Linear Canonical Transform Research Articles

Biquaternion linear canonical stockwell transform and associated uncertainty principles

The intersection of geometric methods and signal processing has led to significant advancements in various fields, including physics, engineering, and mathematics. In recent years, the biquaternion domain, a generalization of complex numbers with a geometric interpretation, has emerged as a promising tool for signal analysis. This paper introduces the Biquaternion Linear Canonical Stockwell Transform (BiQLCST), a novel fusion of advanced mathematical frameworks that offers enhanced signal analysis capabilities and unveils new uncertainty principles, bridging the gap between complex transformations and uncertainty analysis in high-dimensional spaces. First, we established the various fundamental properties, including linearity, shift, modulation, parity, orthogonality relation, reconstruction formula and Plancherel’s theorem. Heisenberg uncertainty principle associated with Biquaternion Linear Canonical Stockwell Transform (BiQLCST) is also established. Toward the end, some potential applications of the BiQLCST are presented.

A novel synchrosqueezing transform associated with linear canonical transform

Synchrosqueezing transforms have aroused great attention for its ability in time–frequency energy rearranging and signal reconstruction, which are post-processing techniques of the time–frequency distribution. However, the time–frequency distributions, such as short-time Fourier transform and short-time fractional Fourier transform, cannot change the shape of the time–frequency distribution. The linear canonical transform (LCT) can simultaneously rotate and scale the time–frequency distribution, which enlarges the distance between different signal components with proper parameters. In this study, a convolution-type short-time LCT is proposed to present the time–frequency distribution of a signal, from which the signal reconstruction formula is given. Its resolutions in time and LCT domains are demonstrated, which helps to select suitable window functions. A fast implementation algorithm for the short-time LCT is provided. Further, the synchrosqueezing LCT (SLCT) transform is designed by performing synchrosqueezing technique on the short-time LCT. The SLCT inherits many properties of the LCT, and the signal reconstruction formula is obtained from the SLCT. Adaptive selections of the parameter matrix of LCT and the length of the window function are introduced, thereby enabling proper compress direction and resolution of the signal. Finally, numerical experiments are presented to verify the efficiency of the SLCT.

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.

Joint time-vertex linear canonical transform

The emergence of graph signal processing (GSP) has spurred a deep interest in signals that naturally reside on irregularly structured data kernels, such as those found in social, transportation, and sensor networks. Recently, concepts and applications related to time-varying graph signal analysis have matured, linking temporal signal processing techniques with innovative tools in GSP. In this paper, similar to the extension of the graph fractional Fourier transform to the graph linear canonical transform, we define the joint time-vertex linear canonical transform (JLCT) and its properties. This transformation extends the joint time-vertex Fourier transform (JFT) and fractional Fourier transform (JFRFT), broadening the Fourier analysis in both time and vertex domains into the domain of the linear canonical transform (LCT). This offers an enhanced set of the LCT analysis tools for joint time-vertex processing. Applications of the JLCT in dynamic mesh denoising, clustering, and energy compactness demonstrate that JLCT can enhance regression and learning tasks, and can refine and improve the performance of the JFT and the JFRFT.

Optical double-image cryptosystem based on a joint transform correlator in a linear canonical domain

In this work, we present a new optical double-image encryption method based on a joint transform correlator (JTC) and the linear canonical domain for the simultaneous authentication of two images. This new extension of the encryption system overcame the vulnerability of the method based on the JTC and the conventional 4f-optical processor in the Fourier domain. Although the simultaneous authentication process is satisfied in the Fourier domain, the data content is partially disclosed in false validation. Therefore, we introduce a quadratic phase encryption system of the linear canonical transform (LCT) domain in this method. The linear canonical transform domain adds more degrees of freedom to the security method due to the six LCT orders. In addition, the double-image encryption scheme became secure against intruder attacks, and it was difficult to recognize confidential information during the negative validation process. A cryptanalysis is performed in terms of a chosen-plaintext attack (CPA) and chosen-ciphertext attack (CCA). Numerical simulations demonstrate the feasibility, security, and effectiveness of the proposed system.

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 linear canonical shearlet transform

In this paper, we introduce the continuous and discrete version of the linear canonical shearlet transform (LCST). The authors begin with the definition of the LCST and then establish a relationship between the linear canonical transform (LCT) and the LCST. Next, the paper derives several basic properties like Parseval’s Formula, inversion formula, and the characterization of the transform’s range. In addition to the continuous version of the transform, the authors also present a discrete version of the LCST. This discrete version allows for practical implementation and efficient computation of the transform in digital signal processing systems. Lastly, the paper establishes a frame condition for the discrete LCST, which thereby helps in establishing the reconstruction formula for the discrete LCST. The paper ends with a conclusion.

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.

Spacetime Linear Canonical Transform and the Uncertainty Principles

Spacetime Linear Canonical Transform and the Uncertainty Principles

Nonuniform sampling and reconstruction of Diracs signal associated with linear canonical transform and its application

Sampling and reconstruction play a critical role in signal processing. The non-ideal sampling conditions motivate the development of the sampling theory. In this paper, associated with multiple non-ideal conditions, we discuss the nonuniform sampling and reconstruction of nonbandlimited signal in the linear canonical transform (LCT) domain with finite samples. The Diracs signal is nonbandlimited in the LCT domain but has the finite rate of innovation property. The sampling of the Diracs signal in the LCT domain is analyzed firstly. Secondly, the reconstruction of the signal with finite nonuniform samples is discussed, including two cases where the nonuniform sampling instants are known or unknown. Finally, the numerical experiment verifies the effect of the reconstruction algorithm, and the potential applications and generalized analysis indicate the value of the research.

Infrared spectral super-resolution model with linear canonical transforms regularization for spectral signals

Infrared spectral super-resolution has achieved great success for spectral signals under the noise-free case. However, the random noise and band overlap restricted the super-resolution performance in the real scenarios. Infrared spectral data acquisition is slow because weak signals (band overlap and random noise) limits its wide applications. To address this problem, a novel spectral super-resolution model is proposed with fast linear canonical transform (LCT) regularization, which can effectively improve the resolution of noisy and low-resolution spectrum. To reveal the sparsity difference between the observed infrared spectra and the real one, the LCT tool is utilized to distinguish the various coefficients distributions. The distribution of the real Infrared spectra is sparser than the observed infrared one, thus, an infrared spectral restoration model is proposed to regularize the sparsity distribution of the observed spectra by L0-norm. Experimental results on simulated and real spectral signals show that the proposed algorithm can effectively preserve the details of infrared spectrum and suppress random noise.

Chirp cyclic moment for chirp cyclostationary processes: Definitions and estimators

In high-mobility wireless communications and radar systems, a higher carrier frequency and higher moving speed/acceleration of terminals enhance the impact of the time-varying Doppler effect. This effect alters the cyclostationarity of the transmitted signal and characterizes the received signal as a chirp cyclostationary (CCS) signal. The generalized cyclic statistics of CCS signals are defined based on Fourier analysis, which is either zero-value or non-bandlimited, and is thus inefficient. In this study, the linear canonical transform (LCT), a generalized form of the Fourier transform, was used to address these issues. The moment function is a k-dimensional variable comprising a single time variable and k−1 lag variables. The chirp cyclic moment and the time-varying canonical spectrum were designed as the LCT of the moment with respect to the time and lag variables, respectively. Furthermore, an estimator of the chirp cyclic moment was defined, which is demonstrated to be asymptotic unbiased, asymptotic mean-square-consistent and asymptotic complex normal. Applications of the chirp cyclic moment in radar parameter estimation and electrocardiogram signal characterization were verified using simulated and real-life data.

Hermite interpolation theorems for band-limited functions of the linear canonical transforms with equidistant samples

We establish convergence analysis for Hermite-type interpolations for L^{2} ( mathbb {R})-entire functions of exponential type whose linear canonical transforms (LCT) are compactly supported. The results bridges the theoretical gap in implementing the derivative sampling theorems for band-limited signals in the LCT domain. Both complex analysis and real analysis techniques are established to derive the convergence analysis. The truncation error is also investigated and rigorous estimates for it are given. Nevertheless, the convergence rate is O(1/sqrt{N}), which is slow. Consequently the work on regularization techniques is required.