TEL AVIV UNIVERSITY Gaddi Blumrosen

m m R R ( hˆ m )
ˆ
h/ h h
hh ˆ
1
hh
hˆ
/ ˆ
hh h hh
hh ˆ
1
hh hhˆ
R R R R R
(2.35)
We have to obtain first, the terms, R R R m , m . R R m , m were
2 2
already obtained above according to (2.32), (2.34) and equal to, R R b R ,
m
. For obtaining R , R , which are identical due to symmetric
h m ah hˆ
LOS
hh ˆ hhˆ,
hh ˆ
similar calculation show that R R .
Rhˆ , h hhˆ
, hh,
hˆ
hˆ
, h hˆ
hh, hh ˆ ˆ, h hˆ
(2.36)
Now we can substitute the obtained terms and obtain the expression of the first two
channel conditional moments:
2 2 2 2 1
m
/ ˆ ah ˆ
LOS b est hRT( b hRT)
( h ahLOS
)
hh
(2.37)
2 2 2 2 2 2 1
2 2
R bRbR( b R ) b R
Note that RT is not always full rank, and thus inverse for RT
can not always be
obtained.
For the case of uncorrelated fadings, i.e. I , we obtain:
(2.38)
(2.39)
To conclude this section, let us look at 3 interesting cases.
The first one, is LOS, where, a=1, b=0. The distribution then becomes, a constant,
h / hˆ
~ CN(
ah
ˆ
LOS ( 1
est ) est
h,
0)
(2.40)
On the other extreme case of Rayleigh fading, b=1, a=0, the distribution then
becomes:
ˆ ˆ 2 2
/ ~ ( est , h ( 1 est ))
(2.41)
With Perfect channel estimation, i.e. , the distribution becomes:
(2.42)
This indicates that the real channel is the estimated one, as expected.
h CN h h
est
1
h / hˆ
h ~ CN(
E(
hˆ
), 0)
CN(
hˆ
, 0)
hh
hˆ
hˆ
(( ˆ ( ˆ H
R E h E h)) ( h E( h)))
considerations, we will use the covariance definition,
and obtain:
hh ˆ
R E(( bhˆ R ) ( bh R )) b E(( R ) hˆ h R ) b R
hh/ hˆ
h/ hˆ
hh/ hˆ
1/ 2 H 1/ 2 2 1/ 2 H H 1/ 2 2 2
Ray T Ray T T Ray Ray T esthT hh ˆ hhˆ
h T est h T h T est h T
m ah (1 ) hˆ
LOS
2 2 2 hest R b(1 ) I
est est
RT N
R NR
and the conditional channel distribution is:
/ hˆ
~ CN(
ah ( 1
) ˆ 2 2 2
h,
b ( 1
) I)
h LOS est est h est
h
T