An acoustic reconstructing method in prolate spheroidal coordinate system is proposed in this paper. The toolbox transparently handles the inherent antisymmetrization of wavefunctions and. The wave equation in spheroidal coordinates and its solutions. A detailed analysis and study of a prolate hemispheroidal dielectric resonator antenna dra excited by a rectangular slot aperture are developed. Note the odd order tapers are normalized so that the slope at the centre is positive, in accordance with slepian 1978 and thomson 1982. In writing this chapter, an unpublished summary by gerald c. The algorithm developed here for calculating the prolate spheroidal function is compact, fast, and efficient, and compared well with other results in the literature. Use features like bookmarks, note taking and highlighting while reading prolate spheroidal wave functions of order. On the use of discrete prolate spheroidal windows for. 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. Ecg logger the ecg logger project is a wearable cardio monitor for longterm up to 24h ecg data acquisition a. This paper introduces the fermifab toolbox for manyparticle quantum systems. Lauchle for having computed the curves shown in figs.
Software for computing the spheroidal wave functions using. A ray tracing algorithm using the discrete prolate. We now consider a football shaped prolate spheroidal surface. I have two points ax1, y1, z1, bx2, y2, z2, minor axis lengtha and major axis lengthb. Code for computing generalized prolate spheroidal functions gpsf for the 1d version, see. Construct the first four discrete prolate spheroidal sequences of length 512. Timevariant channel estimation using discrete prolate spheroidal sequences abstract. 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. Acoustic reconstructing method in prolate spheroidal. The quantities r, theta and phi are spherical coordinates that satisfy the three equations. The output variable, status, is 0 if there is success. Furthermore, efficient algorithms for calculating the prolate spheroidal wave functions are developed using the software package matlab. Spheroidal coordinates the prolate spheroidal coordinate system, shown in fig.
Semianalytical computation of acoustic scattering by. Matlab software, the best results are obtained by using the solvers ode45 and ode1. 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. Coordinates of prolate spheroid is used to solve various partial differential equations. With the help of the method of similarity transformations, an approach is considered that makes it possible to find particular solutions of the grosspitaevskii equation with a nonstationary coefficient of nonlinearity in prolate spheroidal coordinates. Software for computing the spheroidal wave functions. Numerical algorithms for the computation of generalized prolate spheroidal functions the latex source is available and is being updated at in the doc folder.
Finally, a suggestion for the use of discrete prolate spheroidal windows for spectral analysis is provided. Prolate spheroidal coordinates from wolfram mathworld. The fermifab toolbox for fermionic manyparticle quantum systems. Threedimensional timeharmonic acoustic multiple scattering problems are considered for a finite number of prolate and oblate spheroidal objects adjacent to flat surfaces. The described algorithm exploits the properties of the prolate spheroidal coordinates. Im currently trying to solve laplaces equation outside of a prolate spheroid. Wave propagation by spheroids is modelled by the method of separation of variables equipped with the addition theorems in the spheroidal coordinates. Run the command by entering it in the matlab command window. The validity of this algorithm in the extreme parameter regime, up to c 2 10 14, is demonstrated. Now imagine stretching or contracting the sphere by a ratio of ar in the. The following matlab program calculates equation 2. A prolate spheroid is centered at the origin, its focal axis is aligned with the axis, and its interfocal distance is. The software simulates the motion of the solar system by different numerical methods.
Bandlimited signal extrapolation using prolate spheroidal. Prolate spheroidal wave functions, fourier analysis and uncertainty i. Complete solutions to the oblate spheroidal angular equation and its. Although the earth is very round, it is an oblate spheroid rather than a perfect. Computer facilitated generalized coordinate transformations of. These functions, also known as slepian prolate functions, were bandlimited and exhibited interesting orthogonality relations. In this paper, we describe both, metho ds for computing them, and. A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving. Image conditions and addition theorems for prolate and.
In terms of the prolate spheroidal coordinates defined in the main text, the prolate spheroid is determined by. Prolate spheroidal coordinates can also be considered as a limiting case of. We propose and analyze a lowcomplexity channel estimator for a multiuser multicarrier code division multiple access mccdma downlink in a timevariant frequencyselective channel. A finite difference construction of the spheroidal wave. Remove discrete prolate spheroidal sequences from database. 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. Prolatespheroidalcoords, prolatespheroidalcoords3d. Laplaces equation in prolate spheroidal coordinates with axial symmetry s. Discrete prolate spheroidal or slepian sequence database. 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. Solution of axisymmetric potential problem in oblate.
Hackman acoustics brancl naval coastal systems center, panama city, florida 32407 received 28 june 1983. Pdf software for computing the spheroidal wave functions. The scalar field has to vanish far from the spheroid. A spheroid is also known as ellipsoid which is formed by rotating the ellipse with major axis or minor axis. Line of sight intersection with oblate spheroid matlab. Mathematical tools for bandlimited approximation applied mathematical sciences book 187 kindle edition by osipov, andrei, rokhlin, vladimir, xiao, hong. Here is the y coordinate of prolate spheroid calculator to find y coordinate prolate spheroid with the known values of other two coordinates. Compute discrete prolate spheroidal slepian sequences for use as timebased smoother. Oblate spheroidal coordinates are a threedimensional orthogonal coordinate system that results from rotating the twodimensional elliptic coordinate system about the nonfocal axis of the ellipse, i. A fast and simple finite difference algorithm for computing the spheroidal wave functions is described. Change a point in prolate spheroidal to spherical coordinates, specifying a parameter for prolate spheroidal coordinates. Rotation about the other axis produces prolate spheroidal coordinates. Boyd 8 provided the algorithms and matlab codes for computing the pswfs, eigen.
In this work, the dyadic greens functions pertaining to a magneticcurrent source located in a dielectric spheroid is derived and expressed in a form. A secondary goal of this project is to develop methodology for cross. Ieee institute of electrical and electronics engineers. This matlab function creates a database of discrete prolate spheroidal dpss or slepian sequences and saves the results in dpss. Thus, the two foci are transformed into a ring of radius in the xy plane. By incorporating mapping functions into matlab programs, you can.
If the lineofsight does not intersect with the spheroid, lat and lon contain nan s. The transition matrix for acoustic and elastic wave scatte ring in prolate spheroidal coordinates roger h. 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. The dmodels software package provides matlab functions and scripts to 1 compute internal and surface deformation, internal and surface strain, and surface tilt due to a pressurized source or rectangular dislocation in a homogenous, isotropic, elastic, flat halfspace, and 2 invert gps data for spherical, spheroidal and silllike pressure sources. The third set of coordinates consists of planes passing through this axis. You clicked a link that corresponds to this matlab command. Determinant of a matrix involving the prolate spheroidal. Use them to create a database of slepian sequences, dpss. The numerical methods for this code are described in. The transition matrix for acoustic and elastic wave. Since helmholtz equation is separable in prolateoblate spheroidal coordinates, its exact solution for the scattered.
Transform coordinates on the sphere of radius r to corresponding values in the stereographic projection. Firstly, the entire acoustic field in prolate spheroidal coordinate system is divided by some infinite domains and every domain has some nodes. I need to draw a prolate spheroid with a and b as their focii points. Download it once and read it on your kindle device, pc, phones or tablets.
Mccdma is based on orthogonal frequency division multiplexing ofdm. Rotation about the other axis produces oblate spheroidal coordinates. Just multiply 3 with the volume separately and multiply 4 with pi, x coordinate and y coordinate and then divide both the. Oblate spheroidal coordinates are the natural choice for the translation of any ellipsoid parallel to a principal axis. The helmholtz differential equation is separable in prolate spheroidal coordinates see also helmholtz differential equationprolate spheroidal coordinates, latitude, longitude, oblate spheroidal coordinates, spherical coordinates. Discrete prolate spheroidal slepian sequences matlab dpss. For example, oblate and prolate spheroids being surfaces of revolution can be more easily conformed to most districts of human body e. This approach uses the tridiagonal method and exploits symmetry. Load discrete prolate spheroidal sequences from database. This section addresses how mapping toolbox software accurately models. The prolate left and oblate right spheroidal coordinate systems. They are normalized versions of the solutions to helmholtz wave equation 1 in spheroidal coordinates.
It is mainly concerned with the representation of symbolic fermionic wavefunctions and the calculation of corresponding reduced density matrices rdms. Compute discrete prolate spheroidal sequences in slp. The three colored surfaces are isosurfaces for 12 red. Then pressure functions are determined by requiring the assumedform solution to satisfy the pressure boundary condition at the measured points. Solutions of the grosspitaevskii equation in prolate.