The Balian-Low Theorem for the Windowed Quaternionic Fourier Transform

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

In this paper we show how to construct a certain class of orthonormal bases in starting from one or more Gabor orthonormal bases in . Each such basis can be obtained acting on a single function with a set of unitary operators which operate as translation and modulation operators in suitable variables. The same procedure is also extended to frames and wavelets. Many examples are discussed.

Gabor’s approach to signal analysis has been formalized and generalized in several ways. The approach in this chapter integrates signal reconstruction and the uncertainty principle. The signal reconstruction is in terms of frame decompositions and sampling theory, with an emphasis on local sampling and effective irregular sampling theorems and algorithms. The uncertainty principle is formulated in terms of the uncertainty principle for variances in the context of Gabor and wavelet systems; and is developed to include general weighted uncertainty principle inequalities. Examples include dissections of the information plane, real-time sampling, auditory models, stationary systems, the Plancherel-Pólya and Boas theory, aliasing, zero-crossings, entropy inequalities, uniqueness theory and logarithmic integrals, local uncertainty principle inequalities, and the role of Riesz transforms and Ap - spaces. The theory unifies some of these notions in the parlance of wavelets and coherent states. The methods are from harmonic and functional analysis.

The windowed linear canonical transform is a natural extension of the classical windowed Fourier transform using the linear canonical transform. In the current work, we first remind the reader about the relation between the windowed linear canonical transform and windowed Fourier transform. It is shown that useful relation enables us to provide different proofs of some properties of the windowed linear canonical transform, such as the orthogonality relation, inversion theorem, and complex conjugation. Lastly, we demonstrate some new results concerning several generalizations of the uncertainty principles associated with this transformation.

The quadratic‐phase Fourier transform (QPFT) is a neoteric addition to the class of integral transforms and embodies a variety of signal processing tools like the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic‐phase Fourier transform to quaternion‐valued signals, known as the quaternion quadratic‐phase Fourier transform (Q‐QPFT). We initiate our investigation by studying the QPFT of 2D quaternionic signals, and later on, we introduce the Q‐QPFT of 2D quaternionic signals. Using the fundamental relationship between the Q‐QPFT and quaternion Fourier transform (QFT), we derive the inversion, Parseval's, and Plancherel's formulae associated with the Q‐QPFT. Some other properties including linearity, shift, and modulation of the Q‐QPFT are also studied. Finally, we formulate several classes of uncertainty principles (UPs) for the Q‐QPFT like Heisenberg‐type UP, logarithmic UP, Hardy's UP, Beurling's UP, and Donoho–Stark's UP. This study can be regarded as the first step in the applications of the Q‐QPFT in the real world.

As information technology continues to advance rapidly, the field of signal processing is confronted with increasingly complex high-dimensional data processing. Traditional linear canonical transforms (LCT) show certain limitations when dealing with high-dimensional data. Therefore, researchers have begun to explore hypercomplex number fields, which can represent rotations in three-dimensional space more naturally and accurately, and have demonstrated significant advantages in signal processing tasks. Transformations based on quaternions and octonions, such as the quaternion Fourier transform (QFT) and the octonion Fourier transform (OFT), have been widely applied in areas such as signal detection, pattern recognition, and time-frequency analysis. Against this backdrop, this paper investigated the quaternion offset linear canonical transform (QOLCT) and its extension in the octonion domain—the offset octonion linear canonical transform (OOCLCT). First, the paper established the modulatory and differentiation properties of the QOLCT, which were missing in existing literature, laying a solid theoretical foundation for subsequent research. Then, based on the relationship between the QOLCT and the OOCLCT, multiple classical uncertainty principles within the framework of the OOCLCT were derived. These principles included the Heisenberg uncertainty principle, the sharp Hausdorff-Young inequality, the Matolcsi-Szucs uncertainty principle, and the Benedicks-Amrein-Berthier uncertainty principle. Through the derivation and verification of these uncertainty principles, this paper not only deepened the understanding of the OOCLCT theory but also provided new mathematical tools and methods for high-dimensional signal processing. Finally, we discussed the future research directions of the OOCLCT, providing a reference for subsequent studies.

2 - The Formalism of Quantum Mechanics

Uncertainty principle and sparse reconstruction in pairs of orthonormal rational function bases

An elementary proof of a basic uncertainty principle concerning pairs of representations of R/sup N/ vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main contribution in this paper is the improvement of an important result due to Donoho and Huo (2001) concerning the replacement of the l/sub 0/ optimization problem by a linear programming (LP) minimization when searching for the unique sparse representation.

Discrete quaternion linear canonical transform

After reviewing the standard uncertainty relations due to Heisenberg (1927), Robertson (1929), and Schrodinger (1930), as well as the relations of Deutsch, and of Maassen and Uffink-including the so-called entropic relations-the author presents a complete account of the uncertainty relationship between complementary aspects in terms of superspace geometry, an approach not hitherto employed. Two incompatible properties A= Sigma alpha Aalpha mod alpha )( alpha mod and N= Sigma nNn mod n)(n mod belong to a pair of complementary aspects defined by two orthonormal bases ( mod alpha )) and ( mod n)) in the Hilbert space H. If the state is mod psi >, then P( alpha )= mod ( alpha mod psi ) mod 2 is the probability of obtaining the value Aalpha in a measurement of A, and P(n)= mod (n mod psi ) mod 2 is the probability of obtaining the value Nn in a measurement of N. The two aspects are characterised, relative to mod psi ), by the numbers (so-called purities): pi = Sigma alpha P( alpha )2 and pi = Sigma n P(n)2, both <or=1. He gives a complete characterisation of the uncertainty relationship between A and N (more precisely: between their aspects) in terms of the range of joint values of ( pi , pi ) for arbitrary initial states (pure as well as mixed). A theorem of Lenard is given an alternative proof, employing only elementary (superspace) geometry. The results depend on two angles, phi m=minimal angle, and phi M=maximal angle between the two aspects. Exact expressions for phi m and phi M are obtained in terms of the overlap matrix Lambda =( Lambda alpha n)=( mod ( alpha mod n) mod 2). As a corollary he finds the uncertainty relation for a pure state mod psi ) pi + pi <or=1+1/g+1-1/g cos phi m (where g=dim H), and a sharper one for mixed states. pi + pi =2 is obtainable if and only if the intersection of the aspects holds a pure state. If phi m= pi /2 (maximal incompatibility), then pi + pi <or=1+1/g is a special case of a stronger relation: Sigma mu g=o pi ( mu )=2, which one obtains for g+1 maximally incompatible aspects by means of a thereom of Ivanovic.

The novel Hausdorff–Young inequalities associated with the linear canonical transform (LCT) are derived based on the relation between the Fourier transform and the LCT in p-norm space (0<p<∞). Uncertainty relations for Shannon entropy and Renyi entropy based on the derived Hausdorff–Young inequality are yielded. It shows that these relations are functions of the transform parameters (a, b, c, d). Meanwhile, from the uncertainty relation for Shannon entropy the Heisenberg's uncertainty relation in LCT domains is derived, which holds for both real and complex signals. Moreover, the Heisenberg's uncertainty principle for the windowed fractional Fourier transform is obtained. Finally, one review of the uncertainty relations for the LCT and other transforms is listed in tables systematically for the first time.

Vertex-frequency analysis (VFA) is a useful technique in graph signal processing to extract the correspondence between frequencies and vertices. VFA can be calculated by the windowed graph Fourier transform (WGFT) and the localized graph Fourier transform (LGFT). However, since the inability of the classical graph Fourier transform (GFT) to generate orthogonal bases for directed graphs, the two approaches are confined to undirected graphs. In order to address the absence of VFA on directed graphs, we extend the fundamental concepts of VFA, shifted vertex window function and band-pass transfer filters in spectral domain to directed graphs. We first propose WDGFT based on the shifted window function in vertex domain. The relevant properties, reconstruction formulas, and simulation results of vertex-frequency representation are given. Then, we propose the localized directed graph Fourier transform (LDGFT), where the band-pass transfer filters can capture the localized spectral domain. The transfer functions satisfying the reconstruction condition are discussed. The vertex-frequency representation obtained by LDGFT is provided through numerical examples. Furthermore, we present a polynomial approximation technique to decrease computational cost. The LDGFT can be calculated without any matrix decomposition. Finally, we evaluate the proposed directed VFA framework with two kinds of applications, clustering and malfunction detection. We demonstrate that the proposed VFA framework is a powerful tool for directed graph signal processing.

State-of-the-art data analysis tools have to deal with high-dimensional data. Fortunately, the inherent dimensionality of data is often much smaller, as it has an internal structure limiting its degrees of freedom. In most cases, this structure can be approximated using a graph, i.e, a set of nodes connected by edges. Based on this idea, graphs have been largely used in semi-supervised and unsupervised learning. The canonical assumption when incorporating a graph prior is to assume that the signal is smooth with respect to the graph, i.e, that connected pieces of data have similar values. Nevertheless, this hypothesis is insufficient to characterize more complex relations between the data and its structure, the graph. To this end, the field of Graph Signal Processing (GSP) extends the graph smoothness hypothesis to a more general assumption. Its key idea is to think of the signal as a sum of harmonic modes, which are obtained from the eigen-decomposition of the Laplacian, an operator representing the second order derivative. Thus, the essence of GSP resides in the Fourier transform defined by the projection on the Laplacian eigenvectors basis, which allows us to filter graph signals. However, GSP suffers from a counterintuitive irregularity. Contrarily to the classical Fourier basis, the graph Laplacian eigenvectors are not spreading uniformly on the vertex set and can be highly concentrated in a region of the graph. This non-uniformity results in a lack of any intuitive translation operator and consequently poses challenges to generalize some of the classical signal processing theory, e.g., stationarity, uncertainty principles, or sampling. In this thesis, we answer these difficulties by accepting that each node is unique and should be treated specially according to its surrounding structure. This is achieved using an operator called localization that shifts a kernel defined in the spectral domain around a specific node while adapting to the local graph structure. In the first part of the thesis, we focus on harmonic analysis in GSP. We start by making use of the norm of localized kernels to capture the spectral local features of the graph vertices. To illustrate the relevance of this analysis we then introduce structural clustering, an algorithm that groups nodes according to the role they play in the graph. Then, we bound the vertex-frequency concentration of graph filter banks atoms. Because of the irregular structure of graphs, we construct a local uncertainty principle that is, again, driven by the norm of a localized kernel. In the second part of the thesis, we tackle GSP problems from a machine learning perspective. We use the localization operator to extend the notion of stationarity to graph signals. It consists of a graph signal probabilistic framework allowing us to optimally solve inverse problems. Finally, we show how the graph total variation can be used in semi-supervised learning and prove its consistency with respect to the underlying graph manifold.

Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms