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Hausdorff–Young’s inequality establishes the boundedness of the Fourier transform from L p to L q spaces for 1 ≤ p ≤ 2 and q = p ′ , where p ′ denotes the Lebesgue-conjugate exponent of p. This paper extends this classical result by characterizing the L p - L q boundedness of metaplectic operators, which play a significant role in harmonic analysis. We demonstrate that metaplectic operators are bounded on Lebesgue spaces if and only if their symplectic projection is either free or lower block triangular. As a byproduct, we identify metaplectic operators that serve as homeomorphisms of L p spaces. To achieve this, we leverage a parametrization of the symplectic group by Dopico and Johnson. We use our findings to provide boundedness results within L p spaces for pseudodifferential operators with symbols in Lebesgue spaces, and quantized by means of metaplectic operators. These quantizations consists of shift-invertible metaplectic Wigner distributions, which are essential to measure local phase-space concentration of signals. Using the factorization by Dopico and Johnson, we infer a decomposition law for metaplectic operators on L 2 (R 2 d) in terms of shift-invertible metaplectic operators, establish the density of shift-invertible symplectic matrices in Sp (2 d , R) , and prove that the lack of shift-invertibility prevents metaplectic Wigner distributions to define the so-called modulation spaces M p (R d) . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the discrete octonion linear canonical transform (DOCLCT) is presented. According to the definition of DOCLCT, its shift transform and modulation transform are explored. In addition, based on a new convolution operator, we obtain the convolution theorem of DOCLCT. The DOCLCT can be derived by the three-dimensional (3D) linear canonical transform. So, the fast algorithm of 3D DOCLCT is discussed. Finally, Heisenberg's uncertainty principle associated with DOCLCT is established. [ABSTRACT FROM AUTHOR]

This paper is to introduce new distributions associated with the linear canonical transform (LCT), which can be regarded as the generalization of the Wigner distribution (WD) and ambiguity function (AF). Moreover, some basic properties of the proposed distributions are also presented. Namely, the shift properties, the conjugation symmetry property, and the marginal properties are proved; the Moyal formula and the anti-derivative property are developed; and the relationship with the short-time Fourier transform (STFT) is also formulated. Furthermore, the newly defined distributions are shown to be valuable and effective in the detection of linear frequency-modulated (LFM) signals. They have better detection performance than the LW, and the LA but are also more flexible than the WDL, and the AFL. [ABSTRACT FROM AUTHOR]

This paper studies a new distribution associated with the linear canonical transform (LCT), which is general to the scaled Wigner distribution (SWD) and abbreviated as SWDLC. Furthermore, some basic properties of the proposed distributions, such as shift property, conjugation symmetry property, marginal property, Moyal formula, anti-derivative property, and relationship with the short-time Fourier transform (STFT), are also presented in detail. The proposed new distribution has more advantages in the number of parameters than some known distributions. Besides, it shows not only flexibility but also superiority over SWD, especially in a cross-term reduction in the instantaneous frequency estimation of bi-component linear frequency modulation (LFM) signals. Furthermore, the parameter conditions for better angle and distance resolutions are lucidly demonstrated. In addition, numerical simulations clearly also show the superiority of the new distribution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish several versions of Paley–Wiener-type theorems for the canonical Fourier–Bessel transform, which is a Bessel type of linear canonical transform indexed by a matrix parameter m ∈ SL(2, ℝ). [ABSTRACT FROM AUTHOR]

In this article, we propose a new class of transform called linear canonical Stockwell transform (LCST) and obtain its basic properties along with the inner product relation and reconstruction formula. We characterize the range of the transform and show that its range is the reproducing kernel Hilbert space. We also develop a multiresolution analysis (MRA) associated with the proposed transform together with the construction of the orthonormal basis for L2(ℝ)$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the heat semigroups S ν m - 1 (t) t ≥ 0 related to Δ ν m - 1 given by Δ ν m - 1 = d 2 d x 2 + 2 ν + 1 x + 2 i a b x d dx - a 2 b 2 x 2 - 2 i ν + 1 a b and we study some of its important properties. In the present paper, several uncertainty principles for the canonical Fourier–Bessel transform are given, including the Beurling, Gelfand–Shilov and Cowling–Price uncertainty principles. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Many generalizations of the Fourier transform (FT) have been introduced in the literature, including fractional Fourier transform (FrFT) and linear canonical transform (LCT). This paper aims to extend the discrete combined Fourier transform (DCFT) to the FrFT domains and investigate its applications in image enhancement. The DCFT of a signal is a linear combination of the forward and inverse DFTs of the signal. The discrete combined fractional Fourier transform (DCFrFT) introduced in this work can be seen as a generalization of the of DCFT to the FrFT domains. Simulation results of the proposed DCFrFT and the original signal recovery are given to validate the proposed approach. To present an example of application of the proposed combined transform, we propose a methodology for image enhancement in which the root filtering (RF) technique is extended to the DCFrFT domains. Here the proper selection of the angle parameter of the proposed DCFrFT and the RF parameter are crucial in the enhancement process. For this purpose, simulation results for different values of the RF parameter are carried out and its appropriate value is chosen based on the subjective evaluation. It is seen here that the enhancement approach in the proposed transform domain can improve degraded images when an appropriate RF parameter is used. [ABSTRACT FROM AUTHOR]

The ambiguity function (AF) and Wigner distribution (WD) play an important role not only in non-stationary signal processing but also in radar and sonar systems. In this paper, we introduce modified ambiguity function and Wigner distribution associated with quadratic-phase Fourier transform (QAF, QWD). Moreover, many various useful properties of QAF and QWD are also proposed. Marginal properties and Moyal’s formulas of these distributions have elegance and simplicity comparable to those of the AF and WD. Besides, convolutions via quadratic-phase Fourier transform are also introduced. Furthermore, convolution theorems for QAF and QWD are also derived, which seem similar to those of the classical Fourier transform (FT). In addition, applications of QAF and QWD are established such as the detection of the parameters of single-component and multi-component linear frequency-modulated (LFM) signals. [ABSTRACT FROM AUTHOR]

We define a new class of linear canonical wavelet transform (LCWT) and study its properties like inner product relation, reconstruction formula and also characterize its range. We obtain Donoho-Stark’s uncertainty principle for the LCWT and give a lower bound for the measure of its essential support. We also give the Shapiro’s mean dispersion theorem for the proposed LCWT.

Abstract 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.

We define a new class of linear canonical wavelet transform (LCWT) and study its properties like inner product relation, reconstruction formula and also characterize its range. We obtain Donoho-Stark's uncertainty principle for the LCWT and give a lower bound for the measure of its essential support. We also give the Shapiro's mean dispersion theorem for the proposed LCWT. [ABSTRACT FROM AUTHOR]

We establish convergence analysis for Hermite-type interpolations for L 2 (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 / N) , which is slow. Consequently the work on regularization techniques is required. [ABSTRACT FROM AUTHOR]

In today's world achieving safe and efficient image transmission is a major issue. The image is encrypted by using some encryption technique and each algorithm has its strengths and weaknesses no single encryption mechanism can get the maximum security with minimum execution time. This paper proposes that it is possible to develop a Real-Time encryption algorithm system that uses multiple encryption algorithms in one system and provides adequate means of efficiency with security. In this system, we introduce a key vault that holds information of multiple encryption algorithms with a unique key. For every input image, the system selects a random algorithm for encryption from the key vault, encrypts the image with the selected encryption algorithm, and attached the key generated in the key vault to the encrypted image. At the receiver end first, we find out the key to identify the algorithm by which the image is encrypted, then get encryption algorithm details and perform decryption accordingly. We can increase the rate of its difficulty to improve its resistance against attacks by adding more and more encryption algorithms. It is quite impossible to break the proposed encryption technique as it randomly changes the encryption algorithm for each input image in randomly at run-time. The proposed cryptosystem can resist most of the existing attacks, as proved by using performance metrics and results. The aim is to provide a more secure image encryption system with reasonable computational complexity. This project has been implemented, key vault in ASP.NET and transforms and masks in MATLAB (2019a). Experiments confirm the feasibility and efficiency of the proposed real-time encryption system. [ABSTRACT FROM AUTHOR]

The Linear Canonical Transform (LCT) is a mathematical transform that generalizes several well-known transforms, including the Fourier transform, the fractional Fourier transform, and the Fresnel transform. It provides a unified framework for understanding and representing a wide range of linear and linear-like transforms, allowing for the analysis and manipulation of signals in various domains. Recently, Gao et al. extended the notion of LCT to octonion domains and showed its efficacy in precisely representing the non-transient octonion-valued signals. However, the octonion LCT exhibits limitations in effectively localizing the frequency characteristics of non-transient octonion-valued signals. As such, it is imperative to introduce the Octonion Windowed Linear Canonical Transform (OCWLCT) and explore its fundamental characteristics. We delve into the inversion formula and the orthogonality relation for the onedimensional OCWLCT. Additionally, we derive the inversion formula for the three-dimensional Octonion Windowed Linear Canonical Transform (OCWLCT). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we have defined the Schwartz type spaces SΔ,α,A, SΔ,β,B, SΔ,α,AΔ,β,B. We have studied the mapping properties of LCT between these spaces. [ABSTRACT FROM AUTHOR]

In this paper, sampling Theorems for perfect signal reconstruction using samples of a signal taken below the Nyquist rate involving nonlinear signal reconstruction technique are presented. The presented results are in contrast with the linear time-invariant filtering based signal reconstruction in the celebrated Shannon sampling theory. It is shown that using kth order nonlinearity in the signal reconstruction system, the required sampling rate for perfect signal reconstruction can be reduced by the same factor k for positive odd values of it. The extensions of the proposed sampling expansion to the fractional Fourier transform and linear canonical transform domains are also derived. Simulation results of the proposed technique are given to validate the proposed approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The linear canonical deformed Hankel transform is a novel addition to the class of linear canonical transforms, which has gained a respectable status in the realm of signal analysis. Knowing the fact that the study of uncertainty principles is both theoretically interesting and practically useful, we formulate several qualitative and quantitative uncertainty principles for the linear canonical deformed Hankel transform. Firstly, we derive Hardy’s and Miyachi’s uncertainty principles associated with the proposed transform. Next, we formulate the Heisenberg uncertainty principle via many approaches. We culminate our study by formulating several concentration-based uncertainty principles, including the Donoho-Stark and local inequalities for the linear canonical deformed Hankel transform. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we introduce a novel scaling Wigner distribution by intertwining the merits of fractional instantaneous auto-correlation and the linear canonical transform. We initiate the study by investigating the fundamental properties of the scaling Wigner distribution, including the nonlinearity, marginal, shifting, conjugate-symmetry, convolution, and anti-derivative properties. Subsequently, the Moyal's formula is also examined in detail. To testify the efficiency, the proposed distribution is employed for the detection of single-component, bi-component, and tri-component linear frequency modulated signals. The simulation results lucidly demonstrate that the scaling Wigner distribution performs exceptionally well in comparison with the conventional Wigner distribution. Nonetheless, owing to the higher degrees of freedom of the linear canonical transform, the proposed distribution benefits in cross-term reduction while maintaining a perfect time–frequency resolution and clear auto-terms angle resolution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a translation operator associated to the canonical Fourier-Bessel transform F ν m and study some of its important properties. We derive a convolution product for this transform and as application we study the heat equation related to Δ ν m - 1 given by Δ ν m - 1 = d 2 d x 2 + 2 ν + 1 x + 2 i a b x d dx - a 2 b 2 x 2 - 2 i ν + 1 a b. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of quaternion linear canonical S-transform (Q-LCST) which is an extension of the linear canonical S-transform and study the uncertainty principles associated with the Q-LCST. First, we propose the definition of Q-LCST and then study the fundamental properties of Q -LCST including linearity, shift, modulation, orthogonality relation and reconstruction formula. Second, we derive the associated Heisenbergs uncertainty inequality and the corresponding logarithmic version for Q -LCST. Finally, some potential applications of the Q-LCST are introduced. [ABSTRACT FROM AUTHOR]

A generalization of Mallat'as classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we are interested in the dual wavelets whose construction depends on nonuniform multiresolution analysis associated with linear canonical transform. Here we prove that if the translates of the scaling functions of two multiresolution analyses in linear canonical transform settings are biorthogonal, so are the wavelet families which are associated with them. Under mild assumptions on the scaling functions and the wavelets, we also show that the wavelets generate Riesz bases. [ABSTRACT FROM AUTHOR]

The ripplet transform is a refinement of the ordinary curvelet transform based on the non-parabolic scaling law for resolving two-dimensional singularities. However, the ripplet transform offers flexibility only in the angular window which is not feasible for optimizing the concentration of the ripplet spectrum. Therefore, there arises a fundamental question “whether it is possible to induce flexibility in both the radial and angular windows of the ordinary ripplet transform”. The aim of this study is to address this issue by constructing a new family of ripplet waveforms based on the linear canonical transform having certain extra degrees of freedom for resolving higher dimensional singularities. To begin with, we examine the fundamental aspects of the novel ripplet transform including the formulation of reconstruction and Rayleigh’s energy formulae. In the sequel, we obtain a Heisenberg-type uncertainty inequality associated with the novel ripplet transform. Towards the culmination, we introduce and study a new class of localization operators associated with the novel ripplet transform. [ABSTRACT FROM AUTHOR]

The classical phase retrieval problem aims to recover an unknown function from the Fourier magnitudes. The linear canonical transform C A 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 | C A f | , A ∈ Γ , of multiple linear canonic transforms, where Γ 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 | C A f | , | C A Q f | , | C A (f + Q f) | and | C A (f + i Q f) | , where Q f (t) = t f (t). 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 | C A f | and | C A Q f |. [ABSTRACT FROM AUTHOR]

We study the inversion formula for recovering a signal from its windowed linear canonical transform. Different from the known inversion formula, where a double integral is invoked, we show that every signal can be recovered from its windowed linear canonical transform with a univariate integral. Moreover, we show that the integral involved is convergent almost everywhere on R as well as in L p (R) for all 1 < p < ∞. Furthermore, we also obtain an inversion formula for functions in L 1 (R) with the method of Cesàro summability. [ABSTRACT FROM AUTHOR]

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) K M 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. [ABSTRACT FROM AUTHOR]

In this paper, we define the Wigner-Ville distribution function (WVDF) and corresponding Weyl operator in the linear canonical transform (LCT) domain. Further, we examine Moyle identity for the WVDF and investigate some of its properties. Moreover, we discuss the boundedness and compactness of Weyl operator on the L p space in the LCT domain. [ABSTRACT FROM AUTHOR]

Abstract Linear canonical transform as a general integration transform has been considered into Wigner-Ville distribution (WVD) to show more powerful ability for non-stationary signal processing. In this paper, a new WVD associated with linear canonical transform (WVDL) and integration form of WVDL (IWVDL) are presented. First, the definition of WVDL is derived based on new autocorrelation function and some properties are investigated in details. It removes the coupling between time and time delay and lays the foundation for signal analysis and processing. Then, based on the characteristics of WVDL over time-frequency plane, a new parameter estimation method, IWVDL, is proposed for linear modulation frequency (LFM) signal. Two phase parameters of LFM signal are estimated simultaneously and the cross term can be suppressed well by integration operator. Finally, compared with classical WVD, the simulation experiments are carried out to verify its better estimation and suppression of cross term ability. Error analysis and computational cost are discussed to show superior performance compared with other WVD in linear canonical transform domain. The further application in radar imaging field will be studied in the future work.

The linear canonical transform(LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we in this paper construct associated wavelet frames. To be precise we introduce wavelet frames whose construction depends on the nonuniform multiresolution analysis associated with linear canonical transform(LCT-NUMRA) whose translation set is not necessarily a group. The translation set is taken for elements in Λ = {0, r/N} + 2 Z, N ≥ 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime and Z is the set of all integers. Furthermore we establish necessary and sufficient condition for such nonuniform wavelet frames associated with linear canonical transform. [ABSTRACT FROM AUTHOR]

The present study is the first of its kind, aiming to explore the interface between the ridgelet and linear canonical transforms. To begin with, we formulate a family of linear canonical ridgelet waveforms by suitably chirping a one-dimensional wavelet along a specific direction. The construction of novel ridgelet waveforms is demonstrated via a suitable example supported by vivid graphics. Subsequently, we introduce the notion of linear canonical ridgelet transform, which not only embodies the classical ridgelet transform but also yields another new variant of the ridgelet transform based on the fractional Fourier transform. Besides studying all the fundamental properties, we also present an illustrative example on the implementation of the linear canonical ridgelet transform on a bivariate function. [ABSTRACT FROM AUTHOR]

This paper presents a novel and elegant convolution structure for the multi-dimensional linear canonical transform involving a pure multi-dimensional kernel obtained via a general 2 n × 2 n real, symplectic matrix M with n (2 n + 1) independent parameters. The primary intention is to develop the convolution theorem associated with the novel linear canonical convolution. The convolution structure is subsequently invoked to establish the sampling theorem for the band-limited signals in the multi-dimensional linear canonical domain. The validity and efficiency of the sampling procedure are demonstrated via a lucid example. Besides, the Heisenberg's and Beckner's uncertainty principles associated with the multi-dimensional linear canonical transform are also studied in detail. Finally, we study and design the multiplicative filter in the multi-dimensional linear canonical domain by utilizing the proposed multi-dimensional convolution structure. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of linear canonical wave packet transform (LCWPT) based on the idea of linear canonical transform (LCT) and wave packet transform (WPT). Parseval's identity and some properties of LCWPT are discussed. The inversion formula of LCWPT is formulated. Moreover, the composition of LCWPTs is defined and some properties are studied related to it. The LCWPTs of Mexican hat wavelet function are obtained. [ABSTRACT FROM AUTHOR]