lays of all devices and both intersymbol interference and Hence. the signal to noise plus interference ratio intercarrier interference will be present (SNIR) is given by We indicate with Hi the frequency response of the transmit channel from antenna k to antenna e of the m-th OFDM subcarrier. With Gk( we indicate frequency re- OOu+E>N N, sponse of the interference channel from the k-th interfer- ence antenna to the l-th receive antenna of the m-th OFDM III. TRANSMIT GAIN SELECTION subcarrier. Perfect knowledge of the useful and interfer ence frequency responses is assumed According to the information available at the transmit Before transmission, the coded data is scaled by ter and the overall complexity of the device, different cri- he complex gain a( m), for each transmit antenna teria for the choice of the transmit gains may be consid- 1, 2, .., N and each OFDM subcarrier m 0.1.M-1. In order to set a constraint on the transmit As a first option we investigate the minimization of the total power, it must be interference(MI), regardless of the noise. However, this choice may decrease the power of the useful signal at the detection point and hence in general we consider as cost (1) function the maximization of the SNIR(MSNIR) t=1 As a reduced complexity solution we consider also the Since the choice of the transmit gain is independent of the choice of transmit gains with equal amplitude(EA)or subcarrier, in the following we will omit the index(m) equal phase (power adaptation, EP). For both cases we The data signal is coded by space-time block coding, dopt the MSNIR criterion. according to the schemes of [6, 7], and at the receiver A. Minimum interference(MI) maximum ratio combining(MRC) of the received signals If the interference is the limiting factor for the co transmit gains. In particular, by indicating with r(@)the nication, a reasonable target for the choice of the transmit received signal at time t on the antenna g, the k-th trans- gains is the minimization of the residual interference. In mitted signal of the s-th block is obtained by linear pro- order to minimize(4)under constraint(1), we apply the Lagrange multiplier method. Let's indicate with fm the inverse function of Et,1.e =∑∑H()05(k)r,() t=1q=1 By defining the matrix B with entries where for each k, Eg(k) is a permutation function of the indexes (1, 2,..., N and 5q(k) depend on th (7) code. For example, for orthogonal design codes Sq(k) -1,+1, [7]. In the following, without loss of general- ity we will assume s=0 and we will drop the indexes(s) and the vector a=[a1,02,.aN,I collecting the Nt and(k) transmit gains, the interference power (5)can be written After the MRC, from(2)the power of the useful signal in the quadratic form E∑∑a,HrA a2=(∑a2∑H Then the minimization problem is solved by the following while by indicating with i (r) the interference signal re- linear system ofequations ceived at time t on the r-th receive antenna, the power of the residual interference is Ba+λa=0 under the constraint(1). From(9) we conclude that the E minimization of the interference is archived when a is the eigenvector of B corresponding to the minimum eigen value of B We indicate with u the noise variance on each antenna Note that if the minimum eigenvalue of B is zero, then of each subchannel, before combining the interference can be completely canceledlays of all devices and both intersymbol interference and intercarrier interference will be present. We indicate with H (m) k,` the frequency response of the transmit channel from antenna k to antenna ` of the m-th OFDM subcarrier. With G (m) k,` we indicate frequency re￾sponse of the interference channel from the k-th interfer￾ence antenna to the `-th receive antenna of the m-th OFDM subcarrier. Perfect knowledge of the useful and interfer￾ence frequency responses is assumed. Before transmission, the coded data is scaled by the complex gain α (m) t , for each transmit antenna t = 1, 2, . . . , Nt and each OFDM subcarrier m = 0, 1, . . . , M−1. In order to set a constraint on the transmit total power, it must be X Nt t=1 |α (m) t | 2 = 1 . (1) Since the choice of the transmit gain is independent of the subcarrier, in the following we will omit the index (m). The data signal is coded by space-time block coding, according to the schemes of [6, 7], and at the receiver maximum ratio combining (MRC) of the received signals is applied, according to the channel coefficients and the transmit gains. In particular, by indicating with r (q) t the received signal at time t on the antenna q, the k-th trans￾mitted signal of the s-th block is obtained by linear pro￾cessing as u˜ (s) k = X Nt t=1 X Nr q=1 H∗ ²t(k),qα ∗ ²t(k) δt(k)r (q) s+t , (2) where for each k, ²q(k) is a permutation function of the indexes {1, 2, . . . , Nr} and {δq(k)} depend on the code. For example, for orthogonal design codes δq(k) ∈ {−1, +1}, [7]. In the following, without loss of general￾ity we will assume s = 0 and we will drop the indexes (s) and (k). After the MRC, from (2) the power of the useful signal is σ 2 u = ÃX Nt t=1 |αt| 2X Nr r=1 |Ht,r| 2 !2 . (3) while by indicating with i (r) t the interference signal re￾ceived at time t on the r-th receive antenna, the power of the residual interference is σ 2 i = E   ¯ ¯ ¯ ¯ ¯ X Nt t=1 X Nr r=1 i (r) t α ∗ ²tH∗ ²t,rδt ¯ ¯ ¯ ¯ ¯ 2   . (4) We indicate with σw the noise variance on each antenna of each subchannel, before combining. Hence, the signal to noise plus interference ratio (SNIR) is given by Γ = σ 2 u σ 2 wσu + E·¯ ¯ ¯ PNt t=1 PNr r=1 i (r) t α∗ ²tH∗ ²t,rδt ¯ ¯ ¯ 2 ¸ . (5) III. TRANSMIT GAIN SELECTION According to the information available at the transmit￾ter and the overall complexity of the device, different cri￾teria for the choice of the transmit gains may be consid￾ered. As a first option we investigate the minimization of the interference (MI), regardless of the noise. However, this choice may decrease the power of the useful signal at the detection point and hence in general we consider as cost function the maximization of the SNIR (MSNIR). As a reduced complexity solution we consider also the choice of transmit gains with equal amplitude (EA) or equal phase (power adaptation, EP). For both cases we adopt the MSNIR criterion. A. Minimum interference (MI) If the interference is the limiting factor for the commu￾nication, a reasonable target for the choice of the transmit gains is the minimization of the residual interference. In order to minimize (4) under constraint (1), we apply the Lagrange multiplier method. Let’s indicate with fm the inverse function of ²t, i.e. ²fm = m . (6) By defining the matrix B with entries [B]`,m = X Nr r=1 X Nr q=1 E h i (r)∗ f` i (q) fm i H∗ m,rδfmH`,qδf` , (7) and the vector α = [α1, α2, . . . αNt ] collecting the Nt transmit gains, the interference power (5) can be written in the quadratic form E   ¯ ¯ ¯ ¯ ¯ X Nt t=1 X Nr r=1 i (r) t α ∗ ²tH∗ ²t,rδt ¯ ¯ ¯ ¯ ¯ 2   = α ∗Bα . (8) Then the minimization problem is solved by the following linear system of equations Bα + λα = 0 , (9) under the constraint (1). From (9) we conclude that the minimization of the interference is archived when α is the eigenvector of B corresponding to the minimum eigen￾value of B. Note that if the minimum eigenvalue of B is zero, then the interference can be completely canceled