Search

We present different sampling methods for the approximation of functions on the sphere. In this note we focus on Fourier methods on the sphere based on spherical harmonics and on the double Fourier sphere method. Further longitude-latitude transformation is combined with Fourier extension to allow the use of bi-periodic Fourier series on the sphere. Fourier extension with hermite interpolation is introduced and double Fourier sphere method is discussed shortly.

The paper shows relations between two different frequency representations of n-dimensional signals (n = 1, 2, 3): the Cayley-Dickson Fourier transform and the Hartley transform. The formulas relating the n-D complex Fourier transform and the n-D Hartley transform are presented. New formulas relating the Quaternion and Octonion Fourier transforms and respectively the 2-D and 3-D Hartley transforms are developed. The paper is illustrated with examples of Hartley transforms of 1-D, 2-D and 3-D Gaussian signals.

We introduce two generalizations of the exponential function and, hence, two generalizations of the Fourier transform and series. Each generalization depends on a real, nonnegative parameter less than or equal to one, but reduces to the standard exponential function when the parameter is equal to one. For this reason, the corresponding transforms are called fractional Fourier transforms. In this talk we examine sampling theorems of bandlimited functions in these two fractional Fourier transform domains.

The fractional Fourier transform has been proved useful both in theory and in applications of signal processing as well as optics. We present a single channel sampling theorem for signals in shift-invariant spaces associated with the fractional Fourier transform domain.

A range of interesting electromagnetic systems like antennas, extraordinary transmission and absorbers have been shown to have certain bandwidth limitations given by a family of sum-rules. Common for the systems are that they are passive, linear and time-translational invariant. In this paper we shortly review the extension from one-dimensional passive systems to multi-dimensional systems, aiming towards constraining the system properties. The most well-known case of system constraints follows from multi-dimensional passivity, where a Schwartz-kernel representation theorem maps Borel-measures with a growth condition to the (complexified) Fourier transform of the transfer function.

This paper gives a new property for the Infinite Hartley Transform known as the Duality Theorem. The Infinite Hartley Transform is an integral transform and is closely reminiscent to the famous Infinite Fourier Transform. Unlike the former, the latter has not been much renowned in the mathematical world or its applications' domain. Most of the properties of the Infinite Hartley Transform and the Infinite Fourier Transform are quite similar but some differences do persist. A derivation of the Duality Theorem corresponding to Infinite Hartley Transform is presented. The utility of Duality Theorem lies in computing the continuous time-domain function from the Hartley frequency domain and vice versa. This has the tenacity to recede sufficient labour involved in integration. This in turn leads in the saving of precious computation time and implementation cost to a maximum limit. This Duality Theorem can be used in diverse fields such as Signal Processing, Image Processing, and Communication Systems, where it is common to encounter cases involving the continuous — time and the continuous frequency signals to have the same aspect or resemblance.

In this work, we show by means of the technique inspired by the Banach-Steinhaus Thm., that typically the Fourier transform of an integrable signal decays arbitrarily slowly toward the infinity, and has an arbitrary weak worst continuity/smoothness behaviour. However, the corresponding characterization can only be given weakly by means of the limit superior. Those statements gives therefore a tightening of the famous Riemann-Lebesgue's Lemma. Furthermore, we give a construction of functions, whose Fourier transform decays slowly than an arbitrary given decay rate. Inspired by that, we are also able to give an alternative proof of the strong divergence of the Shannon sampling series [1] for signals in the Paley-Wiener space PW1ωg, band-limited to an arbitrary ωg ∊ R+. The corresponding construction of signals is stronger than the existent one given by Boche and Farell, and gives a new insight into the divergence phenomenon of the Shannon sampling series

Finite-Rate-of-Innovation (FRI) sampling theory prescribes a procedure for exact recovery of Dirac impulses from linear measurements in the form of orthogonal projections of streams of Dirac impulses onto the subspace of Fourier—bandlimited functions. This enables recovery of a continuous time sparse signals at sub-Nyquist rates. In many cases, the transform domain of interest may be more general than the Fourier domain. Recent work has extended FRI sampling theory to the spherical Fourier Transform, fractional Fourier Transform and the Laplace Transform. In this paper, we develop a broad FRI framework applicable to a general class of transformations that includes Fourier, Laplace, Fresnel, fractional Fourier, Bargmann and Gauss—Weierstrass transforms, among others. For this purpose, we consider the Special Affine Fourier Transform (SAFT) which parametrically generalizes a number of well known unitary transforms linked with signal processing and optics. We first derive a version of Shannon's sampling theory based on the convolution structure tailored for the SAFT domain. Having identified the subspace of SAFT—bandlimited functions, we apply FRI sampling theory to the SAFT and study recovery of sparse signals, thus providing a unified view of FRI sampling theory for a large class of disparately studied operations.

The paper is devoted to the study of Whittaker-Kotelnikov-Shannon sampling theorem as a tool for transforming the basic object of Fourier analysis — a harmonic signal of limited duration which a priori does not satisfy the theorem conditions. Ambiguity of the transform result is shown, effects of shifting and overlapping in frequency domain are analyzed, and formulas that determine the limits of theorem applicability are obtained.

We describe the norm of Fourier operator from Lp(T) to lq(ℤ), and from lp(ℤ) to Lq(T) for all p, q ≥ 1, motivated by the problem of finding sharp uncertainty principles expressed in terms of Renyi entropies.

Using his formulation of the potential theoretic notion of balayage and his deep results about this idea, Beurling gave sufficient conditions for Fourier frames in terms of balayage. The analysis makes use of spectral synthesis, due to Wiener and Beurling, as well as properties of strict multiplicity, whose origins go back to Riemann. In this setting and with this technology, with the goal of formulating non-uniform sampling formulas, we show how to construct frames using pseudo-differential operators. This work fits within the context of the short time Fourier transform (STFT) and time-frequency analysis.

The Fast Fourier Transform can be determined in Complex field and Galois field. The paper suggests the architecture for finding Fast Fourier Transform over a Galois field. This method uses the advantage of Cyclotomic decomposition. Basically decomposition of the original polynomial into a sum of linearized polynomial is done and then evaluated at a set of basis points. The Fast Fourier Transform methods can be capably used in implementations of discrete Fourier transforms over finite field, which have extensive applications in cryptography and error control codes. The method is becoming popular because of its low computational complexity. In this paper the hardware design and implementation of Cyclotomic fast Fourier transform architecture over finite field GF(24) is described.

Operator sampling considers the question of when operators of a given class can be distinguished by their action on a single probing signal. The fundamental result in this theory shows that the answer depends on the area of the support S of the so-called spreading function of the operator (i.e., the symplectic Fourier transform of its Kohn-Nirenberg symbol). |S| 1 it is impossible. In the critical case when |S| = 1, the picture is less clear. In this paper we characterize when regular operator sampling (that is, when the probing signal is a periodically-weighted delta train) is possible when S is a parallelogram of area 1.

The problem of maximizing the energy of a signal bandlimited to E 1 = [−σ, σ] on an interval T 1 = [−τ, τ] in the time domain, which is called the energy concentration problem, was solved by a group of mathematicians, D. Slepian, H. Landau, and H. Pollak, at Bell Labs in the 1960s. The goal of this article is to solve the energy concentration problem for the n-dimensional special affine Fourier transformation which includes the Fourier transform, the fractional Fourier transform, the Lorentz transform, the Fresnel transform, and the linear canonical transform (LCT) as special cases. The solution in dimensions higher than one is more challenging because the solution depends on the geometry of the two sets E 1 and T 1 . We outline the solution in the cases where E 1 and T 1 are n dimensional hyper-rectangles and discs.

In this paper we introduce the two-modular Fourier transform of a binary function f : R → R defined over a finite commutative ring R = F 2 [X]/ϕ(X), where F 2 [X] is the ring of polynomials with binary coefficients and ϕ(X) is a polynomial of degree n, which is not a multiple of X. We also introduce the corresponding inverse Fourier transform. We then prove the corresponding convolution theorem.

This work considers the problem of super-resolution. The goal is to resolve a Dirac distribution from knowledge of its discrete, low-pass, Fourier measurements. Classically, such problems have been dealt with parameter estimation methods. Recently, it has been shown that convex-optimization based formulations facilitate a continuous time solution to the super-resolution problem. Here we treat super-resolution from low-pass measurements in Phase Space. The Phase Space transformation parametrically generalizes a number of well known unitary mappings such as the Fractional Fourier, Fresnel, Laplace and Fourier transforms. Consequently, our work provides a general super- resolution strategy which is backward compatible with the usual Fourier domain result. We consider low-pass measurements of Dirac distributions in Phase Space and show that the super-resolution problem can be cast as Total Variation minimization. Remarkably, even though are setting is quite general, the bounds on the minimum separation distance of Dirac distributions is comparable to existing methods., 10 Pages, short paper in part accepted to ICASSP 2015

Complex-exponential basis functions found a wide application in various fields of science and technology [1]. However, the basis of these functions has the disadvantages associated with its complex character. A worthy alternative to complex-exponential basis is Hartley basis, it takes only real values. This article focuses on the basis of spectral analysis in Hartley basis, Hartley transform and Fourier transforms are considered, their dependences are also considered. The similarities and differences of Hartley and Fourier transforms are examined and the possibility of replacing the Fourier transform into the Hartley transform is considered.

We briefly introduce the quaternion linear canonical transform (QLCT), which is a generalization of the quaternion Fourier transform (QFT) to the linear canonical transform (LCT) domain. We show that the QLCT can be reduced to the quaternion Fourier transform (QFT). We then derive the inverse transform of the QLCT using the properties of the QFT. We finally provide an example which describes the relationship between the QFT and the QLCT.

The sampling theorem of Whittaker, Ogura, Kotelnikov, Someya, and Shannon (WOKSS's sampling theorem) appears in most books on information theory, and is well known. WOKSS's sampling theorem must exclude periodic functions. This paper first introduces a sampling theorem for periodic functions, and gives its physical meaning; this theorem is effective for functions that can be expressed as a finite Fourier series in the same way as WOKSS's theorem deals with a band-limited signal. Since many signals such as an Orthogonal Frequency Division Multiplexing (OFDM) signal can be expressed by periodic functions, the sampling theorem can play an important role in a communications area. Extending the theorem, this paper proposes a sampling theorem for periodic functions with no minus frequency component of a finite Fourier series, and shows its example. In addition, we examine the OFDM signal generation and termination by applying sampling theorems for periodic functions including the proposed one.

In the paper we revisit some known remarkable formulas and their idempotent versions highlighting the role of functional spaces. More precisely, we focus on the space where the analogue of the Fourier transform takes place, the Legendre transform in ℝmin and the concave conjugate transform in ℝmax, then we give the analogues of some classical theorems.

In this paper, we proposed an analysis of Eigen functions of the n-Dimensional Fourier transform. We extended the theory developed in our previous publication [1]. We applied our extended theory of systematic construction of Eigen function concept for sine and cosine functions. We explained the concept in a clear way through example construction of Eigen functions of 1-D and 2-D Fourier transform for real even case with negative Eigen value. PP Vaidyanathan discussed in his paper [2] that construction of complex Eigen function f1+ jf2 is possible when both functions f1 and f2 are either real even or real odd. But he didn't discuss what will happen when f1 and f2 are different. We subjected this concept and derived the necessary expressions for complex Eigen functions when function f1 is even and function f2 is odd vice versa.

Exploiting the recent framework of normal factor graphs, this paper presents a transparent exposition of the Krawtchouk transform and its relationship to the Fourier transform and the MacWilliams identities. Such treatment of the subject is believed to be more accessible to wider audience of coding theory. Refereed/Peer-reviewed

A two-dimensional quaternion Fourier transform (QFT) defined with the kernel e − i+j+k/√3 ω · x is proposed. Some fundamental properties, such as convolution theorem and Plancherel theorem are established. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

In this paper, a multiuser communication technique based on the discrete fractional Fourier transform (DFrFT) is discussed. Eigenvectors of the DFrFT transform matrix are used as user sequences, which are transmitted over a real adder channel. Compared to other transforms used in the same context, the advantage of using the DFrFT is that the sequences can be generated from a systematic procedure and the number of generated subspaces, which determines the maximum number of simultaneous users of a scheme, is arbitrary. After describing the basic idea of our approach, we discuss some aspects related to its practical implementation and present preliminary simulation results.

We outline a method that brings the elegant, unifying geometry of orthogonal function expansions to the teaching of Fourier Analysis in our gateway course on Signals and Systems at UC Berkeley. Our approach starts with discrete-time periodic signals. Their straightforward representation as finite-dimensional Cartesian vectors provides a gentle ingress into the more abstract Euclidean vector spaces that inform the Fourier decompositions of richer signal types. As we describe how a signal fragments into its elemental frequencies, we are careful with the mathematics but we do not let rigor eclipse clarity; plausible reasoning often suffices. We sequence the topics and develop the theory to reduce algebraic clutter and promote geometric insight into the progressively nuanced world of frequency decompositions nestled in the beautiful heart of Fourier Analysis.

In this paper, various theoretical analysis about Fourier transformation in the Clifford geometric algebra have been proposed. It is believed that these results in this paper are also valuable in solving efficiently classes of very complex engineering problems arising in practical engineering applications.

A new transform, called the generalized fractional Fourier transform (gFrFT), is proposed. Originally, the eigenfunctions of the fractional Fourier transform (FrFT) are known as the Hermite Gaussian functions (HGFs). Besides, in optics, the HGFs are generalized to be the generalized Hermite Gaussian functions (gHGFs) and their adjoint functions (AgHGFs). Therefore, we can define the gFrFT by the eigenvalues of the FrFT and the eigenfunctions (gHGFs/AgHGFs) in the analysis or synthesis step. Four types of the gFrFT are defined and discussed. The integral forms of the gFrFTs are derived and they are closely related to some popular transforms, such as the Fourier transform (FT), the FrFT, and the complex linear canonical transform (CLCT). We can also extend the FT and the FrFT to the standard and elegant versions. Finally, some properties of the gFrFT are discussed.

In this paper, we extend the concept of Eigen functions of the Fourier transform discussed by P. P. Vaidyanathan [9]. But for a scaling factor, if there is no change in the function after the application of the Fourier transform, we term them as Eigen functions of the Fourier transform. Gaussian is the well known Eigen function. But, there are various other functions that can be developed systematically. The theory developed by P. P. Vaidyanathan for 1-D signals is extended for 2-D case in this paper. We also presented extensions for the complex signal case.

It has recently been shown that the model in experimental design can be expressed in terms of an orthonormal system. In this case, the model is expressed by using Fourier coefficients instead of the effect of each factor. As there is an abundance of software for calculating the Fourier transform, such a system allows for a straightforward implementation of the procedures for estimating the Fourier coefficients by using Fourier transform. However, Fourier coefficients themselves do not provide a direct representation of the effect of each factor, and the relation between the Fourier coefficients and the effect of each factor has not yet been clarified. In this paper, we present theorems of the relation between the Fourier coefficients and the effect of each factor. By using these theorems, the effect of each factor can be easily obtained from the computed Fourier coefficients. Therefore, with the aid of an orthonormal system, it is possible to easily implement the estimation procedures as well as to understand how each factor affects the response variable in the model.

The Euler's formula ej θ = cos θ + j sin θ has been proved to be useful in simplifying the evaluation of complex and hyper-complex fourier transform by turning the multiplications of quaternion directly into matrices's addition. In this paper we point out that while further considering the formula equation, the above analytic computations can be proceeded farther, thus the numerical process is simplified. We also illustrate the relationship of this formula with the rotation in physics system, which is helpful for us to do the rotation when discussing the image filters using the quaternion operators. Additionally, we use a special algebraic summation convention Σ m A m B m = Ai Bi to reduce the complexity of non-commutivity in calculating the quaternion Fourier transform and the discrete quaternion Fourier transform. This technique is expected to simplify further the corresponding numerical programme.

Fourier transform is a significant tool in image processing and pattern recognition. By introducing hypercomplex number, hypercomplex Fourier transform [1] treats signal as vector field and generalizes conventional Fourier transform. Inspired from that, hypercomplex polar Fourier analysis is proposed in this paper. This work extends conventional polar Fourier analysis [5]. The proposed method can handle hypercomplex number represented signals like color image. The hypercom-plex polar Fourier analysis is reversible that means it can be used to reconstruct image. The hypercomplex polar Fourier descriptor has rotation invariance property that can be used for feature extraction. Due to the noncommutative property of quaternion multiplication, both left-side and right-side hypercomplex polar Fourier analysis are discussed and their relationships are also established in this paper. The experimental results on image reconstruction, rotation invariance and color plate test are given to illustrate the usefulness of the proposed method as an image analysis tool.

The fractional Fourier transform (FRFT) which is a generalization of the Fourier transform (FT) has been shown to be a powerful analyzing tool in signal processing. Many properties of the FT including the generalized sampling theorem have been extended to the FRFT. However, the existing extensions of the classical generalized sampling theorem for the FRFT are derived from the lowpass signal viewpoint, and the implementation of those extensions is inefficient for the effect of spectral leakage. In this paper, we propose a new generalized sampling theorem associated with the FRFT from the fractional bandpass signal viewpoint. The proposed theorem which is constructed by the ordinary convolution in the time domain can overcome the effect of spectral leakage and is easy to implement in practical applications.

Fourier analysis has many applications in various science and technology. In most problem researchers have to analyze functions (data), which has some singularities. This makes some difficulties in Fourier analysis of singular functional. In these, harmonic analysis in the spaces of distributions can be applied. Recently (see for instance [9]-[11]) interest in spectral expansions of distributions increased and number of research papers were published. Present work it devoted to convergence/summation and regularization of Fourier series of distributions in different topologies. In multidimensional case, convergence essentially depends on methods of summation, i.e. on the definition of partial sums. Even “good” defined partial sums may not supply convergence of Fourier series and in this case, some regularization of the partial sums is required.

The two dimensional fractional Fourier transform (FrFT), which is a generalization of the Fourier transform, has many applications in several areas including signal processing and optics. Signal processing and pattern recognition algorithms make extensive use of convolution. In pattern recognition, convolution is an important tool because of its translation invariance properties. Also convolution is a powerful way of characterizing the input-output relationship of time invariant linear system. In this paper the convolution theorem for two dimensional fractional Fourier transform in the generalized sense is proved.

Based on the fractional Fourier transform, the sampling theorem of the 2-D signals is established.

The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform (FT), has emerged as a very efficient mathematical tool for signals which are having time-dependent frequency component. FRFT has an advantage over other transforms being used in the application areas like: signal processing and optics. Many properties of this transform are already known, but an extension of convolution theorem of Fourier transform is still not having a widely accepted closed form expression. In the literature of recent past different authors have tried to formulate convolution theorem for FRFT, but none have received acclamation because their definition do not generalize very nicely the classical result for the FT. A new convolution theorem for FRFT is proposed in this article which is supposed to be a better realizable proposition.

In this paper, a finite field version for the fractional Fourier transform is introduced. We show that, in some aspects, the finite field fractional Fourier transform (4FT) is in perfect analogy with the discrete fractional Fourier transform. On the other hand, we consider some definitions and properties from the finite field context which allow us to discuss particularities of the 4FT.

In this paper, we discuss a concept of the mixed Fourier transformation, when signals are transformed to the time-frequency domain, where the difference between the time and frequency is disappeared. Both cases of continuous-time and discrete-time signals are considered, and properties of the mixed transformations are described. The concept of mixed Fourier transform includes the fractional power of the discrete Fourier transform (DFT) which is referred to as a discrete version of the fractional Fourier transform. Mixed Fourier transformations can be used for calculation of roots of the Fourier and identity transformations. Examples of such root transforms for signal processing and image encryption are given.

to confirm the useful information better, the changing magnitude of the value must be considered. Secondly, the effect of filter is related to the choice and the input parameters of filter, but it is so difficult to confirm the choice and the input parameters of filter due to the complicated wave of the real time series. So this de-noising method will be complicated to apply. According to this shortage, here proposes a kind of de-noising method based on event. This method directly utilizes the mathematical features of curve such as the extremum slope and curvature to de-noise. The principle of this method is simple and easy to operate. In order to prove the validity and advantage, Here first it researches on the de-noising method based on Fourier transform and makes experiment based on real time series of the stock with two methods. The result proves that the new method can de-noise effectively, at the same time it is easy to operate and can keep the original characters of the time series furthest. II、 DE-NOISING METHOD BASED ON FOURIER TRANSFORM Fourier transform is one of the basic and common-used signal processing method. If let {at :t=…,-1,0,1,…} denote one infinite series of real value variables, then Fourier transform can be defined as complex value function below

In this paper, a unified framework for orthonormal decomposition of the array manifold is derived. An equivalence between spherical harmonics and 2-D Fourier basis functions is established. Consequently, a link between spherical harmonic spectra and 2-D Fourier spectra is found. Moreover, the spherical harmonic and 2-D Fourier decompositions of the array manifold vector, i.e. Wavefield Modeling and 2-D Effective Aperture Distribution Function (EADF) are shown to be equivalent. The rotation of functions on the 2-D Fourier domain is discussed.

In order to reconstruct discontinuous signal with the least error by using finite information, this paper combines the concept of all-phase data processing and the conventional Fourier approximation. By utilizing the coefficients acquired by discrete Fourier transforming the samples of signal and the high harmonic information, all-phase Fourier reconstruction is formed. Theoretical deduction and experimental research show that the waveform reconstructed by all-phase Fourier method has less error than those constructed by conventional continuous Fourier integral approximation and the discrete Fourier approximation.

The uniform sampling theorem and the reconstruction formulae associated with the fractional Fourier transform (FrFT) have been deduced in the literature, but the nonuniform sampling is yet to receive attention in the fractional Fourier domain. This paper focus on a special kind of non-uniform sampling process associated with the fractional Fourier transform. A nonuniform sampling model is first introduced and then, based on a useful lemma, the reconstruction formula for the nonuniform sampled signal points associated with the fractional Fourier transform is deduced. In addition, the classical results in the Fourier domain can be seen as a special case of our work.

In this paper, under weaker condition, we give the sufficient and necessary condition that phi(x) is a scaling function of L2(RS), we give results of the whole support of wavelets, these results are characterized by some equalities and inequalities. We have improved completely Long's results.

The Fourier spectrum characters of the almost perfect binary array pair are studied, which is based on the concept of the almost perfect binary array pair. The distributions of the weights, existent range and the imbalance of almost perfect binary array pair are given by using this spectrum characters.

It is well-known that Fourier approximation, FA and fast Fourier transformation, FFT are applied in many fields. In order to use them to data mining, we present the transfer, which is a mapping from an interval to another. Moreover, some techniques constructing transfer is also given in the paper.

In terms of the fractional Fourier transform and the generalized Hilbert transform, in this note, we prove the kernel function K-p (u,t) of the inverse fractional Fourier transform is a generalized analytic signal. Since there is a close relation between analytic signals and Bedrosian theorem, the generalized Bedrosian theorem is provided in the fractional Fourier domain

A summary of results from linear algebra pertaining to orthogonal projections onto subspaces of an inner product space is presented. A formal definition and a sufficient condition for the existence of a fractional transform given a unitary periodic operator is given. Next, using an orthogonal projection formula the class of weighted discrete fractional Fourier transforms (WDFrFTs) is shown to be completely determined by four integer parameters. Particular choices of these parameters yield the Dickinson-Steiglitz [1] and Santhanam-McClellan [2] WDFrFTs. Another choice gives a WDFrFT which agrees with any eigenvector decomposition-based DFrFT [3] for terms of degree less than four. Applications of the proposed algorithm to chirp filtering is discussed.

The class of generalised Hadamard transforms includes the Fourier, generalised, discrete Fourier, Walsh-Hadamard, complex Hadamard and reverse jacket transforms. The generalised Hadamard transforms may by partly classified by signal length, by group of entries in the transform matrix and by a recently introduced third parameter, the jacket width of the transform matrix. Here we introduce a weighted trace map, which realises the Fourier transform as an exponential weighted sum of Galois ring traces. We give examples of Fourier transforms with jacket width 0, jacket width 1 and maximum jacket width (half the signal length). We show the Fourier transforms of length 4k with entries in {plusmn1, plusmni} obtained using the weighted trace map from the Galois ring GR(4,k) have jacket width 2k-1

When using spectral transforms, particular values or features of the spectral coefficients may give information about the properties of the function being transformed. The present paper shows that this is also the case with Fourier transforms on non-Abelian groups. The fact that then some of the spectral coefficients are matrix-valued adds new possibilities to recognize some structural patterns of the spectrum and associate them to properties of the function.