I.V. Demichev1

1 Military University of Radio Electronics (Cherepovets, Russia)

1 vure@mil.ru

The use of quaternion algebra in solving a number of electrodynamic problems allows one to remove certain limitations of vector calculus. At the same time, the implementation of operations widely used in signal processing, as well as optimal algorithms for their calculation (Fourier transform, convolution, correlation, etc.) does not allow one to obtain an adequate result due to the non-commutativity of quaternion multiplication, which requires searching for new approaches in implementing mathematical calculations. The article discusses the features of calculating the direct and inverse Fourier transform of quaternion functions based on the Cayley-Dickson expansion, and provides the concept of a two-sided Fourier transform. A formulation of the convolution theorem for quaternion functions is presented, showing the need to perform additional transformations of input and output data when calculating convolution and correlation in the frequency domain. Expressions for calculating convolution and correlation of quaternion functions are obtained. The features of calculating the convolution and correlation of quaternion functions for discrete time based on widely used algorithms of the complex discrete Fourier transform are considered. The adequacy of the obtained results of calculating the convolution and correlation is assessed by means of simulation modeling and comparison of calculations in the time and frequency domains. The proposed approach to calculating the convolution and correlation of quaternion functions not only expands the capabilities of hypercomplex processing of electromagnetic fields, but also allows its implementation based on known optimized algorithms for calculating the Fourier transform, which opens up new possibilities in the field of radio engineering.

Demichev I.V. Methods for calculating convolution and correlation of quaternion functions. Electromagnetic waves and electronic systems. 2024. V. 29. № 6. P. 46−53. DOI: https://doi.org/10.18127/j15604128-202406-06 (in Russian)

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