Transmit Gain Optimization for Space Time block Coding Wireless Systems with Co-channel Interference Nevio benvenuto and Stefano Tomasin Dipartimento di Elettronica e Informatica, Universita di Padova Via G Gradenigo 6A-35131, Padova(Italy) Tel:++39-0498277654,Fax:++39-0498277699 E-mail: nb, tomasin/@dei. unipd it ABSTRACT Flex project [8]. In these networks the devices are orga- The combination of Orthogonal Frequency divisio nized into synchronous piconets which can potentially in- terfere with each other and thus limit considerably the net- multiplexing and space-time block coding is a promising work throughput technique for wireless broadband transmission. In a sce- nario where other devices generate interference, we pro- In a STBC OFDM system, according to the particular pose a scheme where the transmit gains of each OFDM condition of both the channel and the interfering signals, subchannel are adaptively chosen. As a design criteria adaptation of the antenna gains could be done for each of we consider both the minimization of the interference and the OFDM subcarriers. However, a fully optimized sys- maximization of the signal to interference plus noise ra- tem turns out to be exceedingly complex, hence we fo- io at the detection point. As a particular case we con- cus our investigation only on the transmit gain adaptation sider also the situation of varying only the amplitude or Although in general, the transmit gains assume complex the phase of the gains. Indeed, it turns out that when in- alues, to limit complexity we also consider cases where terference Is present, an important role is played by the gains have the same phase or the same amplitude. More- phase of the transmit gains, and for the case of two trans- over, in order to limit complexity, the receiver adopts max- mit antennas we derive the optimum phase of the transmit imum ratio combining whose optimization depends only gains,under the assumption of equal amplitudes. As per- on the channel and not on the interference signal. Within formance measure we used the achievable bit rate of the this framework, we consider two cost functions for the various solutions for a broadband indoor system denoted choice of the transmit gains, namely minimization of the Windflex(European Project). Performance was compared power of the interference at the receiver(MI), and maxi- also with the system capacity obtained by a novel close mization of the signal to noise plus interference ratio form expression In order to have an upper bound on the system perfor- mance we derive a novel expression of the capacity of a L INTRODUCTION system with two receive antennas with adaptive transmit gains. In fact, previous results are limited to the system pacediversity has been recently considered with a where each transmitted signal is the linear combination of growing interest for its ability to significantly improve the all space-coded data [11] performances of wireless communications in non disper sive fading channels. In particular, space-time block cod- indeed there is a significant tradeoff between performance that benefits from spatial diversity. First introduced and computational complexity of the various solutions by Alamouti for a communication system with up to two receive and transmit antennas [6], STBC was further gen I SYSTEM DESCRIPTION eralized for a larger number of antennas [71 An OFDM wireless system is considered, where data broadband communications, where the transmission chan- and transmitted by Nt transmit antennas. The receiver is nel is dispersive. The benefits of both spatial and fre- equipped with Nr receive antennas and it receives both quency diversity can be easy achieved by the combination the useful signal and interference generated by N inter- of STBC and orthogonal frequency division multiplexing ferers. We assume that the interferers use OF DM and are (OFDM)[9], which divides the broadband channel into a synchronous with the useful transmitter. Hence, by number of orthogonal signals, which are modulated on suming that the cyclic prefix [9] is sufficiently long, the equally spaced subcarriers. The combined use of sTBc transmission and the interference channels are flat on each and OFDM has been recently considered for the deploy- OFDM subcarrier. Note that if the interference signals are ment of wireless indoor networks in the European Wind- not synchronous, the cyclic prefix may not absorb the de-
Transmit Gain Optimization for Space Time Block Coding Wireless Systems with Co-channel Interference Nevio Benvenuto and Stefano Tomasin Dipartimento di Elettronica e Informatica, Universita di Padova ` Via G. Gradenigo 6A - 35131, Padova (Italy) Tel: ++39-0498277654, Fax: ++39-0498277699 E-mail: {nb, tomasin}@dei.unipd.it ABSTRACT The combination of Orthogonal Frequency Division Multiplexing and space-time block coding is a promising technique for wireless broadband transmission. In a scenario where other devices generate interference, we propose a scheme where the transmit gains of each OFDM subchannel are adaptively chosen. As a design criteria we consider both the minimization of the interference and maximization of the signal to interference plus noise ratio at the detection point. As a particular case we consider also the situation of varying only the amplitude or the phase of the gains. Indeed, it turns out that when interference is present, an important role is played by the phase of the transmit gains, and for the case of two transmit antennas we derive the optimum phase of the transmit gains, under the assumption of equal amplitudes. As performance measure we used the achievable bit rate of the various solutions for a broadband indoor system denoted Windflex (European Project). Performance was compared also with the system capacity obtained by a novel close form expression. I. INTRODUCTION Space diversity has been recently considered with a growing interest for its ability to significantly improve the performances of wireless communications in non dispersive fading channels. In particular, space-time block coding (STBC) is attractive as a simple and effective technique that benefits from spatial diversity. First introduced by Alamouti for a communication system with up to two receive and transmit antennas [6], STBC was further generalized for a larger number of antennas [7]. At the same time, the need of high bit rates favors broadband communications, where the transmission channel is dispersive. The benefits of both spatial and frequency diversity can be easy achieved by the combination of STBC and orthogonal frequency division multiplexing (OFDM) [9], which divides the broadband channel into a number of orthogonal signals, which are modulated on equally spaced subcarriers. The combined use of STBC and OFDM has been recently considered for the deployment of wireless indoor networks in the European WindFlex project [8]. In these networks the devices are organized into synchronous piconets which can potentially interfere with each other and thus limit considerably the network throughput. In a STBC OFDM system, according to the particular condition of both the channel and the interfering signals, adaptation of the antenna gains could be done for each of the OFDM subcarriers. However, a fully optimized system turns out to be exceedingly complex, hence we focus our investigation only on the transmit gain adaptation. Although in general, the transmit gains assume complex values, to limit complexity we also consider cases where gains have the same phase or the same amplitude. Moreover, in order to limit complexity, the receiver adopts maximum ratio combining whose optimization depends only on the channel and not on the interference signal. Within this framework, we consider two cost functions for the choice of the transmit gains, namely minimization of the power of the interference at the receiver (MI), and maximization of the signal to noise plus interference ratio. In order to have an upper bound on the system performance we derive a novel expression of the capacity of a system with two receive antennas with adaptive transmit gains. In fact, previous results are limited to the system where each transmitted signal is the linear combination of all space-coded data [11]. Simulation results for the Wind-Flex scenario show that indeed there is a significant tradeoff between performance and computational complexity of the various solutions. II. SYSTEM DESCRIPTION An OFDM wireless system is considered, where data of each subcarrier is coded by a space-time block code and transmitted by Nt transmit antennas. The receiver is equipped with Nr receive antennas and it receives both the useful signal and interference generated by N interferers. We assume that the interferers use OFDM and are synchronous with the useful transmitter. Hence, by assuming that the cyclic prefix [9] is sufficiently long, the transmission and the interference channels are flat on each OFDM subcarrier. Note that if the interference signals are not synchronous, the cyclic prefix may not absorb the de-
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 canceled
lays 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 response of the interference channel from the k-th interference antenna to the `-th receive antenna of the m-th OFDM subcarrier. Perfect knowledge of the useful and interference 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 transmitted signal of the s-th block is obtained by linear processing 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 generality 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 received 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 transmitter and the overall complexity of the device, different criteria for the choice of the transmit gains may be considered. 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 communication, 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 eigenvalue of B. Note that if the minimum eigenvalue of B is zero, then the interference can be completely canceled
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]1
B. Maximum signal to noise plus interference ratio (MSNIR) The minimization of the interference can lead to poor performance when the interference has a similar propagation characteristic of the useful channel, since the resulting received useful signal may also be particularly attenuated. 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) under 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 transmit 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 system of equations Aβ + λβ = 0 , (14a) X Nt t=1 |βt| 2 = 1 . (14b) Hence, first we need to find the eigenvector β corresponding 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 criterion 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 corresponding 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 solution, 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)
V CAPACITY CONSIDERATIONS As an upper bound on the performance of a STBC with laptive transmit gains, we give the capacity that can be achieved by a multi antenna system with adaptive transmit gains and when interference is present Lets define the matrix H having as entries (Hk, n) for k=1,2, 1. 2,..., Nt, and lets denote with Lets also indicate with T the Nt X N diagonal matrix F> Ri the Nr x Nr autocorrelation matrix of the interference having as entries anI From [1], the capacity of the considered multi antenna system is given by detTRi+IN +rHTT H tfr+INI Since the denominator of C in(20)does not depend on Fig. 1. Achievable bit rate as a function of the signal to T, the maximization of C with respect to T yields the interference ratio(SIR), for different transmit selec- following problem sNR at the channel ax log2 det(IN, +r(R,+HTTHH)I)(21a) is 10dB trace T1 (21b) where A is the Lagrange multiplier. When In[11] Farrokhi et al. computed the matrix T that solve ]1-[q]2 the above problem in the case T is not constrained to be diagonal. In this general case, (21)can be rewritten as the transmit gains that maximize the capacity are given by max log2det(IN+THTT H)(22) P=1+1-Q22(260) and the solution is attained by diagonalizing HTTH a22 2Q]22-Q Hence, by indicating with H= vwU the SVD ofH, (26b) 2 det[Q1 is T=UhE where E is a diagonal matrix with entries If(25)is not satisfied, by indicating with k computed according to the water-filling principle [11] argmaxp[Qlp. p) we set ak= 1, while the other gain is Unfortunately, when we force T to be diagonal, the ma- zero ix HTTH H cannot be diagonalized and for the a sys- ex Note that, since only TTH is present in the capacity pression(20), the phases of the transmit tem with any number of transmit antennas there is no a affect the capacity close solution to the problem, to the authors knowledge However, for the interesting case of Nt=2 and a general V. PERFORMANCE COMPARISON number of receive antennas, we derive the transmit gains that maximizes the capacity For the performance comparison we consider the chan 4s. By using the property det[I+AB]=det[I+BA nel model obtained by the measurements of the indoor equation(22)can be rewritten project [8]. An OFDM system with 64 subcarriers and a max log2(det[I2+QTT,(23) cyclic prefix of length 8 was simulated on a line of sight e is a 2 x 2 matrix with entries Q]n,m, mean rms delay spread of 27 ns and an average SNR at m,n=1, 2. By applying the Lagrange multiplier metho the channel output of 10 dB. As a performance measure to(23 )under the constraint(1), we obtain the system of we use the bit rate that can be achieved by the system, ng, namely ABR [Q 12.20%+det[Qla112a:+ Aa*=0,(24b) log2(1 +Im
IV. CAPACITY CONSIDERATIONS As an upper bound on the performance of a STBC with adaptive transmit gains, we give the capacity that can be achieved by a multi antenna system with adaptive transmit gains and when interference is present. Let’s define the matrix H having as entries {Hk,n} for k = 1, 2, . . . , Nr, n = 1, 2, . . . , Nt, and let’s denote with Ri the Nr×Nr autocorrelation matrix of the interference. Let’s also indicate with T the Nt × Nt diagonal matrix having as entries {αn}. From [1], the capacity of the considered multi antenna system is given by C = log2 det[ΓRi + INr + ΓHT T HHH] det[ΓRi + INr ] ,(20) Since the denominator of C in (20) does not depend on T , the maximization of C with respect to T yields the following problem max T log2{det[INr + Γ(Ri + HT T HHH)]} (21a) trace T T H = 1 . (21b) In [11] Farrokhi et al. computed the matrix T that solve the above problem in the case T is not constrained to be diagonal. In this general case, (21) can be rewritten as max T log2{det(INr + ΓHT T ˜ HH˜ H )} (22) and the solution is attained by diagonalizing HT T ˜ HH˜ H . Hence, by indicating with H˜ = V W U the SVD of H˜ , the optimum transmit matrix that maximizes the capacity is T = U HΞ where Ξ is a diagonal matrix with entries computed according to the water-filling principle [11]. Unfortunately, when we force T to be diagonal, the matrix HT T ˜ HH˜ H cannot be diagonalized and for the a system with any number of transmit antennas there is no a close solution to the problem, to the authors’ knowledge. However, for the interesting case of Nt = 2 and a general number of receive antennas, we derive the transmit gains that maximizes the capacity. By using the property det[I + AB] = det[I + BA], the equation (22) can be rewritten as max T log2{det[I2 + QT T H]} , (23) where Q = H˜ HH˜ is a 2 × 2 matrix with entries [Q]n,m, m, n = 1, 2. By applying the Lagrange multiplier method to (23) under the constraint (1), we obtain the system of equations [Q]1,1α ∗ 1 + det[Q]|α2| 2α ∗ 1 + λα∗ 1 = 0 (24a) [Q]2,2α ∗ 2 + det[Q]|α1| 2α ∗ 2 + λα∗ 2 = 0 , (24b) 0 2 4 6 8 10 12 14 16 18 20 50 100 150 200 250 SIR [dB] ABR [Mbit/sec] CAPACITY MSNIR MI MSNIR (EA) MSNIR (EP) FIXED TX GAINS Fig. 1. Achievable bit rate as a function of the signal to interference ratio (SIR), for different transmit selection schemes. The average SNR at the channel output is 10dB. where λ is the Lagrange multiplier. When |[Q]1,1 − [Q]2,2| det[Q] ≤ 1 (25) the transmit gains that maximize the capacity are given by |α1| 2 = 1 2 + [Q]1,1 − [Q]2,2 2 det[Q] (26a) |α2| 2 = 1 2 + [Q]2,2 − [Q]1,1 2 det[Q] . (26b) If (25) is not satisfied, by indicating with k = argmaxp {[Q]p,p} we set αk = 1, while the other gain is zero. Note that, since only T T H is present in the capacity expression (20), the phases of the transmit gains do not affect the capacity. V. PERFORMANCE COMPARISON For the performance comparison we consider the channel model obtained by the measurements of the indoor radio channel at 17 GHz for the Wind-Flex European project [8]. An OFDM system with 64 subcarriers and a cyclic prefix of length 8 was simulated on a line of sight channel, with a transmission bandwidth of 50 MHz, a mean rms delay spread of 27 ns and an average SNR at the channel output of 10 dB. As a performance measure we use the bit rate that can be achieved by the system, assuming perfect channel loading and coding, namely ABR = 1 T M X−1 m=0 log2 (1 + Γm), (27)
of the performance, when compared to a scheme with no daptation of the transmitter ACKNOWLEDGMENT We would like to thank Philips Research, Monza, Italy, for continuing suppo REFERENCES 1 G.J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Person. Commun., vol 6, no 3, pp. 311-335, June 1998 0.t [2] E. Telatar, Capacity of multi-antenna Gaussian channels, Europ. Trans. on Telcomm, vol. 10, no 6 pp.585-595,Nov.Dec.1999 Fig. 2. Complementary cdf of the achievable bit rate [3]JH. Winters, Switched diversity with feedback for for different transmit gains selection schemes. The dpsk mobile radio systems, IEEE Trans on Vehic. average snR is 10dB. while the average siR is 5 Tech, vol. VT-32, pp. 134-150, Feb. 1983 [4]R. w. Heath Jr, S. Sandhu and A. Paulraj, " Antenna selection for spatial multiplexing systems with linear receivers,IEEE Commun. Letters, in press on the m-th OFDM subcarrier. We considered a system [5].H. Winters. "Optimum combining in digital mo- with Nt= Nr=2 and N= 2 In the figures we indicate with sIr the signal to in- Sel Areas in Commun., voL SAC-2, no 4, pp 528- 539, July the power transmitted by the useful device and the over- [6S. M Alamouti, A simple transmit diversity tech- all power transmitted by the interfering devices, while the nique for wireless communications, IEEE Trans.on transmission channel is assumed to have unitary gain on Commun., voL. 16, no. 8, pp. 1451-1458, Oct. 1998 [7 V. Tarokh, H. Jafarkhani and A.R. Calder- Fig. I shows the ABR as a function of the sIR. For eference, we also plot the performance of the system with bank,"Space-time block codes from an ortho 83 lesing, IEEE Trans. on Info. Theory, vol. 45, no 5 fixed transmit gains, a1 =a2=1/v2, indicated with the pp.1456-1467,July1999 label Fixed Tx gains. From the figure we observe [8] J.L. Garcia and M. Loberia,"Channel characteriza- hat for a sir of 10 db both the ea and the ep solutions tion and model, Wind-Flex Report IST-1999-10025 ttp://www.vttfi/ele/research outperform by about 3 dB the Fixed Tx gains tech- nique, while being only 1 dB poorer than the optimum /els/projects/ indflexdeliverables. htm, DIlI. 1, Dec 2000. Fig. 2 shows the complementary cumulative distribu- on function(ccdf)of the ABR for some schemes, in a 91Y.(G)Li, J. C. Chuang and N.R. Sollenberger scenario with a SIR of 5dB Transmit diversity for OFDM systems and its im pact on high-rate data wireless networkS, IEEE J. L CONCLUSIONS on Sel. Areas in Commun., vol. 17, no. 7, pp. 1233- Transmit gain optimization has been derived for STBC [10] M. Abramowitz and K. Stegun(ed ) Handbook of systems with a multiple transmit and receive antennas, Mathematical Functions. New York. 1970 when co-channel interference is present. The results hold [11] E.R. Farrokhi, G.J. Foschini, A. Lozano, and for a receiver device using maximum ratio combining and R.A. Valenzuela,"Link-optimal blast processing with perfect knowledge of the channel and interference at with multiple-access interference, in Proc. of vehic. the transmitter. Various criteria for the design of the trans echn. Conf. (vTC00), Boston, vol. 1, pp 87-91 were investigated. A close form of Sep.2000 the capacity of this system has been derived for the case of two transmit antennas. Simulations performed on a Wind Flex scenario shows that a simple system as the equal amplitude gain method yields a significant improvement
70 80 90 100 110 120 130 140 150 160 170 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ABR [Mbit/sec] ccdf CAPACITY MSNIR MI MSNIR (EA) MSNIR (EP) Fig. 2. Complementary cdf of the achievable bit rate for different transmit gains selection schemes. The average SNR is 10dB, while the average SIR is 5dB. where Γm is the SNIR after the combining at the receiver on the m-th OFDM subcarrier. We considered a system with Nt = Nr = 2 and N = 2. In the figures we indicate with SIR the signal to interference ratio at the transmitter, i.e. the ratio between the power transmitted by the useful device and the overall power transmitted by the interfering devices, while the transmission channel is assumed to have unitary gain on average. Fig. 1 shows the ABR as a function of the SIR. For reference, we also plot the performance of the system with fixed transmit gains, α1 = α2 = 1/ √ 2, indicated with the label Fixed Tx gains. From the figure we observe that for a SIR of 10 dB both the EA and the EP solutions outperform by about 3 dB the Fixed Tx gains technique, while being only 1 dB poorer than the optimum MSNIR solution. Fig. 2 shows the complementary cumulative distribution function (ccdf) of the ABR for some schemes, in a scenario with a SIR of 5dB. VI. CONCLUSIONS Transmit gain optimization has been derived for STBC systems with a multiple transmit and receive antennas, when co-channel interference is present. The results hold for a receiver device using maximum ratio combining and with perfect knowledge of the channel and interference at the transmitter. Various criteria for the design of the transmit gains were investigated. A close form expression of the capacity of this system has been derived for the case of two transmit antennas. Simulations performed on a WindFlex scenario shows that a simple system as the equal amplitude gain method yields a significant improvement of the performance, when compared to a scheme with no adaptation of the transmitter. ACKNOWLEDGMENT We would like to thank Philips Research, Monza, Italy, for continuing support. REFERENCES [1] G. J. Foschini and M. J. Gans, ”On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Person. Commun., vol. 6, no. 3, pp. 311–335, June 1998. [2] E. Telatar, ”Capacity of multi-antenna Gaussian channels,” Europ. Trans. on Telcomm., vol. 10, no.6, pp. 585–595, Nov.-Dec. 1999. [3] J. H. Winters, ”Switched diversity with feedback for dpsk mobile radio systems,” IEEE Trans. on Vehic. Tech., vol. VT-32, pp. 134–150, Feb. 1983. [4] R. W. Heath Jr., S. Sandhu and A. Paulraj, ”Antenna selection for spatial multiplexing systems with linear receivers,” IEEE Commun. Letters, in press. [5] J. H. Winters, ”Optimum combining in digital mobile radio with co-channel interference,” IEEE J. on Sel. Areas in Commun., vol. SAC-2, no. 4, pp. 528– 539, July 1984. [6] S. M. Alamouti, ”A simple transmit diversity technique for wireless communications,” IEEE Trans. on Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [7] V. Tarokh, H. Jafarkhani and A. R. Calderbank, ”Space-time block codes from an orthogonal desing,” IEEE Trans. on Info. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [8] J. L. Garcia and M. Loberia, “Channel characterization and model,” Wind-Flex Report IST-1999-10025, http://www.vtt.fi/ele/research /els/projects/ windflexdeliverables.htm, DIII.1, Dec. 2000. [9] Y. (G.) Li, J. C. Chuang and N. R. Sollenberger, ”Transmit diversity for OFDM systems and its impact on high-rate data wireless networks,” IEEE J. on Sel. Areas in Commun., vol. 17, no. 7, pp. 1233– 1243, July 1999. [10] M. Abramowitz and K. Stegun (ed.), Handbook of Mathematical Functions, New York, 1970. [11] F.R. Farrokhi, G.J. Foschini, A. Lozano, and R.A. Valenzuela, “Link-optimal blast processing with multiple-access interference,”in Proc. of Vehic. Techn. Conf. (VTC’00), Boston, vol. 1, pp. 87–91, Sep. 2000
