By Jan Zarzycki

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**Additional resources for Nonlinear prediction ladder-filters for higher-order stochastic sequences**

**Example text**

N) ( re=n+ i) -~ L x'l n,m+3 + "'" -~ L x'n-I L x)n n + l ) m - i "~ n + l , m termination: L x)n n+l)m" We notice that e a c h is associated (N=3) 'local' order-update with 'label-update' index-set recursions The index-set n e a r ladder-filter step. T h e is p r e s e n t e d recursions, 6dgorithms step for the b a c k w a r d of the third-order in Fig. 3. derived presented 'Iota/' structure index-sets in this section, will underly in the s u b s e q u e n t paragraphs nonliof this chapter. 2.

Of the set of M - D {M} Z N the M - D trarzsfer func- of the generalized z-polyno- spectral flxnctions of the hi- 3. G E N E R A L I Z E D NONLINEAR In flue previous prediction problem tly introduced LADDER-FILTERS chapter we for higher-order algebraically a n d mains. Algebraic showed solution of the problem, algorithh~, w a s (i983a, b); Z a r z y c k i (1983); in Z a r z y c k i in the s p a c e s : can in time- a n d be in Z a r z y c k i derivation equivalen- in frequency-dogenera/iza- o/qd D e w i l d e of the nonlinear predicti- of the Volterra functional polynomials (1984~a-e).

N~m Now let Ix = n~m where ljl for "~ [ (ki,k2) ~kl;j. I [llX n,m 21x ] = [i. n,m Jl 1.. 11b) 50 with I0 and 20 ctively, w h o s e being the o n e - domains Lx respectively. 13~) n,m X,V G X , V ~ --b~X'V°HX'V-G x'v Fn, m ' n,m j ]~x,v n,m n,m n,m ntm According (expressed _o,N KN,N+ 1 ° will b e D b ~ % v .. b x'v . Lx . Similarly, e a c h n~m nsm will b e introduce v precisely the s p a c e of the s u b s p a c e will b e F x . 14~) 51 B-forward (for v = O .... 15b) 52 Let 0jl denote the zero-entry with 'coordinate' the L-forward ^ n-lh order approximation error, c o r r e s p o n d i n g to the estimate ix , will n,o be expressed as x =A p l x+i,n-I i - I - [0 An,o --~ ;n-l,n x x o (since the estimate x ~n,o).