For example, for the Gaussian correlation function 2 C(r.1-) = exp [_ x ; y2] we obtain in .NET

Implementation data matrix barcodes in .NET For example, for the Gaussian correlation function 2 C(r.1-) = exp [_ x ; y2] we obtain
For example, for the Gaussian correlation function 2 C(r.1-) = exp [_ x ; y2] we obtain
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(9.2.36)
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(9.2.37)
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In calculating higher-order rough surface scattering, we also need to calculate higher moments. If we assume that f(r.1-) is a Gaussian stationary process, then
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(j(ru)f(ru)f(ru)) = 0 (J(ru)f(ru)f(ru)f(ru)) = (J(ru)f(ru)) (J(ru)f(ru)) + (J(ru)f(ru))(j(ru)f(ru))
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(9.2.38)
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In the spectral domain, (9.2.38) and (9.2.39) give
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(F(ku)F(ku)F(ku)F(ku ))
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(9.2.39)
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(9.2.40)
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9 l-D RANDOM ROUGH SURFACE SCATTERING
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2.2 Characteristic Functions For a Gaussian random process, the characteristic functions can be readily calculated. For a Gaussian rough surface f(x), the probability density function (pdf) for f is
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1 p(f) = /'iffh exp - 2h2 ]
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(9.2.41)
The joint pdf for f(X1) =
fl and f(X2)
= 12 is
p(fl, h) = 27rh2Y!1 _ C2 exp -
fr-2Ch12+fi] 2h2(1 _ C2)
(9.2.42)
where C is the covariance function defined by (j(x1)f(X2)) = h 2C(X1 - X2) Thus the characteristic functions are
(9.2.43)
(e iv (h-!2))
i: 1:
df p(f) e dfl
= exp( --2-)
h2v2
(9.2.44)
d12p(fl, h) eiv (fI-!2)
(9.2.45)
= eXP[-h v (1- C)]
Let f(X1) = fl, f(X2) = 12, .. , f(x n ) = fn. Also let f = row vector = (fl, 12, .. , fn), where t = tran~pose and 1 is the corresponding column vector. The covariance matrix is A of dimension n x n
(9.2.46)
Then
1 [ 1-t=-1_] p(fl, , fn) = V(27r)n6. exp -2f A f
=-1 _ _
(9.2.47)
where A is the inverse of A and 6. is the determinant of A. The characteristic function is
(eivlh+iv2!2+.-+ivnfn) =
exp[-~ ~lJiVjAij]
(9.2.48)
For example, for the n = 2 case,
A = h2 [
C(X1 - X2)
C(X1 - X2)]
(9.2.49)
(9.2.50)
= h 2 (1 _
1C2) [1 -C]
3 Small Perturbation Method
and ~ = h4 (1 - C 2 ). Thus
(eivdl+iv2h) = exp [-~h2(vr
+ v~ + 2V1V 2C)]
(9.2.51)
Small Perturbation Method
The scattering of electromagnetic waves from a slightly rough surface can be studied using a perturbation method [Rice, 1963J. It is assumed that the surface variations are much smaller than the incident wavelength and the slopes of the rough surface are relatively small. The small perturbation method (SPM) makes use of the Rayleigh hypothesis to express the reflected and transmitted fields into upward- and downward-going waves, respectively. The field amplitudes are then determined from the boundary conditions. The extended boundary condition (EBC) method may also be used with the perturbation method to solve for the scattered fields. In the EBC method, the surface currents on the rough surface are calculated first by applying the extinction theorem. The scattered fields can then be calculated from the diffraction integral by making use of the calculated surface fields. Both perturbation methods yield the same expansions for the scattered fields, because the expansions of the amplitudes of the scattered fields are unique within their circles of convergence.
3.1 Dirichlet Problem for One-Dimensional Surface
We first illustrate the method for a one-dimensional random rough surface with height profile z = f(x) and (f(x)) = o. The scattering is a twodimensional problem in x-z without y-variation. Consider an incident wave impinging on such a surface with Dirichlet boundary condition (Fig. 9.3.1). This is the same as a TE electromagnetic wave with an electric field in the y-direction impinging upon a perfect electric conductor. Let
(9.3.1)
where kix = k sin Oi, kiz = k cos Oi. In the perturbation method, one uses the height of the random rough surface as a small parameter. This is based on the assumption that kh 1, where h is the rms height. For the scattered wave, we write it as a perturbation series.
(9.3.2)
9 l-D RANDOM ROUGH SURFACE SCATTERING
'l/Jinc
z = f(x)
Plane wave impinging upon a one-dimensional rough surface with height
z=f(x).
The boundary condition at z = f(x) is
'l/Jinc + 'l/Js = 0 (9.3.3) The zeroth-order scattered wave is the reflection from a flat surface at z = O. Thus
(9.3.4)
We let the scattered field be represented by
'l/JsCr) =
dk x eikxx+ikzz'l/Js(kx)
(9.3.5)
where k z = (k 2 - k;)1/2. The Rayleigh hypothesis [Millar, 1973] has been invoked in (9.3.5) as the scattered wave is expressed in terms of a spectrum of upward plane waves only. To calculate 'l/Js(k x ), we match boundary condition of (9.3.3).
eikiXX-ikizf(x)
dk x eikxx+ikzf(x)'l/Js (k x ) = 0
(9.3.6)
Perturbation theory consists in expanding the exponential functions of exp( -ikizf(x)) and exp(ikzf(x)) in power series assuming that kzf 1. In the spectral domain, we also let
'l/Js(k x ) = 'l/J~O)(kx)