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FUNDER - Theoretical Background |
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SYSTEMS - FUNCTIONAL REPRESENTATION FUNCTIONALS Functionals map functions to numbers (or to other functions). They can be understood as a generalization of multivariate functions. Consider the vectors
and
where the elements of
Vectors
Increasing the number of discrete-time points, for a fixed time interval
where
the response,
Note that in this approach the emphasis is on the relationship between the input and the output, and not between the mechanism of the system and the output. Extremely complex systems can be represented by eq. (4). LINEAR SYSTEMS
A simple way to illustrate the meaning of functionals is to derive a
general one
for linear systems.
First, we recall that any signal can be constructed as
a superposition of deltas.
Say that we know the response to a unit impulse at time
or, in functional notation,
Using eq. (9) of the Dirac's delta page, the response to any excitation is
From the additivity of linear systems, the response becomes
Furthermore, applying the homogeneity property, we have that
Finally, substituting the impulse response, eq. (7), a general functional of the input results:
This expression further simplifies if the system is time-invariant, which requires that
For this important class of systems, eq. (11) becomes a convolution product:
The above derivations show that one need not know the mechanism of a linear system for being able to predict its response to any excitation. Furthermore, if the system is time-invariant, the only information required for completely characterising its response is an impulse-perturbation experiment at any time. The following pages give an approach of this kind for nonlinear systems.
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are
and
time points of the input,
,
and output,
,
of a system, respectively.
,
vectors tend to functions. In the continuous limit, the function
becomes a functional. Introducing the notation
.
The generalization to
systems with more than one input is obvious.
,
