An important theorem for the singular value decomposition of variable fractional-delay specification

摘要

This paper proves a very important theorem for the singular value decomposition (SVD) of variable fractional-delay (VFD) design specification. We show that the VFD design specification can be decomposed into the outer-products of vectors with mirror-image symmetry or mirror-image anti-symmetry, thus the difficult VFD filter design problem can be reduced to the sub-problems involving constant one-dimensional (1-D) filter designs and 1-D polynomial approximations. If the resulting vectors from the SVD are mirror-image symmetrical, then constant 1-D filters with symmetrical coefficients and 1-D polynomials with even degrees are used to approximate those vectors. Conversely, if the resulting vectors from the SVD are mirror-image anti-symmetrical, then constant 1-D filters with anti-symmetrical coefficients and 1-D polynomials with odd degrees are used. Since such sub-problems are much easier to solve than the original VFD filter design, the important theorem provides an elegant way for simplifying the VFD filter design problem as easier sub-problems and leads to an indirect and efficient design approach.