Figure 10.4: MIMO Channel with Beamforming.
Solving for the optimal beamforming vector
We wish to choose the beamforming vector c to maximize the average SNR given by
E[SNR] = P c
c = 1.
We need this constraint in order to satisfy the transmit power constraint. But the solution to this
optimization problem is simply the unit norm principal eigenvector (the eigenvector corresponding to the
maximum eigenvalue) of the positive definite matrix E[H
For i.i.d. fades, i.e. when the channel fades between any transmit-receive antenna pair are independent
and identically distributed, E[H
H] is a multiple of the identity matrix. Thus without loss of generality,
we could choose c = [1, 0, 0,
· · · , 0]
. So for i.i.d. fades there is no gain from using multiple transmit
antennas. However the magnitude of the average received SNR is directly proportional to the number of
receive antennas. Hence multiple receive antennas improve average received SNR with i.i.d. fading.
Each row of the channel matrix H is an n-dimensional random vector. Let the covariance matrix for the
row be denoted by K
. For independent fades between all transmit-receive antenna pairs, the K
all diagonal matrices. E[H
is also a diagonal matrix. Again the principal eigenvector is
c = [ 0, 0,
· · · , 0, 1, 0, · · · , 0 ]
. This again corresponds to using just one transmit antenna alone. It can
easily be verified that the transmit antenna is the one that has the highest sum of average channel power
gains to all the receive antennas. Again, multiple receive antennas improve the received SNR.