S. L. Zlobin

New matrix recurrent fast Hartley transform algorithms (FHT) of radixes 2 and 4 with time decimation and with frequency decimation with natural sequence of input and output information are described in the article. At present in digital signal processing (DSP) Hartley transform is widely used, which suits well the processing of real arrays (one-dimensional, two-dimensional and multidimensional). Hartley transform is closely connected with Fourier transform, but allows to do without complex numbers arithmetic. By definition discrete Hartley transform (DHT) is intended for real arrays processing and is simply connected with discrete Fourier transform. Hartley transform is a modification of Fourier transform in the plane of the real variable. Unlike Fourier transform, complex arithmetic is absent in Hartley transform; data processing is fulfilled only in the real numbers field. Many processes require DSP application on the real time scale. Therefore the researches of the fast algorithms of Hartley transform calculation - FHT algorithms, acquired particularly vital interest. Most often because of the convenience of the apparatus and program realizations FHT algorithms of radixes 2 and 4 are used in DSP practice. A number of FHT matrix algorithms of radixes 2 and 4 with time decimation and with frequency decimation were elaborated with the help of system approach to the synthesis of the structure of matrix algorithms of any radix. The algorithms formulas were obtained with the help of DHT shift theorem and DHT extension theorem. Matrix formulas of radixes 2 and 4 algorithms with time decimation and with frequency decimation are given in the article. Besides, non-matrix iterative, recurrent formulas of these algorithms are given too. In some cases these formulas can be more convenient for the programming of the obtained algorithms in practice. Matrix formula entries of new FHT matrix recurrent algorithms of radixes 2 and 4 with time decimation of increased fast-action are presented in the article. Formulas for the basic operations of the algorithms ? «butterflies» are given for all acquired algorithms. Formulas for the computation of complete number of addition operations and of multiplication operations and tables of the comparative characteristics of the algorithms are given too. Matrix diagrams of the algorithms are presented as graphic interpretation. It is demonstrated analytically, and as a result of computation experiment also, that the speediest is the new FHT matrix algorithm of radix 4 with time decimation of increased fast-action.