A note on optimum linear multivariable filters

The explicit solution for the optimum linear physically realizable multivariable filter involves the factorization of a power-spectra matrix into two matrices, one having all its poles in the left-half p-plane and the other having all its poles in the right-half p-plane. No general method of accomplishing this factorization has previously been available. This note contributes a method of factorizing any power-spectra matrix in the required manner. As a result, the explicit solution for the optimum filter is obtainable in a number of cases not previously solvable without resort to implicit methods. In the course of developing the factorization method it is shown that it is always possible to obtain a physically realizable multivariable system which will transform any given set of signals into an equal number of incoherent white-noise signals. Similarly it is shown that a physically realizable multivariable shaping filter may always be found to transform a set of incoherent white-noise signals into an equal number of signals with any desired power-spectra matrix.