Slepian Functions Research Articles

Orthogonal signals with jointly balanced spectra: Application to cdma transmissions

This paper presents a technique for generating orthogonal bases of signals with jointly optimized spectra, in the sense that they are made as close as possible. To this end, we propose a new criterion, the minimization of which leads to signals with close energy inside a set of prescribed subbands. Starting with the case of a single subband, we illustrate it by building orthogonal signals with maximum energy concentration in time and in frequency, with the same energy rate outside a fixed frequency interval or a fixed time interval, by resorting to Slepian sequences or Slepian functions, respectively. Then, we present spectrum balancing in a set of frequency intervals. We apply this method to Slepian sequences and Slepian functions, as well as to Walsh-Hadamard codes. On these examples, we point out a number of nice properties of the so-built orthogonal families that are of interest for signaling applications. PACS: signal processing techniques and tools; modulation techniques

On Spectral Approximations by Generalized Slepian Functions

We introduce a family of orthogonal functions, termed as generalized Slepian functions (GSFs), closely related to the time-frequency concentration problem on a unit disk in D. Slepian (19). These functions form a complete orthogonal system in L 2α (−1,1) withα(x) = (1− x) α ,α>−1, and can be viewed as a generalization of the Jacobi polynomials with parameter (α,0). We present various analytic and asymp- totic properties of GSFs, and study spectral approximations by such functions.

On Spectral Approximations by Generalized Slepian Functions

On Spectral Approximations by Generalized Slepian Functions

Spatiospectral concentration in the Cartesian plane

We pose and solve the analogue of Slepian's time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.

Signal Reconstruction from Nonuniformly Spaced Samples Using Evolutionary Slepian Transform-Based POCS

We consider the reconstruction of signals from nonuniformly spaced samples using a projection onto convex sets (POCSs) implemented with the evolutionary time-frequency transform. Signals of practical interest have finite time support and are nearly band-limited, and as such can be better represented by Slepian functions than by sinc functions. The evolutionary spectral theory provides a time-frequency representation of nonstationary signals, and for deterministic signals the kernel of the evolutionary representation can be derived from a Slepian projection of the signal. The representation of low pass and band pass signals is thus efficiently done by means of the Slepian functions. Assuming the given nonuniformly spaced samples are from a signal satisfying the finite time support and the essential band-limitedness conditions with a known center frequency, imposing time and frequency limitations in the evolutionary transformation permit us to reconstruct the signal iteratively. Restricting the signal to a known finite time and frequency support, a closed convex set, the projection generated by the time-frequency transformation converges into a close approximation to the original signal. Simulation results illustrate the evolutionary Slepian-based transform in the representation and reconstruction of signals from irregularly-spaced and contiguous lost samples.

Effective band-limited extrapolation relying on Slepian series and ℓ1 regularization

We consider a rather simple algorithm to address the fascinating field of numerical extrapolation of (analytic) band-limited functions. It relies on two main elements: namely, the lower frequencies are treated by projecting the known part of the signal to be extended onto the space generated by “Prolate Spheroidal Wave Functions” (PSWF, as originally proposed by Slepian), whereas the higher ones can be handled by the recent so-called “Compressive Sampling” (CS, proposed by Candès) algorithms which are independent of the largeness of the bandwidth. Slepian functions are recalled and their numerical computation is explained in full detail, whereas ℓ1 regularization techniques are summarized together with a recent iterative algorithm which has been proved to work efficiently on so-called “compressible signals”, which appear to match rather well the class of smooth bandlimited functions. Numerical results are displayed for both numerical techniques and the accuracy of the process consisting of putting them all together is studied for some test-signals showing a quite fast Fourier decay.

Truncation-Error Reduction in 2D Cylindrical/Spherical Near-Field Scanning

We introduce a near-field to far-field transformation for two-dimensional cylindrical/spherical scanning that significantly reduces angular-truncation errors. After examining the limitations of the traditional multipole-based expansion of truncated scan data, we consider an alternative expansion based on Slepian functions and show how far-field values can be extracted from the resulting expansion coefficients. We compare the performance and computational cost of the new transformation with those of the traditional one.

Constraints on upper mantle viscosity from the flow‐induced pressure gradient across the Australian continental keel

The thickness of continental lithosphere varies considerably from tectonically active to cratonic regions, where it can be as thick as 250–300 km. Embedded in the upper mantle like a ship, when driven to move by a velocity imposed at the surface, a continental keel is expected to induce a pressure gradient in the mantle. We hypothesize that the viscosity of the asthenosphere or the shear coupling between lower lithosphere and asthenosphere should control this pressure effect and thus the resulting dynamic topography. We perform three‐dimensional finite element calculations to examine the effects of forcing a continental keel by an imposed surface velocity, with the Australian region as a case study. When the upper mantle is strong but still weaker than the lower mantle, positive dynamic topography is created around the leading edge, and negative dynamic topography is created around the trailing edge of the keel, which is measurable by positive and negative geoid anomalies, respectively. For a weak upper mantle the effect is much reduced. We analyze geoidal and gravity anomalies in the Australian region by spatiospectral localization using Slepian functions. The method allows us to remove a best fit estimate of the geographically localized low spherical harmonic degree contributions. Regional geoid anomalies thus filtered are on the order of ±10 m across the Australian continent, with a spatial pattern similar to that predicted by the models. The comparison of modeled and observed geoid anomalies places constraints on mantle viscosity structure. Models with a two‐layer mantle cannot sufficiently constrain the ratio of viscosity between the upper and lower mantle. The addition of a third, weak, upper mantle layer, an asthenosphere, amplifies the effects of keels. Our three‐layer models, with lower mantle viscosity of 3 × 1022 Pa s, suggest that the upper mantle (asthenosphere) is 300 times weaker than the lower mantle, while the transition zone (400–670 km depths) has a viscosity varying between 1021 and 1022 Pa s.

A new generalization of the PSWFs with applications to spectral approximations on quasi-uniform grids

We define a new family of generalized prolate spheroidal wave functions (GPSWFs), which extends the prolate spheroidal wave functions of order zero (PSWFs or Slepian functions; Slepian and Pollak, 1961 [45]) to real order α > − 1 , and also generalizes the Gegenbauer polynomials to an orthogonal system with an intrinsic tuning parameter c > 0 . We show that the GPSWFs, defined as the eigenfunctions of a Sturm–Liouville problem, are also the eigenfunctions of an integral operator. We present a number of analytic and asymptotic formulae for the GPSWFs and the associated eigenvalues, and introduce efficient algorithms for their evaluations. Moreover, we derive a set of optimal results on the GPSWF approximations featured with explicit dependence on the parameter c. As an important application, we implement and analyze the GPSWF spectral methods for elliptic-type equations. We illustrate that the presence of c provides flexibility to design high-order approximations on quasi-uniform grids, and endows the GPSWFs with some favorable advantages over their polynomial counterparts.

Spatially Restricted Integrals in Gradiometric Boundary Value Problems

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

Implications of postseismic gravity change following the great 2004 Sumatra‐Andaman earthquake from the regional harmonic analysis of GRACE intersatellite tracking data

We report Gravity Recovery and Climate Experiment (GRACE) satellite observations of coseismic displacements and postseismic transients from the great Sumatra‐Andaman Islands (thrust event; Mw ∼9.2) earthquake in December 2004. Instead of using global spherical harmonic solutions of monthly gravity fields, we estimated the gravity changes directly using intersatellite range‐rate data with regionally concentrated spherical Slepian basis functions every 15‐day interval. We found significant step‐like (coseismic) and exponential‐like (postseismic) behavior in the time series of estimated coefficients (from May 2003 to April 2007) for the spherical Slepian functions. After deriving coseismic slip estimates from seismic and geodetic data that spanned different time intervals, we estimated and evaluated postseismic relaxation mechanisms with alternate asthenosphere viscosity models. The large spatial coverage and uniform accuracy of our GRACE solution enabled us to clearly delineate a postseismic transient signal in the first 2 years of postearthquake GRACE data. Our preferred interpretation of the long‐wavelength components of the postseismic gravity change is biviscous viscoelastic flow. We estimated a transient viscosity of 5 × 1017 Pa s and a steady state viscosity of 5 × 1018–1019 Pa s. Additional years of the GRACE observations should provide improved steady state viscosity estimates. In contrast to our interpretation of coseismic gravity change, the prominent postearthquake positive gravity change around the Nicobar Islands is accounted for by seafloor uplift with less postseismic perturbation in intrinsic density in the region surrounding the earthquake.

Sampling theorem, bandlimited integral kernels and inverse problems

We describe a novel approach based on the sampling theorem for studying eigenvalue problems associated with bandlimited integral kernels of convolution type. Two sets of functions biorthogonal to the eigenfunctions, one over the infinite interval and the other over a finite interval, are constructed and several identities satisfied by them are derived. The sampling theorem-based approach to the eigenvalue problem is further extended to construct the singular functions associated with the integral operator. It is shown that for the special case of the sinc-kernel, the eigenfunctions, the two biorthogonal sets and the singular functions reduce to the angular prolate spheroidal functions (or Slepian functions). Two methods are discussed for treating the inverse problem associated with bandlimited kernels—one employing the eigenfunctions and the biorthogonal sets and the other employing the singular functions. Numerical examples are included to illustrate the computation of eigenfunctions, biorthogonal sets and the singular functions and their application to the estimation of inverse solution.

Spherical Slepian functions and the polar gap in geodesy

The estimation of potential fields such as the gravitational or magnetic potential at the surface of a spherical planet from noisy observations taken at an altitude over an incomplete portion of the globe is a classic example of an ill-posed inverse problem. We show that this potential-field estimation problem has deep-seated connections to Slepian's spatiospectral localization problem which seeks bandlimited spherical functions whose energy is optimally concentrated in some closed portion of the unit sphere. This allows us to formulate an alternative solution to the traditional damped least-squares spherical harmonic approach in geodesy, whereby the source field is now expanded in a truncated Slepian function basis set. We discuss the relative performance of both methods with regard to standard statistical measures such as bias, variance and mean squared error, and pay special attention to the algorithmic efficiency of computing the Slepian functions on the region complementary to the axisymmetric polar gap characteristic of satellite surveys. The ease, speed, and accuracy of our method make the use of spherical Slepian functions in earth and planetary geodesy practical.

Wavelet Like Behavior of Slepian Functions and Their Use in Density Estimation

Slepian functions (Prolate Spheroidal Wave Functions) are obtained by maximizing the energy of a σ-bandlimited function (normalized with total energy 1) on a prescribed interval [−τ, τ]. The solution to this problem leads to an eigenvalue problem λf(t) = {sinσ(t − x)/π(t − x)}f(x)dx, whose solutions, in turn, form an orthogonal sequence {ϕ n }. This sequence is a basis of the Paley-Wiener space B σ of σ-bandlimited functions. For σ = π, integer translates of the Slepian functions of order 0, {ϕ0(t − n)} form a Riesz basis of the same space. Furthermore, by using ϕ0 as a scaling function we can construct a wavelet theory based on them. Two methods of density estimations thus naturally arise; one based on the orthogonal system {ϕ n } and the other on the scaling functions {ϕ0(t − n)}. The former gives more rapid convergence, while the latter avoids Gibbs phenomenon, is locally positive, and allows the use of thresholding methods. Both approaches exhibit a strong localization property.

Spatiospectral Concentration on a Sphere

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

Slepian functions on the sphere, generalized Gaussian quadrature rule

Denote by K the operator of ‘time–band–time’ limiting on the surface of the sphere and consider the problem of computing singular vectors of K. This problem can be reduced to a simpler task of computing eigenfunctions of a differential operator, if a differential operator, which commutes with K and has a simple spectrum, can be exhibited. In Grünbaum et al (1982 SIAM J. Appl. Math. 42 941–55) such a second-order differential operator commuting with K on the appropriate subspaces was constructed. In this paper, this algebraic property of commutativity is used to produce an efficient numerical scheme for computing a convenient basis for the space of singular vectors of K. The basis forms an extended Chebyshev system, and a generalized Gaussian quadrature rule for such a basis is presented.