Protsko I. Enhancement of the Efficient Computation of Real Harmonic Transforms Based on Cyclic Convolutions

The thesis is devoted to the research and development of an efficient approach to the computation of real discrete harmonic transforms based on cyclic convolutions and to the solution of the problems related to the analysis and synthesis of multiversion efficient algorithms for the computation of real transforms with cosine, sine, casine basis functions. The scientific and technical problems of the analysis and synthesis of efficient algorithms for computation of DCT, DST, DHT and their types using cyclic convolutions is resolved; the usage of a hashing array, which is formed via the cyclic of substitution of the basis matrix arguments, is proposed; the technique of the analysis of block cyclic structure of basis discrete harmonic transform using hashing arrays is developed. In result of analysis the efficient methods of the computation of the discrete transforms of Fourier class based on cyclic convolutions, the general approach to the synthesis of efficient algorithms for the computation of real harmonic transforms and their types is developed. The peculiarities of synthesis algorithms for efficient computation of each of the four types of DCT, DST, DHT are determined. The efficient usage for the synthesis of a hashing array compared with the primitive elements of cyclic groups is proved. The computations of cyclic convolutions for sequences of data with the repetitive groups of the harmonic coefficients, which occur in the proposed algorithms, reduce computational complexity of the algorithms. The computing structures that perform real discrete harmonic transforms based on cyclic convolution and the components of the efficient computation of fast cyclic convolutions are developed. The software implementation of the synthesis of fast algorithms for the computation of real discrete harmonic transforms based on cyclic convolutions is described.