Final report on link level and system level channel models - Winner

WINNER D5.4 v. 1.4
respectively, where F
Gumbel( x,ν ,ς ) and Logistic ( x,ν,ς )
distributions defined in Section 5.4.3, and Q −1
( x)
variables i.e.
Q
F are the CDF of the Gumbel and Logistic
1
x
∫
−∞
( x) = exp⎜
⎟dt
2π
is the inverse of the CDF for Gaussian random
⎛ − t
⎜
⎝ 2
2
⎞
⎟
⎠
. (3.5)
In Table 3.3, the so-called position ν and scale ς parameters for the distributions are listed, except for
Scenario A1 (with 6 instead of 4 parameters) which is listed in Table 3.5. This means that if the largescale
parameter c is log-Gumbel or log-Logistic, the transformed distribution will have zero mean, and
unit variance, i.e., µ
c
= 0 and R c, c ( 0) = 1.
This can be understood by noting that the mean and variance
are taken into account already in the transformation. For log-normal distributions, we use the
transformation
~
= g( s) = log ( s)
(3.6)
s
10
s
g
=
−1
(
~
~ s
s ) = 10
with the exception of shadow-fading (or sometimes called log-normal shadowing, LNS) where we use
~
= g( s) = 10log ( s)
(3.8)
s
10
s =
−
g
1
(
~ 0.1 s
s ) = 10
~
in order to get the transformed shadow-fading in dB scale. For a log-normal distributed parameter c , the
mean
µ and standard deviation ( 0)
c
R are the mean ν and standard deviation ς listed in Table 3.3.
c,c
For normally distributed bulk parameters no transformation is required (i.e. the transformed and
untransformed value are identical) and thus the mean and
in Table 3.3.
µ and standard deviation ( 0)
c
c,c
(3.7)
(3.9)
R are listed
In Table 3.5, the cross-correlation between the transformed parameters are listed for scenario A1, and in
Table 3.2 for the other scenarios. In teRMS of R ( 0)
, the cross-correlation between parameters r and c is
given by
c r , c
r,
r
r,
c
( 0)
( 0) R ( 0)
R
= . (3.10)
R
Thus by combining the cross-correlation and variance information, the matrix R ( 0)
can be derived. In
Table 3.3, a correlation distance ∆ is listed for each large-scale parameter. The correlation distance is
based on fitting of a single exponential exp( − ∆r / ∆)
to the auto-correlation function of the transformed
large-scale parameter. This value is based on measurements or literature or a combination thereof.
However, since the true auto-correlation actually follows the equation (*) of Section 4.1.4.1.4, i.e.
2
E { ( x , y ) s( x y )} = R( ∆r)
, ( ) ( ) 2
R
s
1 1 2,
2
⎛
⎜
⎝
⎛
⎜
⎝
c,
c
∆ r = x
(3.11)
2 − x1
+ y2
− y1
∆r
⎞ ⎛ ∆r
⎞⎞
⎟
K
⎜ ⎟⎟
(*). (3.12)
λ1
⎠ ⎝ λm
⎠⎠
0.5
0.5,T
( ∆r) = R ( 0) diag⎜exp⎜−
⎟,
,exp⎜−
⎟⎟R
( 0)
. This means
c,
c , will be a mixture of the m exponentials of (*). However,
they are selected in a way that the results are roughly the same as the single exponential. The values of the
“eigenvalue auto-correlation distances” λ
1,
K ,λm
are listed in Table 3.4. Note that there is no one-to-one
mapping between any of the lambda parameters and any of the large-scale parameters. The correlation
distance ∆ is included to allow a more easy interpretation of the auto-regressive characteristics of the
model.
0.5
T 0.5
5
where R ( 0)
is obtained from the eigendecomposition R( 0) = EΛE
as R ( 0) = EΛ
0.
that each autocorrelation function, R ( ∆r)
The justification for the expression (*) is that it produces a model from which it is computationally simple
to generate data, and which at the same time gives a fit to experimental auto-correlation functions which
is typically equally good as the single exponential modelling.
The derivation of some of parameters λ
1,
K ,λm
for each scenario, and in some case also other
parameters, are given in Section 5.4.13 below.
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