WINNER II pdf - Final Report - Cept

WINNER II D1.1.2 V1.2
(⋅) g i
(⋅) g j
si
s~i
s~j
s
j
g −1 i
( ⋅)
g −1 j
( ⋅)
Figure 3-7 Correlations of LSP are introduced in transformed domain.
−1
In cases when mapping s g ( s )
i
i
~
= is unknown, necessary relations between LSP and transformed
domain can be established using knowledge about cumulative density distribution (cdf) of
i
such cases s
i
can be generated from s~
i
using expression:
−1
~ −1
s g s = F
( ~
i
( ) ( F ( ~ ~ s )).
s i s
i
s , (s)
Fs
i
. In
= (3.6)
where F~ s i
s ) is cdf of Normally distributed process that can be calculated using Q-function (or erf/erfc).
In simpler cases, e.g. when LSP is log-normally distributed it is possible to use known mappings:
s =
−
g i
~ s =10
~ g = log
1
~ s
( )
( s) ( s)
s
i 10
(3.7)
= (3.8)
As a correlation measure cross-correlation coefficient is used, expression (3.4). Above is explained that
for one link (single position of MS) inter-dependence of multiple control parameters can be described
with correlation coefficient matrix. Additionally if parameters of intra-site links are correlated according
to distance between MS positions, then correlation matrix gets additional dimension that describes
changes in correlations over distance, Figure 3-6. This means that for each pair of TLSP we can define
cross-correlation coefficient dependence over distance, as in expression (3.5):
~ r s
(
k , l
k l
C
~ r ~ s
k l
ρ ~ r ~
k s
( d
l k,
l
) =
(3.9)
Cross-variances C ~ d ) are calculated from measurement data using knowledge about positions of
MS during measurement, and in general have exponential decay over distance.
If each link is controlled by M TLSPs, and we have K links corresponding to MS locations at positions
k = 1..
, then it is necessary to correlate values for N= M·K variables.
( )
k
y k
x , , K
Generation of N Normally distributed and correlated TLSPs can be based on scaling and summation of N
independent zero-mean and unit variance Gaussian random variables,
ξ
[ ξ ( x , y ),
K,
( x y )] T
N
( , y)
=
1 1 1
ξ
N
x ,
N
N
C
r~
r~
( d
k k
k,
l
C
)
~ s ~ s
l l
. Using matrix notation that can be expressed:
~
s ( x,
y)
= Q ξ NxN N
( x,
y)
(3.10)
This will ensure that final distribution is also Gaussian. Scaling coefficients have to be determined in such
way that cross-variances C ~ d )
values. If element
~ r s
(
k,
l
k l
,
d
C
i , j
of matrix
NxN
scaling matrix can be calculated as:
2
2
k, l
( xk
− xl
) + ( yk
− yl
)
= are corresponding to measured
C represents cross-variance between TLSPs s~
i
and s~
j , then
Q
NxN
= C NxN
(3.11)
This approach is not appropriate for correlation of large number of parameters, since dimensions of
scaling matrix are increasing proportionally to the total number of TLSPs in all links (squared dependence
in number of elements). For that reason it is more convenient to generate separately the influence of LSP
cross-correlation and exponential auto-correlation.
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