Octonion Short-Time Fourier Transform for Time-Frequency Representation and Its Applications

The octonion Fourier transform (OFT) is a useful tool for signal processing and analysis. However, due to the lack of time localization information, it is not suitable for processing signals whose frequencies vary with time. In this paper, we utilize octonion algebra to propose a new method for time-frequency representation (TFR) called the octonion short-time Fourier transform (OSTFT). The originality of the method is based on the quaternion short-time Fourier transform (QSTFT). First, we generalize the QSTFT to the OSTFT by substituting the quaternion kernel function with the octonion kernel function in the definition of the QSTFT, and the physical significance of the OSTFT is presented. Then, several essential properties of the OSTFT are derived, such as linearity, inversion formulas, time-frequency shifts and orthogonality relations. Based on the classic Fourier convolution operation, the convolution theorem for the OSTFT is derived. We apply the relationship between the OFT and OSTFT to establish Pitt's inequality and Lieb's inequality for the OSTFT. According to the logarithmic uncertainty principle of the OFT, the logarithmic uncertainty principle associated with the OSTFT is investigated. Finally, an application in which OSTFT can be used to study linear time varying (LTV) systems is proposed, and some potential applications of the OSTFT are introduced.