By Dr. Herbert Arndt (auth.), L. Collatz, G. Meinardus, H. Werner (eds.)
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Additional resources for Numerical Methods of Approximation Theory, Vol. 7 / Numerische Methoden der Approximationstheorie, Band 7: Workshop on Numerical Methods of Approximation Theory Oberwolfach, March 20–26, 1983 / Tagung über Numerische Methoden der Approximationstheorie Obe
N of Q(m) (z) is _SCm) and that Q(m-l) (z) n n n zn in Q(m) (z) is _a(m) n n n Remark that the constant term _Q(m) (z)/S(m). n n The coefficient of ALGORITHM A2 Define a projection operation for formal Laurent series '" k lla:b( ~'" sk z ) = b r sk z k a Furthermore we introduce the notations 0+ and 0_ as follows. Any Laurent- series S(Z) with ll_"':m+l S(z) = 0 is denoted by o+(zm) and if llm+l:'" S(z) =0, m then by O_(z). '" fk z k (F(z) Q(m) (z» n llm+l:'" (F (z) Q~m) (z» 37 p(m) (z) n R(m) (z) IT n m: co (F(z) ~(m) zm+n + 0 (zm+n+1) n + Q(m) (z» n It is not difficult to see that (P,P) and (R,R) obey the same recursions as (Q,Q).
K hk . h. ) ~ L f -~ k e ika Thus to obtain a least squares predictive filter we have to solve the following mathematical problem : Given a function F such F(e -ia) F(e ia ) that (even) (Fl) ia F(e ) ;;;. (H2) This formulation is also applicable for general ARMA (Autoregressive-moving average) filters. e. where the approximation for H is a general rational 35 function. Since in F we have only information on the square of H we should decide in the general ARMA case where we want the zeros of H to make the problem uniquely solvable.
Run+n +m+ 1 (1) where the multiplication in Rq is performed cornponentwise. Q-P) all the terms of degree less than run+n+m+1 disappear. The shift of the degrees in P(x) and Q(x) over run can be motivated as follows. Condition (1) is equivalent with the following two systems of equations. B xnrn+n = A xnrn+n o nrn+n nrn+n with Bnm+J. xnrn+j = 0 for j - wi th '1c xk 0; 0 for k < > vx E nf m o. A solution of (lb) can be computed by means of the following determinants in Rq; these formulas are direct generalizations of the classical formulas for the solution of a homogeneous system : 62 Bnm xnm = C x n + 1- m C x n + 2- m n+l-m n+l Cn+ 1 x n+2-m n+m-l Cn+m- 1 x -Cn+ 1 Cn +m- 1 x n+m-l n+l x C n+l-m xn+ 1- m n+m - Cn+m x I j th coltDln in B xnm replaced by this col\Jlln nm j=l, ...
Numerical Methods of Approximation Theory, Vol. 7 / Numerische Methoden der Approximationstheorie, Band 7: Workshop on Numerical Methods of Approximation Theory Oberwolfach, March 20–26, 1983 / Tagung über Numerische Methoden der Approximationstheorie Obe by Dr. Herbert Arndt (auth.), L. Collatz, G. Meinardus, H. Werner (eds.)