A localized beam representation of high-frequency wave fields using a Wilson basis

Authors


Abstract

[1] A rigorous orthogonal basis is described for the representation of high-frequency wave fields using beam superpositions parameterized by both spatial position and spatial direction of each beam. This representation has many desirable features not found in conventional phase-space-based expansions such as the discrete or continuous Gabor schemes. The representation is highly efficient in the geometrical optics limit, that is, when it is strongly localized in the phase-space.

1. Introduction

[2] A common situation in the study of wave propagation in complex imaging environments consists of the specification of a wave field on a particular plane followed by its propagation forward to a new observation space, where one wishes to compute the field. Examples are abundant in the aperture theory of antennas, GPR applications, optical systems composed of sequences of imaging components, tomographic reconstruction, etc.

[3] Classically this is dealt with by Fourier Transformation [Goodman, 1968] (representation of the field by a superposition of plane waves), since the forward propagation of plane waves in free space is relatively easy to describe mathematically. However, this is not always objectively the most efficient way to represent the propagation, since the reconstruction of the plane wave superposition at the observation plane may involve much destructive interference between spectral plane waves to produce what is actually a highly localized field distribution in real space. This is particularly true in the case of high-frequency fields which propagate essentially along well-defined trajectories described by simple geometrical optics. Intuitively it would be much more efficient to describe such situations by superpositions of spectral elements which are themselves strongly localized to the trajectories which dominate the GO description of the field. Thus one is led to consider superpositions of collimated beams rather than of plane waves. In order to describe the effects of truncation to a finite set of beams it is necessary to have a properly formulated mathematical theory. As an additional feature one would like the theory to describe asymptotic behavior as the frequency of the wave field becomes very large.

[4] An early step in this direction was taken by Gabor [1946], who used a fundamental representation of the form

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for the expansion of an arbitrary function u(x), xR where x is a suitably scaled domain variable. The coefficients in the expansion are given by the biorthogonal projection formula

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with C = 1.599. More recently, the use of the windowed Fourier transform

equation image

has been extensively studied, with U the coefficient function and g:RR an arbitrary windowing function which vanishes sufficiently rapidly at x → ±∞. When g is gaussian, g(x) = (2ν)1/4exp[−νπx2], this is also called the continuous Gabor transform. Here the phase-space parameters (m, n) ∈ R2 are real, not restricted to a discrete grid Z2 as in the discrete Gabor case.

[5] Several facts are known about the discrete and continuous Gabor transforms. For the discrete case, it is known that the sampling of the phase-space afforded by the lattice of discrete points (m, n) ∈ Z2 is optimum (one coefficient per unit of phase-space area), but that the series tends to be very unstable in the sense that small perturbations of the coefficients can produce large fluctuations of the represented function. This can be remedied by oversampling. More problematic is the fact that the coefficients in the expansion are obtained from a complicated expression involving inner products of the given function u on a dual space of biorthogonal functions which are very discontinuous, given explicitly by equation (2), and the inner products are difficult to calculate in closed form even for relatively simple functions. The continuous Gabor transform does not suffer these problems, but on the other hand is highly redundant due to overcompleteness (i.e., there exist uncountably many different sets of coefficient functions for the same function u).

[6] Many cases are known of functions u whose Gabor coefficients, both discrete and continuous, are highly localized for parameters (m, n) near to a lower dimensional manifold in the phase-space; this is particularly true for functions of geometrical optics type

equation image

where a and s are smooth functions and k is a large parameter, in which case the manifold is called the Lagrange submanifold Λ and is given by the locus of points (x, ξ) satisfying

equation image

[7] Localization of the Gabor coefficients near a lower-dimensional manifold should imply that the function u is very efficiently computed by restricting the superpositions (1) or (2) to parameter points in phase-space that are close to Λ, and neglecting contributions from points that are far from the Λ [Arnold, 1995]. However, there are no systematic rules known for characterizing the asymptotic errors involved in this truncation of the phase-space, nor how to construct a robust sampling scheme for a discrete representation of the superposition which avoids both the instability of the discrete Gabor representation and the redundancy of the continuous Gabor representation.

[8] It is most convenient in describing the theory of these representations to normalize the physical length x to some scale length L and the wave number ξ to 2π/L; this is done by the replacements x/Lx, ξL/2π → ξ, which is equivalent to measuring length in units of L. This modifies the GO form of (5) to

equation image

in the normalized variables. The scale length is selected so that L is intermediate between the scale on which a and σ change and that of one local cycle of the function exp[2πiσ(x)].

[9] A simple example of the construction of a Lagrange submanifold is furnished by a point source located at (x, z) = (0, −a), for which we require the distribution of field over the observation space z = 0. The field of this scalar point source at the observation space is described (asymptotically for large ka) by

equation image

[10] This has a natural phase-amplitude decomposition, with a phase function proportional to the ray length from the source point (0, −a) to the observation point (x, 0):

equation image

[11] The local spatial frequency for this phase function is

equation image

and the Lagrange submanifold, which is the graph of the function ξ(x), is an S-shaped curve with asymptotes ξ → ±k/2π as x → ±∞.

[12] A second example of the construction of a Lagrange submanifold is shown in Figure 1. This particular example is for the Gauss–Hermite function u(x) = Hq((2π)1/2x)exp(−πx2), chosen because it represents generic characteristics of smooth Lagrange manifolds quite well. The Gauss–Hermite function satisfies the ordinary differential equation

equation image

with boundary conditions that u vanishes exponentially as x → ±∞. These boundary conditions can only be satisfied for integer values of q. Using the WKB method, the function u can be represented for large values of q by

equation image

with

equation image

The last equation can be integrated to give

equation image
Figure 1.

Lagrange manifold Λ for the Gauss–Hermite function of order q.

[13] Equation (12) defines the local frequency ξ = dxσ(x) = (R2x2)1/2, which in turn defines the curve

equation image

in the phase-space (x, ξ). This curve is the Lagrange submanifold for the Gauss–Hermite function. The total area enclosed by this curve in phase-space is πR2 = q + 1/2; it follows that an increase of the integer q by 1 increases the area enclosed by the Lagrange submanifold by 1. By a result of Daubechies [1988], if the Lagrange submanifold of some general function u(x) is completely enclosed by the circle of radius Rmax, then the integer qmax = πRmax2equation image is the order of the last significant term in the orthogonal expansion in Gauss–Hermite functions of the general function u(x):

equation image

[14] The Lagrange submanifold defines a multiple-valued phase function

equation image

where the function ξ(x) is the assignment of frequency to position specified by the Lagrange submanifold. Integration of (16) once by parts leads to a second function

equation image

where x(ξ) is the inverse function to ξ(x). The identity

equation image

is called the Legendre transform. In this case the functions x(ξ) and ξ(x) are multiple-valued, but there is a unique assignment of points on the Lagrange submanifold. Each branch of the multiple-valued function ξ(x) joins smoothly onto another branch such that the Lagrange submanifold is the union of all the branches.

2. Gabor Frames

[15] The concept of a Gabor frame arises from more general superpositions of the form:

equation image

where g(x) is some prescribed function with suitable decay at x → ±∞, and α and β are real constants. By scaling the space variable x it is always possible to regard one of these constants as prescribed and equal to 1, so we shall consider here β = 1. The remaining parameter α is called the oversampling parameter. In the case α = 1 we recover the classical Gabor expansion (1); for α < 1 the expansion is oversampled, and for α > 1 the expansion is undersampled. The definition of a Gabor frame is then that an expansion of the form (3) is a Gabor frame for the Hilbert space L2(R) if

equation image

for all functions uL2(R), where

equation image

∥.∥ represents the L2(R) norm, and A and B are fixed constants such that 0 < AB < ∞. The frame operatorS is defined by its action on any function u:

equation image

[16] It can be shown that if A = B then the frame operator is proportional to the identity, S = AI, and the frame is then called tight. The frame operator is a bounded operator in L2(R), and therefore S is invertible, and the inverse S−1 exists.

3. Wilson Expansion

[17] In principle a scheme proposed by Daubechies et al. [1991], known as a Wilson basis, has all the requirements for a proper discretization of the phase-space distribution. In this scheme, the Gabor scheme (1) is modified to

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with the basis functions given by

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[18] The basic wavelet in this representation is the single function ϕ given by

equation image

with an arbitrary function g(x). This is a frame expansion for the function ϕ. An alternative expression for the functions wmn(x) is then obtained by substituting the frame expansion (25) in (24) and interchanging orders of summation, to obtain

equation image

with coefficients Wmn rs given by

equation image

[19] The coefficients Ars are constructed so as to secure the orthogonality of the set (24), such that

equation image

[20] They have been computed [Daubechies et al., 1991] for a number of candidates g, with very good convergence properties of the terms of the series (25) as the indices (r, s) spread away from (0, 0). An example of the fundamental wavelet ϕ(x) computed from the frame expansion (25), from which the orthogonal functions are constructed in (24), is shown in Figure 2, using a gaussian function g(x) = (2ν)1/4exp[−νπx2] with parameter ν = 0.5.

Figure 2.

Prototype function ϕ(x) for the orthogonal Wilson basis constructed from a Gaussian frame expansion with ν = 0.5.

[21] It is a very desirable property that the functions wmn are an orthonormal basis of L2(R). It follows directly from the orthogonality that the coefficients Umn are

equation image

and therefore that the coefficients Umn are linearly related to samples on a rectangular grid of the windowed Fourier transform coefficients

equation image

[22] Thus (23) can be regarded either as an orthogonal expansion (with the w-functions as a basis) or, by interchanging orders of summation, as a frame expansion in terms of the fundamental g-functions. The final frame expansion for the wave field u(x) is

equation image

with coefficients given by

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with

equation image

where the overbar and tilde notations distinguish the frame coefficients and the sampled CG coefficients. The linear operator T whose discrete elements are Trs rs is a template for linear combinations of sampled CG coefficients to form frame coefficients, and depends only on the choice of g-functions, not on the function to be expanded, u(x); hence it can be precomputed and stored as a look-up table.

[23] When the function u is the initial distribution of a wave field at z = 0 in the space (x, z), and it is required to propagate the field forward to a parallel observation space at z ≠ 0, the fundamental g-functions are simply propagated forward. If these functions are chosen to be gaussian, then the propagation is that of a gaussian beam. The expansion remains orthogonal at the new observation space.

[24] The windowed Fourier transform coefficients (30) are quite easy to compute for many geometrical optics cases if g is gaussian, g = (2ν)1/4exp[−νπx2]. For example, for a simple Fresnel wave

equation image

the result is

equation image

which is sharply peaked on the line n = κm in the phase-space; this line coincides with the Lagrange submanifold from (5).

[25] As another example, for a simple aperture (−a, a) illuminated by a plane wave at normal incidence, so that

equation image

one finds that

equation image

which can be expressed in terms of two Fresnel integrals of complex arguments. The Lagrange submanifold Λ for this problem is the union of the lines m = −a, m = a and {−a < m < a, n = 0}. Simple asymptotics shows that the coefficients U(m, n) are localized near these line segments, with gaussian decay away from the Lagrange manifold. Using sampled values of these coefficients to determine the final coefficients of the discrete expansion (23), one finds that for a given observation point x only a few phase-space sample points contribute significantly to the reconstructed value of u(x), and those sample points are clustered near a segment of the Lagrange submanifold over x.

[26] The construction of wave fields by this method can be summarized as follows. The field u(x, 0) on the initial space z = 0 is expanded initially in the continuous Gabor (CG) representation (3). This representation is closely related to the geometrical optics description of the field, and consists of a continuous distribution of coefficients over the entire phase-space which are nevertheless strongly localized near the Lagrange manifold. The CG representation is then reduced in redundancy by selecting only those CG elements which reside on points of a regular discrete twofold-oversampled grid

equation image

[27] The coefficients of these discrete elements are determined by linear superposition of CG coefficients sampled from the same grid according to (32). To construct the field u(x, z) at some forward plane these discrete spectral elements are propagated forwards as beams, retaining the same coefficients, and then reassembled by superposition into the total field at the observation plane.

4. A Special Case

[28] We recall that the spacing of the beam centers by 1 unit in the normalized spatial variable x corresponds to some physical length L. If the free-space wave number is k0, then spatial frequencies for which ξ > k0L/2π excite evanescent waves which decay when propagated forwards from the initial aperture at z = 0. Consequently, any beams in the phase-space beam superposition originating from mesh points (m, n) for which nL−1 > k0/2π decay rapidly away from the initial plane, and can be neglected when the field is reconstructed at the observation space. If we choose the scaling parameter L so that all frequency mesh points other than n = 0 are in the evanescent part of the phase-space, then it is only necessary to retain the beams launched with n = 0 in the beam superposition. These particular n = 0 beams correspond to directions of their beam axes normal to the initial space. This condition requires at least L < 2π/k0 = λ, where λ is the free-space wavelength of the field. To reduce aliasing in the propagating part of the phase-space from the spectral tails of n = ±1 beams whose spectra are centered in the evanescent part of the phase-space, this condition should be strengthened to L < λ/2. Under these conditions it is admissible to retain only n = 0 beams in the beam superposition.

[29] If this reduced sampling is applied to the critically sampled (α = 1) Gabor expansion, then the resulting nonevanescent part of the initial field has the form

equation image

which is a simple superposition of translated Gaussians with certain weighting coefficients, and a condition L < λ/2 is required on the spacing of the centers of the Gaussians. An expansion of exactly this form has been used [Maciel and Felsen, 1989, 2002] (V. Galdi et al., Two-dimensional scattering by moderately rough dielectric interfaces via narrow-waisted Gaussian beams, submitted to Radio Science, 2002) for the propagation of a wave initially specified over a line z = 0 in a 2-dimensional space. In these analyses, the exact form for the coefficients was replaced by an approximation obtained by simply sampling the initial wave at the points x = mL, giving

equation image

[30] This expansion has been shown, by numerical comparison with reference solutions, to be accurate when the condition L < λ/2 is met. However, it is not easy to justify the simple sampling form of the coefficients from the biorthogonal projection formula (2) required by the Gabor expansion.

[31] The same principle may be applied to oversampled Gabor frames. In the particular case of oversampling parameter α = 0.5, the orthogonal Wilson basis has additional properties over the regular frame expansion. Removal of evanescent beams from the frame expansion (26) results in the approximation

equation image

for the basic wavelet function, and removal of the harmonic terms n ≠ 0 from the orthogonal expansion (23) approximates the function u as

equation image

[32] These evanescent contributions are removable for all n ≠ 0 if the scale parameter satisfies L < λ/2, as before. It is now easy to justify a sampling approach to the determination of the coefficients, since from (29) we have

equation image

if u(x) is sufficiently slowly varying around the extent of the localized function wm0(x) = ϕ(xm). The “slowly varying” condition is equivalent to the requirement on the field u(x) that its Fourier spectrum be essentially localized around frequency ξ = 0, or alternatively that the wave be essentially paraxial. In terms of the fundamental beam superposition we have, by combining (23), (40) and (41), that

equation image

which is a superposition of gaussian functions centered on the spatial points of a grid with spacing αL. Since α = 0.5 for the Wilson basis, this spatial grid is doubly oversampled with respect to the critically sampled Gabor grid discussed in the previous paragraph. Equation (43) can also be written as

equation image

which represents the function u as a superposition of two functions, each of which is a critically sampled Gabor series on interleaved grids. Both of these functions have the form of the Maciel-Felsen ansatz, though we do not recover exactly that form because here the frame grid is doubly oversampled. It is probable that further analysis at a more quantitative level will reveal a more exact connection between the two representations, based on the selection of a g-function which makes the sequence Ar0 rapidly convergent, so that the r-sum in (43) can be replaced approximately by the single term r = 0; at the present time precise mathematical data for this conclusion do not yet exist.

5. Conclusion

[33] The orthogonal Wilson basis described by Daubechies et al. [1991] is very well-suited to a rigorous description of the expansion of high-frequency wave fields by means of beam summations. The scheme generates a rectangular grid in phase-space on which the continuous Gabor transform is sampled, the resulting representation therefore being a superposition of translated and modulated window functions. The orthogonality of the representation is preserved under propagation of the fields between any parallel observation spaces. When the field is of geometrical optics type, significant localization of the coefficients in the beam expansion occurs around the Lagrange manifold in phase-space, which is the locus of pure geometrical optics.

[34] Clearly, a crucial property of the Wilson basis is the decay rate of the coefficients Ars in the frame representation (25). Not much is known of an analytical nature about these coefficients, although it is likely that the decay is exponential in both directions for gaussian g-functions. This property should be studied further, as it determines the necessary extent in phase-space of the template given by the linear operator T such that the frame representation (31) be numerically accurate with only a small number of coefficients having indices (r, s) near the Lagrange submanifold in phase-space.

[35] The extension of these expansions to 2-dimensional spaces, to permit wave tracking in 3-dimensional spaces, is straightforward; all that is required is the extension of the basis functions to g(x1, x2), and the introduction of vector-valued coefficients.

[36] Since the frame basis functions g(x) are arbitrary, subject to fast enough decay, it is possible to adapt them for different purposes. The gaussian basis used here is useful for general-purpose expansions. If the gaussian beam basis for g-functions is used, their paraxial propagation in free space is relatively easy to describe analytically; nonparaxial propagation in free space can be described by integrals that are computed and stored in look-up tables. Thus the gaussian g-function basis is adequate for many applications in antenna imaging systems, in which free-space propagation is interrupted by focussing or aperturing elements. However, for tracking the beams in complex environments it may be desirable to introduce as g-functions the complex-source-point (CSP) functions

equation image

where b is the displacement parameter of a source placed at the complex point z = ib. These CSP basis functions can be propagated forward into a complex medium by simple ray tracing and analytic continuation of elementary formulae from geometrical optics, provided that the medium does not vary strongly on the scale of the beam parameter L. This method has been used in an application [Maciel and Felsen, 2002].

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