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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
eqnarray121
the impulse response
equation148
is preserved by the action
equation474
that is,
eqnarray478
where T is a nonsingular tex2html_wrap_inline1130 matrix. This corresponds to the change of coordinates of the state vector
equation488
When we choose the filter parameters tex2html_wrap_inline1254 we can exploit this symmetry to obtain choices which, although having the same signal properties, have superior numerical or computational properties. Since these realizations have the same impulse response, they are called observationally equivalent.

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 tex2html_wrap_inline1120 actually is used at some point, and that the distribution of tex2html_wrap_inline1120 vectors actually fills up d-dimensions. However, it can happen that a minimal realization of an impulse response involves coordinates for tex2html_wrap_inline1120 in which the vectors tex2html_wrap_inline1120 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 tex2html_wrap_inline1120 have similar scaling.

It is impossible to obtain this property without any information about tex2html_wrap_inline1092. However, if it is known that tex2html_wrap_inline1092 is relatively noisy, then it is possible to get good results by choosing the coordinates for tex2html_wrap_inline1120 in such a way that tex2html_wrap_inline1120 would be spherically symmetrically distributed if the tex2html_wrap_inline1092 were independent and identically distributed. It turns out that this means choosing tex2html_wrap_inline1280 so that Stein's equation is satisfied
equation506
and determining c from the system
equation511
where
equation521
Note that
equation528
but that thanks to the Stein equation, we have
eqnarray537
which converges as long as A is stable. (We are only interested in stable A, so we can always use this). Since
equation567
it follows that tex2html_wrap_inline1288 is column unitary, and we can solve for the least squares approximation to c by
equation571
This corresponds to expanding the impulse response in the orthogonal functions given by the columns of tex2html_wrap_inline1288, and the `Fourier' coefficients of the impulse response are the components of c.


remark580

We have not exhausted the possible choices of coordinates for tex2html_wrap_inline1120, 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
equation599
and we use this choice to reduce A to a triangular matrix. A system where tex2html_wrap_inline1280 satisfy the Stein equation with A triangular is called a Triangular Input Balanced (TIB) system. This choice is particularly good since in addition to the good numerical properties implied by the Stein equation, it is possible to compute tex2html_wrap_inline1318 in tex2html_wrap_inline1320 operations, as opposed to the usual tex2html_wrap_inline1322 for general A.




next up previous
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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