Transmit Gain Optimization for Space Time Block Coding Wireless Systems with Co-channel Interference Nevio Benvenuto and Stefano Tomasin Dipartimento di Elettronica e Informatica, Universit`a di Padova Via G. Gradenigo 6A - 35131, Padova (Italy) Tel: ++39-0498277654, Fax: ++39-0498277699 E-mail: fnb, tomasing@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 sce- nario where other devices generate interference, we pro- pose 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 ra- tio at the detection point. As a particular case we con- sider also the situation of varying only the amplitude or the phase of the gains. Indeed, it turns out that when in- terference is present, an important role is played by the phase of the transmit gains, and for the case of two trans- mit antennas we derive the optimum phase of the transmit gains, under the assumption of equal amplitudes. As per- formance 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 disper- sive fading channels. In particular, space-time block cod- ing (STBC) is attractive as a simple and effective tech- nique 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 gen- eralized for a larger number of antennas [7]. At the same time, the need of high bit rates favors broadband communications, where the transmission chan- nel is dispersive. The benefits of both spatial and fre- quency 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 deploy- ment of wireless indoor networks in the European Wind- Flex project [8]. In these networks the devices are orga- nized into synchronous piconets which can potentially in- terfere with each other and thus limit considerably the net- work 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 sys- tem turns out to be exceedingly complex, hence we fo- cus 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. More- over, in order to limit complexity, the receiver adopts max- imum 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 maxi- mization of the signal to noise plus interference ratio. In order to have an upper bound on the system perfor- mance 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 inter- ferers. We assume that the interferers use OFDM and are synchronous with the useful transmitter. Hence, by as- suming 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 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 fi(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 NtX t=1 jfi(m)t j2 = 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 = NtX t=1 NrX q=1 H??t(k);qfi??t(k)–t(k)r(q)s+t ; (2) where for each k, ?q(k) is a permutation function of the indexes f1;2;::: ;Nrg and f–q(k)g depend on the code. For example, for orthogonal design codes –q(k) 2 f?1;+1g, [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 2u = ?N tX t=1 jfitj2 NrX r=1 jHt;rj2 !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 2i = E 2 4 flfl flfl fl NtX t=1 NrX r=1 i(r)t fi??tH??t;r–t flfl flfl fl 23 5 : (4) We indicate with w the noise variance on each antenna of each subchannel, before combining. Hence, the signal to noise plus interference ratio (SNIR) is given by Γ = 2u 2w u +E ?flfl flPNtt=1PNrr=1 i(r)t fi??tH??t;r–t flfl fl 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 ?t, i.e. ?fm = m: (6) By defining the matrix B with entries [B]‘;m = NrX r=1 NrX q=1 E h i(r)?f‘ i(q)fm i H?m;r–fmH‘;q–f‘ ; (7) and the vector fi = [fi1;fi2;:::fiNt] collecting the Nt transmit gains, the interference power (5) can be written in the quadratic form E 2 4 flfl flfl fl NtX t=1 NrX r=1 i(r)t fi??tH??t;r–t flfl flfl fl 23 5 = fi?Bfi: (8) Then the minimization problem is solved by the following linear system of equations Bfi+?fi = 0; (9) under the constraint (1). From (9) we conclude that the minimization of the interference is archived when fi 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. 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 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 SNIRΓ 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 c2, Γ is multiplied by c. Hence, in order to find the solution under the constraint (1) first a set of trans- mit gains f?fitg which maximize Γ is found and then (1) is satisfied by setting fit = ?fitPN t t=1j?fitj2 : (10) In order to maximize (5) we minimize its denominator 2w ?N tX t=1 j?fitj2 NrX r=1 jHt;rj2 ! + E 2 4 flfl flfl fl NtX t=1 NrX r=1 i(r)t fi??tH??t;r–t flfl flfl fl 23 5 under the constraint that the numerator is a constant, i.e. NtX t=1 j?fitj2 NrX r=1 jHt;rj2 = 1: (11) Now, by defining the vectorfl = [fl1;fl2;::: ;flNt] with entries fln = ?fin vu utNrX r=1 jHn;rj2 (12) and the matrix A with entries [A]‘;m = [B]‘;mqP Nr r=1jH‘;rj2 ; (13) the Lagrange multiplier method yields the following sys- tem of equations Afl +?fl = 0; (14a) NtX t=1 jfltj2 = 1: (14b) Hence, first we need to find the eigenvector fl corre- sponding to its minimum eigenvalue of A, then the coef- ficients f?fing can be computed by (12). Lastly, in order to satisfy the constraint (1), we normalize f?fing 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 u 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 NrX r=1 jHq;rj2 ; 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 ffitg 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]fl +?fl = 0; (16) where A and fl are defined by (13) and (12), respectively. The linear system (16) must be solved under the constraint (1). In this case, the solution fl 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. fit = e j t pN t ; 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 1 = 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)+2j[B]1;2jcos( 1 +\[B]1;2): (18) Hence the solution is 1 = cos?1 2j[B] 1;2j [B]1;1 +[B]2;2 ? ?\[B]1;2: (19) 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 fHk;ng 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 ffing. From [1], the capacity of the considered multi antenna system is given by C = log2 det[ΓRi +INr +ΓHTT 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 maxT log2fdet[INr +Γ(Ri +HTTHHH)]g (21a) traceTTH = 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 maxT log2fdet(INr +Γ ?HTTH ?HH)g (22) and the solution is attained by diagonalizing ?HTTH ?HH. Hence, by indicating with ?H = VWU the SVD of ?H, the optimum transmit matrix that maximizes the capacity is T = UHΞ 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 ma- trix ?HTTH ?HH cannot be diagonalized and for the a sys- tem 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 maxT log2fdet[I2 +QTTH]g; (23) where Q = ?HH ?H 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;1fi?1 +det[Q]jfi2j2fi?1 +?fi?1 = 0 (24a) [Q]2;2fi?2 +det[Q]jfi1j2fi?2 +?fi?2 = 0; (24b) 0 2 4 6 8 10 12 14 16 18 2050 100 150 200 250 SIR [dB] ABR [Mbit/sec] CAPACITYMSNIR MIMSNIR (EA) MSNIR (EP)FIXED TX GAINS Fig. 1. Achievable bit rate as a function of the signal to interference ratio (SIR), for different transmit selec- tion schemes. The average SNR at the channel output is 10dB. where ? is the Lagrange multiplier. When j[Q]1;1 ?[Q]2;2j det[Q] ? 1 (25) the transmit gains that maximize the capacity are given by jfi1j2 = 12 + [Q]1;1 ?[Q]2;22det[Q] (26a) jfi2j2 = 12 + [Q]2;2 ?[Q]1;12det[Q] : (26b) If (25) is not satisfied, by indicating with k = argmaxpf[Q]p;pg we set fik = 1, while the other gain is zero. Note that, since only TTH 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 chan- nel 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 = 1T M?1X m=0 log2(1+Γm); (27) 70 80 90 100 110 120 130 140 150 160 1700 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ABR [Mbit/sec] ccdf CAPACITYMSNIR MIMSNIR (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 in- terference ratio at the transmitter, i.e. the ratio between the power transmitted by the useful device and the over- all 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, fi1 = fi2 = 1=p2, 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 tech- nique, while being only 1 dB poorer than the optimum MSNIR solution. Fig. 2 shows the complementary cumulative distribu- tion function (ccdf) of the ABR for some schemes, in a scenario with a SIR of 5dB. VI. 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