Fourier Transform of a Single-Frequency Octonion

Abstract

Vadim Sovetov

A method for Fourier transforms in 8D space of a single-frequency octonion for 8 different pulses is presented. The octonion Fourier transform is calculated using the fundamental matrix of the octonion by integrating the product of the signal vector with this matrix. The inverse Fourier transform uses the transposed fundamental matrix multiplied by the spectrum vector. Spectra of different pulses were obtained. It is shown that the pulse spectra are formed using the theorem of shifting spectra along the frequency axis in accordance with their shift in time. In this case, the shift of the signal vector elements is performed using a direct and quadrature shift matrix. As is known, the octonion is used to form a MIMO system with 8 inputs and 8 outputs. Therefore, the spectra of each pulse are located on each axis of spatial coordinates of 8D space. In this case, we obtain the sum of the spectra of different pulses as elements of the output vector. Since the sum spectra are formed from the spectra of individual pulses shifted along the frequency axis in accordance with their time shift, the sum spectra have a wider frequency band and are more resistant to interference. The spatial coordinate axes are orthogonal, since the octonion-based fundamental matrix is orthogonal, therefore, the total spectra on the axes will be orthogonal and the pulses are separated during the inverse transformation.

PDF