QUATERNION DOMAIN DISCRETE FOURIER TRANSFORM OF THE PULSES VECTOR

Abstract

Abstract— At present, technologies of digital representation of signals are becoming more widespread in information transmission systems. The discrete Fourier transform (DFT) is used to obtain the spectrum of such signals. The DFT is represented as a finite sum of signal samples multiplied by a discrete complex exponent.

Modern communication systems use vector signals and many-input – many-output technologies (MIMO). These technologies can significantly increase the capacity of communication channels. As a mathematical model of a MIMO channel, multidimensional orthogonal matrices are usually used. An information vector of pulses are suppled at the input of the channel matrix. The DFT of multidimensional signals is written as a multiple sum of samples for a separate variables, that multiplied by the corresponding discrete complex exponent. The DFT of multidimensional signals in matrix form is written as a sum in the sample area, and the vector of samples is multiplied by the matrix complex exponent from the inverse diagonal matrix of the number of samples. This representation uses the same imaginary unit and, in despite of the multidimensional nature of the signals, they are considered on the same complex plane.

The purpose of the work is to present a technique for the discrete Fourier transform of a four-dimensional pulse vector, the kernel of which is a quaternion in the matrix representation. This technique considers a 4-dimensional signal in 4D quaternion space with three imaginary units and one scalar in which the MIMO signal is formed.

The exponential function from the quaternion provides a conformal mapping of 4D hypercomplex space into 4D hypercomplex space with a constant rotation radius. The exponential function of a quaternion is represented as a fundamental matrix of a quaternion carrier with discrete counts for time and frequency. Using the discrete fundamental matrices of the quaternion carrier, formulas have been obtained for the direct and inverse discrete Fourier transform of the four-dimensional impulses vector.

Formulas have been obtained for calculating the spectra of the elements of the pulse vector with the same and different cyclic shifts in time and frequency. The Parseval’s identity of the quaternion discrete Fourier transform has been proven.

The calculations of vector spectra of various pulses have been presented: rectangle, sawtooth, sine, cosine with different time shifts. The trajectory of movement of quaternion scalar values in time in 3D has been shown for various cyclic shifts and projections of trajectories onto 4 orthogonal planes.