Generalization of the fast Hartley transform in single-base notation systems

The article proposes an original scalar method for synthesizing new algorithms of the generalized fast Hartley transform in a single-base notation system with an arbitrary radix. The proposed method extends the practical application area of the digital signal spectral processing in real time information management systems of different applications. The authors define the conditions under which fast algorithms exist in the generalized Hartley systems with the Paley, Hartmut and Hadamard functions sequence. The analytical descriptions of fast algorithms are given for various levels of different techniques of decimating the input signal and its spectrum. They are made for each type of Hartley system ordering. It is shown that all the developed fast algorithms are easily programmable iterative computational processes of a unified structure with some initial conditions. The latter are presented in the form of few point direct discrete Fourier transforms in the basis of the normal Hartley functions. The computational complexity of the developed fast algorithms is evaluated and the formulae for estimating the number of actual addition and multiplication operations are obtained. The complexity comparative assessment of both fast and direct algorithms of the Hartley spectrum generalized analysis is performed, which confirms the effectiveness of the obtained results.

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