TAILIEUCHUNG - The levanson - durbin algorithm

The numerical stability of the Levinson-Durbin algorithm for solving the Yule-Walker equations with a positive-definite symmetric Toeplitz matrix is studied. Arguments based on the analytic results of an error analysis for fixed-point and floating-point arithmetics show that the algorithm is stable and in fact comparable to the Cholesky algorithm. Conflicting evidence on the accuracy performance of the algorithm is explained by demonstrating that the underlying Toeplitz matrix is typically ill-conditioned in most applications | APPENDIX THE LEVINSON-DURBIN ALGORITHM The Levinson-Durbin algorithm is an order-recursive method for determining the solution to the set of linear equations a . A-l where is a p X p Toeplitz matrix is the vector of predictor coefficients expressed as ap2 - appi and b f is a p-dimensional vector with elements 1 M2 . Kp For a first-order p 1 predictor we have the solution 4 l đ 1 l 0 A-2 The residual mean square error MSE for the first-order predictor is ộ 0 -ữ 1 ộ l MO 1 - ơĩ. A-3 In general we may express the solution for the coefficients of an mth-order 879 880 DIGITAL COMMUNfCATIONS predictor in terms of the coefficients of the m - l th-order predictor. Thus we express am as the sum of two vectors namely I A-4 where the vector dm_j and the scalar km are to be determined. Also may be expressed as m IK . 0 0 J A-5 where t m is just the vector h fi. I in reverse order. Now r j 11 r - 1 r - 11 _ F - I 1 L b7 Wu 0 J lk J U I From A-6 we obtain two equations. The first is the matrix equation But d 1 Hence A-7 simplifies to bOT-idm. 0 This equation has the solution dOT-i A-Ó A-7 A-8 A-9 But is just Ị -1 in reverse order. Hence the solution in A-9 is simply am . I in reverse order multiplied by -kw. That is dm_ -km ữni- i t 2 A-10 La i The second equation obtained from A-6 is the scalar equation A-ll We eliminate dm-j from A-ll by use of A-10 . The resulting equation gives us km. That is .a ộ 0 a _ j 1 X A-12 APPENDIX A THE LEVJNSON-DUR0IN ALGORITHM 881 where I is the residual MSE given as d 0 A-13 By substituting A-10 for dm_t in A-4 we obtain the order-recursive relation a k Ik- k a . k 1 2 . m - 1 m 1 2 . tp A-14 and km The minimum MSE may also be computed recursively. We have K - ẳ A-15 1 Using A-14 in A-15 we obtain i 0 - s fl A-16 But the term in square brackets in A16 is just the numerator of k in A-12 . Hence. -a2m l-4 A-17

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