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

WINNER D5.4 v. 1.4
{ }
( τφθϕϑ , , , , ) ( τφθϕϑ , , , , )
P = E P s
(4.5)
The average PD 3 S in (4.4) is the average per channel segment, which may differ from one segment to
another. In order to relate (4.4) to the average PD 3 S over segments (4.5), all bulk parameters except the
random powers can be pulled out of the expectation to arrive at:
P ( τφθϕϑ , , , , )
{ (, , , , )| ,..., , ,..., , ,..., , ,..., , ,..., }
= ∫ E P τφθϕϑ φ φ θ θ ϕ ϕ ϑ ϑ τ τ
s 1 L 1 L 1 L 1 L 1 L
L
∏
i=
1
f( φ, θ , ϕ , ϑ, τ ) dφdθdϕ dϑdτ
i i i i i i i i i i
(4.6)
L
= ∑ E{ pi
| τφθϕϑ , , , , } f( τφθϕϑ , , , , )
i=
1
{ τφθϕϑ}
= LE p| , , , , f(, τφθϕϑ , , , ).
From (4.6), the over segments (global) average PD 3 S (i.e., ( , , , , )
P τφθϕϑ ) is equivalent to the
conditional expected power of the multipath components multiplied by the joint double-directional-delay
probability density function.
The power spectrum in each dimension is obtained by integration over other dimensions. Thus, power
delay spectrum P ( τ ) , power azimuth-departure-angle spectrum P ( φ)
, power elevation-departure-angle
spectrum P ( θ ) , power azimuth-arrival-angle spectrum P ( ϕ)
, power elevation-arrival-angle spectrum
P can be derived as:
( ϑ)
( ) ( , , , , )
P τ P τφθϕϑ dφdθdϕdϑ
= ∫∫∫∫
(4.7)
( ) ( , , , , )
P φ P τφθϕϑ dτdθdϕdϑ
= ∫∫∫∫
(4.8)
( ) ( , , , , )
P θ P τφθϕϑ dφdτdϕdϑ
= ∫∫∫∫
(4.9)
( ) ( , , , , )
P ϕ P τφθϕϑ dφdθdτdϑ
= ∫∫∫∫
(4.10)
( ) ( , , , , )
P ϑ P τφθϕϑ dφdθdϕdτ
= ∫∫∫∫
. (4.11)
The corresponding marginal probability density functions (pdf) of parameters of each domain can be
derived from:
( ) ( , , , , )
f τ f τφθϕϑ dφdθdϕdϑ
= ∫∫∫∫
(4.12)
( ) ( , , , , )
f φ f τφθϕϑ dd τ θdϕdϑ
= ∫∫∫∫
(4.13)
( ) ( , , , , )
f θ f τφθϕϑ dφdτdϕdϑ
= ∫∫∫∫
(4.14)
( ) ( , , , , )
f ϕ f τφθϕϑ dφdθdτdϑ
= ∫∫∫∫
(4.15)
( ) ( , , , , )
f ϑ f τφθϕϑ dφdθdϕdτ
where f ( τ ), f ( φ)
, f ( θ ), f ( ϕ)
, f ( ϑ)
, and ( , , , , )
= ∫∫∫∫
, (4.16)
f τφθϕϑ are the pdf of path delays, the pdf of
azimuth departure angles, the pdf of the elevation departure angles, the pdf of the azimuth arrival angles,
the pdf of the elevation arrival angles, and the joint double-directional-delay probability density function
of argument parameters, respectively. Similarly, the P ( τ ) , P ( φ)
, P ( θ ) , P ( ϕ)
, P ( ϕ)
can be expressed
as:
( ) { }
P τ = LE p| τ f( τ)
(4.17)
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