Abstract:
The one-dimensional quaternion fractional Fourier transform (1DQFRFT) introduces a fractional-order parameter that extends traditional Fourier transform techniques,
providing new insights into the analysis of quaternion-valued signals. This paper presents
a rigorous theoretical foundation for the 1DQFRFT, examining essential properties such
as linearity, the Plancherel theorem, conjugate symmetry, convolution, and a generalized
Parseval’s theorem that collectively demonstrate the transform’s analytical power. We
further explore the 1DQFRFT’s unique applications to probabilistic methods, particularly
for modeling and analyzing stochastic processes within a quaternionic framework. By
bridging quaternionic theory with probability, our study opens avenues for advanced
applications in signal processing, communications, and applied mathematics, potentially
driving significant advancements in these fields.
