Quaternion Fourier Transform on Quaternion Fields and Generalizations

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations. I thank my family and FTHD organizer S.L. Eriksson. Soli Deo Gloria     

A Modified Uncertainty Principle for Two-Sided Quaternion Fourier Transform

This paper proposes a new uncertainty principle for the two-sided quaternion Fourier transform. This uncertainty principle describes that the spread of a quaternion-valued function and its two-sided quaternion Fourier transform (QFT) are inversely proportional. We obtain a tighter lower bound about the product of the spread of quaternion signal in the QFT domain. As a consequence, we show that the quaternionic Gabor filters minimize the uncertainty.     

Herglotz's theorem and quaternion series of positive term

The present paper first introduces the notion of quaternion infinite series of positive term and establishes its several tests. Next, we give the definitions of the positive‐definite quaternion sequence and the positive semi‐definite quaternion function, and we extend the classical Herglotz's theorem to the quaternion linear canonical transform setting. Then we investigate the properties of the two‐sided quaternion linear canonical transform, such as time shift characteristics and differential characteristics. Finally, we derive its several basic properties of the quaternion linear canonical transform of a probability measure, in particular, and establish the Bochner–Minlos theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Windowed Fourier transform of two-dimensional quaternionic signals

In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral representation of the quaternionic Fourier transform (QFT), we derive several important properties such as reconstruction formula, reproducing kernel, isometry, and orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic Gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic Gabor basis. We apply the QWFT properties and the (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT. Finally, we briefly introduce an application of the QWFT to a linear time-varying system.     

Quantum Fourier transform revisited

The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank‐1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix‐2 QFT decomposition to a radix‐d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.     

THE FOURIER TRANSFORM FOR HOMOGENEOUS VECTOR BUNDLES OVER QUATERNION UNIT DISK

61. IntroductionLet G be a connected noncompart semisimple Lie group with finite center and K amaimal compact subgroup of G, and X = G/K the associated Riemannian symmetricspace of noncompact type.Let (Vr, f) be an irreducible unitary representation of K, and E' be the homogeneousvector bundle over G/K associated with the given representation f. It is well krmwn that across section j 6 F(E") mad be identified with a vector-vained function f: G - V. whichi. right-K-cowiaat of type TI i.…     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.     

Sharper uncertainty principles in quaternionic Hilbert spaces

The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms.     

The Quaternion Domain Fourier Transform and its Properties

So far quaternion Fourier transforms have been mainly defined over \({\mathbb{R}^2}\) as signal domain space. But it seems natural to define a quaternion Fourier transform for quaternion valued signals over quaternion domains. This quaternion domain Fourier transform (QDFT) transforms quaternion valued signals (for example electromagnetic scalar-vector potentials, color data, space-time data, etc.) defined over a quaternion domain (space-time or other 4D domains) from a quaternion position space to a quaternion frequency space. The QDFT uses the full potential provided by hypercomplex algebra in higher dimensions and may moreover be useful for solving quaternion partial differential equations or functional equations, and in crystallographic texture analysis. We define the QDFT and analyze its main properties, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships.     

Uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform

ABSTRACT

In this paper, we present some new elements of harmonic analysis related to the right-sided multivariate continuous quaternion wavelet transform. The main objective of this article is to introduce the concept of the right-sided multivariate continuous quaternion wavelet transform and investigate its different properties using the machinery of multivariate quaternion Fourier transform. Last, we have proven a number of uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform.     

Uncertainty principle for measurable sets and signal recovery in quaternion domains

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time‐limiting data. Copyright © 2017 John Wiley & Sons, Ltd.     

Uncertainty principle for the two-sided quaternion windowed Fourier transform

In this paper, we study the quaternion windowed Fourier transform (QWFT) and prove the Local uncertainty principle, the Logarithmic uncertainty principle and Amrein Berthier for the QWFT, the radar quaternion ambiguity function and the quaternion Wigner transform.     

Uncertainty principles associated with quaternionic linear canonical transforms

In the present paper, we generalize the linear canonical transform (LCT) to quaternion‐valued signals, known as the quaternionic LCT (QLCT). Using the properties of the LCT, we establish an uncertainty principle for the two‐sided QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion‐valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternionic signal minimizes the uncertainty. Copyright © 2016 John Wiley & Sons, Ltd.     

线性正则变换域的复能量密度函数

线性正则变换是经典Fourier变换的广义形式,目前在非平稳信号的参数检测与估计方面取得了优异的应用效果,但线性正则变换理论体系还不完善.探讨了线性正则变换相关的复能量密度函数的基本概念,并详细推导研究了其基本的数学性质与特点,在上述理论的基础上,通过仿真实验来验证所得到结论的准确性.为其在实际应用中发挥更大的作用奠定了基础.     

Structure preserving quaternion full orthogonalization method with applications

This article proposes a structure-preserving quaternion full orthogonalization method (QFOM) for solving quaternion linear systems arising from color image restoration. The method is based on the quaternion Arnoldi procedure preserving the quaternion Hessenberg form. Combining with the preconditioning techniques, we further derive a variant of the QFOM for solving the linear systems, which can greatly improve the rate of convergence of QFOM. Numerical experiments on randomly generated data and color image restoration problems illustrate the effectiveness of the proposed algorithms in comparison with some existing methods.     

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.     

Time–frequency analysis of bivariate signals

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. We show that an adequate quaternion Fourier transform permits to build relevant time–frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, a bivariate counterpart of the usual analytic signal of real signals is introduced, called the quaternion embedding of bivariate signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time–frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.     

Inversion of the Short-Time Fourier Transform Using Riemannian Sums

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂ and modulation spaces. In the present note the Riemannian sums of the inverse short-time Fourier transform are investigated. Under some conditions on the window functions we prove that the Riemannian sums converge to f in the modulation spaces and inWiener amalgam norms, hence also in the L_p sense.     

p-adic数域Q_p上的窗口Fourier变换

提出了 p-adic数域 Qp上的窗口 Fourier变换和逆变换 ,详细地讨论了 p-adic模函数和阶梯函数的窗口 Fourier变换和逆变换 ,最后给出了 p-adic模函数和阶梯函数的窗口 Fourier变换定理 .     

Inversion formulas for the short-time Fourier transform

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of ƒ converges to ƒ in Wiener amalgam norms, hence also in the L_p sense for L_p functions, and pointwise almost everywhere.