Short time biquaternion quadratic phase Fourier transform

Abstract

In this paper, we propose a novel signal representation method based on biquaternions, termed the Short-Time Biquaternion Quadratic-Phase Fourier Transform (ST-BiQQPFT). This method provides a more efficient and comprehensive approach to signal analysis. The key contribution of this work lies in the introduction of the ST-BiQQPFT, a new mathematical framework that leverages biquaternionic representations to model the phase information of signals in a more robust manner. We rigorously develop the properties of the ST-BiQQPFT, including linearity, anti-linearity, boundedness, and shift and scaling properties. Additionally, we establish key results such as Parseval’s theorem and a characterization of the transform’s range. Furthermore, we derive several uncertainty principles, including Heisenberg-type, Nazarov-type, and logarithmic inequalities, to analyze the trade-off between time–frequency localization and phase sensitivity inherent in the ST-BiQQPFT. This framework significantly enhances the analysis of signals with complex phase structures, making it particularly well-suited for applications in quantum mechanics, image processing, and other domains requiring high-precision time–frequency representations of non-stationary signals. The paper also includes a comparative analysis, contrasting the ST-BiQQPFT with other methods, such as the Quaternion Fourier Transform (QFT) and the Short-Time Quaternion Quadratic-Phase Fourier Transform (ST-QQPFT), to highlight its relative advantages in terms of signal representation quality and computational efficiency. Moreover, we conduct a comprehensive performance evaluation of the proposed ST-BiQQPFT, including runtime and memory usage analyses, which demonstrate its computational efficiency and scalability.