Cybernetics or Control and Communication in the Animal and the Machine by Norbert Wiener

Author:Norbert Wiener
Language: eng
Format: mobi, pdf
Published: 2009-11-18T23:31:00+00:00

a(T) f (t - T) dT (4.03)

where a(T) does not effectively vanish for negative values of -r, it means that we have no longer a true operator on f (t), determined uniquely by its past. There are physical cases where this may occur. For example, a dynamical system with no input may go into permanent oscillation, or even oscillation building up to infinity, with an undetermined amplitude. In such a case, the future of the system is not determined by the past, and we may in appearance find a formalism which suggests an operator dependent on the future.

The operation by which we obtain Expression 4.02 from f (t) has two important further properties: (1) it is independent of a shift of the origin of time, and (2) it is linear. The first property is expressed by the statement that if

g(t) = f000 a(T) f (t - T) dT (4.04)

then

a(T)f(t + a - T) dT g(t + a) = f000 (4.05)

The second property is expressed by the statement that if

g(t) = Afl(t) + Bf2(t) (4.06)

then

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