Although some computer programs for evaluating the pswfs are available 10,14, many are not. The spheroidal functions also appear as eigenfunctions of finite analogs of fourier transforms. When applying the fourier transform to extract spectral information from a signal, we assume that each fourier coefficient is a reliable representation of the amplitude and relative phase of the corresponding component frequency. As such, they play an important role in signal processing, fourier analysis, and approximation theory. Prolate spheroidal wave functions, fourier analysis and uncertainty iii. Computational harmonic analysis references page winter, 2002. We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on \l2\mathbb r\. Prolate spheroidal wave functions, fourier analysis and uncertainty i bell. Prolate spheroidal wave functions, fourier analysis and uncertaintyiv.
Discrete prolate spheroidal slepian sequences matlab dpss. The prolate functions themselves are computed by a legendregalerkin discretization of the prolate differential equation. International journal of imaging systems and technology, 1262. The functions have applications in electromagnetism, acoustics, quantum mechanics and finite fourier analysis.
The generalized prolate spheroidal wave functions has been recently defined by gupta 4 as the solution of the differential equation 2. Optimal data acquisition in fmri using prolate spheroidal wave functions. Ball prolate spheroidal wave functions in arbitrary dimensions. Recently, pswfs have been becoming increasingly popular in various areas in which such functions occur. Compared to existing differentiation schemes based on orthogonal polynomials, the new class of differentiation schemes requires fewer points per wavelength to achieve the same accuracy when it is used to approximate derivatives of bandlimited functions. Prolate spheroidal wave functions, fourier analysis and uncertaintyv. In two earlier papers in this series, the extent to which a square. Some comments on fourier analysis, uncertainty and modeling. It was used to study the generalized prolate spheroidal wave functions and their connection to energy concentration problems. Pollak, prolate spheroidal wave functions, fourier analysis and uncertaintyiii. The prolate functions are then defined by legendre series whose coefficients are the eigenfunctions of the matrix eigenproblem. Its inverse fourier transform, gx, has maximal sig. Recently, pswfs have been becoming increasingly popular in various areas in which such functions occur this includes physics e. The dimension of the space of essentially time and bandlimited signals, bell system tech.
Computer programs for this purpose have been developed by many. Inthis paper, the approximationpropertiesofthe prolatespheroidalwavefunctionsoforderzeropswfsarestudied, andasetofoptimal. Some comments on fourier analysis, uncertainty and. The singular functions for the problem of recovering a time limited function from its fourier transform in a certain band of frequencies are given, in the simplest case, by the prolate spheroidal wave functions. A discrete time series has associated with it an amplitude spectrum which is a periodic function of frequency. These finite transforms are more flexible than the fift. Based on the pswf, the finite fractional fourier transform was developed. Adapted wavelet analysis from theory to software, a k peters, ltd. Prolate spheroidal wave functions pswf provide significant advantage in ultra wideband uwb pulseshapes design because of the time limited nature. Calculation of spheroidal wave functions sciencedirect.
Then, i will prove that they are also the eigenfunctions of an appropriate timefrequency limiting operator. They lead to the optimal solution of the concentration problem once posed by claude e. Computational harmonic analysis references page winter, 2002 course. The new signals are a generalization of the prolate spheroidal wave functions also known as slepian functions to quaternionic space, which are called quaternionic prolate spheroidal. Spatial smoothing in fmri using prolate spheroidal wave. The eigenvalues of secondorder spectral differentiation. In this paper, for digital signal processing application, we derive. Continuous prolate spheroidal wave functions pswfs were proposed by slepian and are useful for analyzing the characters of the finite duration continuous fourier transform. Lazhar dhaouadi submitted on 18 jul 2007 v1, last revised 9 apr 2008 this version, v2. Software for computing the spheroidal wave functions using. The schemes are constructed via the prolate spheroidal wave functions pswfs.
Prolate spheroidal wave functions, an introduction to the slepian series and its properties. The multitaper method overcomes some of the limitations of conventional fourier analysis. We extend the theory of pswfs for the finiteextension fractional fourier transform, the finiteextension linear canonical transform, and the finiteextension offset linear canonical transform. Mathematical tools for bandlimited approximation applied mathematical sciences book 187 kindle edition by osipov, andrei, rokhlin, vladimir, xiao, hong. The present paper finds a new kind of hypercomplex signals whose energy concentration is maximal in both time and frequency under quaternionic fourier transform.
Gumerov, and ramani duraiswami abstract the spheroidal wave functions, which are the solutions to the helmholtz equation in spheroidal coordinates, are notoriously di cult to compute. Download it once and read it on your kindle device, pc, phones or tablets. A cosine wave is also a sine wave but with 90 phase shift. They also satisfy a discrete orthogonality relation. Prolate spheroidal wave functions pswf provide significant advantage in ultra wideband uwb pulseshapes design because of the time limited nature and wide bandwidth of such orthogonal signal. Prolate spheroidal wave function as exact solution of the. Prolate spheroidal wave functions pswfs are known to be useful for analyzing the properties of the finiteextension fourier transform fift. Prolate spheroidal wave functions, fourier analysis and uncertainty. Spectral methods based on prolate spheroidal wave functions. Featured on meta feedback on q2 2020 community roadmap. Discretetodiscrete prolate spheroidal wave functions and. The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property.
The crux of the question seems to be in fourier transform and i am lost just because i have never came across dual cube. In mathematics, the prolate spheroidal wave functions pswf are a set of orthogonal. Fourier expansion of generalized prolate spheroidal wave. Prolate spheroidal wave functions, fourier analysis, and. Journal of classical analysis volume 11, number 1 2017, 121 doi. Technical report tr1519, department of mathematics, yale university. Feb 01, 20 prolate spheroidal wave functions, fourier analysis and uncertainty i by d. This capability allows the solution of the helmholtz equation when the spheroid is large compared with the wavelength. Properties of the functions are derived and several applications to the representation of signals are made.
I guess what you are talking about is the product of second order moments of the function and its ft has an lower bound, which is termed as principle of uncertainity for reasons. Software for computing the spheroidal wave functions using arbitrary precision arithmetic ross adelman, nail a. Generalized prolate spheroidal wave functions for offset. Prolate spheroidal wave functions in qfourier analysis. The fmri ac tivation map was obtained with the stimulate software.
The prolate functions are then defined by legendre series whose coefficients. Inthis paper, the approximationpropertiesofthe prolatespher. Prolate spheroidal wave functions in qfourier analysis 707. Prolate spheroidal wave functions, fourier analysis and uncertainty, iii, bell system tech. Prolate spheroidal wave functions, fourier analysis and. Slepianprolate spheroidal wave functions, fourier analysis and uncertaintyiv. Soleski,a new friendly method of computing prolate spheroidal wave func tions and wavelets, appl. Wager2 1 department of statistics, columbia university, new york, ny, 10027 2 department of psychology, columbia university, new york, ny, 10027 address. We prove a weak version of hardys uncertainty principle using properties of the prolate spheroidal wave functions. Sep 04, 2016 the present paper finds a new kind of hypercomplex signals whose energy concentration is maximal in both time and frequency under quaternionic fourier transform.
The objective is the use of large values of mode number and spheroidal parameter. A weak version of hardys uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time. This paper investigates the extent to which a time series can be concentrated on a finite index set and also have its spectrum. Osa generalized prolate spheroidal wave functions for. Prolate spheroidal wave functions in q fourier analysis authors. It was used to study the generalized prolate spheroidal wave functions and their connection to. Analysis of spectral approximations using prolate spheroidal wave functions lilianwang abstract. See also other links at wikipedia article on prolate spheroidal wave functions course notes on zeropadding and interpolation by james mcclellan, georgia tech. Spatial smoothing in fmri using prolate spheroidal wave functions martin a. Osipov, certain upper bounds on the eigenvalues associated with prolate. Uncertainty principles, prolate spheroidal wave functions. Prolate spheroidal wave functions, fourier analysis and uncertainty i by d. Prolate spheroidal wave functions pswfs are the eigenfunctions of the bandlimited operator in one dimension. On the prolate spheroidal wave functions and hardys.
Pollak, prolate spheroidal wave functions, fourier analysis and uncertainty. The lct was first extended to the clifford analysis setting in 38. Prolate spheroidal wave functions, because of their many remarkable properties. Download pdf prolate spheroidal wave functions of order. Martin lindquist 1255 amsterdam ave, 10th floor, mc 4409. Another property that gives them their appeal is that their fourier transform over. Nonetheless, the pswfs exhibit the unique properties to form an orthogonal basis of l 2. We develop a prolate spheroidal wave function pswf filter designed to eliminate truncation artifacts and compare its performance to the standard gaussian filter in simulations and analysis of fmri data on a visualmotor task. Use features like bookmarks, note taking and highlighting while reading prolate. Prolate spheroidal wave functions, fourier analysis and uncertainty i. That is why in signal processing, the fourier analysis is applied in frequency or spectrum analysis. Jun 17, 2019 thus, the term frequency and sine are almost synonymous. The dimension of the space of essentially time and bandlimited signals. This fact was unraveled by david slepian and his collaborators at bell lab in 1960s.
Prolate spheroidal wave functions, an introduction to the. Prolate spheroidal wave functions, fourier analysis, and uncertainty. In the literature, the prolate spheroidal wave functions pswfs are often regarded as mysterious set of functions of l 2. This is done by first computing the prolate nodal basis, and the appropriate quadrature and weights that replace the legendrelobatto grid points, quadrature weights, and. In this paper, we introduce the prolate spheroidal wave functions pswfs of real order. All results in this paper were generated from a c program written by the first. Spectral analysis of the finite hankel transform and. Prolate spheroidal wave functions and applications. Boyd considers the use of prolate spheroidal wave functions. Spatial smoothing in fmri using prolate spheroidal wave functions. These are of great utility in disciplines such as geodesy 28 or cosmology. Therefore, when you talk about sine and cosine functions, you are taking in terms of frequencies. Thus, the term frequency and sine are almost synonymous.
Without the motivation in terms of wavelets, i would agree with you and cite hardys version of the uncertainty principle as a prime example. Prolate spheroidal wave functions, fourier analysis, and uncertainty v. Prolate spheroidal wave functions and their properties. However one does frequently see arguments in harmonic analysis where something resembling what the op is describing is used. A new friendly method of computing prolate spheroidal wave. Prolate spheroidal wave functions whose domain is a portion of the surface of the unit sphere are more generally called slepian functions see also spectral concentration problem. Prolate spheroidal wave functions slepian functions are special functions that are most localized in both spatial and frequency domain, simultaneously.
A two dimensional prolate spheroidal wave function 2dpswf method is developed to address these problems. These are of great utility in disciplines such as geodesy or cosmology. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and bandlimit. The dimension of space of essentially timeand bandlimited signals, bell syst. Other results include a general perturbation scheme for differential equations and the reduction to two dimensions of the case of functions of d 2. A tutorial on fourier analysis fourier series gaussianwaves. The theory developed in the preceding paper is applied to a number of questions about timelimited and bandlimited signals. Support of function and support of its fourier transform. Prolate spheroidal wave functions and their properties computation of the pswfs by flammers method uniform estimates of the pswfs and their derivatives applications of the pswfs prolate spheroidal wave functions from pde point of view pswfs as eigenfunctions of a di erential and an integral operator some properties of the pswfs spheroidal. The magic of the prolate spheroidal functions in various.
May, 2014 we prove a weak version of hardys uncertainty principle using properties of the prolate spheroidal wave functions. Spectral analysis of the finite hankel transform and circular. Abstract generalized prolate spheroidal functions gpsf are the eigen functionsofthetruncatedfouriertransform, restrictedtoddimensional balls in the spatial domain and frequency domain. This paper discusses the calculation of prolate spheroidal wave functions for integral mode numbers and real spheroidal parameter. Prolate spheroidal wave functions associated with the. Two dimensional prolate spheroidal wave functions for mri. Finally, we investigate some properties, such as the spectral concentration problem. We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for. Despite their useful properties in many applications, gpsfs are often replaced by crude approximations. On the evaluation of prolate spheroidal wave functions and.
340 840 675 216 1479 1039 1623 225 372 994 466 315 1465 651 622 1642 1096 1183 1024 1578 1196 288 30 1019 1394 1120 124 1286 1228 915 265 969 926 614 833 1268 891 1471 265 332 104