Linear Canonical Domain Research Articles

Wave packet frames in linear canonical domains: construction and perturbation

Wave packet frames in linear canonical domains: construction and perturbation

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.

Security augmenting of optical cryptosystem based on linear canonical transform domain using a full phase encoding technique

For the purpose of alleviating the vulnerability of double random phase encryption system in the linear canonical transform domain, a novel approach for optical security and cryptographic systems is presented. This proposed system uses a fully phase encoding technique to augment the security of encryption system in the linear canonical transform. The first step in this system involves phase encoding of the initial amplitude image to be ciphered and then modulated by the phase masks. The decryption process of image is the reversal operation of the encryption method. The effectiveness and sensitivity of our proposed cryptosystem for the encryption secret keys are verified. The resistance of our method against occlusion attacks is investigated. Moreover, the results demonstrate that the fully phase-based optical cryptosystem is more secure and robust than the amplitude-based scheme in a linear canonical domain.

Linear canonical Hankel domain based Stockwell transform and associated Heisenberg’s uncertainty principle

Linear canonical Hankel domain based Stockwell transform and associated Heisenberg’s uncertainty principle

Clifford-valued linear canonical transform: Convolution and uncertainty principles

Linear canonical transform is a three-parameter class of linear integral transformation, which has found many applications in optics, filter design, speech, image, video, and signal processing due to its flexibility in representing information within the linear canonical domain. In this article, we extend the linear canonical transform to the class of Clifford-valued signals and introduce a novel transform coined as Clifford-valued linear canonical transform. The primary analysis encompasses the derivation of fundamental properties of the proposed transform such as translation and scaling covariance, inner product relation and inversion formula. Subsequently, we investigate the convolution, continuity, and differentiation theorems for the Clifford-valued linear canonical transform. A non-homogeneous partial differential equation has also been solved in the Clifford framework as an application of the differentiation theorem. Finally, we culminate our investigation by deriving several uncertainty principles for the proposed transform.

Scaled Wigner distribution in the offset linear canonical domain

The scaled Wigner distribution is obtained from the classical Wigner distribution by replacing the instantaneous auto-correlation with fractional instantaneous auto-correlation which is parameterized by a constant k∈Q+. In this paper, we introduce the scaled Wigner distribution in the offset linear canonical domain (SWDOLC). A natural magnification effect characterized by the extra degrees of freedom of the offset linear canonical transform (OLCT) and by a factor k on the frequency axis enables the SWDOLC to have flexibility so that it can be used in cross-term reduction. We initiate our investigation by studying the fundamental properties of the proposed transform, including the marginal, conjugate-symmetry, shifting, scaling, inverse and Moyal’s formulae by using the machinery of SWDOLC and operator theory. Moreover, the convolution and correlation properties for the proposed transform are studied. The crux of the paper lies in the applications of SWDOLC for the detection of single-component and bi-component linear-frequency-modulated (LFM) signals.

Forward Scatter Radar Meets Satellite: Passive Sensing of Aerial Target Using Satellite Communication Waveforms

The problem of single-channel reception of global positioning system (GPS) communication waveforms makes passive sensing of aerial target difficult because of forward scatter. This paper proposes a novel aerial target passive sensing method based on linear canonical transformation (LCT) using the forward scattered satellite communication waveforms. The proposed method firstly preprocesses the received signal based on the characteristics of the traditional satellite tracking loop and the forward scattered satellite communication waveforms to effectively suppress the interference of the direct wave through DC removal. Then, the Gaussian noise and multipath interference in the channel are suppressed by applying a rectangular window to its linear canonical domain. Finally, aerial target sensing is performed based on the peak value of signals in the linear canonical transform domain. The characteristic signal is constructed by analyzing the satellite communication waveforms. Combining the linear canonical transform with the matched filter (MF) to estimate the target parameter. Simulation results show that the proposed method can effectively perform the aerial target sensing by using satellite communication waveforms in the forward scatter scenario.

Non-Separable Linear Canonical Wavelet Transform

This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in L2(Rn). The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, we formulate several uncertainty inequalities, including the Heisenberg’s, logarithmic, and Nazorav’s inequalities for the proposed transform in the linear canonical domain. The obtained results are reinforced with illustrative examples.

Sparse discrete linear canonical transform and its applications

The linear canonical transform (LCT) is a powerful tool for non-stationary signals in signal processing. In this paper, we propose a sparse discrete linear canonical transform (SDLCT) algorithm to solve the high sampling rate and large data calculation of non-stationary signals. In detail, we introduce permutation of spectra and window function to ensure the sparsity of the frequency domain. To get rid of the redundant data of the signal and improve the processing efficiency, we compress the high-dimensional signal into the low-dimensional signal by down-sampling. Then the low dimensional signal is mapped to a linear canonical domain. The above processes effectively reduce the amount of data and computation for signal processing. Then, we apply the SDLCT to the pulse compression of the linear frequency modulation (LFM) signal. The comparison between SDLCT and LCT shows that SDLCT is hardly affected by noise, it can accurately obtain the target positions at a low signal-to-noise ratio (SNR). Furthermore, we apply this algorithm to moving target detection for synthetic aperture radar. The comparison between LCT, sparse Fourier transform (SFT), sparse discrete fraction Fourier transform (SDFrFT), and SDLCT shows that SDLCT has stronger signal detection performance and can detect all the signals that affect each other.

Uncertainty principles in linear canonical domains

The linear canonical transform has been found various applications in several areas, such as signal processing and optics. Several uncertainty principles were established for this transform in the last years. However, most results only focused on the uncertainty relations in two different time-frequency domains. In this paper, we study the uncertainty principle for measurable sets associated with several different linear canonical domains. We first give uncertainty relations for three different linear domains and then generalize it to n different linear canonical domains. It is shown that the new lower bound is tighter than the classical result, in some special cases.

Jittered Sampling in Linear Canonical Domain

Linear canonical transform (LCT) based sampling has been attracted much attention in signal processing community. A variety of sampling theorems associated with the LCT are therefore derived. However, the random jittered sampling in linear canonical domain still remains unresolved. Starting from the Shannon's result for the LCT, a biased interpolated estimate for bandlimited signals in linear canonical domain from jittered samples is proposed. The inverse filter technique is then applied to correct the bias, giving rise to an unbiased estimator whose interpolation noise variance approaches zero as the jitter becomes less and less pronounced. To be specific, for bandlimited stationary stochastic signals in linear canonical domain the interpolation noise variance is increased, and this result is also verified through some examples and simulations.

Linear Canonical Wigner Distribution Based Noisy LFM Signals Detection Through the Output SNR Improvement Analysis

In our previous work, we addressed a unified representation problem on Wigner distribution (WD) in linear canonical transform (LCT) domains by introducing a kind of closed-form instantaneous cross-correlation function type of Wigner distribution (CICFWD) that can be regarded as the WD's closed-form representation in the linear canonical domain. We then discussed the application of CICFWD in the detection of linear frequency-modulated (LFM) signals through the numerical simulation analysis approach, and it turns out that there is a causation between LCT free parameters and the detection performance. The main contribution of this paper is to provide a mathematical model for this causality in order to disclose the intrinsic mechanism of non-stationary signal detection performance improvement triggered by LCT free parameters. We first revisit the definition of CICFWD, and establish some related theories including its essential properties and discretization. We then formulate a mathematical theory foundation of the output signal-to-noise ratio (SNR) improvement, an output SNR inequality relation between the traditional WD and the CICFWD for a general noisy signal, and discuss how to solve this inequality to come out LCT free parameters. Within the well-established output SNR improvement analysis framework, we also explore LCT free parameters selection results on LFM signals added with white noise. Finally, numerical experiments are carried out to validate the correctness of strategies on determination LCT free parameters and the feasibility of output SNR improvement analysis approach.

Choi–Williams distribution in linear canonical domains and its application in noisy LFM signals detection

Due to the requirement of signal representation’s flexibility for strong noise interference in non-stationary signal processing, the traditional Choi–Williams distribution (CWD) is extended to the linear canonical domain by using a well-established closed-form instantaneous cross-correlation function (CICF) type of linear canonical transform (LCT) free parameters embedded approach. The derived CICF type of CWD (CICFCWD) unifies all of the linear canonical Choi–Williams distributions, and can be considered as the CWD’s closed-form representation in linear canonical domains. The CICFCWD of a discrete-time signal defined by the sampling equation is discussed for the requirement of digital signal analysis and processing. To break through the limitation that the existing research techniques are limited to quantitative analysis based on numerical simulations, a qualitative analysis to output signal-to-noise ratio (SNR) inequality between the traditional CWD and the CICFCWD for a general noisy signal is introduced, including the inequality modeling and solving. Within this output SNR improvement analysis framework, the strategies on determination LCT free parameters for linear frequency-modulated (LFM) signals added with white noise are developed. Numerical experiments are also carried out to validate the correctness of LCT free parameters selection results and the feasibility of output SNR improvement analysis approach.

An asymmetric hybrid watermarking mechanism using hyperchaotic system and random decomposition in 2D Non-separable linear canonical domain

An asymmetric hybrid watermarking mechanism using hyperchaotic system and random decomposition in 2D Non-separable linear canonical domain

Electromagnetic Signal Classification Based on Deep Sparse Capsule Networks

In complex electromagnetic environments, electromagnetic signal classification rates are low as long time have to be the cost to extract features. To cope with the issue, in this paper, an electromagnetic signal classification method is proposed based on deep sparse capsule networks. In the proposed method, received signals are frequency reduced and sampled processing first. Subsequently, a cross ambiguity function based on linear canonical transformation, a cross ambiguity function based on linear canonical domain, and higher-order spectrum are estimated, respectively. The maximum value of each section of the cross ambiguity function is combined with the maximum value of equally spaced cross sections of higher order amplitude spectrum to obtain the two-dimensional feature information. Finally, electromagnetic signals are classified by the deep sparse capsule networks. The simulation results show that the proposed method not only has good classification performance but also can automatically get a hierarchical feature representation by learning. Moreover, the corresponding time cost can be effectively reduced.

An Algorithm of DOA Estimation in Linear Canonical Transform Domain

An algorithm of DOA estimation based on linear canonical transform is proposed. In this method, the observed signals are first transformed to linear canonical domain, through selecting appropriate transformation parameters, characteristics of the observed signal appear to be energy-concentrated in the corresponding linear canonical domain, and by that, the linear canonical domain steering vector is defined, then root-MUSIC is used to estimate DOA of impinging signals. Simulation examples are provided showing the effectiveness of the presented technique.

Convolution theorems for the linear canonical transform and their applications

As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.