Near orthogonal discrete quaternion Fourier transform components via an optimal frequency rescaling approach

Abstract

: The quaternion-valued signals consist of four signal components. The discrete quaternion Fourier transform is to map these four signal components in the time domain to that in the frequency domain. These four signal components in the frequency domain are called the discrete quaternion Fourier transform components. There are a total of 16 inner products among any two discrete quaternion Fourier transform components. The total orthogonal error among the discrete quaternion Fourier transform components is defined based on these 16 inner products. This study aims to find the optimal quaternion number in the discrete quaternion Fourier transforms so that the total orthogonal errors among the discrete quaternion Fourier transform components are minimised. It is worth noting that finding the optimal quaternion number in the discrete quaternion Fourier transform is equivalent to finding the optimal rescaling factors. Since the discrete quaternion Fourier transform components are expressed in terms of the high-order polynomials of the trigonometric functions of the rescaling factors, this optimisation problem is non-convex. To address this problem, a two-stage approach is employed for finding the solution to the optimisation problem. The comparison results show that the authors proposed method outperforms the existing methods in terms of achieving the low total orthogonal error among the discrete quaternion Fourier transform components.