АНАЛІЗ СПЕКТРАЛЬНИХ МЕТОДІВ ОБРОБКИ КОДОВИХ ПОСЛІДОВНОСТЕЙ В ЦИФРОВИХ КАНАЛАХ ЗВ’ЯЗКУ

The classical Herzel algorithm for calculating the forward and inverse discrete Fourier transform for elements of the finite field GF(2m) is considered, as well as its modification. It is shown that the modified algorithm for finding error positions in a code sequence belongs to the class of fast algorithms for calculating the discrete Fourier transform rather than to the class of semi-fast ones. An analysis of existing methods for decoding code sequences of cyclic binary codes is carried out, which allow to identify and correct errors that arise as a result of the action of external interference. Means for calculating the polynomial of error locators in the time and spectral domains are presented to assess the efficiency of the work. The main advantages of the spectral approach are determined, which makes it possible to use the forward Fourier transform in finite Galois fields. It is shown that decoding BCH codes in the spectral domain using the Fourier transform for finding errors and correcting codewords significantly speeds up decoding, especially for long codes. A method for calculating spectral components with the definition of the syndrome vector using the polynomial of the remainder from division by the minimal polynomial is considered. The algorithm makes it possible to obtain the value of the polynomial at all points of the finite Galois field for a smaller number of addition and multiplication operations, due to the fact that the remainder polynomial is calculated once for each connected class. An algorithm for finding n–2t spectral components of the error vector through the known t coefficients of the polynomial of error locators and the known 2t syndrome components calculated at the first stage of decoding is proposed. It is confirmed that the proposed spectral method for decoding BCH codes based on the use of the fast Fourier transform in Galois fields allows to speed up the calculation of syndromes and finding the polynomial of error locators without iterations, namely, using the Fourier transform instead of the Berlekamp-Massy algorithm.