Download Advances in Electronics and Electron Physics, Vol. 75 by Peter W. Hawkes (Ed.) PDF

By Peter W. Hawkes (Ed.)

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Example text

0165. Slepian, (1978) gives asymptotics of the eigenvalues and eigenfunctions, proving, in particular, that as W -, 0 and N -+ co,so that 7tNW -+ c, we have &, @uk(N, w ;wx) $k(c,x), (144) where &(c, x) is the PSWF of order k and Ak the corresponding eigenvalue. If we recall the behaviour of the eigenvalues of PSWF, this property shows that when the number of sampling points in a fixed interval is large, the number of singular values nearly equal to one is approximately equal to the number of sampling points corresponding to the Nyquist rate.

But when the function f has a bounded support, its Fourier transform is an analytic function and therefore can be uniquely recovered from its value in a finite sector. We have again the problem of restoring a function f from limited values of its Fourier transform. An explicit inversion formula for the transform (69) was already obtained by Radon (1917), as we recalled in the Introduction. Here we sketch an approach which is the basis of the algorithms currently used in practical applications. If we introduce the formal adjoint R# of the Radon transform, also called the back projection operator, (R#g)(x)= then the equation g = Rf lS, s(e7 (x7 0)) dB7 can be replaced by R'g = R'Rf.

The same mathematical problem must be solved if one applies Rytov approximation (Devaney, 1981). The basic point is the introduction of the complex phase function $(r) = In u(r) - ik(s,, r). mdu~V(r). (57) It is obvious now that if one takes the 2D Fourier transform of exp[ik(~,,r)]$'~)(r)over any plate of constant z that does not intersect the scatterer, one again gets values off on the surface of the Ewald sphere. A short discussion of the limits of validity of the two approximations has been given by Devaney (1 98 1).

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