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efficient, but is capable of producing correlated shadow fading at points laid out on
any arbitrary geographical grid.
It has been widely reported in the literature [
5
] that the spatial correlation of the
shadow fading at two nearly locations
r
i
and
r
j
is known to be of the form,
xx
cov(
xx
)
|
r r
-
|
D
d
æ
ö
æ
ö
i
j
i
j
i
j
ij
ρ
=
=
=
exp
-
ln 2
=
exp
-
ln 2
ç
÷
ç
÷
ij
2
2
σ
σ
d
d
è
ø
è
ø
50%
50%
where
=
σ
, we have taken advantage that the mean value of the shadow
fading is by definition zero and we are using the 50% correlation distance,
x xr
()
i
i
,
d
50%
e
-
correlation distance
d
which would result in the omission of the
ln 2 factor as we saw earlier.
If we consider a grid consisting of
n
receiver points arbitrary positioned in space
around a transmitter, generating a vector,
˜ , of
n i.i.d.
Gaussian random variables
to produce an instance of the shadow fading at all the receivers would fail to cap-
ture the fact that elements of
˜ that are spatially separated by distances smaller than
50
d
are correlated and thus their values should not be
independent
but
correlated
when ensemble averaged. The desirable spatial correlation properties can be
captured using a mathematical transformation that converts a vector
˜, of
n
i.i.d.
Gaussian random variables with a unit standard deviation to a suitably correlated
vector of Gaussian variables with the correct standard deviation
d
σ
.
We can use the two-point correlation equation given above to construct the
nn
1
rather than the
´
c
=σρ
. The Cholesky decomposition [
9
] of
C
=
LU
, where
C
=
LU
, yields the transformation matrix
L
that can be used to trans-
form
x
to
X
,
2
covariance matrix,
C
, such that
ij
dB ij
x
L
=
The desirable correlation properties the
n
´
vector of Gaussian deviates,
X
, can
1
be shown by considering
xx
T
=
Lxx L
TT
=
L xx
T T
L
and since by definition
x
is an i.i.d. Gaussian process of unit variance and zero mean,
T
xx
=
I
where
I
is the unit
n
´
matrix, we have,
xx
T
=
LIL
T
=
LL
T
=
C
which shows that the vector
X
has the desired covariance structure. Note that the
Cholesky decomposition into the form used above is possible by virtue of the fact
that all the matrix elements of
C
are real and positive. An example grid and the
corresponding correlated shadow fading field are shown in Fig.
7.3
.
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