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Harmonic differential calculus and filtering in Galois fields

The spaces of functions having a finite abelian group as domain and a Galois field as co-domain have found interesting application in digital signal processing: the group characters form an orthogonal basis in such a space, and the corresponding Fourier transform can be used to perform fast convolution with no computation error. This note considers the problem of optimality in designing signal processors in Galois fields; and tries to elucidate the significance of operations in the frequency domain related to the respective Galois-field Fourier transform. This is done by introducing a class of operators on the signal space such that the characters are their eigenfunctions, in the same way that the complex exponentials are the eigenfunctions of the classical newtonian differentiator. Starting from this definition (originally introduced by J.E. Gibbs in connection with the Walsh functions), we show that the harmonic differentiators thus defined have many properties in common with the classical differentiator. Applications of harmonic differential calculus to a 'harmonic' state-space analysis of finite-valued signal processors are investigated. The note also examines discrete Laplace (z-) transforms in Galois fields.