Proof of unitarity of multidimensional discrete Fourier transform

Abstract

The multidimensional discrete Fourier transform (MD-DFT) plays an important role in a growing number of signal processing applications. The fundamentals of its applicability as a unitary transform between discrete periodic sequences defined on multidimensional lattices stand on the Hermitian orthogonality of the vectors defining the MD-DFT matrix. A proof of the consistency of the MD-DFT formulation was first provided by Bernardini and Manduchi in 1994 using the Smith normal form theorem of integer matrices. In this reported work, a new proof is provided based on the nullity of the cardinal function on the nonzero cardinal points. © The Institution of Engineering and Technology 2013.