By Dr. sc. nat. M. Göossel (eds.)
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Extra info for Nonlinear Time-discrete Systems: A General Approach by Nonlinear Superposition
Sample text
O, O < k < t . Thez we show ak+ 1 = ... i I = 0 if normal = a k = I, i I > k, (1 m 2) (24) We suppose that Eq. (24) is not true. m+l = 1 , i ~ 2, m + I > k and 1 minimal. Then we have gt~0, .... ,1,Xm+1,0,.. ,0) = dlXm+ I + Xm+ I Xm+ 1 + d2, dl,d 2 ~ ~0,I} in contradiction to Eq. (23). i I = 1 if il~ k, (1 ~ 2). (25) We suppose that Eq. (25) is not true. Let k > 2 and let without loss of generality al... 1 -- 0, 2 ~ 1 ~ k, and 1 minimal. ,O) -- x I + x 2 in contradiction to Eq. (23). If 1 > 2, we have g t ( x l , x 2 , ~ , O , .
0), ~ (I 0 ... 0),... ol), ,I(O 1 . . I ) . . . O I ) . Such linear automata for which the input-output-behaviour can be described by a finite set of periodic inpulse responses are of special interest. I_~. Let ~ be a non-time-dependent automaton with z(0) N O. Since l(Oil O R = 0 i ~(I OG~, i ~ O, the input-output-behaviour of ~ is determined by A ( 1 0 m ) . If ~ is finite, the impulse response ~(I 0 m ) is periodic. The realizations LSC I or LSC H (cf. Figs. 3 and 4) and the corresponding linear automata L I or ~ can be formed as described before.
Periodic Every M > i can be uniquely represented as M : i + N m+ K with O ~K