Real Discrete Fractional Fourier, Hartley, Generalized Fourier and Generalized Hartley Transforms With Many Parameters

TLDR: This paper proposes reality-preserving fractional versions of the discrete Fourier, Hartley, generalized Fouriers, and generalized Hartley transforms, which have random outputs and many parameters and thus are very flexible.

Abstract: Real transforms require less complexity for computations and less memory for storages than complex transforms. However, discrete fractional Fourier and Hartley transforms are complex transforms. In this paper, we propose reality-preserving fractional versions of the discrete Fourier, Hartley, generalized Fourier, and generalized Hartley transforms. All of the proposed real discrete fractional transforms have as many as $O(N^{2})$ parameters and thus are very flexible. The proposed real discrete fractional transforms have random eigenvectors and they have only two distinct eigenvalues 1 and $-$ 1. Properties and relationships of the proposed real discrete fractional transforms are investigated. Besides, for the real conventional discrete Hartley and generalized discrete Hartley transforms, we propose their alternative reality-preserving fractionalizations based on diagonal-like matrices to further increase their flexibility. The proposed real transforms have all of the required good properties to be discrete fractional transforms. Finally, since the proposed new transforms have random outputs and many parameters, they are all suitable for data security applications such as image encryption and watermarking.