B. Maximum signal to noise plus interference ratio Hence, first we need to find the eigenvector B corre- sponding to its minimum eigenvalue of A, then the coef- The minimization of the interference can lead to poor ficients an) can be computed by(12). Lastly, in order to performance when the interference has a similar propaga- satisfy the constraint(1), we normalize an) by (10) tion characteristic of the useful channel. since the result- Note that if the minimum eigenvalue is null. then there ing received useful signal may also be particularly atten- s no interference at the decision point and the MSNIR cri- uated. Hence we consider here the more general target of terion is equivalent to the maximization of au as given maximizing the SNIR T under the constraint(1) by(5). In this case, T is maximized by allocating all the By applying the Lagrange multiplier method to()un- power to the transmit antenna t with the maximum value der the constraint(1)a non-linear system of equations is of obtained. In order to find a solution we observe that by multiplying all transmit gains by a constant real positive JHq, 12, 9=1,2,., Nt value c2, r is multiplied by c. Hence, in order to find the solution under the constraint(1)first a set of trans We examine now two particular cases for the transmit mit gains at) which maximize r is found and then(1)is ga satisfied by setting (10) =1||2 When only the gain amplitude adaptation is considered, this is equivalent to assume that at are real numbers. In In order to maximize(5)we minimize its denominator this case, we maximize (5) under the constraint (1)and we consider only the real solution for the transmit gains Hence, the transmit gains that solves the problem is the a∑2∑|P)+ solution of the linear system of equations t=1 Re[AB+ AB=0 E∑∑,A where A and B are defined by(13)and (12), respectively t=1r=1 The linear system(16)must be solved under the constraint under the constraint that the numerator is a constant. i.e (1). In this case, the solution B is the eigenvector corre- sponding to the minimum eigenvalue of Rela ∑∑|12=1 D. Equal amplitude(EA) t=1 We consider here the adaptation of only the phase of the Now, by defining the vector B=[1,B2,……,BN:]wth ransmit gains,1.e entries t=1.2 (12) From(3)we note that by forming an equal gain amplitude. the power of the received user signal is independent of the transmit gains and the MI and the MSNIR criteria yield and the matrix a with entries the same solution. Additionally, from( 8)we have that it Ble is not restrictive to set 01=0 Now, by imposing the constraint(17)to( 8), we obtain a problem which in general does not have a close form so- lution, to the authors knowledge. However, a close form the Lagrange multiplier method yields the following sys- solution for the case Nt =2 is straightforward. From tem of equations (8), the interference power is minimized by minimizing ([B]1,1+[B2)+21B1,2|cos(61+∠B12).(18) Hence the solution ∑lP=1 81=cos 21B1 B]1,+[B ∠[B]1B. Maximum signal to noise plus interference ratio (MSNIR) The minimization of the interference can lead to poor performance when the interference has a similar propaga￾tion characteristic of the useful channel, since the result￾ing received useful signal may also be particularly atten￾uated. Hence we consider here the more general target of maximizing the SNIR Γ under the constraint (1). By applying the Lagrange multiplier method to (5) un￾der the constraint (1) a non-linear system of equations is obtained. In order to find a solution we observe that by multiplying all transmit gains by a constant real positive value c 2 , Γ is multiplied by c. Hence, in order to find the solution under the constraint (1) first a set of trans￾mit gains {α˜t} which maximize Γ is found and then (1) is satisfied by setting αt = α˜t PNt t=1 |α˜t| 2 . (10) In order to maximize (5) we minimize its denominator σ 2 w ÃX Nt t=1 |α˜t| 2X Nr r=1 |Ht,r| 2 ! + E   ¯ ¯ ¯ ¯ ¯ X Nt t=1 X Nr r=1 i (r) t α ∗ ²tH∗ ²t,rδt ¯ ¯ ¯ ¯ ¯ 2   under the constraint that the numerator is a constant, i.e. X Nt t=1 |α˜t| 2X Nr r=1 |Ht,r| 2 = 1 . (11) Now, by defining the vector β = [β1, β2, . . . , βNt ] with entries βn = ˜αn vuutX Nr r=1 |Hn,r| 2 (12) and the matrix A with entries [A]`,m = q [B]`,m PNr r=1 |H`,r| 2 , (13) the Lagrange multiplier method yields the following sys￾tem of equations Aβ + λβ = 0 , (14a) X Nt t=1 |βt| 2 = 1 . (14b) Hence, first we need to find the eigenvector β corre￾sponding to its minimum eigenvalue of A, then the coef- ficients {α˜n} can be computed by (12). Lastly, in order to satisfy the constraint (1), we normalize {α˜n} by (10). Note that if the minimum eigenvalue is null, then there is no interference at the decision point and the MSNIR cri￾terion is equivalent to the maximization of σu as given by (5). In this case, Γ is maximized by allocating all the power to the transmit antenna t with the maximum value of X Nr r=1 |Hq,r| 2 , q = 1, 2, . . . , Nt . (15) We examine now two particular cases for the transmit gains. C. Equal phase (EP) When only the gain amplitude adaptation is considered, this is equivalent to assume that {αt} are real numbers. In this case, we maximize (5) under the constraint (1) and we consider only the real solution for the transmit gains. Hence, the transmit gains that solves the problem is the solution of the linear system of equations Re[A]β + λβ = 0 , (16) where A and β are defined by (13) and (12), respectively. The linear system (16) must be solved under the constraint (1). In this case, the solution β is the eigenvector corre￾sponding to the minimum eigenvalue of Re[A]. D. Equal amplitude (EA) We consider here the adaptation of only the phase of the transmit gains, i.e. αt = e jθt √ Nt , t = 1, 2, . . . , Nt. (17) From (3) we note that by forming an equal gain amplitude, the power of the received user signal is independent of the transmit gains and the MI and the MSNIR criteria yield the same solution. Additionally, from (8) we have that it is not restrictive to set θ1 = 0. Now, by imposing the constraint (17) to (8), we obtain a problem which in general does not have a close form so￾lution, to the author’s knowledge. However, a close form solution for the case Nt = 2 is straightforward. From (8), the interference power is minimized by minimizing the cost function ([B]1,1 + [B]2,2) + 2|[B]1,2| cos(θ1 + ∠[B]1,2). (18) Hence the solution is θ1 = cos−1 µ 2|[B]1,2| [B]1,1 + [B]2,2 ¶ − ∠[B]1,2. (19)