Improving the Energy Efficiency of 5G and 6G Systems with Linear Recurrent Sequence Synchronization Methods in the Vilenkin-Crestenson Function Basis

The synchronization of a multivariate linear recurrent sequence of maximum period can be achieved using the generalized fast Fourier transform based on the VilenkinCrestenson function. This is done by rearranging the sequence's symbols according to the increasing values of the elements of a given multiplicative group within an extended Galois field, formed based on an irreducible primitive polynomial. This permutation transforms the original sequence into a discrete Vilenkin-Crestenson function without a zero symbol. To reveal the correspondence between the obtained function and the cyclic shift of the transformed linear recurrence sequence, consider methods of transforming circulant matrices of linear recurrence sequences into Vilenkin-Crestenson function matrices. Using the proposed method achieves a fourfold increase in the number of elementary arithmetic operations. Using ternary linear recurrent sequences and ternary computers reduces the computational complexity of the algorithm by one and a half to two times compared to the usual fast Fourier transform. In total, this is about a 4.5- to 6-fold improvement over synchronizing binary Msequences using the fast Adamar transform.