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

WINNER D5.4 v. 1.4
Depending on the measurement capabilities and the scenario requirements some of the parameters may be
neglected. Let us denote the vector of these elements s ( x, y)
where we think of s( x, y)
as a stochastic
s x, y be m .We call
multivariate process where x and y is the position of the user. Let the size of ( )
s ( x, y)
“the large-scale parameter vector” or “large-scale vector” for short.
The angles in ( x, y)
unit-less (i.e. not in dB). The value of s ( x, y)
in two positions s ( ) and s(
)
s are taken to be degrees, the delay-spread in seconds, and the remaining ones are
x , y x , y 1 1
2 2 can be used to
represent two users or the same user which is at the two positions at two different time instants. When
1
2
two base-station sites are involved, two different vectors s ( x, y)
and s ( x, y)
characterize the link to the
two sites and the position x, y , respectively.
In the first three sections following, we further elaborate this model in teRMS of distribution, autocorrelation,
and inter-base-station correlation. In the fourth section we describe how an evolving channel
can be generated based on the model.
The model obtained here is similar to the ideas in [Alg02]. Here however, we consider system level
variables with different correlation distances as well as non log-normal variables. We also analyze crosscorrelation
functions an issue which is completely overlooked in [Alg02]. Furthermore, we discuss
extension to include inter-site correlations in Section 4.1.4.2.3 below.
4.1.4.2.1 Distributions
Based on measurements and literature, we have found transfoRMS g ( s)
for each element of ( x, y)
~
that the transformed vector ( x , y) g( s( x,
y)
)
s such
s = is a vector of Gaussian random variables for each scenario.
We assume that the elements of this vector are jointly Gaussian. We have also found the inverse
transform, such that we can easily generate samples of the distribution by generating a vector of random
−
s x , y = g
1 ~
s x,
y .
variables and then perform the inverse i.e. ( ) ( ( ))
4.1.4.2.2 Auto-correlations
In order to facilitate generation of multiple realizations of ~ s ( x, y)
in several
positions x = x , y = y 1 1 , x = x , y = y
~ 2 2 , … need to know the mean and the auto-correlation of the vector
s ( x, y)
i.e.
= {
~ s ( x, y)
} , R( r ) = Ε (
~
s( x , y ) − µ )( ~
s( x y ) − µ )
µ Ε
{ }
T
∆
1 1
0,
0
2
, where ( ) ( ) 2
∆ r = x
, (4.36)
1 − x0
+ y1
− y0
where we have assumed that the autocorrelation is only a function of the distance between any two points.
The correlation matrix contains the cross-correlation functions between all element variables. In order to
arrive at a model which is usable also when simulating cases involving many positions we have impose a
structure on R ( ∆r)
namely
R
⎛
⎜
⎝
⎛
⎜
⎝
∆r
⎞ ⎛ ∆r
⎞⎞
⎟
K
⎜ ⎟⎟
(*) (4.37)
λ1
⎠ ⎝ λm
⎠⎠
0.5
0.5,T
( ∆r) = R ( 0) diag⎜exp⎜−
⎟,
,exp⎜−
⎟⎟R
( 0)
0.5
T 0.5
5
where R ( 0)
is obtained from the eigendecomposition R( 0) = EΛE
as R ( 0) = EΛ
0. .
Using a model with this structure we can simulate realizations of ( x, y)
m independent Gaussian random processes, ?( x, y) [ ξ ( x,
y) ( x y)
] T
1
Kξ
m
,
variance one. The autocorrelation of process ξc ( x,
y)
is given by exp( − ∆r / λ c
)
dimensional “maps” can be performed with a filtering operation. Following the generation of ( x, y)
transformed large-scale vector is obtained as
~ s by first generating
= , each one with mean zero,
0.
( x,
y) = R ( ∆r) ?( x y) + µ
~ 5
s ,
. Generating such two
? the
. (4.38)
Thus we have in Section fitted model parameters µ andλ , K,λ
to our measurements.
Note that the resulting effective autocorrelation function for each large-scale parameter is not exponential,
as commonly found in literature, but rather a sum of weighted exponential functions.
4.1.4.2.3 Multi-site cross-correlations
The justification for introducing the cross-correlations of the large-scale parameters is that the links
between a pair of base-stations and a mobile-station is that 1) there may be many common scatterers in
the close proximity of the MS 2) common shadowing objects of two mobiles located close in angle and 3)
1
m
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