Complex and hypercomplex discrete Fourier transforms based on matrix exponential form of Euler's formula

We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of different researchers can be unified into a single theoretical framework based on a matrix exponential version of Euler’s formula e = cos θ + j sin θ, and a matrix root of −1 isomorphic to the imaginary root j. The transforms thus defined can be computed numerically using standard matrix multiplications and additions with no hypercomplex code, the complex or hypercomplex algebra being represented by the form of the matrix root of −1, so that the matrix multiplications are equivalent to multiplications in the appropriate algebra. We present examples from the complex, quaternion and biquaternion algebras, and from Clifford algebras Cl1,1 and Cl2,0. The significance of this result is both in the theoretical unification achieved, and also in the scope it affords for insight into the structure of the various transforms, since the formulation is such a simple generalization of the classic complex case. It also shows that hypercomplex discrete Fourier transforms may be evaluated numerically using standard matrix arithmetic packages, which is of importance in providing a reference implementation for testing implementations based on hypercomplex libraries.