immediately leads to recurrence relations for bases ofJackson-Szegö spaces, in the following way (see refs 4 and 5). From any basis(gr, g~) of 9k construct the 2-vector

(3.7)

The inclusion property 9k + 1c ffl9k can be interpreted as an identity

(3.8)

424 Philips Journalof Research Vol.44 No.4 1989


where Mk(z) is a well-defined 2 x 2 matrix whose entries are members of .rI'i.e. reallinear combinations of 1, cos e and sin e (for z = eiO). Note that 9k isa subspace of .r19k+l; this implies that Mk(z) is unimodular: its determinantis a non-zero real 'number.

We shall explain how Mk(z) can be constructed explicitly by use of theeven-degree Szegö representation (2.14). To that end we introduce thenotations

L\(z) = [~ ~l (3.9)

We have Zkgk(Z) = Vka2k(Z) and ak(z) = RkL\(z)ak_1 (z), by eqs (2.14) and (2.11).Applying each of these identities twice and using eq. (3.8) we obtain theremarkable expression

II

(3.10)

In view of the structure of the matrices involved in the right-hand side of eq.(3.10), it can easily be checked that the entries of Mk(z) actually belong to the.r1space. Furthermore, it is obvious from eg. (3.10) that Mk(z) is unimodular.Note that an iterated use of eqs (3.8) and (3.9) yields a 'constructive proof'of eq. (3.6).

Let us now consider the question of selecting a particular orthogonal basisof the Jackson-Szegö space 9k. A preliminary observation about 'change ofbases' may be useful. A change of 9k basis can be expressed by a transformation

(3.11)

where Qk is any non-singular real matrix of order 2. In terms of the Szegörepresentation (2.14) this amounts to changing the parameter matrix Vk intothe parameter matrix V ~= Qk Vk• Concerning Grammian matrices, we obtainthe relation G~= QkGkQL in agreement with eq. (3.5). Thus, we can obtainan orthonormal basis g~(z) of 9k from any given basis gk(Z) of 9k by factorizingits Grammian matrix (3.2) as Gk = HkHL for a suitable 2 x 2 matrix Hk' andby setting Qk = Hk 1. (In particular, an orthonormal basis is transformed intoanother orthonormal basis if and only if the transformation matrix Qk isorthogonal. )Jackson's approach consists in orthogonalizing the 'standard trigonometrie

sequence' 1, cos e, sin e, cos 2e, sin 2e, ... , in this precise order"). Inagreement with eq. (2.2), let gZ(z) denote the 2kth element in the corresponding

Philips Journa! of Research Vol.44 No.4 1989 425

g2(z) = d2z-k[a2k_1 (z) + za2k-1 (z)],gHz) = id~z-k[a2k(z) - a2k(z)],

(3.12)


orthogonal basis of f/", with 1~ k ~ n < 00. It is characterized, among thenon-zero members of ~, by the fact that the coefficient of sin kB in the Fourierexpansion of gk(eiO) is equal to zero. This means that the parameter u2 in theodd-degree Szegö representation (2.12) of g2(z) is real. If g~(z) is orthogonalto g2(z) in ~, then it follows from eq. (3.1) that the parameter v~ in theeven-degree Szegö representation (2.14) of gHz) is imaginary. As a result, theorthonormal basis of ~ that results from Jackson's approach consists of thetrigonometrie polynomials

where d2 and d~ are appropriate real numbers (which can easily be determinedby use of eq. (3.1)). It is seen that the Jackson basis has a remarkably simpleexpression in terms of the Szegö orthogonal polynomials on the unit circle.Note that this basis can be computed efficiently (from the trigonometriemoménts Ck of the measure dil) by way of the Levinson algorithm based oneq. (2.11).

An alternative choice for the orthogonal basis has been proposed by Szegö 5).It requires the polynomials zkg2(z) and -i zkg~(Z) to be the symmetric andantisymmetrie components of Wka2k(Z), for a suitable complex number Wk.This amounts to the conditions Wk= 2v2 and Wk= - 2i v~; hence the constraintis

V~ = i v2. (3.13)

It follows from eqs (3.1) and (2.13) that g2(z) and g~(z) are orthogonal if andonly if the parameter v2 satisfies

( 0/-0)2 - /Vk Vk = P2k P2k' (3.14)

when P2k =I: O. (Otherwise v2 is arbitrary.) The resulting orthogonal basis(g2, g~) seems to be more complicated than that given in eqs (3.12). Besides,the actual computation of the basis by means of eq. (3.14) may encounternumerical accuracy problems when the reflection coefficient P2k is close to. zero. Let us emphasize an interesting property of Szegö's basis: the zeros ofg2(ei9) and g~(eiO) interlace 5). A third type of orthogonal basis will be describedin Sec. 4.<, Let us add a few explanations about the important special case where themeasure dil is symmetric (with respect to the point n). Equivalently, this meansthat all trigonometrie moments ck are real. As a consequence, the reflection

426 Pbilips Journalof Research Vol.44 No.4 1989

PhilipsJournal of Research Vol. 44 No.4 1989 427


coefficients Pk and the prediction coefficients ak,j are real. Jt is quite natural,in this case, to impose that g~(ei8)be an even trigonometrie polynomial (withcosine terms only) and that g~(eiO)be an odd trigonometrie polynomial (withsine terms only). Such functions are often referred to as cosine polynomialsand sine polynomials, respectively. The constraint just mentioned is perfectlysound, since cosine and sine polynomials are mutually orthogonal providedthe measure is symmetric. It determines the orthogonal basis of.9k in a uniquemanner (within normalization); it amounts exactly to the fact that v~ is realand v~ is imaginary. The Jackson and Szegö bases clearly have the propertyin question. (Note that we could make the alternative choice, requiring g~ tobe odd and g~ to be even; this is simply a matter of notation.)

Each of these cosine and sine polynomial systems, [g~(z)]r'=o and [g~(z)]r'=I'can be transformed into a system of orthogonal polynomials on the realinterval [-1, 1]by means of a simple change of variable 1,3). Note that thetwo formulas (3.12) involve only Szegö polynomials of odd and even degree,respectively. By removing this restriction, and thus considering (real)polynomials proportional to ak(z) +Qk(Z) or to ak(z) - Qk(Z) for all k, we canconstruct two [-1,1] orthogonal polynomial systems, into which theaforementioned systems are embedded (roughly speaking) 8,10). Let us stressthe fact that the symmetric measure assumption is crucial in this matter; thecorrespondence between unit circle orthogonal polynomial systems and realinterval orthogonal polynomial systems does not extend to general positivemeasures (i.e. to general positive definite Hermitian Toeplitz matrices).

4. Connection with the split Levinson algorithm

The split Levinson algorithm is a modification of the classical Levinsonalgorithm that constitutes a more economical solution method for thefixed-order linear prediction problem (2.9) 8,9,11-13). It is based on aremarkably simple recurrence formula for (complex) symmetric (orself-reciprocal) polynomials Pk(Z) of the form

(4.1)

for some well-defined non-zero complex numbers Yk' The connection with theodd-degree Szegö representation (2.12) is obvious. Note however that the splitLevinson algorithm involves the polynomials (4.1) for all degrees k (from 1up to n + 1), and not only those of odd degree as in eq. (2.12).The coefficients Yk= Pk(O) in eq. (4.1) are determined in such a way that the......, -,


symmetric polynomials Pk(Z) satisfy a three-term recurrence relation

(4.2)

with suitable complex coefficients (J,k= Yk +Jv; In fact, the general splitLevinson algorithm (for complex data ck) depends on two unit-modulusparameters 13); these will be denoted '0 and cv1• The predietor polynomialak(z) can be recovered from the symmetric polynomials Pk(Z) and Pk + 1(z) byway of the formula

(4.3)

The Ä. coefficients in eq. (4.3), termed Jacobi parameters 12.13.17), are positivereal numbers; they can be determined with the help ofthe recurrence formula

(4.4)

The split Levinson algorithm computes the sequence of symmetricpolynomials Pk(Z) by way of eq. (4.2), for 1 ~ k ~ n + 1, and yields the desiredpredietor polynomial an(z), in a final stage, by use of eq. (4.3) with k = n. Tocomplete the description of the algorithm we have to explain how to initializeit (in terms of the parameters '0 and cv1, with 1'01= Icvd = 1), and how todetermine the recurrence coefficients (J,k in eq. (4.2) from the entries of theToeplitz matrix (2.8). The required information is the following:

(4.5)

k

where 7:k = L ci : j Pk.j,j=O

(4.6)

together with 7:0 = coW2cvi/2 and Ä.1 = 1 (to initialize eq. (4.4». A detailedderivation of the split Levinson algorithm, with the same notations as above,can be found in refs. 13 and 17.

Our main objective in this section is to show how the split Levinsonalgorithm (without its 'final stage') can be used to determine orthogonal bases(gf, gt) for the Jackson-Szegö spaces Bk (and even to obtain orthonormalbases). In view of the results of Secs 2 and 3, we can make the choice

(4.7)

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Philips Journolof Research Vol.44 No.4 1989 429


(4.8)

for any non-zero real number dk, with the definition

(4.9)

The argument is as follows. First, in view of eqs (4.1) and (4.9), bothtrigonometrie polynomials g~(z) and gt(z) are seen to belong to ~, by use ofeqs (2.12) and (2.14) where

u~= Y2k' (4.10)

Next, g~(z) and gt(z) are verified to be orthogonal, by eqs (3.1) and (4.10).In fact, formulas (4.7) and (4.8) yield any orthogonal basis of ~ from a

suitable choice of the parameter' Y2k in eqs (4.1) and (4.9). The specific choiceassociated with the recurrence relation (4.12 turns out to be especiallyinteresting from an algebraic and computational viewpoint. The reason is thatboth trigonometrie polynomials g~(z) and gt(z) can be determined efficiently(for all k) by means of the split Levinson algorithm. This statement iscompletely obvious as regards the g~(z) subsystem, which is produced directlyby the algorithm. Let us now more closely examine the case of the gt(z)subsystem. Substituting eq. (4.3) into eq. (4.9), and using the recurrencerelations (4.2) and (4.4), we readily obtain

(4.11)

With the choice dk = À.2klC>:2kI2, this yields a fairly simple expression for gt (z),namely,

1(z) = P2k+ 1(z) - À.2kÀ.2k+ lZP2k-l (z) .gk • Yl/2(1 Y-l) k

-1"0 -"0 Z Z(4.12)

If we wish to construct an orthonormal basis of ~, we need to know thenorms ofthe trigonometrie polynomials g~(z) and gt(z) in eqs (4.7) and (4.12).Simple computations, using the results of ref. 13, yield the expressions

Ilg~112 = COÀ.2kllC>:2kl-2(1 + À.2kÀ.2k+d,

IIgtl12 = COÀ.2k+1(1 + À.2kÀ.2k+l)·(4.13)


It is interesting to mention the reformulation ofthe split Levinson recurrencein terms of the real trigonometrie polynomials 12)

(4.14)

The three-term recurrence formula (4.2) can be rewritten as

with a" = 2 Re(ak) and a~ = 2 Im(ak). Set (0 = eWo. In view of eqs (4.7), (4.12)and (4.14) we obtain the expressions

(4.16)

These results would become more attractive if we could avoid performingdivisions by the 1 - Co lZ factor in eq. (4.12). Such a simplification is possibleif (and only if) we force all symmetric polynomials Pk(Z) to vanish at the givenpoint Z = (0 (with Kol = 1). In view of eq. (4.5), this amounts to making thechoice COl = -COl, which is called the singular case of the split Levinsonalgorithm 13.17).Let us briefly explain the resulting method. Define the reducedsymmetric polynomials

(4.17)

with Pl(Z)= -i(à'2(1-(01z) in the singular case we are considering. Thetrigonometrie polynomials (4.7) and (4.12) can be written as follows:

(4.18)

The result (4.18) is really interesting since the reduced symmetricpolynomials Pk(Z) can be computed by means of a suitable split Levinsonalgorithm relative to the measure dji defined by

(4.19)

i.e. relative to the Toeplitz matrix whose entries ck are given in terms of the

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original data Cl by

(4.20)

The main relations of the algorithm are eqs (4.2) and (4.6), where p, c, r and()(are replaced by p, ê, i and à. The initial values are p _ 1(z) = 0 and Po(z) = 1,together with L 1= coap and io = Co - (oC i - Note the identity

(4.21)

As for the Jacobi parameters Ak involved in eq. (4.18), they are computableexactly as explained above, i.e. by means of the recurrence (4.4) with the initialvalue Al = 1.A derivation of this 'reduced split Levinson algorithm' (with aslight difference in normalization) can be found in ref. 17.When the data Ck are real (i.e. when the measure dJl is symmetric), it is

natural to restrict the parameters (0 and Wl to the real values 1 and -1.(Recall that the choices (0 = 1,wl = -1 and (0 = -1, Wl = 1allow furthersimplifications, by use of the reduced split Levinson algorithm.) It can easilybe verified that the g~(z) subsystem consists of even trigonometrie polynomialswhen Wl = (0 and of odd trigonometrie polynomials when Wl = - (0 (the g~ (z)subsystem has the opposite property). It is interesting to mention that, forreal data (exclusively), the symmetric polynomials qk(Z) in eqs (4.9) and (4.1i)obeya three-term recurrence relation similar to eq. (4.2),within normalization.

REFERENCESI) G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1959.2) G. Szegö, Math. Z., 9,167 (1921); reprinted in Gabor Szegö, Collected Papers, ed. R. Askey,

Vol. I, Birkhäuser, Boston, pp. 277-302, 1982.3) G. Szegö, Math. Ann., 82, 188 (1921); reprinted in Gabor Szegö, Collected Papers, ed.

R. Askey, Vol. 1, Birkhäuser, Boston, pp. 371-396, 1982.4) D. Jackson, Ann. Math., 34, 799 (1933).S) G. Szegö, Magyar Tud. Akad. Mat. Kutató Int. Közl., 8, 255 (1963); reprinted in Gabor

Szegö, Collected Papers, ed. R. Askey, Vol. 3, Birkhäuser, Boston, pp. 795-815, 1982.6) J. Makhoul, IEEE Proc., 63, 561 (1975).7) N. Levinson, J. Math. Phys., 25, 261 (1946).8) P. Delsarte and Y. Genin, IEEE Trans. Acoust. Speech Signal Process., 34, 470 (1986).9) P. Delsarte and Y. Genin, Philips J. Res., 43, 346 (1988).10) Y. Genin, Proc. 1st Int. Conf. on Industrial and Applied Mathematics, Paris, 1987,

eds J. McKenna and R. Teman, SIAM Publ., Philadelphia, PA, 1988, p. 102.11) H. Krishna and S.O. Morgera, IEEE Trans. Acoust. Speech Signal Process., 35,839

(1987).12) P. Delsarte and Y. Genin, SIAM J. Math. Anal., 19,718 (1988).13) P. Delsarte and Y. Genin, An introduetion to the class of split Levinson algorithms, in

Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, eds G .H. Goluband P. Van Dooren, Springer NATO-ASI Series, to be published.

14J S.O. Morgera, IEEE Trans. Acoust. Speech Signal Process., 37, 436 (1989).


is) U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, University ofNorthCalifornia Press, Berkeley, CA, 1958.

16) N.1. Akhiezer, The Classical Moment Problem, Oliver and Boyd, 1965.17) P. Delsarte and Y. Genin, Tridiagonal approach to the algebraic environment of Toeplitz

matrices, part I: basic results, submitted for publication.

AuthorsP. Delsarte: Ir. degree (Electrical Engineering and Applied Mathematics), University ofLouvain, Belgium, 1965 and 1966; Dr. degree (Applied Sciences), University of Louvain, 1973;Philips Research Laboratory Brussels, Belgium, 1966- . His research interests include algebraiccoding theory, combinatorial mathematics, and the theory and applications of orthogonalpolynomials and Toeplitz matrices.

Y. Genin: Ir. degree (Electrical Engineering), University of Louvain, 1962; Dr. degree (AppliedSciences), University of Liège, Belgium, 1969; Philips Research Laboratory Brussels, 1963- ;Consulting Professor at Stanford University, CA, U.S.A. (1979-1980); Visiting Professor atFacultés Universitaires de Namur, Belgium (1974-1976 and 1984-1985). He currently heads theApplied Mathematics Group at Philips Research Laboratory Brussels. His principal researchinterests concentrate on the mathematical aspects of signal processing, network theory and systemtheory. He is a Fellow of the IEEE Society and is currently serving as an associate editor forthe journals IEEE Transactions on Circuits and Systems, SIAM Journalon Matrix Analysis andApplications, Mathematics of Control, Signals and Systems, and Philips Journalof Research.

432 Philips Journul of Research Vol. 44 No. 4 1989

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