Sampling Theory in Fourier and Signal Analysis: Foundations
Abstract
With much material not previously found in book form, this book fills a gap by discussing the equivalence of signal functions with their sets of values taken at discreet points comprehensively and on a firm mathematical ground. The wide variety of topics begins with an introduction to the main ideas and background material on Fourier analysis and Hilbert spaces and their bases. Other chapters discuss sampling of Bernstein and Paley-Wiener spaces; Kramer's Lemma and its application to eigenvalue problems; contour integral methods including a proof of the equivalence of the sampling theory; the Poisson summation formula and Cauchy's integral formula; optimal regular, irregular, multi-channel, multi-band and multi-dimensional sampling; and Campbell's generalized sampling theorem. Mathematicians, physicists, and communications engineers will welcome the scope of information found here.
... Commonly, B ∞ 0,σ and B p σ are referred to as Bernstein spaces. For a comprehensive introduction, we refer the reader to [11,12]. ...
... are proper. For further details regarding the generalized Shannon equivalence, we refer to [11,12] (Lecture 21, pp. 155-162; Chapter 6, pp. ...
... For a signal f ∈ CB( ), the function f L : R → R, t → rct(t, L) f (t) is an element of L (R) (c.f. [12], Definition 2.1, p. 15). We refer to the mapping ...
This article compares two methods of algorithmically processing bandlimited time-continuous signals in light of the general problem of finding “suitable” representations of analog information on digital hardware. Albeit abstract, we argue that this problem is fundamental in digital twinning, a signal-processing paradigm the upcoming 6G communication-technology standard relies on heavily. Using computable analysis, we formalize a general framework of machine-readable descriptions for representing analytic objects on Turing machines. Subsequently, we apply this framework to sampling and interpolation theory, providing a thoroughly formalized method for digitally processing the information carried by bandlimited analog signals. We investigate discrete-time descriptions, which form the implicit quasi-standard in digital signal processing, and establish continuous-time descriptions that take the signal’s continuous-time behavior into account. Motivated by an exemplary application of digital twinning, we analyze a textbook model of digital communication systems accordingly. We show that technologically fundamental properties, such as a signal’s (Banach-space) norm, can be computed from continuous-time, but not from discrete-time descriptions of the signal. Given the high trustworthiness requirements within 6G, e.g., employed software must satisfy assessment criteria in a provable manner, we conclude that the problem of “trustworthy” digital representations of analog information is indeed essential to near-future information technology.
... For n = 1 this lemma is well-known, see [6], [10], p. 143-145. The main idea appeared already in [17]. ...
... It follows that the equation Rc = g can be re-written as the system of linear equations with respect to f l with matrix A. Condition (10) implies uniqueness of the solution in [L 2 (I)] k . Expanding each f l in the Fourier series in the orthogonal basis E(Z n +u l ), we get a unique vector c ∈ l 2 (Λ) which solves the equation Rc = g. ...
The paper discusses sharp sufficient conditions for interpolation and sampling for functions of n variables with convex spectrum. When n=1, the classical theorems of Ingham and Beurling state that the critical values in the estimates from above (from below) for the distances between interpolation (sampling) nodes are the same. This is no longer true for n>1. While the critical value for sampling sets remains constant, the one for interpolation grows linearly with the dimension.
... The classical sampling theory focuses mainly on samples that are taken from a signal at some specified instances. A typical example is the Whittaker-Shannon-Kotel'nikov sampling theorem [1] which has been extended in various ways (see [2,3] and references therein). ...
... i.e., ker(S * ) ⊆ ker( P); (2) (consistency) P is consistent, ...
This paper considers generalized consistent sampling and reconstruction processes in an abstract separable Hilbert space. Using an operator-theoretical approach, quasi-consistent and consistent approximations with optimal properties, such as possessing the minimum norm or being closest to the original vector, are derived. The results are illustrated with several examples.
... For more details on the value of this space in signal processing, in [7]. It will be established later. ...
... Proof: The inner product of ( ) 2 L functions and, more broadly, the impact of a distribution on a function are both shown here using the notation , . According to the Paley-Wiener-Schwartz Theorem in [7], ψ is the limit to of an entire function of exponential type at most π, as it has a compactly supported Fourier transform. ...
... The elementary fact that {e 2πin·x } n∈Z d forms an orthogonal basis for L 2 [− 1 2 , 1 2 ] d has far-reaching implications in many areas of mathematics and engineering. For instance, the celebrated Whittaker-Shannon-Kotel'nikov sampling theorem is an important consequence of this fact (see e.g., [1]). ...
... For instance, the conjecture is true when Γ is a lattice of R d -in which case the set Λ ⊂ R d can be chosen to be the dual lattice of Γ [4]-and also when S ⊂ R d is a convex set of finite positive measure for all d ∈ N [5]. In particular, it was shown in [6] that there is no exponential orthogonal basis for L 2 (S) when S is the unit ball of R d for d ≥ 2, in contrast to the case d = 1, where the unit ball is simply S = [−1, 1], and E( 1 2 Z) is an orthogonal basis for L 2 [−1, 1]. For more details on Fuglede's conjecture and its recent progress, we refer the reader to [5] and the references therein. ...
We construct a bound set that does not admit a Riesz spectrum containing a nonempty periodic set for which the period is a rational multiple of a fixed constant. As a consequence, we obtain a bounded set V with an arbitrarily small Lebesgue measure such that for any positive integer N, the set of exponentials with frequencies in any union of cosets of NZ cannot be a frame for the space of square integrable functions over V. These results are based on the proof technique of Olevskii and Ulanovskii from 2008.
... For more information on properties of W π h , please see e.g. [4,5,15]. ...
Aveiro Method is a sparse representation method in reproducing kernel Hilbert spaces (RKHS) that gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying RKHS. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro Method. To avoid those difficulties we propose an anew Aveiro Method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so called Pre-Orthogonal Greedy Algorithm (P-OGA) involving completion of a given dictionary. The new method is called Aveiro Method Under Complete Dictionary (AMUCD). The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition, bring available for the classical Hardy and Paley-Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro Method and the greedy algorithm.
In high-dimensional magnetic resonance imaging applications, time-consuming, sequential acquisition of data samples in the spatial frequency domain (k-space) can often be accelerated by accounting for dependencies along imaging dimensions other than space in linear reconstruction, at the cost of noise amplification that depends on the sampling pattern. Examples are support-constrained, parallel, and dynamic MRI, and k-space sampling strategies are primarily driven by image-domain metrics that are expensive to compute for arbitrary sampling patterns. It remains challenging to provide systematic and computationally efficient automatic designs of arbitrary multidimensional Cartesian sampling patterns that mitigate noise amplification, given the subspace to which the object is confined. To address this problem, this work introduces a theoretical framework that describes local geometric properties of the sampling pattern and relates these properties to a measure of the spread in the eigenvalues of the information matrix described by its first two spectral moments. This new criterion is then used for very efficient optimization of complex multidimensional sampling patterns that does not require reconstructing images or explicitly mapping noise amplification. Experiments with in vivo data show strong agreement between this criterion and traditional, comprehensive image-domain- and k-space-based metrics, indicating the potential of the approach for computationally efficient (on-the-fly), automatic, and adaptive design of sampling patterns.
Reconstructing a band-limited function from its finite sample data is a fundamental task in signal analysis. A Gaussian regularized Shannon sampling series has been proven to be able to achieve exponential convergence for uniform sampling. In this paper, we prove that such an exponential convergence can also be achieved for nonuniform sampling by regularization methods. Specifically, it is shown that one can recover a band-limited function by Gaussian or hyper-Gaussian regularized nonuniform sampling series with an explicit exponential convergence rate. The analysis is based on the residue theorem in complex analysis to express the truncation error by a contour integral, and the Laplace method to estimate integrals. Several concrete examples of nonuniform sampling with exponential convergence will be presented.
This study presents a novel algorithm that combines the Lorenz gauge equations with the Fourier domain technique to simulate magnetotelluric responses in three‐dimensional conductivity structures with general anisotropy. The method initially converts the Helmholtz equations governing vector potentials into one‐dimensional differential equations in the wave number domain via the horizontal two‐dimensional Fourier transform. Subsequently, a one‐dimensional finite element method employing quadratic interpolation is applied to obtain three five‐diagonal linear equation systems. Upon solving these equations, the spatial domain fields are obtained via the inverse Fourier transform. This process guarantees the computational efficiency, memory efficiency and high parallelization of the algorithm. Moreover, an anisotropic medium iteration operator guarantees stable convergence of the method. The correctness, competence and applicability of the algorithm are verified using some synthetic models. The results demonstrate that the new method is efficient and performs well in anisotropic undulating terrain and complex structures. Compared to other Fourier domain methods and the latest edge‐based finite element algorithm, the proposed method exhibits superior computing performance. Finally, the impact of the Euler angles on the magnetotelluric responses is analysed.
Recently, in the field of periodic nonuniform sampling, researchers (Wang et al., 2019; Asharabi, 2023) have investigated the incorporation of a Gaussian multiplier in the one-dimensional series to improve its convergence rate. Building on these developments, this paper aimed to accelerate the convergence of the two-dimensional periodic nonuniform sampling series by incorporating a bivariate Gaussian multiplier. This approach utilized a complex-analytic technique and is applicable to a wide range of functions. Specifically, it applies to the class of bivariate entire functions of exponential type that satisfy a decay condition, as well as to the class of bivariate analytic functions defined on a bivariate horizontal strip. The original convergence rate of the two-dimensional periodic nonuniform sampling is given by , where . However, through the implementation of this acceleration technique, the convergence rate improved drastically and followed an exponential order, specifically , where . To validate the theoretical analysis presented, the paper conducted rigorous numerical experiments.
Cardinal series representations for solutions of the Sturm–Liouville equation −y′′+q(x)y=ρ2y,x∈(0,L) with a complex‐valued potential q(x) q(x) are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to ρ in any strip Imρ<C of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential q(x) q(x) as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.
In this article, we study the approximation properties of Durrmeyer-type sampling operators. We consider the composition of generalized sampling operators and Durrmeyer sampling operators. For the new composition operators, we provide the pointwise and uniform convergence, as well as the quantitative estimates in terms of the first-order modulus of continuity and K-functional. Moreover, we investigate the rate of convergence using weighted modulus of continuity. Also, difference estimates are provided for the operators. Additionally, we provide approximation results for the linear combinations of the composition operators. Finally, we discuss the rate of convergence for the operators via graphical example.
It is a well-known fact that the classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas having an application to derive sampling formulas associated with differential or difference problems. The aim in this chapter is to provide, in a unified way, some generalizations of the Kramer result corresponding to various different settings. All these generalizations fit the sampling pattern followed in this book. Besides, these Kramer’s generalizations are profusely illustrated with a significant number of examples showing, in particular, how sampling theory is transversal in the field of mathematics.
This chapter intends to serve as an introduction to sampling theory which deals with the reconstruction of functions through their values on an appropriate sequence of points by means of sampling expansions involving these values. Reproducing kernel Hilbert spaces are suitable spaces for sampling purposes since evaluation functional are continuous and thus, the recovery depends on the frame properties of the reproducing kernel at the sampling points. After an introduction to the basic theory of reproducing kernel Hilbert spaces, we pay attention to the paradigmatic example of Paley-Wiener spaces: their main properties and some generalizations related to sampling aspects are included.
This introductory chapter is devoted to give a historical flavor to sampling theory. Besides, it serves as a motivation for the main aim in the book: the use of frames in sampling theory. For the sake of completeness it includes the basic frame theory needed throughout the book.
The paper presents linear predictors and causal lters for discrete-time signals featuring some di erent kinds of spectrum degeneracy. These predictors and lters are based on approximation of ideal noncausal transfer functions by causal transfer functions represented by polynomials of the Z-transform of the unit step signal.
In this paper, we investigate the approximation properties of exponential sampling series within logarithmically weighted spaces of continuous functions. Initially, we demonstrate the pointwise and uniform convergence of exponential sampling series in weighted spaces and present the rates of convergence via a suitable modulus of continuity in logarithmic weighted spaces. Subsequently, we establish a quantitative representation of the pointwise asymptotic behavior of these series using Mellin–Taylor’s expansion. Finally, it is given some examples of kernels and numerical evaluations.
The aim of this work is to study generalized Shannon sampling operators, where the kernel function is defined via the Fourier transform of a certain even window function. These windows are associated with Hann windows, which are widely used in signal analysis. The basics of the proofs are in the framework of the cosine operator functions in the abstract Banach space. We provide estimates of the approximation error for these sampling operators; in some cases, we can give the upper and lower estimates through the high order of the modulus of smoothness. We also evaluate the asymptotic behavior of the corresponding operator norms.
This chapter studies frequency bandlimited solutions to the cardinal interpolation problem with data that may have polynomial growth or decay. There exist solutions that interpolate the data at the Nyquist rate and grow or decay asymptotically like the data. If the data sequence does not decay, such solutions exist but are not unique. If the data decays, such solutions are unique but may fail to exist without additional restrictions on the data that are characterized here.
We show that the exponential sampling theorem and its approximate version for functions belonging to a Mellin inversion class are equivalent in the sense that, within the setting of Mellin analysis, each can be obtained from the other as a corollary. The approximate version is considered for both, convergence in the uniform norm and in the Mellin–Lebesgue norm. An important tool is the introduction of a Mellin version of the mixed Hilbert transform and its continuity properties. Our chapter extends the analogous equivalence between the classical and the approximate sampling theorem of Fourier analysis.
The paper presents linear predictors and causal filters for discrete-time signals featuring some different kinds of spectrum degeneracy. These predictors and filters are based on approximation of ideal noncausal transfer functions by causal transfer functions represented by polynomials of the Z-transform of the unit step signal.
In this paper, we introduce a family generalized Kantorovich-type exponential sampling operators of bivariate functions by using the bivariate Mellin-Gauss-Weierstrass operator. Approximation behaviour of the series is established at continuity points of log-uniformly continuous functions. A rate of convergence of the family of operators is presented by means of logarithmic modulus of continuity and a Voronovskaja-type theorem is proved in order to determine rate of pointwise convergence. Convergence of the family of operators is also investigated for functions belonging to weighted space. Furthermore, some examples of the kernels which support our results are given.
Recent neuroimaging studies in humans have reported distinct temporal dynamics of gyri and sulci, which may be associated with putative functions of cortical gyrification. However, the complex folding patterns of the human cortex make it difficult to explain temporal patterns of gyrification. In this study, we used the common marmoset as a simplified model to examine the temporal characteristics and compare them with the complex gyrification of humans. Using a brain-inspired deep neural network, we obtained reliable temporal-frequency fingerprints of gyri and sulci from the awake rs-fMRI data of marmosets and humans. Notably, the temporal fingerprints of one region successfully classified the gyrus/sulcus of another region in both marmosets and humans. Additionally, the temporal-frequency fingerprints were remarkably similar in both species. We then analyzed the resulting fingerprints in several domains and adopted the Wavelet Transform Coherence approach to characterize the gyro-sulcal coupling patterns. In both humans and marmosets, sulci exhibited higher frequency bands than gyri, and the two were temporally coupled within the same range of phase angles. This study supports the notion that gyri and sulci possess unique and evolutionarily conserved features that are consistent across functional areas, and advances our understanding of the functional role of cortical gyrification.
We establish convergence analysis for Hermite-type interpolations for L2(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.
The present paper deals with construction of a new family of exponential sampling Kantorovich operators based on a suitable fractional-type integral operators. We study convergence properties of newly constructed operators and give a quantitative form of the rate of convergence thanks to logarithmic modulus of continuity. To obtain an asymptotic formula in the sense of Voronovskaja, we consider locally regular functions. The rest of the paper devoted to approximations of newly constructed operators in logarithmic weighted space of functions. By utilizing a suitable weighted logarithmic modulus of continuity, we obtain a rate of convergence and give a quantitative form of Voronovskaja-type theorem via remainder of Mellin–Taylor’s formula. Furthermore, some examples of kernels which satisfy certain assumptions are presented and the results are examined by illustrative numerical tables and graphical representations.
The periodic nonuniform sampling has attracted considerable attention both in mathematics and engineering although its convergence rate is slow. To improve the convergence rate, some authors incorporated a regularized multiplier into the truncated series. Recently, the authors of [18] have incorporated a Gaussian multiplier into the classical truncated series. This formula is valid for bandlimited functions and the error bound decays exponentially, i.e. ? Ne??N, where ? is a positive number. The bound was established based on Fourier-analytic approach, so the condition that f belongs to L2(R) cannot be considerably relaxed. In this paper, we modify this formula based on localization truncated and with the help of complex-analytic approach. This formula is extended for wider classes of functions, the class of entire functions includes unbounded functions on R and the class of analytic functions in an infinite horizontal strip. The convergence rate is slightly better, of order e??N/? N. Some numerical experiments are presented to confirm the theoretical analysis.
Given an arbitrary sequence of elements ξ={ξn}n∈N of a Hilbert space (H,⟨·,·⟩), the operator Tξ is defined as the operator associated to the sesquilinear form Ωξ(f,g)=∑n∈N⟨f,ξn⟩⟨ξn,g⟩, for f,g∈{h∈H:∑n∈N|⟨h,ξn⟩|2<∞}. This operator is in general different from the classical frame operator but possesses some remarkable properties. For instance, Tξ is always self‐adjoint with regard to a particular space, unconditionally defined, and, when ξ is a lower semiframe, Tξ gives a simple expression of a dual of ξ. The operator Tξ and lower semiframes are studied in the context of sequences of integer translates of a function of L2(R). In particular, an explicit expression of Tξ is given in this context, and a characterization of sequences of integer translates, which are lower semiframes, is proved.
For many decades, Kramer’s sampling theorem has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval [0, 1]. Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.
We present linear integral predictors for continuous-time high-frequency signals with a finite spectrum gap. The predictors are based on approximation of a complex-valued periodic exponential (complex sinusoid) by rational polynomials.
This chapter introduces the important concepts associated with the sampling theorem and the degrees of freedom in a signal as measured by its space-bandwidth product. After developing several important results that will be found useful later in the book, we will also establish a connection between the sampling theorem and the minimum uncertainty prolate spheroidal functions.
In the note is shown that for the d-dimensional Bernstein functions class Bσ,dp,p>0 the Plancherel–Pólya inequality holds with the constant which equals to the product of the constants occuring in the one-dimensional cases. Related truncation error upper bounds are precised in the irregular sampling restoration of functions in several variables.
We continue the work started in a previous article and introduce a general setting in which we define nets of nonlinear operators whose domains are some set of functions defined in a locally compact topological group. We analyze the behavior of such nets and detect the fairest assumption, which are needed for the nets to converge with respect to the uniform convergence and in the setting of Orlicz spaces. As a consequence, we give results of convergence in this frame, study some important special cases, and provide graphical representations.
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