SVD-based theorem for designing variable digital filters

Arbitrary desired variable frequency response can be uniformly sampled to construct a multi-dimensional (M-D) complex array. In this paper, we propose a new method called vector-array decomposition (VAD) for decomposing M-D complex array into the products of complex vectors and real arrays. Based on the VAD, the difficult problem of designing variable digital filters can be reduced to some easier sub-problems that require one-dimensional (1-D) constant filter designs and M-D polynomial approximations. Since 1-D constant filters can be easily obtained by applying the well developed design techniques, and M-D polynomials can be obtained by utilizing least-squares curve-fitting, variable filters can be indirectly designed through solving the easier sub-problems. The VAD-based approach is straightforward and particularly efficient for designing variable filters with arbitrary variable magnitude responses and arbitrary phases.