Discrete Linear Canonical Transform Research Articles

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.

Discrete linear canonical transform on graphs

With the wide application of spectral and algebraic theory in discrete signal processing techniques in the field of graph signal processing, an increasing number of signal processing methods have been proposed, such as the graph Fourier transform, graph wavelet transform and windowed graph Fourier transform. In this paper, we propose and design the definition of the discrete linear canonical transform on graphs (GLCT), which is an extension of the discrete linear canonical transform (DLCT), just as the graph Fourier transform (GFT) is an extension of the discrete Fourier transform (DFT). First, based on the centrality and scalability of the DLCT eigendecomposition approach, the definition of the GLCT is proposed by combining graph chirp-Fourier transform, graph scale transform and graph fractional Fourier transform. Second, we derive and discuss the properties and special cases of GLCT. Finally, some GLCT examples of the graph signals and comparisons with the DLCT are given to illustrate the improvement of the transformation.

Applications of the linear canonical transform to digital image processing.

In this paper, an existing approximation of discrete linear canonical transform (DLCT) is analyzed, and constraints are derived to fulfill some paramount properties as inversibility and additivity or the possibility to perform classical image operations in the frequency domain as image filtering. Giving some special values to the DLCT parameters and taking advantage of the division of the image spectrum in four zones of different significance, an application of image feature classifications is successfully investigated. Also, the required constraints are obtained to determine the suitability of the selected approximation when working with digital images.

Operator theory-based computation of linear canonical transforms

Linear canonical transforms (LCTs) are extensively used in many areas of science and engineering with many applications, which requires a satisfactory discrete implementation. Recently, hyperdifferential operators have been proposed as a novel way of defining the discrete LCT (DLCT). Here we first focus on improving the accuracy of this approach by considering alternative discrete coordinate multiplication and differentiation operations. We also consider canonical decompositions of LCTs and compare them with the originally proposed Iwasawa decomposition. We show that accuracy of the approximation of the continuous LCT with the DLCT can be drastically improved. The advantage and elegance of this approach lie in the fact that it reduces the problem of defining sophisticated discrete transforms to merely defining discrete coordinate multiplication and differentiation operations, by reducing the transforms to these operations. As a result of systematic investigation of possible parameters and design choices, we achieve a DLCT that is both theoretically satisfying and highly accurate.

Blind Image Watermarking in Canonical and Cepstrum Domains Based on 4-Connected t-o’clock Scrambling

Copyright protection of multimedia content is confronted with great challenges such as easy access to the Internet. Digital watermarking is widely applicable technique for copyright protection of multimedia contents. In this paper, a blind symmetric watermarking method in canonical and cepstrum domains based on four-connected t-o’clock scrambling is proposed. Initially, the watermark image is scrambled using the four-connected t-o’clock method to enhance the security. Then, the rotation operation is applied to the host image to extract the region where the watermark bits are embedded. After that, discrete linear canonical transform (DLCT) is applied to the extracted region to obtain the DLCT region. Cepstrum transform (CT) is performed on DLCT region to attain CT region. The CT region is then divided into non-overlapping blocks. The watermark bits are inserted into each block using max-heap and min-heap tree property. Experimental results illustrate that the proposed method shows high robustness against numerous attacks. Moreover, it produces high quality watermarked images and provides high security. Furthermore, it has superior performance to recent methods in terms of imperceptibility, robustness, and security.

Sliding Discrete Linear Canonical Transform

The linear canonical transform (LCT) has been shown to be one of the most powerful tools in signal processing, and in this paper, we propose an adaptive approach for the computation of the discrete LCT (DLCT), termed the sliding discrete linear canonical transform (SDLCT). First, we introduce a scheme for the single-point DLCT, which can effectively calculate a single or a few linear canonical spectra. Second, the SDLCT is proposed based on an iterative algorithm to meet the requirements of online spectral analysis when only a subset of $N$ frequencies are required from an $\tilde{N}\hbox{--}$ point discrete LCT ( $N\leq \tilde{N}$ ). The additivity and reversibility properties of the proposed algorithms are also discussed in detail. Third, the DLCT convolution operation is obtained to reduce the spectral leakage of the proposed algorithm, and time-domain windowing is implemented via frequency-domain convolution. Finally, we present two methods to assess performance with regard to computational complexity and precision and to show the correctness of the derived results.

Modelling the noise influence associated with the discrete linear canonical transform

In this study, the properties of noise after a discrete linear canonical transform (DLCT) are analysed. First, the authors prove that the DLCT of noise can be modelled by a Gaussian distribution with very weak assumptions on the noise in the time domain. Then, the mean and covariance matrix of this Gaussian distribution is derived and the general trend of the noise after DLCT is described. In addition, they find that the properties of noise in the LCT domain are the generalisation of the properties of noise in the Fourier transform domain. What is more, the authors prove that the additive white Gaussian noise (AWGN) is still an AWGN after performing the DLCT. Finally, the simulations are performed to verify the effectiveness of the obtained results.

Discrete Linear Canonical Transform Based on Hyperdifferential Operators

Linear canonical transforms (LCTs) are of importance in many areas of science and engineering with many applications. Therefore a satisfactory discrete implementation is of considerable interest. Although there are methods that link the samples of the input signal to the samples of the linear canonical transformed output signal, no widely-accepted definition of the discrete LCT has been established. We introduce a new approach to defining the discrete linear canonical transform (DLCT) by employing operator theory. Operators are abstract entities that can have both continuous and discrete concrete manifestations. Generating the continuous and discrete manifestations of LCTs from the same abstract operator framework allows us to define the continuous and discrete transforms in a structurally analogous manner. By utilizing hyperdifferential operators, we obtain a DLCT matrix which is totally compatible with the theory of the discrete Fourier transform (DFT) and its dual and circulant structure, which makes further analytical manipulations and progress possible. The proposed DLCT is to the continuous LCT, what the DFT is to the continuous Fourier transform (FT). The DLCT of the signal is obtained simply by multiplying the vector holding the samples of the input signal by the DLCT matrix.

Simulating first order optical systems—algorithms for and composition of discrete linear canonical transforms

The linear canonical transforms (LCTs) are a parameterised group of linear integral transforms. The LCTs encompass a number of well-known transformations as special cases, including the Fourier transform, fractional Fourier transform, and the Fresnel integral. They relate the scalar wave fields at the input and output of systems composed of thin lenses and free space, along with other quadratic phase systems. In this paper, we perform a systematic search of all algorithms based on up to five stages of magnification, chirp multiplication and Fourier transforms. Based on that search, we propose a novel algorithm, for which we present numerical results. We compare the sampling requirements of three algorithms. Finally, we discuss some issues surrounding the composition of discrete LCTs.

Random discrete linear canonical transform.

Linear canonical transforms (LCTs) are a family of integral transforms with wide applications in optical, acoustical, electromagnetic, and other wave propagation problems. In this paper, we propose the random discrete linear canonical transform (RDLCT) by randomizing the kernel transform matrix of the discrete linear canonical transform (DLCT). The RDLCT inherits excellent mathematical properties from the DLCT along with some fantastic features of its own. It has a greater degree of randomness because of the randomization in terms of both eigenvectors and eigenvalues. Numerical simulations demonstrate that the RDLCT has an important feature that the magnitude and phase of its output are both random. As an important application of the RDLCT, it can be used for image encryption. The simulation results demonstrate that the proposed encryption method is a security-enhanced image encryption scheme.

Sampling Rate Conversion in the Discrete Linear Canonical Transform Domain

Sampling rate conversion (SRC) is one of important issues in modern sampling theory. It can be realized by up-sampling, filtering, and down-sampling operations, which need large complexity. Although some efficient algorithms have been presented to do the sampling rate conversion, they all need to compute the N-point original signal to obtain the up-sampling or the down-sampling signal in the time domain. Most of the published papers about the sampling rate conversion require the signal to be band limited in the Fourier transform domain, and there are few paper published related to the SRC in the linear canonical transform (LCT) domain. This paper investigates how to perform the SRC in the discrete linear canonical transform (DLCT) domain for integer and fractional rate conversion. The simulations are performed to verify the correctness of the proposed results.

Fast Discrete Linear Canonical Transform Based on CM-CC-CM Decomposition and FFT

In this paper, a discrete LCT (DLCT) irrelevant to the sampling periods and without oversampling operation is developed. This DLCT is based on the well-known CM-CC-CM decomposition, that is, implemented by two discrete chirp multiplications (CMs) and one discrete chirp convolution (CC). This decomposition doesn't use any scaling operation which will change the sampling period or cause the interpolation error. Compared with previous works, DLCT calculated by direct summation and DLCT based on center discrete dilated Hermite functions (CDDHFs), the proposed method implemented by FFTs has much lower computational complexity. The relation between the proposed DLCT and the continuous LCT is also derived to approximate the samples of the continuous LCT. Simulation results show that the proposed method somewhat outperforms the CDDHFs-based method in the approximation accuracy. Besides, the proposed method has approximate additivity property with error as small as the CDDHFs-based method. Most importantly, the proposed method has perfect reversibility, which doesn't hold in many existing DLCTs. With this property, it is unnecessary to develop the inverse DLCT additionally because it can be replaced by the forward DLCT.

Constraints on additivity of the 1D discrete linear canonical transform.

The continuous linear canonical transform (LCT) can describe a wide variety of paraxial (quadratic phase) first-order optical systems. Digital algorithms to numerically calculate the LCT are therefore important in modeling the field propagations and are also of interest for many digital signal-processing applications. The continuous LCT is additive (and unitary), but discretization can destroy additivity. In this paper, the general constraint sufficient to ensure the discrete LCTs are additive is derived. Often, we wish to decompose the transform into a series of more computationally efficient steps. Having previously discussed the unitarity of such algorithms, in this paper we consider how our additivity constraint applies to the direct method (DM) and spectral method (SM) algorithms. Examples are presented showing how to correct nonadditive calculations and to appropriately choose parameters.

Signal reconstruction from partial information of discrete linear canonical transform

Signal reconstruction, especially for nonstationary signals, occurs in many applications such as optical astronomy, electron microscopy, and x-ray crystallography. As a potent tool to analyze the nonstationary signals, the linear canonical transform (LCT) describes the effect of quadratic phase systems on a wavefield and generalizes many optical transforms. The reconstruction of a finite discrete-time signal from the partial information of its discrete LCT and some known samples under some restrictions is presented. The partial information about its discrete LCT that we have assumed to be available is the discrete LCT phase alone or the discrete LCT magnitude alone. Besides, a reconstruction example is provided to verify the effectiveness of the proposed algorithm.

Discrete Linear Canonical Feature Research of Chirp Signal

The Chirp signal has many advantages that widely applied in communication, sonar, radar and other information processing fields as a common pulse compressional signal. The paper brought out an expression of the kernel of the Linear Canonical Transform (LCT) using its eigenfunctions. According to new expression, LCT can be expressed in terms of a new definition. Based on principle of sampling in time and LCT domains, a new definition of Discrete Linear Canonical Transform (DLCT) was put forward. The paper then proposed how to calculate DLCT of chirp signal in accordance with this new definition. Compared with other algorithms presented recently, it has more approximate results of continuous LCT.

Discrete linear canonical transform computation by adaptive method

The linear canonical transform (LCT) describes the effect of quadratic phase systems on a wavefield and generalizes many optical transforms. In this paper, the computation method for the discrete LCT using the adaptive least-mean-square (LMS) algorithm is presented. The computation approaches of the block-based discrete LCT and the stream-based discrete LCT using the LMS algorithm are derived, and the implementation structures of these approaches by the adaptive filter system are considered. The proposed computation approaches have the inherent parallel structures which make them suitable for efficient VLSI implementations, and are robust to the propagation of possible errors in the computation process.

Derivation and discrete implementation for analytic signal of linear canonical transform

Analytic signal and Hilbert transform associated with linear canonical transform (LCT) have been developed [Opt. Commun.281, 1468 (2008)]. However, the aforementioned paper has some drawbacks; for example, this kind of analytic signal cannot preserve the positive frequency bands of the original signal under LCT operations. In this paper, these drawbacks are pointed out, and a better definition for analytic signal associated with LCT is proposed. In addition, a discrete implementation for the analytic signal associated with discrete LCT is also provided. Several numerical examples, shown with both time and Wigner time-frequency domains, are demonstrated to illustrate the efficiency and accuracy of the proposed LCT analytic signal.

A Discrete Linear Canonical Transform Method on Chirp Signal

A Discrete Linear Canonical Transform Method on Chirp Signal

Generalized Discrete Entropic Uncertainty Relations on Linear Canonical Transform

Uncertainty principle plays an important role in physics, mathematics, signal processing and et al. In this paper, based on the definition and properties of discrete linear canonical transform (DLCT), we introduced the discrete HausdorffYoung inequality. Furthermore, the generalized discrete Shannon entropic uncertainty relation and discrete Rényi entropic uncertainty relation were explored. In addition, the condition of equality via Lagrange optimization was developed, which shows that if the two conjugate variables have constant amplitudes that are the inverse of the square root of numbers of non-zero elements, then the uncertainty relations touch their lowest bounds. On one hand, these new uncertainty relations enrich the ensemble of uncertainty principles, and on the other hand, these derived bounds yield new understanding of discrete signals in new transform domain.

Discrete Linear Canonical Transform of Finite Chirps

Linear canonical transformation (LCT) as an Fourier transform and fractional fourier transform's generalization, is an effective tool for non-stationary signals and has more flexibility, chirp signal can be looked as a typical nonstationary signal. In this letter, we derived closed-form expressions for the DLCT of a finite chirp. It is shown that the DLCT of a finite chirp is zeros at some points, and the number of zeros is related to chirp rate a , the parameter of DLCT α%, and the root of unity N . And if we choose proper DLCT parameters, the finite chirp is again a finite chirp or the original chirp except for a coefficient.