On weak equivalence of linear systems and finite state systems

الملخص EN

It is shown that finite state machines exist which are weakly equivalent to linear systems for some nontrivial definitions of weak equivalence.

Two systems, one linear with state space R'\ the other with finite state spece, operate on the same stationary uncorrelated input sequence u.

The two systems have real valued output sequences y and y.

Notions of weak equivalence are formulated, which involve sets of second moments of the input and two outputs.

Power spectrum equivalence requires the E(ytynJ = E (y,yl+J for all t.

Cross-correlation equivalence requires that E (u:yt+T) = E (u,y,J for all r.

The interdependence of these and other notions of weak equivalence are studied.

The existence of weakly equivalent finite state systems is constructively demonstrated for a standard class of linear systems.