The concept of homogeneous random fields is a generalization of one-
dimensional wide-sense stationary stochastic processes to higher dimensions.
In fact homogeneous random fields are also sometimes referred to as "wide-
sense stationary random processes or fields/'
2.4.4 Spectral Density
The spectral density S
ff
(u,v) of a homogeneous random field f(r) is the
Fourier transform of its autocorrelation
CO
S
ff
(u,v) = j j R
ff
((x,ß)expl-j2n(oa4 + ßv)]d(xdß (111a)
— 00
Similarly, the cross spectral density of two homogeneous random fields f and g
is the Fourier transform of their cross correlation
GO
S
fg
(u,v) = J*J R
fg
{a,ß)zxpl-j2n(au ...
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