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In this paper, the discrete octonion linear canonical transform (DOCLCT) is presented. According to the definition of DOCLCT, its shift transform and modulation transform are explored. In addition, based on a new convolution operator, we obtain the convolution theorem of DOCLCT. The DOCLCT can be derived by the three-dimensional (3D) linear canonical transform. So, the fast algorithm of 3D DOCLCT is discussed. Finally, Heisenberg’s uncertainty principle associated with DOCLCT is established. [ABSTRACT FROM AUTHOR]

The fractional Fourier transform (FRFT) is a generalized form of the Fourier transform (FT), it is another important class of time–frequency analysis tool in signal processing. In this paper, we study the two-dimensional (2D) FRFT in the polar coordinates setting. First, Parseval theorem of the 2D FRFT in the polar coordinates is obtained. Then, according to the relationship between 2D FRFT and fractional Hankel transform (FRHT), the convolution theorem for the 2D FRFT in polar coordinates is obtained. It shows that the FRFT of the convolution of two functions is the product of their respective FRFTs. Moreover, the fast algorithm for the convolution theorem of the 2D FRFT is discussed. Finally, the sampling theorem for signal is explored. [ABSTRACT FROM AUTHOR]

In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between the DOCLCT and the three-dimensional (3-D) discrete linear canonical transform (DLCT) is obtained. Moreover, based on a new convolution operator, we derive the convolution theorem of the DOCLCT. Finally, the correlation theorem of the DOCLCT is established. [ABSTRACT FROM AUTHOR]

The quaternion linear canonical transform (QLCT) has been widely used in color image processing. Biquaternion is a more generalized algebra of quaternion, which has attracted scholars' research interest in recent years. In this paper, a new transform is proposed called the biquaternion linear canonical transforms (BiQLCTs). Due to the noncommutativity of biquaternion algebra multiplication, there are three different types of the BiQLCTs: Left‐sided BiQLCT, right‐sided BiQLCT, and two‐side BiQLCT. The transforms are the extension of the complex linear canonical transforms. Then, the relationships between the three kinds of transforms are obtained. Next, based on the right‐side biquaternion linear canonical transform (RBiQLCT), some general properties of this transform are proved. Moreover, the convolution and correlation theorems of the RBiQLCT are studied. As an application, according to the convolution operator and convolution theorem, the biquaternion linear time‐invariant system is analyzed. Finally, the Heisenberg uncertainty principle for the RBiQLCT is established. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the N-dimensional Heisenberg's inequalities for the windowed linear canonical transform (WLCT) of a complex function. Firstly, the definition for N-dimensional WLCT of a complex function is given. In addition, the N-dimensional Heisenberg's inequality for the linear canonical transform (LCT) is derived. It shows that the lower bound is related to the covariance and can be achieved by a complex chirp function with a Gaussian function. Finally, the N-dimensional Heisenberg's inequality for the WLCT is exploited. In special cases, its corollary can be obtained. [ABSTRACT FROM AUTHOR]

We investigate the octonion short‐time linear canonical transform (OCSTLCT) in this paper. First, we propose the new definition of the OCSTLCT, and then several important properties of newly defined OCSTLCT, such as bounded, shift, modulation, time‐frequency shift, inversion formula, and orthogonality relation, are derived based on the spectral representation of the octonion linear canonical transform (OCLCT). Second, by the Heisenberg uncertainty principle for the OCLCT and the orthogonality relation property for the OCSTLCT, the Heisenberg uncertainty principle for the OCSTLCT is established. Finally, we give an example of the OCSTLCT. [ABSTRACT FROM AUTHOR]

The windowed offset linear canonical transform (WOLCT) can be identified as a generalization of the windowed linear canonical transform (WLCT). In this paper, we generalize several different uncertainty principles for the WOLCT, including Heisenberg uncertainty principle, Hardy's uncertainty principle, Donoho–Stark's uncertainty principle and Nazarov's uncertainty principle. Finally, as application analogues of the Poisson summation formula and sampling formulas are given. [ABSTRACT FROM AUTHOR]