Real-Valued Discrete Fractional Hadamard Transform: Fast Algorithms and Implementations

This paper introduces a new real-valued discrete fractional Hadamard transform (RFHT) designed to address the issues of high computational complexity and large storage demands found in the traditional discrete fractional Hadamard transform (FHT) and its modifications. Additionally, a fast algorithm for the RFHT is developed, and the corresponding computational complexity analysis demonstrates that for sizes ranging from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N = 2$ </tex-math></inline-formula> to 1024, the proposed fast algorithm can reduce the number of multiplications and additions by up to 50.0% and 83.3%, respectively, compared to state-of-the-art fast algorithms. The comparison results show that the RFHT also has lower execution time and power consumption. Furthermore, due to its real-valued property, the RFHT has been applied in an image watermarking system and implemented on real iOS devices, demonstrating enhanced information security. Lower execution and power consumption, reduced storage and transmission requirements, and superior information protection make the RFHT a superior candidate compared to WHT, FHT, and its modifications.