TEL AVIV UNIVERSITY Gaddi Blumrosen

When only one BF (direction) exists, v , 0,
,
0 and the eigenvector is the estimated
channel, the matrix WT
gets the classical meaning of BF (one dimensional BF), and if
the channel estimate is perfect then U X becomes simply X.
3.2.3 Optimal MMSE BF
In this chapter an MMSE approach for BF is introduced. In this approach, a different
weight is induced for each antenna, and forms one vector and thus is sometimes
referred to as one dimensional BF. In order to adopt these weight, (antenna weight
vector), to our system model (antenna weight matrix), the antenna weight matrix has
to be diagonal where its terms are the same as the antenna weights.
In the case of receive diversity (SIMO, similar to [3]), the cost function, obtained for
instance, by sending training sequence (pilot), where H is assumed as being perfectly
estimated, is:
H
2
H H H
R R R
J( W) arg min E W Y X E( W Y X ) ( W Y X )
WR
and in the case of transmit diversity (MISO), the cost function is:
T
(3.6)
(3.7)
From considerations of reciprocallity, reasonable in many channel models, the receive
weights will be the same as the transmit weights. Therefore, with out loss of
generality, we will describe here the optimal weight for receive diversity as were
derived in [3]. Following the method in [5], the gradient:
(3.8)
H
H
Where RY E(
YY ) is the received signal autocorrelation and RYX E(
YX ) is the
cross correlation between received and transmitted signals. The solution of (3.8) is the
Spatial Winner filter:
1
WR
RY
RYX
(3.9)
H
With the power normalization E( XX ) INt
By substituting our system model in (2.45), we obtain:
WR E(( HX N)( HX N) ) E(( HX N) X )
(3.10)
1 1
( Rhh RN)
E( H)
In the case EH ( ) is zero, equal antenna weights are used (from symmetric
considerations).
In the simple case of slow fading, the channel tends to be constant in reservation
period (e.g. LOS) and the location of the reception source is the same as the steering
vector in the source direction. (3.9), then becomes [3]:
(3.11)
When the noise is not directional (like our system model), (3.11) becomes:
1
WR
N
h
(3.12)
1
H
2
H H H
T T T
J( W ) arg min E Y HW X E( Y HW X ) ( Y HW X )
WT
J ( W ) ( E( W Y X ) ( W Y X )) 2 E( YY ) W 2 E( YX )
2R W 2R 0
W
Y R YX
1
RNh R
H 1
N
h R h
R
H H H H H
R R R
H 1
H