Efficient discrete fractional Hirschman optimal transform and its application

All of the existing TV-point discrete fractional signal transforms require O(N²) computation complexity. In this paper, we propose a new discrete fractional signal transform whose computation complexity can be reduced to O(N^1.5). This new transform is a fractional version of a DFT-based signal transform called as the Hirschman optimal transform (HOT) in the literature. Eigenvalues and eigenvectors properties of the HOT are also developed. Moreover, the proposed discrete fractional HOT transform is extended to further reduce the required computation complexity to linear order O(N). As an application example, we apply this new computationally efficient discrete fractional signal transform to encrypt digital images.