Unified Time–Frequency–Phase Analysis of Speech Using Quaternion Linear Canonical Transform

The Quaternion Linear Canonical Transform (QLCT) provides a unified time-frequency-phase representation for multidimensional signals. However, its applications have largely been limited to theoretical studies and image processing, leaving speech an inherently one-dimensional non-stationary signal unexplored. This paper presents a Discrete QLCT (DQLCT) implementation that operates directly on real-valued temporal signals by mapping them into the quaternion domain via analytic signal representation. The transform is derived from first principles, with invertibility proven through discrete exponential orthogonality and validated experimentally on a 9.41 second speech signal. Reconstruction error remains at machine precision (mean$8.36 \times 10^{-13}$) and Itakura-Saito COSH spectral distances stay below$10^{-12}$, confirming lossless forward-inverse transform consistency. This establishes DQLCT as a numerically stable, practically implementable tool for quaternion-based speech analysis.