Linear Canonical Transform Research Articles

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.

Prolate spheroidal wave functions associated with the canonical Fourier–Bessel transform and uncertainty principles

The aim of this paper is to establish an extension of the prolate spheroidal wave functions which is the eigenfunction of the finite Bessel type of linear canonical transform (LCT) so‐called finite canonical Fourier–Bessel transform. We study the concentration problem, uncertainty principles about the essential supports, and signal recovery related to the canonical Fourier–Bessel transform on the space of band‐limited functions.

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

2D Linear Canonical Transforms on Lp and Applications

As Fourier transformations of Lp functions are the mathematical basis of various applications, it is necessary to develop Lp theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this Lp theory for 1≤p<∞. By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on L1(R2). Together with the Plancherel identity and Hausdorff–Young inequality, we establish Lp(R2) multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the L1(R2) signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images.

Optical image encryption based on linear canonical transform with sparse representation

We propose optical image encryption and authentication in the linear canonical transform (LCT) method by using a sparse representation technique. The linearity in the LCT based double random phase encryption (DRPE) system caused the vulnerability of the system. Not only the linearity in the LCT -DRPE system causes the vulnerability of the system but also the LCT order parameter may cause this weakness. Therefore, the encrypted data of double random phase encoding based on LCT is integrated with sparse representation to overcome this weakness. Unlike the traditional DRPE method, only sparse data from the encrypted data is kept for decryption. The numerical simulation proved that the correct authentication results are satisfied despite the use of partially encrypted data. The randomly selective sparse encrypted data reinforces the security of amplitude encoding of DRPE-LCT. Thus, the proposed method added an additional layer to optical security systems. The attacks analysis is implemented in order to show the robustness of the proposed security. In addition, this method has fulfilled the transmission and storage requirements owing to the high reduction of information data.

A Gaussian regularization for derivative sampling interpolation of signals in the linear canonical transform representations

The linear canonical transform (LCT) plays an important role in signal and image processing from both theoretical and practical points of view. Various sampling representations for band-limited and non-band-limited signals in the LCT domain have been established. We focus in this paper on the derivative sampling reconstruction, where the reconstruction procedure utilizes samples of both the signal and its first derivative. Our major aim was to incorporate the reconstruction sampling operator with a Gaussian regularization kernel, which on the one hand is applicable for not necessarily band-limited signals and on the other hand hastens the convergence of the reconstruction procedure. The amplitude error is also considered with deriving rigorous estimates. The obtained theoretical results are tested through various simulated experiments.

The Weyl correspondence in the linear canonical transform domain

The main objective of the paper is to generalize and enrich the Weyl transform by introducing the Weyl correspondence in the linear canonical transform (LCT) domain. In this paper, we propose the linear canonical-Wigner transform in harmonic analysis of phase space along with the admissible Wigner-Ville distribution (WVD) and Weyl transform in the LCT domain and discuss some useful results. Further we establish the relationship between the Wigner-Ville distribution and the Weyl transform in the LCT domain.

Erratum to “Time-Frequency Properties of the Short-Time Linear Canonical Transform and Its Application”

Erratum to “Time-Frequency Properties of the Short-Time Linear Canonical Transform and Its Application”

Phase Retrieval from Linear Canonical Transforms

The classical phase retrieval problem aims to recover an unknown function from the Fourier magnitudes. The linear canonical transform has a more generalized form of the well-known (fractional) Fourier transform and a wide range of engineering applications such as optics and quantum mechanism. In this paper, we consider the linear canonic phase retrieval problem of determining a function from the magnitudes of the linear canonic transforms. We show that a compactly supported function f can be determined, up to a global phase, from the magnitudes of multiple linear canonic transforms, where is a class of real unimodular matrices. It generalizes the results of phase retrieval from multiple fractional Fourier transforms. On the other hand, we show that a compactly supported function f can be determined, up to a global phase, from the interference linear canonic magnitudes and where Moreover, if the ambiguity of conjugate reflection is taken into account, the compactly supported function f can be determined, up to a rotation and conjugate reflection, from the linear canonic magnitudes and

Scaling Wigner Distribution in the Framework of Linear Canonical Transform

Scaling Wigner Distribution in the Framework of Linear Canonical Transform

On Quantum Representation of the Linear Canonical Wavelet Transform

For the efficient identification of quantum states, we propose the notion of linear canonical wavelet transform in the framework of quantum mechanics. Using the machinery of Dirac representation theory and integration within an ordered product of operators, we recast the linear canonical wavelet transform to a matrix element of the squeezing–displacing operator U(μ,s)KM between analyzing vector ⟨ψ| and two-mode quantum state vector |f⟩ to be transformed. We also derive the inner product relation and inversion formula for the linear canonical wavelet transform in the realm of quantum mechanics. Lastly, we present an explicit example for the lucid implementation of linear canonical wavelet transform in identifying the quantum states.

Approximate Hermite Interpolations for Compactly Supported Linear Canonical Transforms

There has been several Lagrange and Hermite type interpolations of entire functions whose linear canonical transforms have compact supports inℝ. There interpolation representations are called sampling theorems for band-limited signals in signal analysis. The truncation, amplitude, and jitter errors associated with the Lagrange type interpolations are investigated rigorously. Nevertheless, the amplitude and jitter errors arising from perturbing samples and nodes are not studied before. The aim of this work is to establish rigorous analysis of their types of perturbation errors, which is important from both practical and theoretical points of view. We derive precise estimates for both types of errors and expose various numerical examples.

Novel windowed linear canonical transform: Definition, properties and application

Linear canonical transform (LCT) has emerged as a powerful tool in nonstationary signal processing. However, it fails to reveal the information of local frequency varying with time due to the global kernel function. In this paper, a novel windowed linear canonical transform (NWLCT) is presented to give a clear time-frequency representation for time-varying signals. Firstly, the definition of NWLCT is proposed by a generalized convolution operator not a sliding window. Some basic properties of NWLCT are derived, such as inverse transform, Parseval theorem and reproducing kernel. Moreover, time-frequency resolution of NWLCT is analyzed theoretically, and the optimal window is derived. Finally, the applications of NWLCT in local spectral analysis for linear frequency modulated (LFM) signals are offered, including mono- and multi-component signals with similar local features. Theoretical and simulation results demonstrate that the proposed NWLCT preserves a crucial physical interpretation similar to classical short-time Fourier transform, which extends low-pass filter banks in the LCT domain. Compared with other time-frequency methods, NWLCT outperforms in improving the energy concentration and being robust to noise in terms of LFM signal processing.

Uncertainty Principles for Wigner–Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

Uncertainty Principles for Wigner–Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

Scaled ambiguity function and scaled Wigner distribution for LCT signals

In this article, a new version of ambiguity function (AF) and Wigner distribution (WD) based on the linear canonical transform (LCT) and the fractional instantaneous auto-correlation are proposed which are coined as scaled ambiguity function and scaled Wigner distribution (SAFL/SWDL). We initiate our investigation by establishing the fundamental relationship between the SAFL and SWDL. Then, the main properties including the conjugate-symmetry, shifting, scaling, marginal and Moyal’s formulae of SAFL and SWDL are investigated in detail. Finally, the applications of the newly defined transforms are performed to show the advantage of the theory.

Octonion Offset Linear Canonical Transform

Octonion Offset Linear Canonical Transform

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.