Qiang Feng

In the fields of signal processing and applied mathematics, the uncertainty principle plays a significant role. Motivated by the theoretical significance of the octonion quadratic-phase Fourier transform (OQPFT), we present a systematic investigation of uncertainty relations for this transform. Firstly, we derive the differential properties of the OQPFT. Secondly, we systematically derive four corresponding uncertainty principles for the OQPFT, covering both deterministic and random octonion signal scenarios. Finally, three sets of simulation experiments using 3D octonion chirped signals are performed, including two Gaussian-modulated cases with different complexity and a constant-coefficient rectangular-window non-Gaussian case for comparative verification. Full three-dimensional spectral observation shows that the OQPFT achieves superior frequency-domain aggregation compared with the octonion Fourier transform (OFT). Further comparative analysis demonstrates that this spectral aggregation advantage is not limited to Gaussian signals, but also holds for non-Gaussian windowed signals. This paper enriches the uncertainty theory in the hypercomplex domain and provides theoretical support for the frequency-domain analysis of high-dimensional quadratic-phase signals.

Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example.

The convolution integral equations are very important in optics and signal processing domain. In this paper, fractional mixed-weighted convolution is defined based on the fractional cosine transform; the corresponding convolution theorem is achieved. The properties of fractional mixed-weighted convolution and Young’s type theorem are also explored. Based on the fractional mixed-weighted convolution and fractional cosine transform, two kinds of convolution integral equations are considered, the explicit solutions of fractional convolution integral equations are obtained, and the computational complexity of solutions are also analyzed.

The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

The quaternion windowed fractional Fourier transform (QWFRFT) is a generalized form of the quaternion fractional Fourier transform (QFRFT), it plays a crucial role in signal processing for the analysis of multidimensional signals. In this paper, we first give the definition of the two-sided QWFRFT and some fundamental properties. Then, the quaternion convolution is proposed, and the relation between the quaternion convolution and the classical convolution is also given. Based on the quaternion convolution of the QWFRFT, relevant convolution theorems for the QWFRFT are studied. Moreover, the fast algorithm for QWFRFT is discussed. Finally, the complexity of QWFRFT and the quaternion windowed fractional convolution are given.

In this paper, two types of fractional Fourier–Laplace convolutions are defined, and the corresponding fractional Fourier–Laplace convolution theorems associated with the fractional cosine transform (FRCT), fractional sine transform (FRST) and Laplace transform (LP) are investigated in detail. The relationship between the fractional Fourier–Laplace convolutions and the existing convolutions is given, and Young's type theorem as well as the weighted convolution inequality are also obtained. As an application for fractional Fourier–Laplace convolution, the filter design and the system of convolution-type integral equations are also considered, the computational complexity for the multiplicative filter is analysed, and explicit solutions for these equations are obtained.

Three kinds of generalised convolution operations of fractional cosine transform and fractional sine transform are investigated, and the corresponding convolution theorems are derived, which can be seen as the generalisation of the classical results. The relationships of these generalised fractional convolution operations are also discussed. Additionally, the potential applications of the derived results in solving two kinds of generalised convolution integral equations are discussed, and the explicit solutions of these convolution‐type integral equations are obtained.