Multi-dimensional to one-dimensional form preserving transformations

This study deals with the structure of multidimensional polynomials under form preserving transformations that translate a N -dimensional polynomial into a lower (N-K) dimensional polynomial (the most important case occurs when K= N-1). A form preserving transformation is one that maintains a one-to-one correspondence between the coefficients of the original polynomial and the transformed polynomial. The authors have applied one such transformation to a 2-D polynomial to set up the normal equations involving a block Toeplitz matrix (for the purpose of least-squares stabilization) using the 1-D technique (IEEE Trans. Acoustics, Speech, and Signal Processing, vol.36, p.414-7, Mar. 1988). The authors extend the results to three- and higher-dimensional cases. Various types of form preserving transformations such as single-polynomial form preservation, two-polynomial (product) form preservation, etc. are studied. Further, the application of N-D to 1-D transformation to perform the convolution of two N-D signals is investigated