Synchronization of Gold sequences based on fast transform in a truncated basis of Walsh–Hadamard functions

Based on the analysis of the structures of isomorphic multiplicative groups of extended Galois fields, it is established that any cyclic shift of a pseudo-random Gold sequence can be transformed into a function belonging to the complete set of analogues of Rademacher functions of the corresponding dimension. This made it possible to develop a new algorithm for fast synchronization of Gold sequences based on the calculation of their discrete convolution using fast spectral transformation in a truncated basis of Walsh–Hadamard functions. The gain of the developed algorithm in terms of the number of arithmetic operations compared to the traditional method of calculating discrete convolution increases with increasing sequence length N and for N=511.1023 is approximately 3.4 times.