Discrete prolate spheroidal slepian sequences matlab. The author has extended his previous results pertaining to spheroidal functions by introducing a new finite transform involving generalized prolate spheroidal functions. Load discrete prolate spheroidal sequences from database. The prolate spheroidal wave equation pswe is transformed, using suitable mappings, into three different canonical forms which resemble the jacobi, laguerre and the hermite differential equations. Compute discrete prolate spheroidal slepian sequences for use as timebased smoother. Spheroidal coordinates the prolate spheroidal coordinate system, shown in fig. Bandlimited signal extrapolation using prolate spheroidal. The spheroidal interpolant function closely resembles a spherical variogram, which has a fixed range beyond which the value is the constant sill. Matlab, crc press, boca raton, fla, usa, 3rd edition, 2009. A significant step in the process is the higher order piecewise polynomial approximation of the overlap integral required for obtaining the expansion coefficients accurately with very high precision.
Threedimensional timeharmonic acoustic multiple scattering problems are considered for a finite number of prolate and oblate spheroidal objects adjacent to flat surfaces. In terms of the prolate spheroidal coordinates defined in the main text, the prolate spheroid is determined by. Prolate spheroidal coordinates are a threedimensional orthogonal coordinate system that results from rotating the twodimensional elliptic coordinate system. Volkmer department of mathematical sciences, university of wisconsinmilwaukee, milwaukee, wisconsin acknowledgements. Potential outside a uniform spheroid university of texas at. They may be defined as eigenfunctions of either a differential operator or an integral operator as observed by slepian in the 1960s.
Uniform asymptotic approximations are obtained for the prolate spheroidal functions ps n mx. Evaluation of q in an electrically small antenna in prolate spheroidal coordinates 2004 cached. A new friendly method of computing prolate spheroidal wave. Software for computing the spheroidal wave functions using arbitrary precision arithmetic ross adelman, nail a. The problem is to construct the angular eigenfunctions of the laplacian in three dimensional, spheroidal coordinates.
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. Just multiply 3 with the volume separately and multiply 4 with pi, x coordinate and y coordinate and then divide both the results of both multiplication. Now imagine stretching or contracting the sphere by a ratio of ar in the. Spatial smoothing in fmri using prolate spheroidal wave.
On the use of discrete prolate spheroidal windows for. S spectral methods based on prolate spheroidal wave functions for hyperbolic pdes. Spatial smoothing in fmri using prolate spheroidal wave functions martin a. Image conditions and addition theorems for prolate and. The transition matrix for acoustic and elastic wave scatte ring in prolate spheroidal coordinates roger h. For example, oblate and prolate spheroids being surfaces of revolution can be more easily conformed to most districts of human body e. Oblate and prolate spheroidal coordinates are widely used in many fields. The wave equation in prolate spheroidal coordinates is 1 where 2 substitute in a trial solution 3 4 the radial differential equation is 5 and the angular differential equation is 6 note that these are identical except for a sign change. The helmholtz differential equation is separable in prolate spheroidal coordinates see also helmholtz differential equationprolate spheroidal coordinates, latitude, longitude, oblate spheroidal coordinates, spherical coordinates. This matlab function creates a database of discrete prolate spheroidal dpss or slepian sequences and saves the results in dpss. Prolate spheroidal coordinates are a threedimensional orthogonal coordinate system that results from rotating a spheroid around its major axis, i. Asymptotics of prolate spheroidal wave functions t.
A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the elliptic cylindrical coordinates about the xaxis, which is relabeled the zaxis. Prolate spheroidal coordinates from wolfram mathworld. The described algorithm exploits the properties of the prolate spheroidal coordinates. Relation of average error in prolate spheroidal wave. Dunster san diego state university san diego, ca 92182 usa abstract. Evaluation of q in an electrically small antenna in prolate spheroidal coordinates, year 2004 share. Just multiply 3 with the volume separately and multiply 4 with pi, x coordinate and y coordinate and then divide both the. A spheroid is also known as ellipsoid which is formed by rotating the ellipse with major axis or minor axis. A ray tracing algorithm using the discrete prolate spheroidal. Let the axis of rotation coincide with the axis, and let the outer boundary of the spheroid satisfy.
Section 2 provides a brief introduction to the prolate spheroidal coordinates and to the prolate spheroidal harmonics. Potential outside a uniform spheroid university of texas. Prolate spheroidal coordinates for modeling mobileto. The third set of coordinates consists of planes passing through this axis. Line of sight intersection with oblate spheroid matlab. However, the corresponding pdfs required quite extensive numerical evaluations. Semianalytical computation of acoustic scattering by. The resulting eigenvalues and eigenfunctions for real and complex spheroidal bandwidth parameter, c, agree with those in the literature from four to more than eleven significant figures. I have two points ax1, y1, z1, bx2, y2, z2, minor axis lengtha and major axis lengthb. This matlab function loads all sequences with length n and timebandwidth product nw in the columns of e and their corresponding concentrations in vector v from the dpss matfile database dpss. The prolate spheroidal coordinate system, shown in fig. Prolate spheroidal wave functions, fourier analysis and uncertainty i. A prolate spheroid is centered at the origin, its focal axis is aligned with the axis, and its interfocal distance is. Monographs tying together many aspects of the theory of spheroidal wave functions were written by strutt, stratton et al.
The most general solution to laplaces equation in these coordinates is. Note the odd order tapers are normalized so that the slope at the centre is positive, in accordance with slepian 1978 and thomson 1982. Finally, a suggestion for the use of discrete prolate spheroidal windows for spectral analysis is provided. Originally, the spheroidal wave functions were introduced by c. Often the choice of an appropriate coordinate system permits a simpler solution to a problem. Pswf prolate spheroidal wave function acronymfinder. In his paper 25, slepian proposed the idea of bandlimited. A spheroid is the solid body produced by rotating an ellipse about a major or minor axis. Citeseerx evaluation of q in an electrically small. Note that there is no \favored length scale in the elementary coordi. In mathematics, the prolate spheroidal wave functions pswf are a set of orthogonal bandlimited functions. Prolate spheroidal wave function how is prolate spheroidal. The and coordinates are dimensionless coordinates based on confocal ellipsoids of revolution and confocal hyperboloids of revolution.
Oblate spheroidal coordinates are the natural choice for the translation of any ellipsoid parallel to a principal axis. Hackman acoustics brancl naval coastal systems center, panama city, florida 32407 received 28 june 1983. These are the prolate, oblate and generalized spheroidal wave functions and to compute the corresponding eigenvalues. Prolate spheroidal coordinates for modeling mobiletomobile. Discrete prolate spheroidal slepian sequences matlab dpss. Although the earth is very round, it is an oblate spheroid rather than a perfect. Hamming, blackman, bartlett, chebyshev, taylor, kaiser. Similarly, the spheroidal interpolant function flattens out when the distance from x is greater than a defined distance, the range. A ray tracing algorithm using the discrete prolate. A finite difference construction of the spheroidal wave. If the lineofsight does not intersect with the spheroid, lat and lon contain nan s. This software is called scattering and runs in matlab. When considering the same modeling problem in prolate spheroidal coordinates, it becomes possible to obtain analytical representations in some special, yet practically relevant cases, and thus simplify the computations of the resulting density functions significantly.
Prolate spheroidal wave functions, because of their many remarkable properties leading to new applications, have recently experienced an upsurge of interest. Solution of axisymmetric potential problem in oblate spheroid. Potential outside a uniform spheroid let us now calculate the gravitational potential generated outside a spheroid of uniform mass density and mean radius. In this work it is shown that in certain cases a prolate spheroidal coordinate system can be advantageous when calculating the delaydependent doppler probability density functions for mobiletomobile wireless channels. The method used is based on the dyadic greens function technique in the spheroidal coordinates.
Oblate spheroidal coordinates are the natural choice for the translation of. A finite difference construction of the spheroidal wave functions. Ipems aim is to promote the advancement of physics and engineering applied to medicine and biology for the public benefit. The overlap and the kinetic integrals in r3 are reduced to the integrals over the rectangular domain in r2, what substantially reduces the complexity of the problem. Helmholtz wave equation 1 in spheroidal coordinates. Illustration of photoacoustic wave production of rbcs and mfc7 cell nuclei, the corresponding oblate spheroidal coordinates system for a rbc and prolate spheroidal coordinates system for a cell.
A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the elliptic cylindrical. We have made our software freely available for download on our webpage. It has a closed quadric surface that is a threedimensional analogue of an ellipse. This matlab function transforms corresponding elements of the twodimensional cartesian coordinate arrays x and y into polar coordinates theta and rho. The wave equation in prolate spheroidal coordinates is 1. Compute discrete prolate spheroidal sequences in slp. A fast and simple finite difference algorithm for computing the spheroidal wave functions is described. The nth chromatic derivative of an analytic function is a linear combination of kth ordinary derivatives with 0 k n, where the coe cients. Software for computing the spheroidal wave functions. Wave propagation by spheroids is modelled by the method of separation of variables equipped with the addition theorems in the spheroidal coordinates. Then, interior and exterior neumann functions and their image systems for the laplace operator in the prolate spheroidal geometry are developed in sections 3 and 4, respectively. Prolate spheroidal wavelets and multidimensional chromatic series expansions devendra kumar1 abstract. Coordinates x,y and z of prolate spheroid can be used to solve various partial differential equations.
I need to draw a prolate spheroid with a and b as their focii points. Prolate spheroid coordinates calculator x coordinate of. Prolatespheroidalcoords, prolatespheroidalcoords3d. Solution of axisymmetric potential problem in oblate. Approximate formulae for certain prolate spheroidal wave functions valid for large.
A ray tracing algorithm using the discrete prolate spheroidal subspace mingming gan1, francesco mani2, florian kaltenberger3, claude oestges4, thomas zemen1 1ftw forschungszentrum telekommunikation wien, vienna, austria. The prolate angular function of the first kind is given by 7 where is an associated legendre. Discrete prolate spheroidal or slepian sequence database. Semianalytical computation of acoustic scattering by spheroids and. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and bandlimit. I thought that it might be convenient to use prolate spheroidal coordinates in this situation, but ive run into a problem when trying to satisfy the condition that the field vanish at infinity. Figure 1 omitted figure 2 omitted figure 3 omitted figure 4 omitted figure 5 omitted 5 conclusions we have studied properties of prolate spheroidal wave functions pswfs that are bandlimited to 12, 12 and approximately timelimited to t, t for a fixed t 0. Simulating photoacoustic waves produced by individual. Radiation pattern of aperture coupled prolate hemispheroidal. In this work, the dyadic greens functions pertaining to a magneticcurrent source located in a dielectric spheroid is derived and expressed in a form. Wager2 1 department of statistics, columbia university, new york, ny, 10027 2 department of psychology, columbia university, new york, ny, 10027 address.
Software for computing the spheroidal wave functions using. J discretetodiscrete prolate spheroidal wave functions and finite duration discrete fractional fourier transform. Download scientific diagram 3d illustation of spherical coordinates, and the. Rotation about the other axis produces the oblate spheroidal coordinates prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match. Martin lindquist 1255 amsterdam ave, 10th floor, mc 4409. A detailed analysis and study of a prolate hemispheroidal dielectric resonator antenna dra excited by a rectangular slot aperture are developed. Image theory for neumann functions in the prolate spheroidal.
The prolate angular function of the first kind is given by. The transition matrix for acoustic and elastic wave. Left ventricular motion reconstruction with a prolate. Niven, which lead to a helmholtz equation in spheroidal coordinates. Image conditions and addition theorems for prolate and oblate. Compare mainlobe widths and sidelobe levels of windows as a function of their size and other parameters. Prolate spheroidal coordinates for modeling mobiletomobile channels article in ieee antennas and wireless propagation letters 14 september 2014 with 18 reads how we measure reads. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. This approach uses the tridiagonal method and exploits symmetry. Prolate spheroidal coordinates are a threedimensional orthogonal coordinate system that results from rotating the twodimensional elliptic coordinate system about the focal axis of the ellipse, i. This chapter is based in part on abramowitz and stegun 1964, chapter 21 by a. A prolate spheroid is a spheroid that is pointy instead of squashed. Pswf is defined as prolate spheroidal wave function very rarely.
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