D. Alpay, A. Chevreuil, Ph. Loubaton * Mathematics Department, Ben-Gurion University. P.O.B. 653, Beer-Sheva, Israel

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1 An extension problem for discrete-time periodically correlated stochastic processes. D. Alpay*, A. Chevreuil**, Ph. Loubaton*** * Mathematics Department, Ben-Gurion University P.O.B. 653, Beer-Sheva, Israel dany@math.bgu.ac.il ** Departement Signal, elecom Paris 46 rue Barrault, Paris Cedex 13, France chevreui@sig.enst.fr *** Universite de Marne la Vallee / Unite de Formation SPI 2 rue de la butte verte, Noisy le Grand Cedex, France loubaton@univ-mlv.fr Corresponding author: Antoine Chevreuil, Departement Signal, elecom Paris 46 rue Barrault, Paris Cedex 13, France 1

2 Abstract In the context of wide-sense stationary processes, the so-called Caratheodory-Fejer problem of extending a nite non-negative sequence of matrices has been much studied. We here investigate a similar extension problem in the setting of wide-sense periodically correlated processes: given the rst N coecients of scalar-valued sequences, we study under which condition(s) it is possible to nd extensions which are the cyclo-correlation sequences of a periodically correlated process with period. Using a result of Gladysev (1961), one shifts the problem to a Caratheodory-Fejer problem with symmetry constraints. he existence of extensions is proved. In non-degenerate cases, the set of all solutions is given in terms of a homographic transformation (3.12) of some Schur function G. he choice G = leads to the maximum entropy solution. he associated gaussian processes are then proved to have a periodic auto-regressive structure. Keywords: Periodically Correlated Processes, Extension of a non-negative sequence, matrixvalued Szego polynomials 2

3 1 Introduction. A centered discrete-time stochastic process (y(n)) n2zz is said to be periodically correlated (PC) with period 2 IN if for each n, the sequence m 7! E(y(n + m)y(m) ) is periodic with period - see Gladysev (1961) for instance. In this case, there exist sequences (R (n)) n2zz ; : : :; (R?1 (n)) n2zz for which E(y(n + m)y(m) ) = X?1 k= R k (n)e 2ikm= (1:1) for each (m; n) 2 ZZ 2. he sequences (R k ) k=?1 are sometimes called the cyclocorrelation sequences of y; they are quite useful in many signal processing problems - see Gardner (1993) and the references therein. As pointed out in Gladysev (1961), the sequences (R (n)) n2zz ; : : :; (R?1 (n)) n2zz correspond through (1.1) to the cyclorrelation sequences of a PC process of period if and only if the matrix-valued sequence (R(n)) n2zz dened by R k;`(n) = R`?k (n)e 2ikn= (1:2) (where `? k stands for its value modulo ) is non-negative, i.e. if it coincides with the autocovariance sequence of some {variate stationary stochastic process, or equivalently, if there exists a hermitian bounded non-decreasing matrix-valued measure dened on [; 2[ for which R(n) = R 2 e?in! d(!). his result, which can be interpreted as a Bochner-like theorem, will often be used in the following. In this paper, we solve the following extension problem which to our knowledge has not be treated in previous works: Problem 1.1 Given nite sequences of scalars (R (n)) n=n ; : : :; (R?1 (n)) n=n, nd necessary and sucient conditions for them to be the rst values of the cyclocorrelation sequences of a PC process of period. Under these conditions, characterize the set of the extended cyclocor- 3

4 relation sequences. Find the extension maximizing the entropy rate of a corresponding Gaussian PC process. As is well known and easily checked, a scalar-valued process y(n) is PC of period if and only if the {variate time-series Y (n) = (y(n ); : : :; y(n +? 1)) is stationary. It is therefore natural to investigate if the above problem is equivalent to a classical covariance extension problem for the multivariate times series Y (n). In order to specify this point, let us rst remark that the knowledge of the cyclocorrelation sequences (R (n)) n=n ; : : :; (R?1 (n)) n=n of a PC process y(n) of period is equivalent to the knowledge of E(y(m + n)y(m) ) for n = : : :N and for each m =? 1. Put N = L + p where p <. hen, it is easy to show that the E(y(m + n)y(m) ) for n N; m =? 1 give the full autocovariance coecients E(Y (m + n)y (m) ) for n = : : :L? 1, as well as the entries (E(Y (m + L)Y (m) )) k;` for k? ` p (1.3) (E(Y (m + L + 1)Y (m) )) k;` for `? k? p (1.4) of the matrices E(Y (m + L)Y (m) ) and E(Y (m + L + 1)Y (m) ). Note that if p =, the last constraint (1.3) is of course irrelevant. Our extension problem is thus equivalent to the following covariance extension problem for {variate stationary processes. Problem 1.2 Given the matrices (S(n)) n=l?1 and the entries S(L) k;l for k? ` p, S(L + 1) kl for `? k? p of two matrices S(L) and S(L + 1), nd necessary and sucient conditions for the existence of a {variate stationary process Y (n) such that E(Y (m + n)y (m) ) = S(n) for n = L? 1, and (E(Y (m + L)Y (m) )) k;l = S(L) k;` for k? ` p, (E(Y (m + L + 1)Y (m) )) k;` = S(L + 1) k;` for `? k? p. If the problem is solvable, characterize the autocovariance sequences of all -variate stationary processes Y (n) which rst autocorrelation coecients satisfy the above requirements. 4

5 One can also connect the PC processes to the stationary ones as underlined in Gladysev (1961); this characterization will give rise to another problem, equivalent to Problem 1.2. Lemma 1.1 If (R(n)) n2zz is a matrix-valued sequence which veries (1.2), then R(n)e?2in= = R(n) (1:5) for each n, where where = C A (1:6) Conversely, if any {sequence (R(n)) n2zz veries (1.5), there exist (scalar) sequences (R (n)) n2zz ; : : :; (R?1 (n)) n2zz for which (1.2) holds. By the Bochner-like theorem of Gladysev (1961), the partial sequences (R (n)) n=n ; : : :; (R?1 (n)) n=n can be extended to cyclocorrelation sequences if and only if the nite matrix-valued sequence (R(n)) n=n dened by (1.2) has a non-negative extension (R(n)) n2zz for which R k;`(n) = R`?k (n)e 2ikn= for some (extended) sequences (R (n)) n2zz ; : : :; (R?1 (n)) n2zz. Using Lemma 1.1, this can be reformulated as the following constrained positive extension problem: Problem 1.3 Let the nite matrix-valued sequence (R(n)) n=n dened from (R (n)) n=n ; : : :; (R?1 (n)) n=n by (1.2). Find a necessary and sucient condition such that there exists a non-negative extension (R(n)) n2zz of (R(n)) n=n satisfying (1.5). When this condition is in force, characterize the set of all extensions. his paper is organized as follows. In section 2, we review some basic facts related to the covariance extension problem. In section 3, we solve Problem 1.3. he maximum entropy extension is nally characterized in section 4. 5

6 2 he standard covariance extension problem. Let us rst recall that a matrix-valued function F (z) is said to be positive real if F (z) is holomorphic in the open unit disk ID and if F (z) + F (z) for z 2 ID. Let (R(n)) n=n be a nite matrix-valued sequence. hen, it is well known - see Delsarte et al. (1979) - that the covariance extension problem of (R(n)) n=n is equivalent to the so-called Caratheodory-Fejer problem - namely to nd the set of all positive real functions F (z) for which F (z) = R() + 2 NX k=1 R(k)z k + o(z N ) It is straightforward to connect this problem to the well-known Riesz-Herglotz theorem which states that F (z) is positive real with F () hermitian if and only it exists a (uniquely dened) matrix-valued hermitian measure on [; 2[, bounded and non-decreasing such that F (z) = ^() + 2 1X k=1 ^(k)z k where ^(k) = R 2 e?ik! d(!). It is well established that the Caratheodory-Fejer problem is solvable if and only if the block-oeplitz matrix? N = (R(i? j)) (i;j)=n (2:7) is non-negative. When moreover? N >, it is possible to characterize the set of all its solutions, thus yielding a parameterization of all non-negative extensions of the sequence (R(n)) n=;n ; more precisely, F (z) is such a solution if and only if it veries F (z) =?1 ~RN (z) + zs N (z)g(z) ~PN (z)? zq N (z)g(z) (2:8) where G(z) is any Schur function (i.e. it is analytic in ID and it veries kg(z)k 1 for all jzj < 1) and P (z), Q(z) (respectively R(z), S(z)) are the right and left Szego polynomials of rst kind (respectively of second kind). As they are widely used in the sequel, we briey recall the denitions. 6

7 Let (A k;n ) k=n and (B k;n ) k=n be the matrices dened by (A ;N ja 1;N j : : :ja N;N )? N = (Ijj : : :j) (2.9) (B N;NjB N?1;Nj : : :jb ;N)? N = (jj : : :ji): (2.1) Let A N (z) = P N k= A k;n z N?k and B N (z) = P N k= B k;n z N?k. he matrices A ;N and B ;N are of course non-negative denite, and thus we can dene the hermitian positive square root of their inverse U N = A?1=2 ;N and V N = B?1=2 ;N. he Szego polynomials of the rst kind associated to? N are built as follows: P N (z) = U N A N (z); Q N (z) = B N (z)v N : he polynomials of the second kind R N (z) and S N (z) are easily deduced from P N (z) and Q N (z) via the relations R N (z) = [P N (z)(r() + 2R (1)z?1 + : : : + 2R (N)z?N )] + S N (z) = [(R() + 2R (1)z?1 + : : : + 2R (N)z?N )Q N (z)] + where \ + " means the polynomial part. Finally, ~ RN (z) and ~ P N (z) denote the reciprocal polynomials of R N (z) and P N (z) (explicitly, ~ RN (z) = z N (R(1=z )) ). Finally, we point out that bitangential versions of this problem (i.e. when the data do not consist in the full matrices (R(n)) but in their value in certain directions) may be considered - see Ball et al. (199) for an exhaustive review. However, the problem of nding all positive real functions F (z) for which both F () and [1 ]F () are given - corresponding to Problem 1.2 for = 2 and N = 1 - is not solvable, as far as we know, by the various methods of interpolation which have been developed to solve Caratheodory-Fejer type problems. 7

8 3 he solution of the constrained extension problem. We now return to the case where (R(n)) n=n is the matrix-valued sequence constructed from the data (R (n)) n=n ; : : :; (R?1 (n)) n=n. As for the classical moment problem, it is easy to describe the solutions in terms of the solutions of a related interpolation problem for positive real functions. More precisely, it is readily seen that the solutions of Problem 1.3 are in one-to-one correspondence with the ones of the following problem: Problem 3.1 Find the set of all positive real functions F (z) for which F (z) = R() + 2 NX k=1 R(k)z k + o(z N )holds; and which satisfy the condition F (ze?2i= ) = F (z): (3:11) his allows us to state and prove the main result of this paper, namely: heorem 3.1 Let (R (n)) n=n ; : : :; (R?1 (n)) n=n be nite sequences, and (R(n)) n=n the matrices given by (1.2); dene next? N as (2.7). Problem 3.1 admits solutions if and only if? N is non-negative. When? N is denite, the set of all extensions is parameterized via the description F (z) = ~ RN (z) + zs N (z)g(z) ~ PN (z)? zq N (z)g(z)?1 (3:12) where G(z) is any Schur function satisfying G(z) = e?2i(n +1)= G(ze?2i= ) (3:13) and where P N ; Q N ; R N ; S N are the Szego polynomials associated to? N. proof of heorem 3.1: we divide the proof into two parts. First we consider the case? N > ; then an approximation theorem is used to study the case? N. 8

9 We rst check that the symmetry veried by the (R(n)) n=n induces particular relations on the Szego polynomials. Let indeed G the (N +1) (N +1) block-diagonal matrix G = diag(; : : :; ) and D the (N + 1) (N + 1) matrix Ddiag(I ; e 2i= I ; : : :; e 2iN= I ), where I denotes the identity matrix of size. From (1.5) and (1.2), it is easily seen that D G? N GD =? N : (3:14) From (2.9), we get (A ;N ja 1;N j : : :ja N;N )D G? N GD = (Ijj : : :j) which immediately yields (A ;N ja 1;N e?2i= j : : :ja N;N e?2in = )? N = (Ijj : : :j) and (A ;N ja 1;N e?2i= j : : :ja N;N e?2in = ) = (A ;N ja 1;N j : : :ja N;N ): his last identity nally means that A N (z) = e 2iN = A N (ze?2i= ): (3:15) Moreover, A ;N veries A ;N = A ;N, thus U N = U N. Consequently P N (z) = e 2iN= P N (ze?2i= ): (3:16) Similarly, one can show that Q N (z) has the same symmetry. Lastly, using the denition of second kind Szego polynomials leads to R N (z) = e 2iN= R N (ze?2i= ) and S N (z) = e 2iN= S N (ze?2i= ). Of course, (3.16) implies PN ~ (z) = PN ~ (ze?2i= ), RN ~ (z) = RN ~ (ze?2i= ). hus, a function F (z) given by (2.8) satises (3.11) if and only if the Schur function G(z) veries the relation G(z) = e?2i(n +1)= G(ze?2i= ). As this class of Schur function is not empty (indeed G = satises (3.13)), the existence of solutions to Problem 1.1 is established, as well as the parameterization of the solutions in the case? N >. 9

10 It remains to establish that Problem (3.1) is still solvable if? N is non-negative singular. A classical approximation argument is used; let indeed (R (n)) n=n ; : : :; (R (n))?1 n=n the sequences dened by R () = R () +, and R k (n) = R k(n) for n 6= or k 6=. It is clear that the block oeplitz matrix? N associated to this perturbed data is given by? N =? N + I, and is thus non-negative denite. he corresponding structured Caratheodory-Fejer problem is thus solvable. Let F (z) be a solution. As F () = R() + I, the family (F ()) <<1 is bounded. By the Helly selection theorem - see Brockwell and Davis (199) - the family (F (z)) <<1 has a convergent subsequence as!. he corresponding limit F (z) is clearly a solution of Problem 3.1, showing that Problem 1.1 is solvable. 4 he maximum entropy solution In this section, we assume that the matrix? N is non-negative denite, and we characterize the extension of the sequences (R (n)) n=n ; : : :; (R?1 (n)) n=n for which the entropy rate of an associated Gaussian periodically correlated process (y(n)) n2zz is maximum. Such an extension will be called a maximum entropy extension. Of course, the entropy rate of a Gaussian periodically correlated process (y(n)) n2zz coincides with the entropy rate of the associated {variate stationary process Y (n) = (y(n ); : : :; y(n +? 1)), and is thus given by - see Cover and homas: H(y) = 1 2 log 2(2e) + 1 Z 2 log 2 det(s(e i! ))d! (4:17) 4 where S(e i! ) stands for the spectral density of Y (n). Let (R(n)) n2zz be the matrix-valued nonnegative sequence dened from the cyclocorrelation sequences of y(n) by (1.2), and denote by F(e i! ) the associated spectral density. If F (z) denotes the positive real function corresponding to the sequence (R(n)) n2zz, it is well known that lim r!1 F (re i! )) exists almost everywhere - the corresponding value is denoted F (e i! )) - and that F(e i! ) = 1=2(F (e?i! )+F (e?i! ) ). In particular, 1

11 the symmetry on F (z) implies that F(e i(!?2= ) ) = F(e i! ). It is moreover pointed out in Gladysev (1961) that F(z) and S(z) can be linked by S(e i! ) = U(e i!= ) F(e i!= )U(e i!= ) (4:18) where U(e i! ) is the unitary matrix-valued function dened by U k;`(e i!= ) = 1 p exp(i2k`=? i`!= ): (4:19) We now show that H(y) can be expressed in terms of F(e i! ). We have for all k = : : :? 1 F(e i(!+2k= ) ) =?k F(e i! ) k : Plugging (4.18) into (4.17) yields Z 2 log det(s(e i! ))d! = = = = Z 2 Z 2 log det(f(e i!= ))d! X?1 Z (`+1)2 `= Z 2 `2 log det(f(e i! ))d! log det(f(e i! ))d! log det(f(e i! ))d!: In other words, the entropy rate of y(n) is H(y) = 1 2 log 2(2e) Z 2 log 2 det(f(e i! ))d!: Hence, in order to characterize the maximum entropy extension of the sequences (R (n)) n=n ; : : :; (R?1 (n)) n=n, it is sucient to maximize R 2 log 2 det(f(e i! ))d! over the set of all spectral densities corresponding to the solution of Problem 3.1. Moreover, it is well known that the maximum of R 2 log 2 det(f(e i! ))d! over the solutions of the unconstrained Caratheodory- Fejer problem corresponds to the solution (2.8) associated to the parameter G(z) =, i.e. F max (z) = RN ~ (z) P ~?1 N (z): (4:2) 11

12 he choice G = moreover clearly satises (3.13), which says that F max is a solution of Problem 3.1. hus, (4.2) is the solution of Problem (3.1) maximizing the entropy. Using well known properties of Szego polynomials - Delsarte et al. (1979) - we have that F max (e i! ) = ~P? N (e?i! ) P ~?1 N (e?i! ). We now establish that any associated Gaussian periodically correlated process y max (n) is periodic autoregressive. For this, it is sucient to show that the spectral density S max (e i! ) of Y max (n) = (y max (n ); : : :; y max (n +? 1)) is the inverse of a matrix-valued trigonometric polynomial of the variable!. We have S?1 max(e i! ) = U(e i!= ) PN ~ (e?i!= ) PN ~ (e?i!= ) U(e i!= ) (4:21) {z } K N (e i!= ) It is clear that (3.16) implies that K N (e i! ) = K N (e i(!?2= ) ). Hence, there exists a family L r (e i! ) of trigonometric polynomials - precisely, L r is the entry number (; r) of K N (:) - so that [K N (e i! )]`;m = L m?`(e i(!?`2= ) ) where `? m stands for its values modulo. We can write that L m (e i! ) = P N r=?n L m (r)e ir!. Setting = 2, the general term (S?1 max ) pq(e i! ) of S max?1 (ei! ) can thus be written 1 X?1 `;m= = 1 ei(p?q)!=?1 e?ip`e ip!= (K N ) l;m (e i!= )e imq e?iq!= X `;m= X = 1 ei(p?q)!=?1 = 1 ei(p?q)!= 2 k= X 4?1 e i(mq?p`) L m?`(e i(!=?`) ) X?1 e ikq e i`(q?p) NX k= r=?n N X `= r=?n 3 e ikq L k (r)e ir!= 5 L k (r)e?i`r+ir!= "?1 X `= e i`(q?p?r) # Moreover P?1 `= ei`(q?p?r) = as soon as r 6= q? p modulo. It follows that where pq = fn : q? p? n 2 [?N; N]g: (S max?1 ) pq(e i! ) = 1 X?1 X e ikq L k (q? p? n )e?in! k= n2 pq 12

13 We deduce that S?1 max is a trigonometric polynomial of the variable!. As (4.21) holds, it means that the {variate process Y max (n) exhibits an auto-regressive structure, which implies that the corresponding process y(n) is a periodic auto-regressive process. References [1] Ball J., Gohberg I. and Rodman L. (199) Interpolation of rational matrix-valued functions. Operator heory : Advances and Applications, Birkhauser Verlag, Basel, vol. 45. [2] Cover. and homas J.A (199) Elements of Information heory. Wiley Interscience. [3] Delsarte P., Genin Y. and Kamp Y. (1979) Schur parametrization of positive block oeplitz systems. SIAM J. Appl. Math, vol. 36, pp [4] Gardner W.A. (1994) Cyclostationarity in communications and signal processing. Edited by W.A. Gardner, IEEE Press. [5] Gladysev E.G. (1961) Periodically random sequences. Sov. Math., vol. 2, pp [6] Brockwell P. and Davis R. (199) ime series: theory and methods. Springer Verlag in Statistics, Springer-Verlag. 13

Counterexamples to witness conjectures. Notations Let E be the set of admissible constant expressions built up from Z; + ;?;;/; exp and log. Here a co

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