TEL AVIV UNIVERSITY Gaddi Blumrosen

4.4.4 OSTBC with linear antenna weighting, based on only channel covariance
feedback (ML sub-optimal)
As we seen from previous chapters, the optimal weights are thus obtained by the
conditioned value of the first two moments. If we assume, as in [31], the following
scenario of no direct measurement of the channel is known to the transmitter, but only
the channel covariance matrix, we can assume, in the case of the Rayleigh channel a
zero mean value , m 0 , and no correlation between the channel covariance
hh | ˆ
approximation and its real value, R R .
hh| hˆ
Consequently, the solution can be substantially simplified, especially for the case of
correlated channel coefficients. Note that although [31] does not use conditional
covariance matrix (measure of CSI quality), it has shown that for this scenario of
Gaussian channel with zero conditional channel mean value, same antenna weights
are derived also for non perfect CSI.
Using those relations and multiplying (4.14) by , we obtain:
l( Z) log det I RhhZ (4.24)
Let us further perform Singular Value decomposition (SVD) over the channel
covariance matrix, , where is the unitary eigenvector matrix and
is the corresponding square root eigenvalue matrix.
H
Rhh Uh DhU h
h U
D h
Let us further define, as in [30], the diagonal matrix D as:
2
f
D W U W U
H
T h T h
(4.25)
2
As in [30], maximization of (4.24) force the matrices RhhZ and DD h f to be diagonal.
Now, by substitution of Rhh Uh
DhU h and Z WTWT
into (4.24) and simple
algebraic manipulation, relying on the symmetry of the matrices R and Z and the
unitary of U , we obtain the same minimization expression as in [30]:
h
2
l( D f ) log det I DhD f
hh
H
(4.26)
Explicit expressions, for the k th beam obtained after power loading according to the
power loading algorithm in [30] in the direction corresponding to the k th eigenvector,
which minimizes (4.26), was found to be:
H
Wk D f Uk
(4.27)
The assumptions above, 0 and R
R , make sense in fast fading
mhh | ˆ
hh| hˆ
scenarios. Since both algorithms (in [25] and in [31]) minimize PEP, we expect the
performance of the system of [25], and [31], to be the same in fast fading channel. In
case of slow fading, if we have strong nonzero conditional channel mean value, as in
Rician fading, we expect the solution in (4.14) to enhance the system performance.
R hh
hh
f
H
hh