Beginning with an external specification of a digital filter, structures which minimize roundoff noise are investigated. After fixing the probability of overflow through an l_{2} scaling procedure, roundoff noise is studied via the internal structure of the filter using a state variable formulation. An output noise variance formula in terms of the internal structure is derived. Conditions for minimizing this output noise are established and realizations which meet these conditions are constructed. A new set of filter invariants called second-order modes are defined and shown to play a definitive role in minimal noise realizations. From these invariants, for example, one can calculate the minimal output noise variance of a given external specification. Numerical results are given which compare these new filter structures with the usual parallel and cascade connections of second-order filters, both theoretically and through simulations. For narrow-band filters, these new structures can be orders of magnitude better (in terms of output noise variance). One drawback of these new structures is a large increase in the number of multipliers needed to realize them. However, by applying the theory to subfilters connected in parallel and cascade, a good compromise between output noise and number of multipliers is obtained.

Synthesis of minimum roundoff noise fixed point digital filters

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

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Computational Methods for Linear Matrix Equations

New properties of the Schwarz matrix associated with a prescribed polynomial are derived. A matrix analogous to the Schuarz matrix is described which is useful in studying the root distribution of a prescribed polynomial relative to the unit circle.

Schwarz matrix properties for continuous and discrete time systems

A previously presented technique for solving the extended Lyapunov matrix equation is applied to the extended discrete Lyapunov matrix equation. The system matrix is first transformed to its Jordan canonical form. Then an explicit solution is obtained for the equivalent equation which results.

A technique for solving the extended discrete Lyapunov matrix equation

The output roundoff noise variance and the scaling of the internal registers are associated with two impulse response sequences. Furthermore, two Lyapunov equations are established, corresponding to those which have been already used for the noise analysis and optimal structure synthesis in the 1-D case. An analytic formula is given for the solution of the Lyapunov equations. Thus the known techniques from the 1-D filters can be directly extended.

On the roundoff noise in 2-D state-space digital filtering

Matrices W i and N_{i}, which result from application of Krylov's algorithm to the matrices A and B related by A = (B + I)(B - I)^{-1}, are shown to be row equivalent, i.e., N_{i} = M_{i}W_{1} . This result is applied to solution of the Lyapunov matrix equation for discrete-time systems, A'LA - L = -K .

http://ieeexplore.ieee.org/iel5/9/24108/01099187.pdf

A note on the matrix equation A' LA-L = -K

Krylov's algorithm is used to unify and extend recent work on relating a matrix to its companion form and the companion matrix to the Schwarz form. A general solution to the "inverse problem of stability" is presented and illustrated.

H. Power

Papers

A note on the matrix equation A' LA-L = -K

Similarity transformations involving the schwarz and companion matrices