Impact of Spatial Correlation and Precoding Design in OSTBC MIMO Systems IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS Volume 9, Issue 11, Pages 3578-3589, November 2010. Copyright c 2010 IEEE. Reprinted from Trans. on Wireless Communications. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the KTH Royal Institute of Technology's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. EMIL BJORNSON,¨ EDUARD JORSWIECK, AND BJORN¨ OTTERSTEN Stockholm 2010 KTH Royal Institute of Technology ACCESS Linnaeus Center Signal Processing Lab DOI: 10.1109/TWC.2010.100110.091176 KTH Report: IR-EE-SB 2010:010 3578 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 11, NOVEMBER 2010 Impact of Spatial Correlation and Precoding Design in OSTBC MIMO Systems Emil Bj¨ornson, Student Member, IEEE, Eduard Jorswieck, Senior Member, IEEE, and Bj¨orn Ottersten, Fellow, IEEE Abstract—The impact of transmission design and spatial spatially limited which leads to a correlated channel [3]–[5], correlation on the symbol error rate (SER) is analyzed for also known as spatial correlation. multi-antenna communication links. The receiver has perfect channel state information (CSI), while the transmitter has either The impact of CSI and spatial correlation on the ergodic ca- statistical or no CSI. The transmission is based on orthogonal pacity has received much attention. For simplicity, the receiver space-time block codes (OSTBCs) and linear precoding. The is usually assumed to have perfect CSI [6], while various types precoding strategy that minimizes the worst-case SER is derived of CSI has been considered at the transmitter [7]–[11]. The for the case when the transmitter has no CSI. Based on this impact of spatial correlation on the capacity was evaluated strategy, the intuitive result that spatial correlation degrades the SER performance is proved mathematically. numerically in [9] (among others), but the relationship was In the case when the transmitter knows the channel statistics, first derived analytically in [10]. It was shown that spatial the correlation matrix is assumed to be jointly-correlated (a correlation decreases the capacity when the transmitter has generalization of the Kronecker model). The eigenvectors of the no CSI or perfect CSI, which is intuitive since correlated SER-optimal precoding matrix are shown to originate from the channels have fewer degrees of freedom and thus less suitable correlation matrix and the remaining power allocation is a convex problem. Equal power allocation is SER-optimal at high SNR. for spatial multiplexing. When the transmitter has statistical Beamforming is SER-optimal at low SNR, or for increasing CSI, this negative effect is however countered by the advantage constellation sizes, and its optimality range is characterized. of having smaller channel variations; in highly correlated A heuristic low-complexity power allocation is proposed and channels, the channel direction is in fact given by the statistics. evaluated numerically. Finally, it is proved analytically that Interestingly, it was proved in [10] that correlation among the receive-side correlation always degrades the SER. Transmit-side correlation will however improve the SER at low to medium transmit antennas improves the capacity in this case. SNR, while its impact is negligible at high SNR. While most previous work considered the ergodic capacity Index Terms—Beamforming, channel state information, requiring Gaussian constellations, this paper considers the MIMO systems, orthogonal space-time block codes, power al- symbol error rate (SER) with practical symbol constellations. location, spatial correlation, symbol error rate. Prior work includes [12] and [13] that made numerical obser- vations on the impact of spatial correlation on error rates. I. INTRODUCTION Herein, we derive an analytical solution to the impact of correlation by analyzing a general class of SER-like functions. N wireless communication, the use of antenna arrays at the This class includes the exact SER for Rayleigh fading channels I transmitter and receiver can greatly improve the spectral with orthogonal space-time block codes (OSTBCs), linear efficiency and system performance. Under the ideal conditions precoding [14]–[18], and uncoded PAM, PSK, or QAM. We of uncorrelated antennas and perfect channel state information use the jointly-correlated model, proposed in [19] and [20], (CSI), it was shown in [1] and [2] that the ergodic capacity to analyze transmission design and the impact of spatial improves linearly as the number of antennas increases at both correlation under more general conditions than the commonly sides. In practice, this fundamental gain is difficult to obtain. used Kronecker model [12], [13], [21]. Our main contributions Firstly, the channel fading makes it costly for the transmitter to are: keep track on the current CSI. Secondly, the scattering is often ∙ Optimal transmission strategies: When the transmitter Manuscript received August 5, 2009; revised February 5, 2010 and June has no CSI, it can protect itself against the unknown 7, 2010; accepted September 17, 2010. The associate editor coordinating the review of this paper and approving it for publication was H.-C. Yang. Rayleigh fading channel by using OSTBCs and equal E. Bj¨ornson and B. Ottersten are with the Signal Processing Laboratory at power allocation in all spatial directions. This precoding the ACCESS Linnaeus Center, KTH Royal Institute of Technology, SE-100 strategy minimizes the worst-case SER (Theorem 1). 44 Stockholm, Sweden (e-mail: {emil.bjornson, bjorn.ottersten}@ee.kth.se). B. Ottersten is also with securityandtrust.lu, University of Luxembourg. When the transmitter has statistical CSI, the eigenvec- E. Jorswieck is with the Communications Laboratory, Dresden University tor structure of the SER minimizing precoder is de- of Technology, D-01062 Dresden, Germany (e-mail: eduard.jorswieck@tu- rived for jointly-correlated systems (Theorem 2). This dresden.de). The research leading to these results has received funding from the Euro- structure reduces the transmission design to a convex pean Research Council under the European Community’s Seventh Framework power allocation problem that can be solved numerically Programme (FP7/2007-2013) / ERC Grant Agreement No. 228044. Parts of or heuristically with low complexity (Strategy 1). At this work were presented at the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Taipei, Taiwan, April 2009. high SNR, the power is allocated equally among the Digital Object Identifier 10.1109/TWC.2010.100110.091176 available eigendirections. Single-stream beamforming in 1536-1276/10$25.00 ⃝c 2010 IEEE BJORNSON¨ et al.: IMPACT OF SPATIAL CORRELATION AND PRECODING DESIGN IN OSTBC MIMO SYSTEMS 3579 N the dominant eigendirection is SER-optimal at low and Precoder Channel Joint medium SNR and this is also the case asymptotically as s OSTBC sˆ WH Detection the constellation size grows (Section V). The SNR range C(s) Y where beamforming is SER-optimal is characterized as a (a) Linear precoded OSTBC MIMO system. function of the constellation size (Theorem 5). nk ∙ Impact of spatial correlation: When the transmitter Equivalent Channel Separate sk HW sˆk has no CSI, it is proven that spatial correlation always y Detection degrades the SER in jointly-correlated Rayleigh fading k systems with OSTBCs (Theorem 3). In the case with (b) Equivalent parallel single-input single-output (SISO) systems, for = 1,..., statistical CSI at the transmitter, correlation between . eigendirections at the receiver also degrades the perfor- Fig. 1. Block model of the original MIMO communication system and its mance. Transmit-side correlation will however improve equivalent parallel structure after receive processing. the performance at low and medium SNR (Theorem 4), while the impact at high SNR is negligible (Section V). The conclusion is that CSI and spatial correlation impacts will be considered). These symbols are coded in an OSTBC the SER in a jointly-correlated system with OSTBCs in a matrix C(s) ∈ ℂ× that fulfills the orthogonality property similar (but non-identical) manner as the ergodic capacity in C(s)C(s) = ∥s∥2I and has the spatial coding block length Kronecker-structured Rayleigh fading systems [10]; statistical . The linear precoder W ∈ ℂ × is used to project CSI at the transmitter can improve the performance by proper the signal into advantageous spatial directions by using the transmission design that adapts to the correlation and turns available transmit-side CSI [16]. Its maximal rank is denoted transmit-side correlation into an advantage. by ≜ min( ,) and the design of W will be considered Notations: We use boldface (lower case) for column vec- in Section III for different CSI. By introducing the power tors, x, and (upper case) for matrices, X. With X , X , constraint ∥W∥2 =1, we make sure that the average transmit ∗ and X we denote the transpose, the conjugate transpose, power allocated per symbol is {∥WC(s)∥2}/ = . and the conjugate of X, respectively. The Kronecker and Observe that