where a is an m x m Vandermonde matrix given by 入1 入 A=222 入;=e,i=1→m and Eq (16.12)can be used to determine Pk>0,k=1-m. The power spectrum associated with Eq (16.11) s(e)=∑P(0- and it represents a discrete spectrum that corresponds to pure uncorrelated sinusoids with signal powers P If the given autocorrelations satisfy T, >0, from Eq (16. 4), every unknown Tg, k2n+ I, must be selected so as to satisfy Tk>0, and this gives rx+1-2≤R2 where 5k=f r)and r=det T, /det t - From Eq.(16.13), the unknowns could be anywhere inside a sequence of circles with center 5k and radius Ri and as a result there are an infinite number of solutions to this problem. Schur and Nevanlinna have given an analytic characterization to these solutions in terms of bounded function extensions. A bounded function p(z)is analytic in z< 1 and satisfies the inequality p(a)s 1 everywhere in z<1 In a network theory context, Youla [1980] has also given a closed form parametrization to this class of solutions. In that case, given ro, ri,..., ru, the minimum phase transfer functions satisfying Eqs. (16. 8)and (16.9)are given by (16.14) P(z)-zp(z)P(z) where p(z) is an arbitrary bounded function that satisfies the inequality(paley-Wiener criterion) ∫nm and r(z) is the minimum phase factor obtained from the factorization -lp(e)2=r(e)2 Further, Pn(z) represents the Levinson polynomial generated from ro -r through the recursion sP(z)=Pn2(2)-xnP2-(2) c 2000 by CRC Press LLC© 2000 by CRC Press LLC where A is an m ¥ m Vandermonde matrix given by and Eq. (16.12) can be used to determine Pk > 0, k = 1 Æ m. The power spectrum associated with Eq. (16.11) is given by and it represents a discrete spectrum that corresponds to pure uncorrelated sinusoids with signal powers P1 , P2 , …, Pm . If the given autocorrelations satisfy Tn > 0, from Eq. (16.4), every unknown rk , k ³ n + 1, must be selected so as to satisfy Tk > 0, and this gives *rk+1 – zk* 2 £ R2 k (16.13) where zk = f T k T – k 1 bk , fk = (r1, r2 , . . ., rk )T , bk = (rk , rk –1 , . . ., r1 ) and Rk = det Tk /det Tk – 1 . From Eq. (16.13), the unknowns could be anywhere inside a sequence of circles with center zk and radius Rk, and as a result, there are an infinite number of solutions to this problem. Schur and Nevanlinna have given an analytic characterization to these solutions in terms of bounded function extensions. A bounded function r(z) is analytic in *z* < 1 and satisfies the inequality *r(z)* £ 1 everywhere in *z* < 1. In a network theory context, Youla [1980] has also given a closed form parametrization to this class of solutions. In that case, given r0 , r1 , . . ., rn , the minimum phase transfer functions satisfying Eqs. (16.8) and (16.9) are given by (16.14) where r(z) is an arbitrary bounded function that satisfies the inequality (Paley-Wiener criterion) and G(z) is the minimum phase factor obtained from the factorization 1 – *r(ejq)* 2 = *G(ejq)* 2 Further, Pn(z) represents the Levinson polynomial generated from r0 Æ rn through the recursion A = Ê Ë Á Á Á Á Á Á ˆ ¯ ˜ ˜ ˜ ˜ ˜ ˜ = =Æ - 11 1 1 1 2 1 2 2 2 2 1 1 2 1 1 ... ... ... ... ... , , – – ll l ll l ll l l q m m m m m m i j ei m i MM M S Pk k k m () ( ) q dq q = - = Â 1 H z z P z z zP z n n r r ( ) ( ) ( )– ( ) ˜ ( ) = G ln ( ) -Ú - [ ] > -• p p q 1 r q 2 * * e d j 1 2 1 1 - s P z P z zs P z n n n nn ( ) = - - ( ) - ( ) ˜