Finding commutative multidimensional downsamplers and upsamplers

The commutativity of multidimensional downsamplers and upsamplers has been discussed very intensively for the past few years. This is due to its importance in sampling structure conversion, e.g., the conversion between conventional television signals and high definition television (HDTV) signals. Among many other results, a test for such commutativity was found to be that the two matrices that define the multidimensional downsampling and upsampling should be commutative and coprime. However, the problem of finding multidimensional downsamplers and upsamplers that satisfy these conditions has remained open. We develop a systematic procedure to solve this open problem