Precoding for Distributed Space-Time Codes in Cooperative Diversity-Based Downlink

Hilde Skjevling David Gesbert Are Hjørungnes Department of Informatics Mobile Communications Department UniK - University Graduate Center University of Oslo Eurécom Institute University of Oslo Oslo, Norway Sophia Antipolis, France Oslo, Norway Email: [email protected] Email: [email protected] Email: [email protected]

Abstract— In this paper, we investigate the cooperative diver- destination (path loss, correlation, etc.) are different. The sity concept for use in MIMO multi-cell networks. We show cooperative diversity scheme ought to be optimized with that, in such networks, cooperative diversity processing must be respect to the channel conditions, and we handle this via optimized to account for the variability of channel conditions across the cooperative devices. This can be done via distributed distributed precoding, optimized based on channel statistics precoding and, in mobile networks, it is based realistically on only (incurs less overhead than instantaneous channels). We channel statistics. The cooperative MIMO correlation matrix make the following contributions: admits a special structure which is used to optimize the precoder. • We investigate algorithms for exact error-rate and low-complexity The collaborative MIMO network is recast as a point- approximated optimization. Gains are evaluated in multi-cell to-point MIMO system, where the channel second-order scenarios with collaborating base stations. statistics admit a special non-Kronecker form. • Upon feedback of the statistics to the transmitters, a I. INTRODUCTION distributed linear precoder is optimized and applied prior MIMO systems yield their best in the case of uncorrelated to transmission. channel matrix elements. Therefore, interest in MIMO net- • We show that, under realistic conditions, the optimal works has recently focused on scenarios that provide additional precoder takes the form of a diagonal precoder, boiling dimensions, yielding independent sources of diversity. This down to a power allocation scheme. includes setups where some or all of the multiple antenna • We express the exact average error probability at the re- elements of the overall MIMO system are distributed over ceiver, as function of the precoder and channel statistics. the network, instead of being localized on a unique device. • We investigate several optimization algorithms for the Prominent examples of this are given by (i) the so-called precoder, including one intuitive equivalent criterion multiuser multi-cell MIMO, where one [1] or more [2], [3], coined the ”maximum diversity criterion”, and provide [4] access points address the data needs of multiple user a justification in the form of a Gaussian approximation terminals simultaneously and in a joint fashion and by (ii) the of the combined cooperative transmitter channel gain. so-called cooperative diversity setup, where multiple devices collaborate to combat the detrimental effects of fading at any II. SIGNAL AND CHANNEL MODEL one particular device. We consider a MIMO-system with L spatially distributed Most cooperative diversity scenarios investigated so far base stations (BS), engaged in downlink communication with a include single-antenna user terminals relaying data between single, receiving mobile unit (MU). Each BS has Mtl antennas, a source terminal and the target destination [5], [6], [7]. This l ∈{0, 1,...,L− 1}, while the receiver is equipped with an includes the use of Space-Time Block Codes (STBC), where L−1 array of Mr antennas. In total, there are Mt = l=0 Mtl the spatial elements of the codewords are distributed over the transmit antennas. Fig. 1 illustrates the described scenario. antennas of the collaborating devices [7], [8], [9]. The synchronized BSs are connected to a central unit (CU) In this paper, we focus on the cooperative processing using via fast optical links. The CU and BSs only have access a distributed orthogonal STBC [10]. We ignore the issues to long-term statistical knowledge of the channel conditions, associated with the relay protocol or consider that such a whereas the MU knows the full, instantaneous downlink protocol is not needed. A practically relevant example is channel. The channel is assumed to be flat fading, but results downlink cooperative diversity in a multi-cell network, where are expected to carry over through an OFDM setting. multiple base stations collaborate to serve one user terminal. Thanks to the more generous antenna spacing at the base All transmitters and receivers may be equipped with mul- station side as well as the large inter-cell spacing, we assume tiple antennas. Because of the large-scale separation of the that the effect of transmit correlation is negligible, i.e., the collaborating devices, the channel conditions to the common outgoing paths from all the Mt antennas are uncorrelated. MU antennas, however, will be correlated, and the amount Supported by the The Research Council of Norway (RCN) and the French Ministry of Foreign Affairs through the Aurora project “Optimization of of correlation is likely to depend on which BS the signal is Broadband Wireless Communications Networks" and RCN project 160637. coming from. Additionally, different BSs may see different

1-4244-0355-3/06/$20.00 (c) 2006 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. Central h × unit ci is the Mr 1 channel from the i-th transmit antenna to R h hH the correlated receiver array. We define ri = E[ ci ci ] as the receive correlation matrix for the i-th transmit path. The BS 1 transmit antennas are assumed to be uncorrelated, so we write the full correlation of the path-loss normalized MIMO matrix BS 0 H . as the MtMr ×MtMr-sized R = E[vec(Hc)vec (Hc)]: M . BS L -1 ... t1 .   R 0 ρj r0 0Mr ×Mr ... Mr ×Mr Mt0 ... ρM t0 0 R 0   Mr ×Mr r1 ... Mr ×Mr  ... MtL−1 R =  . . . .  . (2) ρ0  . . . .  ρMt−1 . . . . 0 0 R ... Mr ×Mr Mr ×Mr ... rM −1 t Importantly, the different transmit antennas experience dif- Receiving MU, Mr antennas ferent correlation matrices upon reception. That is because the Fig. 1. Distributed MIMO-system, with L transmitters and one receiver. signals from different base stations may see a very different angular spread at the terminal. This leads to a practical instance of the non-Kronecker correlation structure evoked path loss and slow fading coefficients to the same MU. We recently in [11]. consider Rayleigh fading channel conditions. The receiver is assumed to know H perfectly, whereas the L Although several forms of cooperation schemes are avail- transmitters only have access to long-term statistics in P and able, including distributed spatial multiplexing, we limit our- R, consistent with a practical multi-cell signaling overhead. selves to the cooperative diversity setup where a space-time For all generalized complex orthogonal designs Gc,we code is applied in a distributed manner over the Mt antennas know that [10] of the L collaborating BSs. We focus on the use of distributed ∗ 2 2 2 GMt T GMt | | | | ··· | | I Orthogonal Space-Time Block Codes (OSTBC), as they lend ( c ) ( c ) = a ( x0 + x1 + + xK−1 ) B , (3) themselves to easier analysis. where IB is the identity matrix of size B and the scalar a de- To compensate/exploit the effect of unequal path loss and pends on the choice of orthogonal design. In the following, we −1 transmit correlation, we apply linear precoding over the coop- let B = M and find that C(x)CH (x)=a K |x |2I . erating transmit antennas before launching the codeword into t l=0 l Mt The symbol-per-period rate of the code is R = K/N, and the channel. We represent the linear precoder by the Mt × B R<1 for all Mt > 2. matrix F ,sotheM ×N received signal Y at the user terminal 2 r Now, we define the scalar α
HF , where · is the writes as F F Frobenius norm. From [12], we know that the OSTBC system Y HFC(x) V = + . (1) with a full complex-valued precoder F and a full channel H × correlation matrix R has an equivalent Single-Input Single- Here, represents the overall Mr Mt MIMO channel. The Mr × N-sized matrix V represents the additive, complex Output (SISO) formulation where the output yk for an input 2 ∈{ − } Gaussian, circularly distributed noise, vij ∼CN(0,σ ) and xk,k 0, 1,...,K 1 is v √ x =[x0,x1,...,x −1] is the vector of M-ary modulated K y = αx + v , (4) symbols. The B × N matrix C(x) is the result of applying a k k k x ∈{ − } chosen orthogonal design to the vector , related to the chosen That is, every input symbol xk,k√ 0, 1,...,K 1 C(x) GB T generalized complex design as =( c ) [10]. A block experiences the same channel gain α and independent, ∼CN 2 model of the transmit/receive chain is shown in Fig. 2. additive noise vk (0,σv/a).

V III. SER EXPRESSIONS

Mt × B Mr × Mt x Z x bi C(x) Y ˆ ˆbi Modulation OSTBC F H MLD Demodulation From the above SISO formulation of the OSTBC, we find that the instantaneous received SNR γ is expressed as K × 1 B × NMt × NMr × N K × 1 2 2
aσx α HF2 aσx Fig. 2. Block model of a linearly precoded MIMO-system. γ 2 = δα = δ F , where δ = 2 . (5) σv σv

The large-scale attenuations associated with each transmit Given that all the symbols xk, where k ∈{0, 1,...,K − 1}, antenna are collected in go through the same channel, the average symbol error rate √ √ √ (SER) of the MIMO system with OSTBC is [12] P = diag ρ0 ρ1 ··· ρM −1 : Mt × Mt , t ∞
| so that the total channel H = HcP decomposes into SER Pr{Error} = Pr{Error γ}pγ (γ)dγ 0 √ √ √ ∞ (6) H = ρ0 h ρ1 h ··· ρ −1 h : M ×M . c0 c1 Mt cMt−1 r t = SERγ pγ (γ)dγ , 0

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. where pγ (γ) is the probability density function (pdf) of γ, Now, the combined channel gain α may be written out −1 Mt 2h 2 while SERγ is the symbol error probability for a chosen bit- as α = i=0 ρifi cj . The receive correlation ma- R R to-symbol mapping, e.g. M-PAM, M-PSK or M-QAM, and trices ri have the following eigen-decomposition: ri = γ V Λ V H h R1/2h h a given SNR-value as shown per (5). As an example, the ri ri ri , so that ci = ri wi , where wi is an Mr- SNRγ for M-PSK is sized vector of complex Gaussian distributed independent ∼CN ( −1) variables, h (0, 1).Now,α may be further developed M π wij M − gPSK γ 1 sin2( ) 2 π into SERγ = e θ dθ , gPSK =sin . (7) π 0 M Mt−1 Mt−1 2hH V Λ V H h 2 h H Λ h α = ρifi wi ri ri ri wi = ρifi ( wi ) ri wi i=0 i=0 IV. OPTIMAL PRECODING −1 −1 Mt Mr F 2 | |2 We want to find the matrix such that the exact SER = ρif λij h , i wij is minimized, under a global peak power constraint for the i=0 j=0 (10) transmitted block Z
FC(x). Note that this sum power where h = V H h ∼CN(0, I ) and λ is the j-th constraint across multiple base stations makes engineering wi ri wi Mr ij R sense. Since, in practice, there will be many users and those eigenvalue of ri . The SNR γ becomes −1 −1 are likely to be symmetrically distributed across the cells, each Mt Mr 2 | |2 base station should distribute its total power optimally across γ = δ ρif λij h . (11) i wij the cell users. i=0 j=0 K−1 2 2 We normalize with a =0 σx =1, where σx is the k k l The instantaneous SNR γ is thus a random variable on the variance of the k-th symbol in the OSTBC. The peak power N−1 | |2 ∈ form γ = l=0 al Zl . When all the coefficients al, l constraint is expressed as {0, 1,...,N− 1}, are different1, we can find the exact pdf of Tr{FFH } = P, (8) γ, and the SER for this scenario. For details, see [13]. Lemma 1: The pdf of γ is where P is a measure of the average power used. Now, the −1 −1 (− ( 2 )) Mt Mr 2 MtMr −2 γ/ δρifi λi 1 (ρif λi ) e j u(γ) problem of minimizing the SER can be expressed as p (γ)= i j γ δ (ρ f 2λ − ρ f 2λ ) Problem 1: i=0 j=0 i i ij k k kl k∈{0,Mt−1} l∈{0,Mr −1} min SER (9) (k,l)=(i,j) {F ∈CMt×Mt | Tr{FFH }=P } Theorem 2: The exact SER is expressed as Because the transmit antennas are assumed to be uncorre- −1 −1 2 ∞ Mt Mr 2 M M −2 (−γ/(δρif λi )) (ρ f λ ) t r e i j lated (unlike the receive antennas), it can be shown that the SERγ ii ij SER= 2 − 2 dγ search for the optimal precoder F can be limited to a diagonal 0 δ =0 =0 (ρifi λij ρkfk λkl ) i j ∈{0 −1} ∈{0 −1} precoder: k ,Mt l ,Mr (k,l)=(i,j) Theorem 1: If (2) holds, the optimal F can be chosen (12) −1 diagonal, with real and non-negative diagonal elements. Mt 2 under the sum power constraint i=0 fi =1. Proof: This is an application of the theorem presented in [12]. For M-PSK mapping, substitution of SERγ allows for Note that the authors of that paper did not consider different integration over γ. path loss specifically, but we observe that by building scaled −1 −1 Mt Mr 2 MtMr −1 R R 1 (ρifi λij ) correlation matrices such that r = ρi ri their theorem can i SER = 2 − 2 be made to hold here as well. In other words, the problem of π =0 =0 (ρifi λij ρkfk λkl ) i j ∈{0 −1} ∈{0 −1} optimal precoding for cooperative diversity is reduced to an k ,Mt l ,Mr (k,l)=(i,j) M optimal allocation of power between the t transmit antennas. (M−1)π More importantly, the implementation of this precoder is fully M 1 × dθ g δρ f 2λ compatible with distributed processing over each cell (rather 0 PSK i i ij 1+ 2 than multi-cell joint processing). sin (θ) We now proceed to find the optimal power allocation, first We note that a simple, closed-form expression of the exact by directly exploiting the SER expression, then, alternatively, SER is not obtainable and also that minimization of the SER 2 by resorting to a modified, simpler criterion. with respect to the power values fi (keeping θ constant), is a difficult task. Nonetheless, it is possible to optimize using the A. Optimal SER Precoder iterative algorithm developed in [12]. A minimization of the SER as it is given in (6) involves the Note that the SER depends on the SNR-related coefficient 2 2 →∞ pdf of the instantaneous received SNR γ. To find this function, δ = aσx/σv.Ifweletδ , with non-zero path-loss ρi and we begin by developing γ for the case of a diagonal precoder 1 2 2 This is the general case, if ρifi λi == ρkfk λk ,for(i, j) =(k, l), F 0 1 −1 j l = diag([f ,f, ... ,fMt ]). We assume a unit power the pdf needs to be modified. However, the case of repeated, equal factors Mt−1 2 constraint, such that i=0 fi =1, i.e., P =1in (8). can be derived from the general case, by the use of the limit operator.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. ∀ eigenvalues λij , (i, j), then it can be shown, using the same than another antenna that sees a higher level of correlation at arguments as in [14], that the optimal power allocation will the receiver. 2 approach the even distribution fi =1/Mt. From the results in Lemma 2, we notice that, unlike the In general, the optimal power allocation depends on both the exact SER, the power allocation after the maximum diversity receive correlation conditions, the path loss and the average principle is independent of the SNR. It turns out this criterion SNR. We now attempt to find the power allocation via another is good for low to moderate values of SNR and shows sub- optimality criterion. optimality at high SNR. B. Closed-Form Precoder for Equal Average SNR 2) Gaussian Approximation Based Precoder: In this sec- tion, we pursue another approach for the optimization of We now assume that slow power control is activated at the the power allocation. Since γ is sum of M M weighted, transmitters. Then the path loss coefficients are set equal (say t r independent random variables, we are motivated by the central ρ =1,i∈{0, 1,...,M − 1}). The correlation matrices i t limit theorem in proposing to approximate γ’s distribution by remain possibly different. In this case a closed-form precoder a truncated Gaussian, where the truncation follows from the can be formulated by generalizing the maximum diversity known positivity of γ. principle introduced in [12], for the case of M =2transmit t With this assumption, the symbol-error-rate in (6) is now antennas. (M−1)π 2 ∞ (γ−mγ ) 1) The Maximum Diversity Principle: ρ =1,i∈ M − gPSK When i C 2 2 − 2 γ SER ≈ √ e σγ e sin (θ) dγdθ , {0, 1,...,Mt − 1}, the SNR γ in (11) simplifies to σγ π 2π 0 0 Mt−1 Mr −1 (15) 2 | |2 where C, given as γ = δ f λij h . (13) i wji i=0 j=0 2 ∞ (γ −mγ ) − 1 1 2 2 √ σγ The SNR γ is a sum of MtMr uncorrelated diversity = e dγ , (16) 2 C 0 σγ 2π branches, each weighted by fi λij . In accordance with the proposed maximum diversity principle, we attempt to spread is the correction factor due to truncation. the sum energy of the signal evenly across all the diversity The moment generating function (mgf) of a real Gaussian- ∼N 2 branches. This is done by making the weights as equal as distributed random variable η (mη,ση) is defined as possible under a sum power constraint. The mean m of the ∞ − 2 tη √1 −(η mη) weights, under this constraint, is Mη(t)=E[e ]= exp 2 +tη dη , −∞ση 2π 2ση −1 −1 −1 Mt Mr Mt δ 2 δ 2 δ (17) m = fi λij = fi = , or in an alternative form MtMr Mt Mt i=0 j=0 i=0 2 2 t {R } ∈{ − } Mη(t) = exp mηt + ση . (18) where we use that Tr ri = Mr,i 0, 1,...,Mt 1 . 2 We propose that the weights be decided according to the We observe that the inner integral of (15) resembles (17), following minimum variance problem. with t = −g / sin2(θ). Using (18), the SER is rewritten as Problem 2: PSK (M−1)π −1 −1 M 2 Mt Mr 2 1 gPSK 1 2 gPSK 2 δ SER ≈ exp −m + σ dθ . − γ 2 γ 2 min fi λij π 0 sin (θ) 2 sin (θ) ≥0 ∈{0 1 −1} Mt−1 2=1 Mt fi ,i , ,...,Mt i=0 fi i=0 j=0 (19) Now, we propose to minimize the integrand 2 gPSK 1 2 gPSK This problem has a simple closed-form solution, stated in exp −m + σ 2 , for any value of θ. γ sin 2(θ) 2 γ sin (θ) Lemma 2 below and proved in [13]. 1 − 2 gPSK This problem reduces to minimizing mγ + 2 σγ sin2( ) . Lemma 2: Each power weight fi is found as θ 2 The mean mγ and variance σγ of γ, derived in [13], are β given as fi = ,i∈{0, 1,...,Mt − 1} , (14) Mr −1 2 λ −1 −1 j=0 ij Mt Mr 2 2 2 2 2 mγ = δMr , and σ = δ (f ) λ . where the constant β can be computed by using the sum power γ i ij −1 =0 =0 Mt f 2 i j constraint i=0 i =1, such that (20) −1 2 Mt β 1 Because mγ is a constant and sin (θ) is always non- ⇒ M −1 2 =1 = β = M −1 1 . negative, the approximate SER in (19) is minimum when the r t 2 i=0 j=0 λi i=0 Mr −1 2 j =0 λ variance σ is minimum, so we optimize j ij γ 2 min σγ , In words, this means that the power is distributed so that M −1 f ≥0,i∈{0,M −1} t f 2=1 a transmit antenna i for which the receive correlation is i t i=0 i 2 negligible, i.e. uncorrelated, will receive more transmit power with the variance σγ as given in (20).

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. The solution to this minimum variance problem is developed This special setting of unequal average SNR and no correla- in [13]. For i ∈{0, 1,...,M − 1},wehave tion can be seen as an equivalent to the optimization problem t presented by Sampath and Paulraj in [16]. There, the authors β 1 fi = −1 ,β= −1 1 . assumed transmit correlation, no receive correlation and equal Mr 2 Mt j=0 λi i=0 Mr −1 2 j =0 λ average SNR. j ij Interestingly, this closed-form precoder is identical to that 2) Modified Waterfilling Solution for Correlated Case: We obtained from the maximum diversity principle and presented now return to the most general case where each transmitter in Lemma 2. This gives insights into the justification of the use experiences a different path loss and receive correlation. of the maximum diversity principle, beyond simple intuition. To find a solution to the optimization problem in (21), we differentiate with respect to f 2 i ∈{0, 1,...,M − 1}, C. Precoder for Unequal Average SNR i t introduce a Lagrange multiplier µ with the power constraint H In the most general case, the different transmit antennas Tr{FF } =1, and find Mt equations on the form may see different average path loss. If these path losses −1 are substantially different and slow power control is not Mr kρiλij activated, the Gaussian approximation above looses credibility + µ =0 for i ∈{0, 1,...,Mt − 1} . 1+kρ f 2λ and another approach should be resorted to. We propose to j=0 i i ij optimize the PEP-based metric shown in [15] For each transmit antenna i ∈{0, 1,...,M − 1},we ad2 t I min R1/2P¯ F ∗F T ⊗I PR¯ 1/2 optimize with respect to the leading term in the sum. We max det MtMr+ 2 ( ) Mr , {F ∈CM ×B } t 4σv ≥ assume that the eigenvalues are ordered, so that λik λil , H under the constraint Tr{FF } =1. for all k ≤ l. The other Mr − 1 terms in the sum, we fix in a With uncorrelated transmitters, it still holds that the optimal constant ci. F can be chosen as diagonal, real and non-negative, hence we −1 Mr must solve a power allocation problem [15]. We introduce the kρiλij ad2 ci = 2 for i ∈{0, 1,...,Mt − 1} . min F substitution k = 4 2 . When using the knowledge of from 1+kρif λij σv j=1 i Theorem 1, the optimization is reduced to the less complex problem Now, the Mt independent equations can be rewritten as −1 −1 −1 Mt Mr Mt 2 2 kρiλi0 max 1+kρifi λij , fi =1, { 2≥0 ∈{0 1 −1}} 2 = −(µ + ci) for i ∈{0, 1,...,Mt − 1} . fi ,i , ,...,Mt =0 =0 =0 i j i 1+kρifi λi0 which is equivalent to Mt−1 2 −1 −1 −1 By imposing the power constraint =0 f =1,we Mt Mr Mt i i 2 2 attempt to find µ. This approach means solving an equation max log 1+kρifi λij , fi =1. {f 2≥0,i∈{0,1,...,M −1}} ≤ i t i=0 j=0 i=0 of degree Mt, analytically feasible for Mt 4. We get the (21) expression Interestingly, the maximization of the above sum of loga- M −1 M −1 rithms resembles the well-known capacity maximization prob- t t 2 − 1 1 lem, for which the solution is the classical water-filling. fi = + =1. µ + ci kρ1λi0 Unfortunately, the above objective function is different in the i=0 i=0 2 fact that each power variable fi appears in Mr termsofthe Of the possible values found for µ, use the one that minimizes sum, making it a totally different problem to solve in general, −1 −1 except in special cases. Mt Mt | 2| 1 1 1) Solution for Uncorrelated Case: fi = + , Assume as a special µ + c kρ λ case that there is no correlation on either side, neither trans- i=0 i=0 i i i0 mit nor receive. In this case, R = I for all i ∈ ri Mr and finally, we update the power values {0, 1,...,Mt − 1}. Then, the optimization problem in (21) is simplified to 2 1 1 ∈{ − } −1 −1 fi = + for i 0, 1,...,Mt 1 . Mt Mt µ + c kρ λ 2 2 i i i0 max Mr log 1+kρifi , fi =1. {f 2≥0,i∈{0,1,...,M −1}} i t i=0 i=0 The above described procedure is repeated until conver- We recognize this problem as one for which the solution is gence in a fixed-point is reached. No formal proof for the the classical water-filling, such that conditions of convergence is available yet, and from the theory −1 on fixed-point iterations [17], we know that non-convergence Mt 2 1 2 may occur. However, using a slightly modified approach, f = µ − ,i∈{0, 1,...,Mt − 1} , f =1. i kρ + i i i=0 detailed in [13], a fixed-point is still obtainable.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. V. R ESULTS

−1 We present simulation results for selected scenarios, assum- 10 ing both equal and unequal average received SNR. In the first case, we show the BER vs SNR for the PEP-based modified

−2 water-filling and the maximum diversity principle. For the case 10

of unequal average SNR, we show the modified water-filling BER results. In both cases, comparisons are made against the trivial

−3 equal power allocation to evaluate the gain of precoding, and 10 also against results using the minimum, exact SER precoding

Equal power allocation, unequal avg. SNR from [12]. PEP−based approach, unequal avg. SNR Minimum exact SER prec. from [12], unequal avg. SNR −4 Two single-antenna BSs transmit cooperatively to one MU 10 0 2 4 6 8 10 12 with Mr =4antennas. The Monte Carlo simulations transmit SNR [dB] 1.5 · 106 bits, over 1500 channel realizations. The receive R 1 R I Fig. 4. Comparison of BER-performance versus SNR for Mt =2, Mr =4, correlation matrices are r0 = Mt and r1 = Mt , where in the case where Rr0 = 1M and Rr1 = IM and for non-equal average 1M is an Mt × Mt matrix of all ones. This corresponds to t t t SNR, realized by setting the path losses as [ρ0,ρ1]=[0.75, 1]. The power the extreme situation of one BS seeing a fully correlated link, allocation based on the PEP-measure outperforms the equal power allocation while the other sees a fully decorrelated link. by close to 1 dB at BER 10−3. The results for the equal average SNR case are shown in REFERENCES Fig. 3. For unequal SNR values, we use [ρ0,ρ1]=[0.75, 1], [1] C. B. Peel, Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “An obtaining the results of Fig. 4. The results show that the Introduction to the Multi-User MIMO Downlink,” IEEE Commun. Mag., iterative algorithm minimizing the PEP gives the same per- Oct. 2004. formance as the maximum diversity-based algorithm, for the [2] S. Shamai and B. M. Zaidel, “Enhancing the cellular downlink capacity via co-processing at the transmitting end.” Vehicular Technology equal SNR case. In both figures, we observe that the equal −3 Conference (VTC), Spring 2001. power allocation is outperformed by almost 1 dB at BER 10 . [3] H. Zhang, H. Dai, and Q. Zhou, “Base station cooperation for mul- Also, the Gaussian and PEP-based approaches obtain the same tiuser MIMO: Joint transmission and BS selection.” Conference on information sciences and systems, Mar. 2004. results as the iterative, minimum, exact SER precoding from [4] B. L. Ng, J. S. Evans, S. V. Hanly, and D. Aktas, “Transmit beamforming [12]. with cooperative base stations.” IEEE Int. Symp. on Inf. Theory (ISIT), Sept. 2005. VI. CONCLUSIONS [5] T. M. Cover and A. A. E. Gamal, “Capacity theorems for the relay We address the problem of distributed space-time coding channel,” IEEE Trans. Inform. Theory, vol. 25, no. 5, Sept. 1979. [6] A. Sendonaris, E. Erkip, and B. Aazhang, “User Cooperation Diversity– for the cooperative downlink cellular network. The channel Part I: System Description, Part II: Implementation Aspects and Perfor- correlation structure is highly non-Kronecker and can be mance Analysis,” IEEE Trans. Commun. compensated by a distributed power allocation. [7] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” Several approaches are presented to find the optimal pre- IEEE Trans. Inform. Theory, vol. 49, no. 10, Oct. 2003. coder, including a closed form precoder based on the max- [8] S. Barbarossa, L. Pescosolido, D. Ludovici, L. Barbetta, and G. Scutari, imum diversity principle and iterative solutions based on “Cooperative wireless networks based on distributed space-time coding.” Proc. of Int. workshop on wireless ad-hoc netw. (IWWAN), June 2004. modified waterfilling. [9] X. Li, “Space-time coded multi-transmission among distributed transmit- ters without perfect synchronization.” IEEE Signal Processing Letters, vol. 11, no. 12, 2004. −1 10 [10] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. on Information Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [11] E. Bonek, H. Özcelik, M. Herdin, W. Weichselberger, and J. Wallace,

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings.