Triangular Input Balanced Realizations

In a realization of the state space form
eqnarray121
the impulse response

is preserved by the action

that is,
eqnarray478
where T is a nonsingular matrix. This corresponds to the change of coordinates of the state vector

When we choose the filter parameters 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 actually is used at some point, and that the distribution of vectors actually fills up d-dimensions. 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

and determining c from the system

where

Note that

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

it follows that 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 , and the `Fourier' coefficients of the impulse response are the components of c.

remark580

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

and we use this choice to reduce A to a triangular matrix. A system where 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 in operations, as opposed to the usual for general A.

Next: Meixner Functions and TIB Up: Recursive Windowing of Time Previous: Comparison With a Rectangular

Fri Jun 27 03:10:38 EDT 1997