Arriving at a good combination of coding and modulation schemes that can achieve good error correction constitutes a challenge in digital communication systems. In this work, we explore the combination of permutation coding (PC) and pulse amplitude modulation (PAM) for mitigating channel errors in the presence of background noise and jitter. Since PAM is characterised with bi-polar constellations, Euclidean distance is a good choice for predicting the performance of such coded modulation setup. In order to address certain challenges facing PCs, we therefore introduce injections in the coding system, together with a modified form of PAM system. This modification entails constraining the PAM constellations to the size of the codeword’s symbol. The results obtained demonstrate the strength of the modified coded PAM system over the conventional PC coded PAM system.
It is well known that the magnitudes of the coefficients of the discrete Fourier transform (DFT) are invariant under certain operations on the input data. In this paper, the effects of rearranging the elements of an input data on its DFT are studied. In the one-dimensional case, the effects of permuting the elements of a finite sequence of length N on its discrete Fourier transform (DFT) coefficients are investigated. The permutations that leave the unordered collection of Fourier coefficients and their magnitudes invariant are completely characterized. Conditions under which two different permutations give the same DFT coefficient magnitudes are given. The characterizations are based on the automorphism group of the additive group ZN of integers modulo N and the group of translations of ZN. As an application of the results presented, a generalization of the theorem characterizing all permutations that commute with the discrete Fourier transform is given. Numerical examples illustrate the obtained results. Possible generalizations and open problems are discussed. In higher dimensions, results on the effects of certain geometric transformations of an input data array on its DFT are given and illustrated with an example.