Complex-coefficient variable filter design using successive vector-array-decomposition

Singular-value-decomposition (SVD) can be efficiently utilized to obtain the optimal vector-array-decomposition (VAD) for simplifying real-coefficient variable digital filter design problem, but the SVD-based VAD methods cannot be utilized to design complex-coefficient variable filters. The paper proposes a successive algorithm for decomposing an arbitrary multi-dimensional complex array into the VAD form, and thus a complex-coefficient digital filter with an arbitrary variable frequency response can be easily obtained through complex-coefficient constant filter design and multi-dimensional polynomial fitting. The new VAD algorithm successively decomposes the complex array and its residual arrays into the vector-array pairs stage-by-stage, and each stage contains an iterative optimization that can be easily solved in a closed-form. Our computer simulations have verified that the successive VAD converges very fast to the optimal solution.