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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 noncommutativ...
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... Biquaternions (complexified quaternions) and split quaternions were given by Bekar and Yayh [7] along with involutions and anti-involutions. This study [17] provided the basic features of biquaternions along with various representations. In the collection of biquaternions, Sangwine [24] derived the roots of -1. ...
The ambiguity function (AF) is crucial not just in non-stationary signal processing but also in radar and sonar systems. The biquaternion ambiguity function extends the standard AF using biquaternion algebra. Several properties like linearity, dilation, translation, shift, boundedness, Moyal's formula, reconstruction formula and inversion identity are thoroughly examined. Additionally, an intriguing correlation between the biquaternion ambiguity function and the biquaternion Fourier transform is illustrated. Based on these facts, we aim to derive multiple versions of uncertainty inequalities related to the proposed biquaternion ambiguity function.
... The BiQFT has garnered more and more interest as a quaternion Fourier transform (QFT) extension [7,19]. More recently, Gao and Li [22] introduced the notion of biquaternionic linear canonical transform and establish its various fundamental properties and applications. Motivated and inspired by the work of Gao and Li [22], we this paper introduce the notion of Biquaternion Linear Canonical Stockwell Transform (BiQLCST), a novel signal processing tool that utilizes biquaternions. ...
... More recently, Gao and Li [22] introduced the notion of biquaternionic linear canonical transform and establish its various fundamental properties and applications. Motivated and inspired by the work of Gao and Li [22], we this paper introduce the notion of Biquaternion Linear Canonical Stockwell Transform (BiQLCST), a novel signal processing tool that utilizes biquaternions. Firstly, we established the various fundamental properties, including linearity, shift, modulation, parity, orthogonality relation, reconstruction formula and Plancherel's theorem. ...
... The definition of an exponential with biquaternion values for the transform's kernel is a crucial step in creating a biquaternion Fourier transform [22]. In the sections that follow, we will primarily focus on exponential kernels with a biquaternion root of −1 [40]. ...
In this paper, we introduce the notion of Biquaternion Linear Canonical Stockwell Transform (BiQLCST), a novel signal processing tool that utilizes biquater-nions. Firstly, we established the various fundamental properties, including linearity, shift, modulation, parity, orthogonality relation, reconstruction formula and Plancherel's theorem. Heisenberg uncertainty principle asociated with Biquaternion Linear Canonical Stockwell Transform (BiQLCST) is also established. Towards the end, some potential applications of the BiQLCST are presented.
... where q 0 , q 1 , q 2 , q 3 ∈ C are complex numbers. When q 0 = 0, then q is the pure biquaternion [27]. ...
... The quaternion conjugate of a biquaternion q ∈ H C is given by [23,27] ...
... The complex conjugate of a biquaternion q ∈ H C is represented by [21,27] q =q 0 +q 1 i +q 2 j +q 3 k, ...
In this paper, the offset linear canonical transform associated with biquaternion is defined, which is called the biquaternion offset linear canonical transforms (BiQOLCT). Then, the inverse transform and Plancherel formula of the BiQOLCT are obtained. Next, Heisenberg uncertainty principle and Donoho-Stark’s uncertainty principle for the BiQOLCT are established. Finally, as an application, we study signal recovery by using Donoho-Stark’s uncertainty principle associated with the BiQOLCT.
... The linear canonical transform (LCT) [1][2][3] is a generalized form of the fractional Fourier transform (FrFT). As a linear integral transform with three parameter class, the LCT is more flexible than the FrFT and is a widely used analytical and processing tool in applied mathematics and engineering [4][5][6][7][8]. For analyzing and processing the non-stationary spectrum of finite-duration signals, Pei and Ding [9] proposed the discrete linear canonical transform (DLCT). ...
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.
