A New Proof of the Unitarity of the Multi-Dimensional Discrete Fourier Transform

The Multi-Dimensional Discrete Fourier Transform (MD-DFT) plays an important role in a growing number of signal processing applications. The fundaments 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 the following we provide a new proof based on the nullity of the cardinal function on the non-zero cardinal points.