Linear Canonical Transform Domain Research Articles

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.

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.

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.

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.

Speech Encryption in Linear Canonical Transform Domain Based on Chaotic Dynamic Modulation

In order to transmit the speech information safely in the channel, a new speech encryption algorithm in linear canonical transform (LCT) domain based on dynamic modulation of chaotic system is proposed. The algorithm first uses a chaotic system to obtain the number of sampling points of the grouped encrypted signal. Then three chaotic systems are used to modulate the corresponding parameters of the LCT, and each group of transform parameters corresponds to a group of encrypted signals. Thus, each group of signals is transformed by LCT with different parameters. Finally, chaotic encryption is performed on the LCT domain spectrum of each group of signals, to realize the overall encryption of the speech signal. The experimental results show that the proposed algorithm is extremely sensitive to the keys and has a larger key space. Compared with the original signal, the waveform and LCT domain spectrum of obtained encrypted signal are distributed more uniformly and have less correlation, which can realize the safe transmission of speech signals.

The Extrapolation Theorem for Discrete Signals in the Offset Linear Canonical Transform Domain

The Extrapolation Theorem for Discrete Signals in the Offset Linear Canonical Transform Domain

A Novel Sub-Nyquist FRI Sampling and Reconstruction Method in Linear Canonical Transform Domain

A Novel Sub-Nyquist FRI Sampling and Reconstruction Method in Linear Canonical Transform Domain

A Secure Asymmetric Optical Image Encryption Based on Phase Truncation and Singular Value Decomposition in Linear Canonical Transform Domain

A new asymmetric optical double image encryption algorithm is proposed, which combines phase truncation and singular value decomposition. The plain text is encrypted with two-stage phase keys to obtain a uniformly distributed cipher text and two new decryption keys. These keys are generated during the encryption process and are different from encryption keys. It realizes asymmetric encryption and improves the security of the system. The unscrambling keys in the encryption operation are mainly related to plain text. At the same time, the system is more resistant to selective plain text attacks; it also improves the sensitivity of decryption keys. With the application of phase truncation, the key space expanded and the security of the cryptographic system is enhanced. The efficacy of the system is calculated by evaluating the estimated error between the input and retrieved images. The proposed technique provides innumerable security keys and is robust against various potential attacks. Numerical simulations verify the effectiveness and security of the proposed technique.

Truncation error estimates for two‐dimensional sampling series associated with the linear canonical transform

The linear canonical transform (LCT) provides a more general framework for many well‐known linear integral transforms, such as Fourier transform, fractional Fourier transform, Fresnel transform, and so forth, in digital signal processing and optics. There has been a great deal of research on sampling expansions of bandlimited signals in the LCT domain. However, results on error estimation and convergence analysis appear to be relatively rare, especially for multi‐dimensional sampling series associated with the LCT. In this paper, we present error estimation and convergence analysis for two‐dimensional (2D) sampling series in the LCT domain. Specifically, we first prove the absolute convergence and uniform convergence of 2D LCT sampling series. Then, we analyze truncation error estimates for uniformly sampling 2D bandlimited signals in the LCT domain, as well as deriving three truncation error bounds. Finally, our theoretical results are demonstrated by numerical examples.

Sampling theorems for bandlimited functions in the two-dimensional LCT and the LCHT domains

Sampling theorems for the two-dimensional linear canonical transform (LCT) in polar coordinates deserve special attention in medical imaging such as computerized tomography and magnetic resonance imaging due to the superiority of solving the traditional non-bandlimited images processing problems. Two types of interpolation formulae that interpolate in both radius and azimuth the angularly periodic and highest frequency limited functions with different bandwidth constraints, the LCT and the linear canonical Hankel transform (LCHT), are derived. The first interpolation formula is essentially equivalent to the latest one for bandlimited functions in the LCT domain while enjoys more concise and general mathematical derivations. Thanks to the consistency of the LCHT's order for bandlimited functions in the LCHT domain, the second interpolation formula outperforms the first one in computational complexity.

Solving Generalized Wave and Heat Equations Using Linear Canonical Transform and Sampling Formulae

Several essential properties of the linear canonical transform (LCT) are provided. Some results related to the sampling theorem in the LCT domain are investigated. Generalized wave and heat equations on the real line are introduced, and their solutions are constructed using the sampling formulae. Some examples are presented to illustrate our results.

Generalized Heisenberg uncertainty relations for complex-valued signals out of Hilbert transform associated with LCT

Heisenberg uncertainty principle is of much signification to physics and signal processing, and the most classical representation of Heisenberg uncertainty principle represents the relationship between time duration and frequency duration in traditional time-frequency domain. Linear canonical transform (LCT) is the generalization of Fresnel transform and fractional Fourier transform (FRFT), and has been used in physical optics and information processing. In this paper, from various aspects six new results of Heisenberg uncertainty principles in LCT domain are obtained for the complex-valued signals out of traditional Hilbert transform (HT) and generalized Hilbert transform (GHT) in the first time to our best knowledge, among which two ones are related with parameters a and b and another two ones are related with c and d, and the last two ones are related with four transform parameters a, b, c and d simultaneously. Since the complex-valued signals out of GHT have the merit of the absence in negative generalized frequency domain, their physical meanings are shown as well, and these results discover the inequalities’ relationships for durations and phase differential for these complex-valued signals. In addition, it has also been shown that any one of these Heisenberg uncertainty principles can turn to classical Heisenberg uncertainty principle in traditional time-frequency domain for the complex-valued signals out of traditional HT.

Generalized Balian–Low Theorem Associated with the Linear Canonical Transform

Linear canonical transform (LCT), which is of importance in solving differential equations and analyzing optical systems, extends the conventional Fourier transform to the time-frequency domain characterized by a parameter matrix $$\mathbf{A }=(a,b;c,d)$$ . The main objective of this paper is to study the Balian–Low theorem in the LCT domain. We first state a Balian–Low theorem associated with the LCT, indicating that if a function’s Gabor system forms a Riesz basis for $$L^2({\mathbb {R}})$$ then it must be poorly localized in either time domain or LCT domains satisfying $$\frac{a}{b}\in {\mathbb {Z}}$$ . We then show that the symmetrically weighted version derived can hold for arbitrary LCT parameters under an assumption that the function is real-valued differentiable. Namely, if a real-valued differentiable function’s Gabor system forms a Riesz basis for $$L^2({\mathbb {R}})$$ , then the product of its spreads in time and any LCT domains must be infinite.

Nonuniform sampling for random signals bandlimited in the linear canonical transform domain

In this paper, we mainly investigate the nonuniform sampling for random signals which are bandlimited in the linear canonical transform (LCT) domain. We show that the nonuniform sampling for a random signal bandlimited in the LCT domain is equal to the uniform sampling in the sense of second order statistic characters after a pre-filter in the LCT domain. Moreover, we propose an approximate recovery approach for nonuniform sampling of random signals bandlimited in the LCT domain. Furthermore, we study the mean square error of the nonuniform sampling. Finally, we do some simulations to verify the correctness of our theoretical results.

Linear canonical Stockwell transform

The linear canonical transform (LCT) has long been recognized as a competent tool for optics and signal processing. However, the LCT cannot reveal the local LCT-frequency contents due to its global kernel and as such makes the LCT incompetent in situations demanding a joint information of time and linear canonical domain-frequency. In this paper, we propose the linear canonical Stockwell transform (LCST) in order to rectify the limitations of the LCT and the Stockwell transform (ST). Firstly, we examine the resolution of the novel transform in the time and LCT domains and then derive some of its basic properties, such as Moyal's formula, inversion formula and provide a characterization of the range. Furthermore, we derive a direct relationship between the well known Wigner distribution and the proposed LCST. In sequel, a discrete version of the linear canonical Stockwell transform is also presented. Finally, we extend the scope of the proposed work by studying the LCST in the realm of almost periodic functions.

Weighted Heisenberg–Pauli–Weyl uncertainty principles for the linear canonical transform

The classical uncertainty principle plays an important role in quantum mechanics, signal processing and applied mathematics. With the development of novel signal processing methods, the research of the related uncertainty principles has gradually been one of the most hottest research topics in modern signal processing community. In this paper, the weighted Heisenberg–Pauli–Weyl uncertainty principles for the linear canonical transform (LCT) have been investigated in detail. Firstly, the Plancherel–Parseval–Rayleigh identities associated with the LCT are derived. Secondly, the weighted Heisenberg–Pauli–Weyl uncertainty principles in the LCT domain are investigated based on the derived identities. The signals that can achieve the lower bound of the uncertainty principle are also obtained. The classical Heisenberg uncertainty principles in the Fourier transform (FT) domain are shown to be special cases of our achieved results. Thirdly, examples are provided to show that our weighted Heisenberg–Pauli–Weyl uncertainty principles are sharper than those in the existing literature. Finally, applications of the derived results in time frequency resolution analysis and signal energy concentrations are also analyzed and discussed in detail.

Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. Many theories for this transform are already known, but the uniform sampling theorem, as well as the sampling rate conversion theory about arbitrary lattices sampling in the LCT domain are still to be determined. Focusing on these issues, this paper carefully investigates arbitrary lattices sampling, the sampling with separable matrices and nonseparable matrices, to obtain uniform sampling theorem and the sampling rate conversion theory in the LCT domain. Firstly, the spectral expression of the discrete-time signal sampled via arbitrary lattice is deduced in the LCT domain. Based on it we propose the alias-free sampling relationship between two matrices and present the perfect reconstruction expressions for bandlimited signals in the LCT domain. Secondly, for further research on discrete signals to obtain sampling rate conversion theory, we define the multidimensional discrete time linear canonical transform (MDTLCT), as well as the convolution for the MDTLCT. Thirdly, the formulas of multidimensional interpolation and decimation via integer matrices in the LCT domain are derived. Then, based on the results of interpolation and decimation, we make analyses of the sampling rate conversion via rational matrices in the LCT domain, including spectral analyses and the formulas in time domain. Finally, simulation results and the potential applications of the theories are also presented.

Uncertainty principle for linear canonical transform using matrix decomposition of absolute spread matrix

We define a matrix, the absolute spread matrix (ASM), whose positive definite property is independent of signals. Based on the relation between the ambiguity function and the linear canonical power spectra, we give a new perspective on a key relation associated with spreads in time, frequency, time-frequency and linear canonical transform (LCT) domains, from which a new LCT uncertainty principle for complex signals using the ASM's rotation orthogonal decomposition is derived. The proposed lower bound of uncertainty product in two LCT domains is tighter than the latest one given by the author. We then obtain the sufficient and necessary condition that gives rise to this stronger result truly, and define a quantitative index to analyse the difference with the existing bound. We also reduce the uncertainty inequality to time and LCT domains. Furthermore, we present the example and simulation results that validate the previous theoretical analyses, and finally we demonstrate the effectiveness of the new proposal through discussing its applications in signal processing and optics.

Filter Design for Constrained Signal Reconstruction in Linear Canonical Transform Domain

The linear canonical transform (LCT), which includes many classical transforms, has increasingly emerged as a powerful tool for optics and signal processing. Signal reconstruction associated with the LCT has blossomed in recent years. However, many existing reconstruction algorithms for the LCT can only handle noise-free measurements, and when noise is present, they will become ill posed. In this paper, we address the problem of reconstructing an analog signal from noise-corrupted measurements in the LCT domain. A general methodology is proposed to solve this problem in which the analog signal is recovered from ideal samples of its filtered version in a unified way. The proposed methodology allows for arbitrary measurement and reconstruction schemes in the LCT domain. We formulate signal reconstruction in an LCT-based function space, which is the span of integer translates and chirp-modulation of a generating function, with coefficients derived from digitally filtering noise corrupted measurements in the LCT domain. Several alternative methods for designing digital filters in the LCT domain are also suggested using different criteria. The validity of the theoretical derivations is demonstrated via numerical simulation.