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Next: Meixner Functions and TIB Up: Recursive Windowing of Time Previous: Comparison With a Rectangular
Triangular Input Balanced Realizations
In a realization of the state space form
After imposing some more or less unobjectionable constraints on the filter, it turns out that any two observationally equivalent systems are related by this type of coordinate change. Note: we should go into minimality, observability and reachability in somewhat more detail here. A system of dimension d is called minimal if it is not possible to realize the impulse response with a system of lower dimension. This means that each component of actually is used at some point, and that the distribution of vectors actually fills up ddimensions. However, it can happen that a minimal realization of an impulse response involves coordinates for in which the vectors are distributed in such a way that the variation in some directions is very small compared to others, i.e. that the ellipsoids of constant probability are very far from spheres. It should not be too surprising that this can expose a finite precision realization of the system to numerical pathologies. It is therefore desirable that all the components of have similar scaling.
It is impossible to obtain this property without any information about . However, if it is known that is relatively noisy, then it
is possible to get good results by choosing the coordinates for in
such a way that would be spherically symmetrically distributed if
the were independent and identically distributed. It turns out that
this means choosing so that Stein's equation is
satisfied
We have not exhausted the possible choices of coordinates for , since
by Shur's decomposition, we can always arrange for A to be triangular with
a unitary choice of T. The unitarity of T preserves the Stein equation
Next: Meixner Functions and TIB Up: Recursive Windowing of Time Previous: Comparison With a Rectangular Fri Jun 27 03:10:38 EDT 1997 
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