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 .