also studied, and the scheme is shown to be robust to channel estimate imperfections, as well. Finally, we focus on antenna selection for

Transcription

1 ABSTRACT SUDARSHAN, PALLAV. Antenna Selection and Space-Time Spreading Methods for Multiple-Antenna Systems. (Under the direction of Prof. Brian L. Hughes.) The use of multiple antennas at the transmitter and receiver can significantly improve the performance of a wireless communication system. In recent years, there has been a lot of interest in deriving efficient receiver architectures and designing signalling and coding schemes that maximize the performance gains of a multi-antenna system. In this dissertation, we focus on two such issues: space-time spreading methods at the transmitter, and antenna selection techniques at the receiver. For a synchronous code-division multiple-access (CDMA) system that employs multiple transmit antennas, we characterize the asymptotic spectral efficiency in terms of the number of users, processing gain, signal to noise ratio (SNR), array size, etc. Using this formula, we design the linear space-time spreading methods that maximize the spectral efficiency. The strategy for optimal spreading sequence allocation across antennas, and across users is also addressed. We show that the system capacity per chip is maximized when each user employs all the spreading sequences allocated to it on each transmit antenna. We then study reduced complexity receiver designs for multiple-antenna systems. A RF pre-processing architecture, that processes the received signal at carrier frequency, followed by selection, and down-conversion is considered. Recent results show that this architecture can outperform conventional antenna selection with the same number of RF chains. We derive the optimum RF pre-processing that is based only on the large-scale parameters of the channel. For a correlated channel, we show that RF pre-processing using channel statistics gives good results, and that instantaneous channel knowledge is not required for pre-processing. A beam pattern based geometric intuition is also developed to justify the performance gains. To accommodate the practical design constraints imposed by current variable phase-shifter technology, a sub-optimal phase approximation is also introduced. We show that this scheme is extremely robust to RF imperfections, such as phase and quantization errors. The impact of imperfect channel estimates on the performance of RF pre-processing is

2 also studied, and the scheme is shown to be robust to channel estimate imperfections, as well. Finally, we focus on antenna selection for multi-access channels. For a multi-user system, we derive the statistics-based selection criteria that maximizes tight bounds on ergodic capacity. Two different receiver architectures are considered, and the performance gain compared to sub-optimal selection is quantified.

3 Antenna Selection and Space-Time Spreading Methods for Multiple-Antenna Systems by Pallav Sudarshan A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Electrical Engineering Raleigh, NC 2004 Approved By: Dr. Hamid Krim Dr. Jack Silverstein Dr. Brian L. Hughes Chair of Advisory Committee Dr. Alexandra Duel-Hallen

4 To my parents ii

5 iii Biography Pallav Sudarshan received the B.E. degree in Electrical Engineering from Indian Institute of Technology - Delhi, India in He worked with Ericsson, India from August 1999 to August In August 2000, he started his PhD degree study at North Carolina State University under the guidance of Prof. Brian Hughes. He defended his dissertation in December His research interests include Signal Processing for Multiple Antennas, Code Division Multiple Access, Information Theory, and Coding Theory.

6 iv Acknowledgements I would like to express my sincere gratitude to my advisor Prof. Brian Hughes, for giving me the opportunity to work with him. I have immensely benefitted from his intuitive approach to solving problems. His insightful thinking and invaluable guidance has made my graduate study experience a truly rewarding one. I thank my committee members Prof. Alexandra Duel-Hallen, Dr. Hamid Krim, and Prof. Jack Silverstein for their useful comments and suggestions. I would also like to thank Dr. Huaiyu Dai for sharing and discussing many exciting ideas with me. I enjoyed the work done in collaboration with him very much. I am thankful to Dr. Jinyun Zhang for providing me with the opportunity to work at Mitsubishi Electric Research Labs, where the final part of this work was done. I am immensely grateful to Dr. Neelesh Mehta for always motivating and encouraging me. I have learnt a lot from his unmitigated enthusiasm towards research. He has been a great guide and mentor to me ever since. I also extend my gratitude to Dr. Andreas Molisch for his valuable suggestions and rich discussions during my work at Mitsubishi. I would like to thank my colleagues Ajith Kamath and Sandeep Krishnamurthy for their unflinching willingness to help. The technical and philosophical discussions with them are one of the most memorable moments of my graduate experience. Thanks are also due to Keyoor Gosalia, Chris Mary James, Xinying Yu and all other friends and colleagues at NCSU, for the nice time I had during my PhD. A special thanks to my close friends Aashish Mehra and Meeta Sharma, who were always there for me. I will forever be indebted to my parents and sister for their unconditional love and support for me. I thank them for always believing in me, encouraging me, and for driving me to bring out the best in me. I truly owe it to them. Finally, I am grateful to my late grandfather, who laid the foundation for most of what I am.

7 v Contents List of Figures List of Tables viii x 1 Introduction Multiple-Antenna Systems Gains in a Multiple-Antenna System Space-Time Codes Multiple-Access Methods Code-Division Multiple-Access Multiuser Detection Capacity of Multiple-Access Systems Antenna Selection Dissertation Overview Optimal Space-Time Spreading Codes for CDMA Systems Introduction Transmission Model A General Model of Linear Modulation MISO Channel Model Spectral Efficiency Random Matrices in the Large System Limit Empirical Eigenvalue Distribution Stieltjes Transform R-Transform Asymptotic Spectral Efficiency for Optimal Detector Parametric Formula for Asymptotic Spectral Efficiency Performance in the Wideband Regime Optimal Space-Time Modulation Open-Loop Upper Bound

8 vi Properties of Optimal Open Loop Spectral Efficiency Conclusions Channel Statistics-Based RF Pre-Processing with Antenna Selection Introduction System Model for Spatial Diversity and Spatial Multiplexing Spatial Diversity Spatial Multiplexing Channel Model Receiver Architectures for Spatial Diversity Full Complexity (FC) Pure Antenna Selection (Ant-Sel) FFT Pre-Processing Followed by Selection Instantaneous Time-Variant Pre-Processing Spatial Diversity: Optimal Time-Invariant Pre-processing Optimal Time-Invariant (TI) L N r Pre-Processing Time-Invariant Phase-Shift (TI-Ph) Pre-Processing Time-Invariant N r N r Pre-Processing Followed by Selection (TI-S) Receiver Architectures for Spatial Multiplexing Full Complexity (FC) Pure Antenna Selection FFT Pre-Processing Followed by Selection Time-Variant Pre-Processing Spatial Multiplexing: Optimal Time-Invariant Pre-Processing Optimal L N r Time-Invariant (TI) Pre-Processing N r N r Time-Invariant Pre-Processing Followed by Selection (TI-S) Performance Analysis Single Cluster Channel Multi-Cluster Channel Robustness Analysis Impact of Phase Quantization and Phase Error Impact of Insertion Loss Impact of Channel Estimation Error Conclusions Statistics-Based Antenna Selection for Multi-Access MIMO Systems Introduction System and Channel Model

9 vii 4.3 Antenna Selection for MMSE-ML Receiver Antenna Selection for ML Receiver Simulation Results MMSE Detector ML Detector Conclusions Conclusions 102 Bibliography 105

10 viii List of Figures 1.1 Multiple-antenna system in a rich scattering environment Spreading in CDMA system Spectral efficiency of different space-time codes for t = 1, 2, 4 and E b /N 0 = 10 db Spectral efficiency as a function of number of spreading sequences per user Block diagram for diversity transmission with RF-baseband design Block diagram for spatial multiplexing with RF-baseband design Antenna arrangement for MEA with N r receive antennas Beam pattern for FFT pre-processing for d/λ = 0.5, and N r = 4 as a function of the azimuth angle Beam pattern for M TI for d/λ = 0.5, N r = 4, and θ r = 45 as a function of the azimuth angle Beam pattern for M TI for d/λ = 0.5, N r = 4, and θ r = 60 as a function of the azimuth angle CDF of SNR for diversity system with N t = 4, N r = 4, L = 1, σ r = 6, d = 0.5λ, T = I Nt, and ρ = 10 db Effect of spatial correlation σ r on SNR for diversity system with N t = 4, N r = 4, L = 1, θ r = 60, d = 0.5λ, T = I Nt, and ρ = 10 db CDF of capacity for spatial multiplexing system with N t = 2, N r = 4, θ r = 45, σ r = 6, d = 0.5λ, T = I Nt, and ρ = 10 db Effect of spatial correlation σ r on CDF of capacity for spatial multiplexing system with N t = 2, N r = 4, L = 1, θ r = 45, T = I Nt, and ρ = 10 db Beam pattern for 2-cluster TI pre-processing for d/λ = 0.5, N r = 4, θ r1 = 45, and θ r2 = 75 as a function of the azimuth angle CDF of SNR for diversity system N t = 4, N r = 4, d = 0.5λ, T = I Nt, ρ = 10 db, for two clusters with θ r1 = 45, θ r2 = 75, and σ r = 6. 81

11 ix 3.13 CDF of capacity for spatial multiplexing system N t = 2, N r = 4, d = 0.5λ, T = I Nt, ρ = 10 db, for two clusters with θ r1 = 45, θ r2 = 75, and σ r = Impact of phase quantization and calibration error: Capacity CDF for ρ = 6 db, θ = 45, and σ r = Impact of insertion loss: Capacity CDF for ρ = 6 db, θ = 45 and σ r = Impact of channel estimation error: Capacity CDF for ρ = 6 db, θ = 45 and σ r = CDF of capacity for MMSE-ML receiver CDF of capacity for ML receiver

12 x List of Tables 3.1 List of abbreviations for different receiver architectures Average SNR (in db) for different receiver architectures Ergodic capacity (in bits/s/hz) for different receivers Ergodic capacity with phase quantization and calibration error for θ = 45 and σ r = Ergodic capacity with insertion loss for θ = 45 and σ r = Ergodic capacity with imperfect CSI for θ = 45 and σ r = Ergodic capacity (in bits/s/hz) for MMSE-ML receiver Ergodic capacity (in bits/s/hz) for ML receiver

13 1 Chapter 1 Introduction The use of wireless links for communication of information began with the invention of telegraph and radio in the late 19 th century. While significant advances have been made in the field of wireless communications in the last 100 years, it is in the last decade or so that the interest in wireless system design has really exploded. There is an exponentially increasing demand for broadband wireless access for applications such as multimedia information transfer on the move and high-speed wireless internet connectivity. These applications require reliable communication at very high data rates, which increases the bandwidth requirements of the system. However, wireless bandwidth is a scarce and expensive resource. Thus, bandwidth and power efficient schemes that can meet the ever increasing demand of high data rates need to be considered. Although communication through the wireless channel is of much practical interest, the wireless channel is plagued by many impairments. The signal strength in ideal free-space wireless transmission decreases with distance according to the power law function [2]. Furthermore, in a typical wireless environment, multiple scatterers are present, which reflect, refract and scatter the signal. The transmitted signal travels

14 2 by multiple paths to the receiver, each with possibly different path lengths, and hence different path delays, phase, amplitudes, and angles of arrival. The receiver thus sees a time-dispersed version of the transmitted signal, from which it can be difficult to recover the original signal. Also, as the signals from different paths superimpose, they might add constructively or destructively. Thus the amplitude of the received signal depends on the geometry of the scatterers. This phenomenon is called multipath fading, and such channels are referred to as fading channels. Since the geometry of the scatterers changes in a random fashion, the fading channel gain is typically modelled as a random parameter. Due to these effects, multipath fading is one of the most prominent effects that limits the performance of a wireless channel. One of the common methods to mitigate the ill-effects of fading is to use diversity. Diversity techniques reduce the fading-related fluctuations by repeated transmission of the same information. Conventional diversity methods include time diversity, where same information is transmitted repeatedly in time, or frequency diversity, where same information is transmitted on multiple frequencies. Although these diversity methods improve the performance of fading channels, they require additional resources, either in form of more transmission time for time diversity, or more bandwidth for frequency diversity. An alternate form of diversity that does not require additional bandwidth and time is space diversity. In this method, multiple antennas are employed at the transmitter and/or receiver. In a rich scattering environment (i.e., the number of scatterers is large), the signal transmitted from each antenna takes a different path to the receive antenna. Then the multipath fading between each pair of transmit and receive antennas will be independent if the antennas are placed sufficiently apart. Each antenna then provides an extra signalling dimension, which can be used to provide diversity.

15 3 These diversity methods improve the reliability of a wireless link and can help achieve higher data rates. The reliability of a communication system is often measured in terms of bit-error-rate (BER), defined as the probability that a transmitted bit of information will be received in error. A fundamental limit associated with the practically achievable data rates in a communication system is channel capacity. Shannon, in his seminal work in 1948 [1], proved that reliable, error-free communication through a link is always possible as long as the rate of information transfer between the transmitter and the receiver is less than a certain threshold, known as channel capacity. An efficient communication system design maximizes the rate, while keeping the BER low. For fading channels, since the fading gain is a random parameter, the maximum throughput will also be a random variable. Therefore, the performance of fading channels is measured in terms of ergodic capacity, defined as the maximum instantaneous data rate averaged over all possible channel realizations. Ergodic capacity indicates, on average, how many bits of information can be transmitted per channel use. In the next section, we give an overview of multiple-antenna systems, and their potential benefits in reducing the BER and increasing the capacity of a wireless system. 1.1 Multiple-Antenna Systems Figure 1.1 shows a multiple-antenna system in a rich scattering wireless communication environment. The transmitter has N t antennas and the receiver has N r antennas. This system is also referred to as multiple-input multiple-output (MIMO) system. Due to the presence of many scatterers, this channel exhibits multipath fad-

16 4 Tx Rx 1 1 Nt Nr Figure 1.1: Multiple-antenna system in a rich scattering environment ing. If the delay between the signals from different paths between a pair of transmit and receive antennas is small compared to the inverse bandwidth of the signal, then it results in frequency non-selective fading, also known as frequency-flat fading. This occurs commonly in indoor wireless channels, where all the scatterers are located close to each other. Consider a frequency flat fading wireless channel. Let h ij denote the overall complex fading coefficient between the j th transmit antenna and i th receive antenna. Then the MIMO channel for this system is typically represented by arranging the coefficients {h ij } in a N r N t matrix, H, called the channel matrix. The N r 1 vector of

17 5 received symbols, y can be written as y = ρ N t Hx + n, (1.1) where x denotes the vector of transmitted symbols and n denotes the receiver noise. In a communication system, the transmitted signal is corrupted by additive thermal noise, generated from different sources like electronic devices, atmospheric effects, etc. The combined effect of all these sources is often modelled as additive white Gaussian noise (AWGN), i.e., the power spectral density of the noise is assumed to be flat. In (1.1), ρ denotes the signal-to-noise ratio (SNR) of the system. SNR is defined as the ratio of total power and noise spectral density. MIMO systems have the potential to achieve enormous capacity gains, which are not possible using single antenna systems. Foschini and Gans [4] quantified the advantages of using multiple antennas. They showed that for uncorrelated fading, i.e., when the channel between each transmit and receive antenna pair fades independently, the capacity increases linearly with min(n r, N t ) for a fixed SNR. This shows the potential of achieving extremely high data rates by using multiple antennas at both ends of the communication system. Motivated by these benefits, multiple-antenna systems have attracted much interest, both in the research community and in the industry. Several emerging communication standards are based on MIMO architecture. For example, n wireless local area network (LAN) standard uses MIMO technology to increase capacity and reliability of indoor wireless channels. As another example, some third generation cellular standards, such as high speed downlink packet access (HSDPA) require the basestation and the handset to support multiple antennas, in order to meet their high-data-rate requirements. In the next section, we will discuss some of the gains achieved in a MIMO system.

18 Gains in a Multiple-Antenna System 1. Multiplexing gain: The multiplexing gain [6] of a MIMO system is defined as C e m = lim ρ log 2 (ρ), where C e is the ergodic capacity of a multi-antenna system [4, 5]. Multiplexing gain gives an idea of the number of linearly independent signalling dimensions that the channel provides. The rank of the channel matrix indicates the spatial multiplexing gain. 2. Diversity gain: Diversity gain, d, [6] is defined as the number of degrees of freedom available to each data stream. Mathematically, log(ber) d = lim, ρ log(ρ) i.e. the diversity gain is the negative of the exponent of the SNR in BER expression. Diversity gain indicates how many independently fading copies of the same signal are available to the receiver. 3. Array gain: Also known as the beamforming gain, array gain refers to the gain in the SNR achieved by using multiple antennas. For example, for a system with one transmit antenna and two receive antennas, the beamforming gain is given by B = h h 2 2, where h 1 and h 2 are the fading coefficients between the transmitter and the receive antennas Space-Time Codes The maximum rate achieved in a MIMO system depends on how the data is distributed over space and time. There has been extensive work on finding space-

19 7 time codes that maximize the data rate of a MIMO system. The broad categories of space-time codes include [7, 8] space-time trellis codes (STTC), space-time block codes (STBC), space-time turbo codes, quasi-orthogonal codes, layered designs, etc. We will look at space-time block codes in more detail. STBC distribute a block of input data symbols over space and time. They are represented in terms of a matrix, whose rows represent time and the columns represent space. Depending upon their design, they provide diversity gain and/or multiplexing gain. The rate of block code is defined as the number of information symbols transmitted per channel use [33]. Next, we describe the structure of a few common STBC. Alamouti Code One of the first STBC was proposed by Alamouti [44], for the case of two transmit antennas. The code matrix is given by X = x 1 x 2 x 2 x 1 where x 1 and x 2 are independent data symbols and () denotes the complex conjugate., The rate of this code is 1, and the diversity order is 2. The Alamouti code has an elegant property that it decouples the MIMO channel into independent scalar channels. This simplifies the receiver structure and makes the Alamouti code very appealing for practical systems. While the Alamouti code is defined for 2 transmit antennas only, codes of higher dimensions having this property were explored in [33]. However, unlike the Alamouti code, these codes have rate less than one.

20 8 BLAST In Bell Labs Layered Space-Time (BLAST) transmission [22], independent data streams are transmitted from each antenna. The code matrix (vector) is given by X = x 1 x 2. x Nt. This scheme has rate N t and diversity order 1. Unlike the Alamouti code, the optimal receiver for BLAST is very complex because different data streams interfere with each other. 1.2 Multiple-Access Methods In a practical system, multiple users communicate simultaneously with a common node, for example, a basestation in the case of cellular networks, or an access point in the case of wireless LANs. This calls for a mechanism to multiplex the information from multiple users, so that the inter-user interference is minimized. Multiplexing can be achieved using time-division multiple-access (TDMA), frequency-division multipleaccess (FDMA) or code-division multiple-access (CDMA), among other methods. In TDMA [2], each user communicates with the common basestation only during the time slot assigned to it, so that it does not interfere with other users. Similarly, in FDMA [2], each user is assigned a frequency slot for communication. In CDMA [23], each user spreads its information sequence over the entire time duration and the entire bandwidth using a unique pseudo-random spreading sequence. The spreading code sequences, known at the receiver, are used to separate the users. In the next section, we discuss the benefits of using CDMA in more detail.

21 9 (a) Data Sequence (b) Chipping Sequence (c) Modulated Data Figure 1.2: Spreading in CDMA system Code-Division Multiple-Access CDMA systems use spreading sequences to spread the information symbols of each user. The spreading sequences can be designed using direct sequence (DS), frequency hopping, or time hopping methods. DS-CDMA involves modulation of the information signal by a digital, discrete-time, discrete-valued spreading sequence, also known as the chipping sequence. The individual chips are ±1, as shown in Fig The rate of chipping sequence is much higher than the symbol rate. This results in spreading of the symbol over a wider bandwidth. Processing Gain The ratio of the transmitted signal bandwidth and the information bandwidth is called the processing gain. For a DS-CDMA system, the processing gain N is equal to the number of chips per symbol. For the example in Fig. 1.2, the processing gain is 16.

22 10 Advantages of CDMA CDMA offers some distinct advantages over other multiple access methods. 1. Multipath: As described earlier, in a wireless channel, the transmitted signal travels through multiple paths, before reaching the receiver. This leads to time dispersion of the signal if each path has a different delay. CDMA can combat multipath interference if the chipping sequences are designed so that time-shifted versions of the sequences have low (or zero) correlation. Rake receivers [24] exploit this property of DS-CDMA very effectively to combat multipath interference. In fact, in a rich scattering environment, use of rake receivers provides a diversity gain to the system. 2. Anti-Jamming: Any narrowband interference that is added to a CDMA signal has a reduced effect after despreading. 3. Privacy: In order to recover the transmitted signal, the receiver needs to know the spreading sequence used. 4. Frequency-selective fading and deep fades: Spread-spectrum methods provide protection against frequency-selective fading. Each symbol occupies the entire bandwidth. Even if the channel exhibits fades at certain frequencies, the signal after decorrelation can be recovered faithfully. FDMA does not provide this kind of protection, since each user is allotted a particular frequency band, and if that frequency band is fading, the entire information of the user is lost. Similarly, CDMA provides protection against time-varying fading, where the channel might be undergoing deep fades.

23 Multiuser Detection In practice, spreading sequences do not have ideal autocorrelation and crosscorrelation properties. After despreading, the interference from other users will not completely cancel out. A CDMA system is thus interference limited, i.e., adding an additional user deteriorates the performance of all the users. Various multiuser detectors have been proposed in the literature [9], that exploit the structure of spreading sequences to suppress the interference from other users. The simplest among them is matched filter receiver. It decorrelates the received signal of the user of interest and treats interference from other users as white noise. This receiver acts like a single-user receiver and does not use the knowledge of the spreading sequences of other users. A decorrelating receiver completely cancels the interference caused by other users. This linear receiver projects the desired users received signal in the null space of the space spanned by interfering spreading sequences. Although the resulting signal is free of interference, the decorrelating receiver can lead to noise enhancement. The performance of this receiver degrades appreciably in the low SNR regime. The minimum mean square error (MMSE) receiver is a linear receiver that uses the information about the received SNRs of all the users. For additive white Gaussian noise, MMSE receiver is the optimal receiver in the mean-square-error sense. MMSE receiver also maximizes the signal-to-interference ratio of multiuser system. All the multiuser detectors mentioned above are linear. The optimal detector is non-linear and yields minimum achievable probability of error in a CDMA system. It uses the complete knowledge of the spreading sequences and amplitudes of all the users and jointly decodes the information of all the users. The complexity of this receiver increases exponentially with the number of users.

24 Capacity of Multiple-Access Systems In the multiuser case, the capacity is defined by a region. For the purpose of illustration, consider a system with two users, communicating with a common receiver in a non-fading AWGN channel. Let R 1 and R 2 be the rates of user 1 and user 2, respectively. Then the achievable capacity region of this multiple-access channel is the closure of the convex hull of the set of points (R 1, R 2 ) satisfying [3]: ( R 1 log 1 + P ) 1, (1.2) N ( 0 R 2 log 1 + P ) 2, (1.3) N ( 0 R 1 + R 2 log 1 + P ) 1 + P 2, (1.4) N 0 where P 1 and P 2 is the power of user 1 and user 2, respectively, and N 0 is the noise power spectral density. 1.3 Antenna Selection In Section 1.1.1, we saw the various gains of using multiple antennas at the transmitter and the receiver. However, the widespread deployment of such systems is hampered by the substantial increase in the complexity associated with MIMO systems. The use of multiple antennas significantly increases the signal processing complexity for transmission, and more so for decoding. Furthermore, each additional antenna element at the transmitter (receiver) requires a separate modulator (demodulator) chain, consisting of power amplifier (low-noise amplifier), up-convertor (down-convertor) and digital-to-analog (analog-to-digital) convertor. These elements add to the cost of the transreceiver.

25 13 Often, the cost of an additional RF chain is much higher than that of an antenna element. This has prompted a lot of interest in the field of antenna selection, where a subset of available antennas are selected for baseband processing and up-conversion (down-conversion). Selecting a subset of antennas reduces the number of RF chains required, as well as simplifies the baseband signal processing, thereby considerably reducing the complexity of a MIMO system. This comes at the expense of performance loss, which under most conditions is usually small [46]. In fact, the diversity order of a judiciously selected antenna subset is same as that of a full complexity system using all the antennas [53]. The antenna selection algorithms proposed in the literature optimize different performance metrics, such as, information theoretic capacity [53 56], and probability of error [51, 52, 57]. Although antenna selection reduces the digital signal processing required by the MIMO system, finding the optimal set of antennas itself can be computationally intensive. This is especially true if the performance metric to maximize is capacity. Low-complexity, near-optimal algorithms to compute the antenna subset have also been proposed in the literature [53, 58]. The antenna selection methods are also classified depending upon whether they use the exact channel knowledge (ECK) [53 55, 57], or statistical channel knowledge (SCK) [50,56,59,87]. While selection based on ECK is more accurate, it requires faster switching in the RF domain, resulting in higher switching losses. SCK-based selection uses the spatial correlation information of the channel to simplify the algorithms for antenna selection.

26 Dissertation Overview In this dissertation, we derive the optimal space-time spreading methods that maximize the spectral efficiency of a multi-antenna CDMA system. We then design efficient, low-complexity receivers for MIMO systems. A pre-processing architecture is introduced, which reduces the number of RF chains required, while maintaining most of the benefits of multiple-antenna systems. Finally we consider antenna selection algorithms for multi-access MIMO systems, and derive the statistics-based optimal selection criteria. In Chapter 2, we determine the fundamental information theoretic performance limits for a MIMO CDMA system. The performance of a CDMA system depends on the particular choice of spreading sequences. To remove this dependence we assume a random spreading model: the chips are randomly chosen to be ±1. Furthermore, we perform an asymptotic analysis, i.e., we let the number of chips, and the number of users in the system approach infinity. For such a system, we characterize the capacity using some recent results from random matrix theory. Such an analysis makes some of the properties of multi-antenna systems very transparent. Using this characterization, we then optimize the spectral efficiency over the class of all linear space-time spreading methods. Specifically, we design the codes for distributing the data over antennas and spreading sequences, that maximize the spectral efficiency. In the process, we also derive some interesting properties of the optimal space-time codes. Next, we focus our attention on the practical aspects of MIMO systems. As described in Section 1.3, employing multiple antennas results in significant increase in complexity. Antenna selection was discussed as a means to reduce the complexity of a MIMO system. Although antenna selection maintains the diversity order of a multi-antenna system, a price is paid in terms of reduced beamforming gain. Re-

27 15 cent approaches [60, 61], involving the use of a pre-processing matrix at the carrier frequency before selection, have shown that most of the beamforming gain can be maintained, in spite of using reduced number of RF chains. These solutions propose the use of either a fixed, channel-independent pre-processing, or an instantaneouschannel-state-based pre-processing. While both these solutions yield performance benefits as compared to pure antenna selection (without pre-processing), they do not exploit the inherent correlation in the channel. In Chapter 3, we propose a channelstatistics-based pre-processing solution for antenna selection that fully exploits the channel correlation information. We also show the robustness of this architecture to RF and channel imperfections. Most of the prior work on antenna selection has focussed on single-user systems. In Chapter 4, we derive the capacity maximizing statistics-based selection criteria for a multi-access system. and demonstrate the advantages of using antenna selection for such a system.

28 16 Chapter 2 Optimal Space-Time Spreading Codes for CDMA Systems This chapter considers the asymptotic spectral efficiency of a general linear spacetime spreading system for CDMA with random spreading sequences, in the limit of large user populations. The resulting parametric formulas for spectral efficiency provide a design criterion for space-time coded CDMA. Using this criterion, we derive optimal space-time codes. 2.1 Introduction Third generation (3G) standards like CDMA2000 and W-CDMA are based on spread spectrum techniques for multi-user communication. These standards have requirements for high data rates, due to increasing demand for broadband wireless access. It has been well established [4], [5] that in rich scattering, fading environments, the use of multiple antennas at the transmitter and receiver helps in achieving substantially higher data rates.

29 17 Several transmit diversity methods have been proposed in the literature for improving the performance of CDMA using multiple antennas at the base station and the handset. Transmit diversity methods are usually grouped into two categories, depending on the transmitter s knowledge of current channel conditions [15]. Closedloop methods, such as transmit beamforming [27] and selection transmit diversity [34], require channel information to be fed back from the receiver to the transmitter. On the other hand, open-loop methods, such as orthogonal transmit diversity [32] and space-time spreading (STS) [29], require no such feedback. Both types of methods play a role in future wideband CDMA systems under different conditions, which depend, in part, on the quality of the channel estimates. Most work on transmit diversity for CDMA has focussed on the downlink [10 14], where spreading codes are orthogonal, and on particular transmit diversity methods. Huang, et al. [10] describe some interesting variants of BLAST for a CDMA system, and calculate the capacity gain for downlink channel. In [11], the spectral efficiency for a CDMA system using orthogonal spreading sequence is calculated for STS and compared with the performance of BLAST. While most of the previous work has focused on orthogonal CDMA systems, relatively little attention has been paid to the use of multiple-antennas in non-orthogonal CDMA systems. It is not clear what is the impact of non-orthogonal spreading sequences, and whether it is possible to significantly improve on these simple transmit diversity methods. Thus, it is natural to examine the fundamental information-theoretic limits of space-time coding methods and to examine the performance of a larger class of transmit diversity methods. However, this is often a difficult task due to the complex dependence of system performance on the spreading sequences assigned to each user. Recently, however, asymptotic analyses of single-antenna CDMA for large-user populations, based on random-spreading sequences, have made some fundamental design tradeoffs much

30 18 more transparent [19, 20, 35, 38, 39]. In particular, the spectral efficiency (capacity per chip) of randomly-spread CDMA is characterized in [39] when all users are received with equal power, and in [35], when users are plagued by frequency-flat fading. Multiple receive antennas are also considered in [35]. The ergodic and outage spectral efficiency of a randomly-spread multi-antenna CDMA system with a BLAST-like transmission was derived in [18]. The authors compared two extreme cases for the allocation of spreading sequences: using same sequences across all the antennas, and using a different sequence on each antenna. However, it is not clear that the best strategy to allocate spreading sequences across the antennas is one of these simple techniques. In this chapter, we characterize the asymptotic spectral efficiency for a general, synchronous CDMA transmission scheme, in which spreading sequences are not tied to a specific transmit antenna. The asymptotic spectral efficiency is derived as a function of the array size, SNR, and the number of users per chip, for large user populations, in presence of frequency-flat fading. Then we derive the conditions that must be satisfied by the space-time codes to maximize the asymptotic spectral efficiency. Using these conditions, we examine some interesting properties of optimal space-time codes and the maximum achievable spectral efficiency. The rest of the chapter is organized as follows. In Section 2.2 we introduce the transmission model and the channel model. The spectral efficiency of a CDMA system is defined in Section 2.3. Section 2.4 gives a brief background on random matrix theory, followed by the derivation of asymptotic spectral efficiency in Section 2.5. The optimal space-time spreading codes are derived in Section 2.6, followed by conclusions in Section 2.7. The following notation is used in the chapter: (.) for the Hermitian transpose of a matrix, () for the Hermitian of a complex number,. for the determinant of

31 19 a matrix, Tr(.) for the trace of a matrix, E X [.] for the expectation with respect to X, and N c (m, σ 2 ) denotes a complex Gaussian random variable with mean m and variance σ Transmission Model A General Model of Linear Modulation We consider a synchronous direct-sequence CDMA system in which K equalenergy users transmit to a common receiver. Each user transmits d independent data streams over t antennas using m spreading sequences. The spreading sequences are not necessarily tied to specific transmit antennas. The only requirement is that the transmitted signal must be linear in the data symbols. The signal transmitted by user k is given by X k = 1 tmn d (x kl A l + x klb l ) C k, (2.1) l=1 where N is the number of chips per spreading sequence, x k1,..., x kd are the data symbols, C k is an m N matrix of spreading sequences, and A l and B l are t m complex modulation matrices, which distribute the data over the spreading sequences and antennas. Assuming that the data symbols are independent, identicallydistributed (i.i.d.) N c (0, 1) complex random variables, and independent of C k, where { } E C k C k = NI m, the matrices are required to satisfy an average energy constraint: { [ ]} E Tr X k X k = 1 tm d l=1 [ ] Tr A l A l + B lb l 1. (2.2) Note that (2.1) can be viewed as a generalization of the linear dispersion codes [16] to CDMA. Next, we represent a few common transmission schemes as particular cases of (2.1).

32 20 Example 1. Orthogonal Transmit Diversity (OTD) [29] In OTD, each user transmits different data streams on each antenna, using different spreading sequences. For t transmit antennas, this scheme can be expressed in the form of (2.1), by choosing d = t, m = t, B l = 0 and A l = t diag(δ 1l,..., δ tl ), l = 1,, d. Here 0 is a matrix with all-zero elements, diag(a,..., z) refers to a diagonal matrix with elements a... z along the main diagonal, and 1 if i = j δ ij = 0 otherwise. Example 2. Space-Time Spreading (STS) [29] In STS, two data symbols x k1 and x k2 are spread over two antennas as follows: 1 X k = x k1c 1 + x k2 c 2 4N x k2 c 1 x k1 c 2 where c 1 and c 2 are spreading sequences of length N. This can be modelled as in (2.1) with m = d = t = 2 and A 1 = 1 0, B 1 = , A 2 = , B 2 = (2.3) Example 3. Independent data, same codes [18] (BLAST) When different data streams are transmitted on each antenna using the same spreading sequence, the transmitted matrix can be written as X k = x k1 c 1 x k2 c. tn. x kt c

33 21 To express it in the form of (2.1), choose d = t, m = 1, B l = 0, and A l = [δ 1l,..., δ tl ], l = 1,, d. This scheme can be considered as a wideband representation of the Bell Labs layered space-time (BLAST) transmission [22]. This scheme (along with a variant that uses adaptive rate control) has also been included in Lucent s 3GPP proposal for high speed downlink packet access (HSDPA) [25]. Example 4. Same data, independent codes [18] In this scheme, the same data symbol is sent on each of the transmit antennas using different spreading sequences. This can be modelled using (2.1) by setting d = 1, m = t, B 1 = 0, and A 1 = ti t, where I t is t t identity matrix MISO Channel Model For the sake of simplicity, we assume that all users employ the same linear modulation scheme. The arguments are easily generalized to unequal powers and different modulation. We assume that perfect instantaneous channel state information (CSI) is available at the receiver. We consider a multiple-input single-output (MISO) channel, i.e. the transmitter has multiple antennas, while the receiver has a single antenna. At the receive antenna, the output of a filter matched to the chip waveform is synchronously sampled. In frequency-flat fading, we can write the samples in vector form as Y = K ρ H k X k + W, (2.4) k=1 where Y is an 1 N vector of received chip samples, H k is a 1 t vector of fading path gains from user k to the receiver, ρ is the signal-to-noise ratio (SNR) per transmission path, and W is a 1 N vector of noise. We assume the elements of W are

34 22 i.i.d.complex Gaussian zero-mean, unit-variance random variables. H k are independent (not necessarily Gaussian) random vectors such that E H k H k { } = t. Note that the results are not changed by allowing H k and C k to vary with time. We can write the channel in a unified matrix form by collecting the data symbols of all users into one Kd-dimensional vector x = [x 1,..., x K ], x k = [x k1,..., x kd ]: Y = ρ N (xh A + x H B ) C + W where H A = diag{a 1,..., A K } and H B = diag{b 1,..., B K } are dk mk blockdiagonal matrices, and H k A 1 A k = 1 H k A 2 mt. H k A d, B k = 1 mt H k B 1 H k B 2. H k B d, C = C 1. C K. (2.5) Recall that the algebra of n m complex matrices can be expressed in terms of 2n 2m real matrices via the correspondence C = R(C) + ji(c) C = R(C) I(C) I(C) R(C), where R(C) and I(C) are the real and imaginary parts of C, respectively. Thus, in real form x k A k + x k B k can be written as R(x k) I(x k ) R(A k) I(A k ) + I(x k ) R(x k ) I(A k ) R(A k ) R(x k) I(x k ) I(x k ) R(x k ) R(B k) I(B k ) I(B k ) R(B k ) The two rows of the resulting matrix are redundant; dropping the second row gives (1/ 2) x k Mk where (with a slight abuse of notation) x k = 2[R(x k ) I(x k )] and M k = R(A k + B k ) I(A k + B k ). (2.6) I(A k B k ) R(A k B k ).

35 23 Thus, the channel can be written in real form as Ȳ = ρ 2N x H C + W, (2.7) where x = [ x 1,..., x K ], W = 2[R(W) I(W)], M 1 O O O H M 2 O = C = O O MK 2.3 Spectral Efficiency C 1. C K. (2.8) Once we have the channel in a composite form, we can calculate the spectral efficiency of (2.7). The capacity region of a multi-access system (2.7) is given by the convex hull of the maximum rate of each user [3]. Thus the maximum achievable sum-rate spectral efficiency in bits/chip is given by η(n, K, m, t, ρ) = 1 2N log I + = 1 2N log I + ρ 2N H C C H (2.9) ρ 2N C H H C, (2.10) where (2.10) follows from the matrix identity I + AB = I + BA. The formula for spectral efficiency above does not provide much insight into the behavior of the multi-antenna CDMA system. For example, the impact of varying the number of users, SNR, and number of antennas is not very clear from (2.10). Furthermore, it does not provide a mechanism to design the codes that maximize the spectral efficiency. Recently, asymptotic analyses of single-antenna CDMA systems for large-user populations, based on random spreading sequences, have made some design tradeoffs much more transparent (see [20, 35, 38]). In this work, we consider the asymptotic

36 24 behavior of the spectral efficiency of a multiple-transmit-antenna system for largeuser populations. In particular, we consider the asymptotic behavior of (2.10) under the following assumptions: 1. The number of users, K, and the number of chips per bit, N, approach infinity, but their ratio is bounded, i.e. K, N, 0 < β = K N < 2. The spreading sequence, c k, has i.i.d. complex elements, and each element has i.i.d. real and imaginary parts with mean zero and variance 1/2. Verdu et al. show in [39], [35] that many fundamental properties of a CDMA system become apparent under the assumptions mentioned above. The performance of a system with random spreading sequences serves as a lower bound to the performance of a CDMA system with optimally designed deterministic spreading sequences. Furthermore, random spreading accurately models CDMA systems where the pseudonoise sequences span many symbol periods [39]. 2.4 Random Matrices in the Large System Limit In this section, we give a brief overview of the main results in the theory of random matrices. In [26], Tulino and Verdu provide an excellent overview of the applications of random matrices to communications. We review some definitions and results from [26] in this section Empirical Eigenvalue Distribution We first define the empirical eigenvalue distribution of a matrix.

37 25 Definition: Empirical Eigenvalue Distribution (edf) For any n n Hermitian matrix A, the empirical eigenvalue distribution, F A, is defined as F A (λ) = 1 n n 1 {λ i λ} where λ 1,..., λ n are the eigenvalues of A and 1{.} is the indicator function. i=1 The edf of a matrix gives a measure of the fraction of eigenvalues that are less than a certain number. Intuitively, this can be thought of as the matrix equivalent of cumulative distribution function (cdf) of a random variable Stieltjes Transform It is often convenient to represent the edf of a matrix in terms of its Stieltjes transform. Definition: Stieltjes Transform Let X be real-valued random variable with distribution F X. Its Stieltjes transform is defined as S F (z) = 1 λ + z df X(λ). From the definition above, we can see that the Stieltjes transform of a matrix can be written as { } 1 S(z) = E, λ + z where the expectation is taken with respect to (w.r.t.) the empirical eigenvalue distribution of the matrix. The reader is referred to [26] for examples of Stieltjes transforms of some common class of matrices, and their applications to communications. Next we state a lemma from [36] which will be useful in characterization of the spectral efficiency.

38 26 Lemma 1. [36, Thrm 1.1]: Consider the n n random matrix Q n = 1 n C np p C n. (a) If C n is an p n random matrix with i.i.d. elements satisfying E {c ij } = 0, E { c ij 2 } = 1; (b) P p is a (possibly random) p p nonnegative-definite Hermitian matrix, independent of C, such that F Pp converges in distribution, almost surely (a.s.), to a fixed distribution G on [0, ) as p ; then, as n and p/n c > 0, F Qn also converges in distribution (a.s.) to a fixed distribution F. This distribution is most easily given in terms of its Stieltjes Transform by the implicit relation S F (z) = z = 1 s c 1 df (λ) (2.11) λ + z λ dg(λ), (2.12) 1 + sλ where for I(z) > 0, S F (z) is the unique solution of (2.12) satisfying I(S F (z)) < R-Transform The R-transform is closely related to the inverse function of S F (z) and plays a central role in the theory of free random variables [40, Sec. 3.2]. For the problem at hand, we will see that R F (s) is easier to calculate than S F (z), and will prove useful in deriving codes that maximize the spectral efficiency. 1 The Stieltjes Transform is defined differently than in [36]. The relationship is S F ( z) = m F (z) = (λ z) 1 df (λ).