In this case, the source node knows the instantaneous CSI
between all the cooperative nodes and the destination node,
i.e., the channel gain matrix H with dimension r×k, and the
correlation information among all the nodes. we want the product of those channel gains to be as large as possible. Due to the fact that the eigenvalues of R are equal to Rii,we have the following properties
?Rii=1 = ? lamda(Rii) = det(Rh.R) =(Rh.Qh Q.R)
= det(Hh.H),
where Qh means hermitian transpose of Q unitary matrix
R upper triangular matrix
it is clear that in order to maximize the product
of the channel gains, ?R1,1?2??R2,2?2 . . . ?RN,N?2, we only need
to maximize the determinant of the corresponding channel
matrix (Hh.H).
To accomplish this, consider the use of a maximal channel
gain (MCG) algorithm as follows: at the (k+1)th step, where
k nodes have already been chosen, and the corresponding
channel matrix H(k) are known, where H(k) is the channel
matrix when k nodes are chosen, we want to select one
additional node S? from the set S containing the remaining
Kk nodes such that
S?= argmax{det((H(k+1))h.(H(k+1))}.where (H(k+1))h means hermitian transpose of H(k+1)
We repeat this until all the K nodes are chosen. Therefore,
at each step, we obtain a selected combination of nodes,
$, with an increasing number of nodes in it. In total, the
algorithm runs K steps, thus the search space for the
previous optimization problem has only K combinations.
Finally, we choose the optimal subset $? which results
in the largest D2 for the cooperative transmission while
meeting the specified total endtoend delay and energy
constraints.
