Canonical Transform Research Articles

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Third-Order Nonlinear Semi-Canonical Functional Differential Equations: Oscillation via New Canonical Transform

Properties of the Octonion Linear Canonical Transform

Abstract The linear canonical transform is a widely utilized integral transform in the field of signal analysis, characterized by three independent parameters. It not only facilitates rotation of the time-frequency plane but also enables expansion and contraction of the frequency domain, thereby playing a crucial role in handling non-stationary signals. In recent years, the linear canonical transform has been extended to octonion domains, which possess a more generalized form and offer greater research potential. This extension effectively harnesses the processing capabilities of the linear canonical transform for non-stationary signals in high-dimensional spaces. In this paper, we explore the definition and the conjugacy of the left side octonion linear canonical transform. Moreover, we thoroughly examine the differential properties of octonion linear canonical transform. The study of these properties is helpful for understanding the characteristics and applications of geometric transformations as well as expanding the scope of mathematical theory and practical applications.

Uniqueness of Short-Time Linear Canonical Transform Phase Retrieval for Bandlimited Signals

Uniqueness of Short-Time Linear Canonical Transform Phase Retrieval for Bandlimited Signals

Sampling Theorems Associated with Offset Linear Canonical Transform by Polar Coordinates

The sampling theorem for the offset linear canonical transform (OLCT) of bandlimited functions in polar coordinates is an important signal analysis tool in many fields of signal processing and optics. This paper investigates two sampling theorems for interpolating bandlimited and highest frequency bandlimited functions in the OLCT and offset linear canonical Hankel transform (OLCHT) domains by polar coordinates. Based on the classical Stark’s interpolation formulas, we derive the sampling theorems for bandlimited functions in the OLCT and OLCHT domains, respectively. The first interpolation formula is concise and applicable. Due to the consistency of the OLCHT order, the second interpolation formula is superior to the first interpolation formula in computational complexity.

Sharper Hardy Uncertainty Relations on Signal Concentration in Terms of Linear Canonical Transform

Sharper Hardy Uncertainty Relations on Signal Concentration in Terms of Linear Canonical Transform

FORMULATION OF THE CORRELATION ON THE REDUCED LINEAR BIQUATERNION CANONIC TRANSFORMATION

This research aims to formulate a correlation theorem on the canonical transformation of reduced linear biquaternion (RBLCT).This research is a literature review conducted at the Department of Industrial Technology, Makassar PSDKU Graphics Engineering Study Programme, Politeknik Negeri Media Kreatif.The definition of reduced biquaternion linear canonical transformation (RBLCT) is obtained by first constructing the definition of reduced biquaternion Fourier transform (FTRB) and investigating its properties by replacing the kernel of Fourier Transform with the kernel of FTRB in the definition of Linear Canonical Transformation (LCT). The proposed Correlation Theorem for RBLCT is an extension of the correlation theorem of linear canonical transform to the domain of RBLCT. A different proof of the Parseval formula for RBLCT is also given whose proof is much simpler by using the conjugacy property of the RBLCT kernel. As an application of the obtained results, the RBLCT correlation theorem is briefly discussed to study frequency-swift filters in general.

Second-Order Noncanonical Delay Differential Equations with Sublinear and Superlinear Terms: New Oscillation Criteria via Canonical Transform and Arithmetic–Geometric Inequality

Second-Order Noncanonical Delay Differential Equations with Sublinear and Superlinear Terms: New Oscillation Criteria via Canonical Transform and Arithmetic–Geometric Inequality

Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

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.

A metaplectic perspective of uncertainty principles in the linear canonical transform domain

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.

Discrete Octonion Linear Canonical Transform: Definition and Properties

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.

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

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.

A Novel Image Encryption Algorithm Based on Compressive Sensing and a Two-Dimensional Linear Canonical Transform

In this paper, we propose a secure image encryption method using compressive sensing (CS) and a two-dimensional linear canonical transform (2D LCT). First, the SHA256 of the source image is used to generate encryption security keys. As a result, the suggested technique is able to resist selected plaintext attacks and is highly sensitive to plain images. CS simultaneously encrypts and compresses a plain image. Using a starting value correlated with the sum of the image pixels, the Mersenne Twister (MT) is used to control a measurement matrix in compressive sensing. Then, the scrambled image is permuted by Lorenz’s hyper-chaotic systems and encoded by chaotic and random phase masks in the 2D LCT domain. In this case, chaotic systems increase the output complexity, and the independent parameters of the 2D LCT expand the key space of the suggested technique. Ultimately, diffusion based on addition and modulus operations yields a cipher-text image. Simulations showed that this cryptosystem was able to withstand common attacks and had adequate cryptographic features.

Oscillation Theorems for Nonlinear Second-Order Delay Differential Equations with Some Sublinear Neutral Terms via Canonical Transform

Abstract New sufficient conditions for oscillation of a second-order nonlinear differential equation with some sublinear neutral terms are established via canonical transform and integral averaging method. Examples are provided to illustrate the significance and novelty of the presented results.

Oscillation results for nonlinear weakly canonical fourth-order delay differential equations via canonical transform

By using canonical transformation technique, we convert the nonlinear weakly canonical fourth-order delay differential equations into strongly canonical equations. Then we apply oscillation results for canonical equations to get similar results for the studied equation. The significance and novelty of our results are illustrated via numerical examples.

Spacetime Linear Canonical Transform and the Uncertainty Principles

Spacetime Linear Canonical Transform and the Uncertainty Principles

Inequalities for the Windowed Linear Canonical Transform of Complex Functions

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.

Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform

Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness in analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical transform (QOLCT), which is the most general form of QFT with additional free parameters. We explore the properties of 2D right-sided QOLCT, including inversion and Parseval formulas, besides its relationship with other transforms. We also examine the convolution and correlation theorems of 2D right-sided QOLCT, followed by several uncertainty principles. Additionally, we present an illustrative example of the proposed transform, demonstrating its graphical representation of a given signal and its transformed signal. Finally, we demonstrate an application of QOLCT, where it can be utilized to generalize the treatment of swept-frequency filters.

Fast Linear Canonical Transform for Nonequispaced Data

The investigations of the discrete and fast linear canonical transform (LCT) are becoming one of the hottest research topics in modern signal processing and optics. Among them, the fast calculation of LCT for non-uniform data is one of key problems. Focus on this problem, a new fast algorithm of the LCT has been proposed in this paper firstly by interpolation and approximation theory. The proposed algorithms can calculate quickly the LCT of the data, whether the input or output data is uniform. Secondly, the complexity and precision of derived algorithms have been analyzed for different situations. Finally, the experimental results are presented to verify the correctness of the obtained results.

Hyperbolic linear canonical transforms of quaternion signals and uncertainty

This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann–Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.

Two-sided Clifford-valued Linear Canonical Transform: Properties and Mustard Convolution

Two-sided Clifford-valued Linear Canonical Transform: Properties and Mustard Convolution