The optimal decomposition (OD) technique for decomposing 2-D magnitude specifications into 1-D ones is proposed. Differing from the conventional matrix decomposition methods such as the singular value decomposition (SVD), the OD of a 2-D magnitude specification matrix results in 1-D magnitude specifications which are always non-negative while the root mean-squared (rms) error in decomposition is minimum. Therefore, the OD is more suitable for designing 2-D digital filters (2DDFs) through designing one-dimensional digital filters (1DDFs) than the conventional matrix decomposition methods. Based on the OD, the problem of designing a recursive 2DDF can be reduced to one of designing a pair of 1DDFs, one 1-input/multi-output, and the other multi-input/ 1-output. Consequently, 2DDFs can easily be obtained by designing 1DDFs using the well-established 1-D design techniques. Three design examples are presented to illustrate that this OD-based 2DDF design technique is extremely efficient.
