Precoding and Beamforming for Multi-Input Multi-Output Downlink Channels

Transcription

1 Precoding and Beamforming for Multi-Input Multi-Output Downlink Channels by Roya Doostnejad A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto c Copyright by Roya Doostnejad, 2005

2 Precoding and Beamforming for Multi-Input Multi-Output Downlink Channels Roya Doostnejad Doctor of Philosophy, 2005 The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto Abstract This dissertation presents precoding and beamforming schemes for multi-user wireless downlink channels when multiple antennas are employed at both the transmitter and the receivers. In the first part of the thesis, we will discuss transmitter processing without channel information which is applicable in both flat and frequency selective (when orthogonal frequency-division multiplexing (OFDM) is applied) fading channels. This leads to methods for designing signature matrices for transmitters that use any combination of the spatial, temporal and frequency dimensions, with good performance provided by low-complexity receivers. In the rest of the thesis, we pose the problem when the channels between the base station and each user are known perfectly at the base station. A non-linear precoding scheme is designed to minimize the mean-squared error between the transmitted and received data with a per-user power constraint. We also develop methods that are able to provide user-specific signal-to-interference-noise ratios (SINRs) with minimal total transmit power, through the extension of a so-called uplink-downlink duality result. Our study indicates that channel knowledge at the transmitter leads to substantial reductions in required power for providing given levels of SINRs to users. ii

3 Acknowledgements I would like to express my sincere gratitude to my supervisors Prof. Teng Joon Lim, and Prof. Elvino S. Sousa for their guidance, advice, and continued support throughout my thesis research. Prof. Lim has provided the key technical insights and contributed tireless editorial effort which has vastly improved the quality of this dissertation. Prof. Sousa has provided me a gentle encouragement and a far-reaching vision of the work. I wish to thank my entire committee: Prof. Frank Kschischang, Prof. Ravi Adve, Prof. Dimitris Hatzinakos, Prof. Bruce A. Francis, and Prof. Murat Uysal of the University of Waterloo for their effort, discussions and constructive comments. In particular, I would like to thank Prof. Kschischang for his invaluable inputs and constant encouragement throughout the course of this research. I also acknowledge the administrative support of Diane B. Silva during these years. I am appreciative of my colleagues in the communication group as well as my friends in Toronto who made this period of my life most enjoyable and beneficial. The financial support of the University of Toronto and Ontario Graduate Scholarships in Science and Technology (OGSST) is also greatly appreciated. I would like to extend my appreciation to the professors from whom I learned a great deal in earlier stages of my studies in Isfahan University of Technology, in particular, Dr. H. Alavi, Dr. A. Doosthoseini, Dr. S. Sadri and Dr. V. Tahani. It is impossible to express the debt that I owe to my late parents. My father who shaped the first stages of my education and has been always a role model for me, and my mother, that if it was not because of her intense care and compassionate support, I would have never been able to come this far. I would also like to thank my siblings, Rezvan, Mehdi and Ahmad, and my in-laws Akbar Abdollahi and Shahla Dardashti who have always been a source of encouragement and drive behind my achievements. At last, my most special thanks goes to my husband, Kambiz Bayat for infinite love, support, patience and devotion, and to my little one for inspiration at the end of this journey. iii

4 To my husband, Kambiz and In memory of my parents. iv

5 Contents Abstract Acknowledgements ii iii List of Tables ix List of Figures xii 1 Introduction Multipath Fading Channels Space-Time Coding System Model Design Criteria Space-time Coding Schemes Space-Time Coding in a Multiuser System Precoding MIMO Single-user Systems MIMO Multiuser Systems Overview of the Thesis Notations v

6 2 Space-Time Multiplexing for MIMO Multiuser Downlink Channels System Model Transmitted Signal Design Assumptions and Goals Spreading Matrix Design Constellation and Power Allocation Receiver Structures Joint ML Detection Multi-Stage Successive Interference Cancellation Comparison With Other STC-CDMA Transceivers Simulations Summary Precoding and Beamforming for MIMO Downlink Channels with Per- User Power Constraints Problem Formulation Signal Model Precoding MMSE Beamforming/Precoding Precoding Matrix Design Optimum Receive Matrix Optimum Transmit Matrix Precoding Ordering Space-Time Spreading Simulation Results Summary vi

7 4 Precoding and Beamforming for MIMO Downlink Channels to Minimize Total Transmit Power Problem Formulation Signal Model Background Joint Power Allocation and MMSE Beamforming Using Uplink/Downlink Duality Uplink-Downlink Duality for MIMO channels Proposed Algorithm Space-Time Spreading Multiple Symbol Transmission to each user Simulation Results Summary Precoding and Beamforming for the Down-link in a MIMO/OFDM System Single User MIMO/OFDM Systems Signal Model Transmit Signal Received Signal SFS Matrix Design with no Channel Information at the Transmitter Comparison With MIMO Multi-Carrier CDMA Schemes SFS Matrix Design with Perfect Channel Knowledge at the Transmitter Simulation Results Summary vii

8 6 Conclusion Summary of Contributions Future Work A Spreading Matrix Design Examples 129 B Proof of Uplink-Downlink Duality in MIMO Multiuser Systems 131 C The Algorithms for Multiple Symbol Transmission to each user 134 Bibliography 137 viii

9 List of Tables 2.1 Comparison between different STC schemes for the downlink in a MIMO multiuser channel The algorithm for precoding and MMSE beamforming The precoding/beamforming algorithm for MIMO-BC channels minimizing total transmit power The error rate performance of TTPC versus PUPC algorithm for t = r = 4, K = 4, SINR = 10(dB) The error rate performance of TTPC versus PUPC algorithm for t = r = 4, G = 4, K = 16, SINR = 10(dB) C.1 The precoding/beamforming for multiple symbol transmission C.2 The space-time precoding/beamforming for multiple symbol transmission. 136 ix

10 List of Figures 1.1 Multiple Access Channel Broadcast Channel Matrix DFE Transmission system model Structure of two-stage SIC Performance of 2-D STSC for different receivers, t = r = 2, G = 2, U = The effect of MAI (number of users) on the achieved diversity with MMSE for t = r = 4, G = The impact of power allocation on the performance of 2-D STSC for t = r = 2, U = The impact of power allocation on the performance of SIC for t = r = 4, G = 4, U = Performance of 2-D STSC in correlated fading channels for t = r = Performance comparison of various schemes for multiuser channel in the downlink for t = r = 2, G = x

11 = Performance of proposed 2D-STSC versus randomly generated ST spreading codes which do not have the zero average MAI property, and Hadamard codes which give zero average MAI but do not satisfy the full-diversity criterion Performance comparison of the proposed space-time coding scheme and rotated constellation (TAST) in a single user system for t = r = 2, G = Block diagram of the matrix DFE Matrix form of the Tomlinson-Harashima precoder The average P e for different number of receive antennas and t = 4, K = 2, z = The performance of space-time spreading for different number of receive antennas, t = 4, G = 4, K = 8, z = Average P e compared with P e for each individual user, t = 2, r = 2, K = 2, z = Uplink-downlink duality these two multi-user channels have the same achievable SINR region for a given sum power constraint Performance of the iterative linear beamforming and the proposed algorithm with MMSE and random initializations, for t = r = 4, K = 4, SINR = 10 db Total transmit power versus the required SINR for different number of transmit/receive antennas for K = Transmit power per user versus the number of active users at the system for r = 4, SINR = 10 db Precoding/Beamforming over space and time for t = r = 4, K = Precoding/Beamforming over space and time for t = r = 4, K = xi

12 5.1 OFDM/MIMO Block Diagram Transceiver structure of MIMO MC-CDMA systems Performance comparison of Space-Frequency Spreading methods for one, two and four tap equal power fading channels for t = r = G = Performance of ML detection versus SIC with and without power control for two-tap channel for t = r = 2, L = 2, N f = The performance of space-frequency spreading compared with MMSE beamforming over flat fading channel t = 2, r = 2, N f = 8, z K = Average P e compared with P e for individual users in space-frequency spreading, t = 2, r = 2, N f = 8, K = 8, z = Precoding/Beamforming over space and frequency for t = r = 2, K = 16, SINR = 10(dB) xii

13 Chapter 1 Introduction The use of antenna arrays at both the transmitter and the receiver has received significant attention as a promising method to provide diversity and/or multiplexing gain over wireless links. Multiple antennas create extra dimensions in the signal space which can be used in different ways. The receiver can be provided with replicas of the same data to increase the reliability of signal transmission which results in spatial diversity gain. The spatial dimensions can also be used to carry independent data streams to increase the data rate which results in spatial multiplexing gain. This collective improvement associated with spatial multiple-input multiple-output (MIMO) channels is based on the premise that in the wireless system with enough separation between antennas in an array, a rich scattering environment provides different channels between each transmit and receive antenna which are statistically uncorrelated to some extent. MIMO techniques were first investigated in a point-to-point or single-user communication link. In a MIMO single-user system with t transmit and r receive antennas, a diversity order of tr can be provided for the system. Also, if the channel is perfectly known at the receiver, capacity scales linearly with min(t, r) relative to a system with just one transmit and one receive antenna. A MIMO system is thus able to provide im- 1

14 Chapter 1:Introduction 2 proved power and bandwidth efficiencies, at the cost of setting up additional antennas. Space-time coding schemes have been designed for MIMO single-user systems to achieve diversity gain [1 3], or achieve high data rates by taking advantage of multiplexing gain of MIMO systems [4, 5], or both [6, 7]. Base Station User 1 User 2 User K Figure 1.1: Multiple Access Channel In many applications ranging from wireless LAN to cellular telephony, multiuser communication is a reality. Therefore, recently researchers have been attracted to investigate the impact and implications of using MIMO systems in multiuser environments. There are two basic multiuser MIMO channel models: the MIMO multiple-access channel (MAC) and the MIMO broadcast channel (BC). In MIMO MAC, a number of users share a common communication channel to transmit their individual signals to a receiver. Such a system is shown in Figure 1.1. In the uplink of a mobile cellular communication system, the users are the mobile transmitters in any particular cell and the receiver is

15 Chapter 1:Introduction 3 the base station of that cell. In MIMO BC, a transmitter sends information to multiple receivers as shown in Figure 1.2. In the downlink of a mobile cellular communication system, the transmitter is the base station and the receivers are the mobile stations. A key difference between single-user, MAC, and BC channels is that in the single-user channel, there is a full collaboration at both sides of transmitter and the receiver, while in the MAC channel there is collaboration only at the receiver, and in the BC channel collaboration exists only at the transmitter. Therefore in the BC channel joint processing between the receivers cannot be supported. Based on this fact, the design of BC channel is proved to be more challenging [8 10]. Base Station User 1 User 2 User K Figure 1.2: Broadcast Channel This thesis is primarily concerned with the design of the transmitter in a MIMO broadcast channel. Assuming no channel information at the transmitter, space-time spreading matrices are designed to maximize diversity gain and spectral efficiency. Assuming per-

16 Chapter 1:Introduction 4 fect channel knowledge at the transmitter, an algorithm based on MMSE beamforming combined with non-linear interference pre-subtraction is proposed which is applicable to a multiuser BC channel with any desired number of transmit/receive antennas. This chapter will provide the basis for the rest of the thesis. Multipath fading and different diversity schemes are explained in the next section. In Section 1.2, a brief review on space-time coding schemes for single-user systems and then the extension to multiuser systems is explained. In Section 1.3, precoding is introduced for both singleuser and multiuser systems when the channel is known perfectly at the transmitter. The overview of the thesis is provided in Section 1.4, and the notations which are used through the thesis are given in Section Multipath Fading Channels The physical characteristics of the wireless channel presents a fundamental technical challenge for reliable communications. This is mainly because of the time varying multipath nature of the channel. Multipath propagation is a result of the propagation of the signal over a number of different paths due to reflections of the signal by mountains, buildings, and other objects. Because of the time variations in the structure of the wireless channel, the nature of the multipath varies with time. This results in signal fading over time. The amplitude variations in the received signal are due to the destructive and constructive addition of multiple signal paths between receiver and transmitter. For a multipath fading channel, we define the time-variant impulse response of the channel as h(t, τ) which is the output of the channel at time t to an impulse applied at time t τ. Since the channel time variations are not predictable, the time variant multipath channel is modelled statistically. The most common statistical fading model is the Rayleigh fading model in which the impulse response of the channel, h(t, τ), is assumed to be a complex

17 Chapter 1:Introduction 5 random variable whose real and imaginary parts are zero-mean statistically independent Gaussian random variables, each having a variance σ 2 τ. Therefore the magnitude of the channel at any instant t, r = h(t, τ), has a Rayleigh distribution P (r) = r σ 2 τ e r2 /2σ 2 τ, r 0. (1.1) The autocorrelation function of h(t, τ) is given by [11] φ h ( t; τ i, τ j ) = 1 2 E [h (t, τ i )h(t + t, τ j )], (1.2) where t is the observation time difference. Since in most radio transmissions the impulse response of the channel for different paths are independent, if we let t = 0, then we have φ h (τ i, τ j ) = φ h (τ i )δ(τ i τ j ). (1.3) In fact φ h (τ i ) represents the average channel output power as a function of the time delay τ i. The different paths have different time delays and different average powers. We call φ h (τ) = 1 2 E [h (t, τ)h(t, τ)], (1.4) the multipath intensity profile of the channel [11]. The range of values of τ that φ h (τ) is nonzero is said the multipath spread of the channel, and the largest value among these delays is defined as the delay spread of the channel which is denoted by T m. In other words φ h (τ) 0, for τ T m. The coherence bandwidth of the channel, ( f) c, is defined as the frequency separation at which two frequency components of the signal undergo independent attenuations by the channel. This parameter will be defined corresponding to T m as ( f) c 1 T m. (1.5) If the bandwidth of the signal, W, that is transmitted through the channel is smaller than the coherence bandwidth of the channel, i.e. W < ( f) c, the channel is called

18 Chapter 1:Introduction 6 flat-fading channel in which all the frequency components of the signal undergo the same attenuation by the channel. In other words, within the bandwidth of the signal, the transfer function of the channel is constant in the frequency variable. In this case, the multipath components in the received signal are not resolvable, and the channel appears as a single fading path. This implies that in flat-fading channels, the received signal is simply the transmitted signal multiplied by the channel coefficient, h, where h is a zero-mean complex-valued Gaussian random process. For a single-antenna system, this can be simply modelled as y = hx + ν, (1.6) where x, and y are the transmitted and received signal respectively, and ν is additive noise which is usually assumed to be Gaussian distributed and independent of x. If the signal bandwidth is such that W > ( f) c the channel is called to be frequencyselective and the signal is severely distorted by the channel. In this case, the multipath components can be resolved in the received signal and therefore the receiver is provided with several independently fading signal paths [11]. Consequently, the frequency-selective channel is modelled as a tapped delay line filter with time-variant tap coefficients. The frequency-selective fading can degrade system performance by causing inter-symbol interference (ISI) which result in an irreducible bit error rate (BER). Time-domain equalization [11] and orthogonal frequency-division multiplexing (OFDM) [11 13] are practical techniques that can be used to resolve ISI. Diversity techniques are based on the fact that if the channel is in a deep fade because of the destructive addition of the multipath signals, errors may occur due to the large channel attenuation. However if we can provide the receiver with several replicas of the same signal transmitted over L independently fading channels, the probability that all the signals fade simultaneously will be reduced considerably. If p is the probability

19 Chapter 1:Introduction 7 that any one signal fades below a threshold level, then p L is the probability that all L independently fading replicas will fade below the threshold level. There are several diversity techniques that can be employed in wireless communication systems to supply to the receiver L independently fading replicas of the same information signal. Diversity techniques which may be used include time, frequency, and space diversity. Time Diversity refers to transmitting the same signal over L different time slots where the separation between successive time slots is enough to make their channels independent. A common example of time diversity is the interleaving of coded symbols over a large block length. Frequency Diversity refers to transmitting the same signal over a large bandwidth, exceeding the coherence bandwidth of the channel. An example of the use of frequency diversity is spread spectrum modulation. In fact, in a frequency-selective fading channel, the receiver is provided with T m W W/( f) c resolvable signal components. By applying either OFDM or time-domain equalization schemes, a frequency diversity of order L W/( f) c can be obtained. Space Diversity refers to transmitting or receiving the same signal over multiple antennas that are separated enough to create independent fading channels. To provide space diversity, multiple antennas are used at the transmitter and/or the receiver. The independent spatial channels provided by multiple antennas can be also used to carry independent data steams to increase the data rate. This latter technique is known as spatial multiplexing. In this thesis, both flat-fading and frequency-selective fading channels are considered. In the latter case, OFDM is applied to resolve ISI and extract the frequency diversity of the channel.

20 Chapter 1:Introduction Space-Time Coding System Model A single-user channel is considered with t transmit and r receive antennas. The transmit symbols s 1,..., s p are encoded to a n t (possibly complex) space-time code matrix C which is transmitted from t transmit antennas over n time slots. The rate of this code is defined as R = p/n symbols/channel-use, (1.7) where again p is the number of data symbols transmitted in n time slots. The t r channel matrix H is defined such that H(i, j), which represents the element in the ith row and the jth column of the matrix H, is the channel gain between the transmit antenna i and receive antenna j. Each channel coefficient has the same variance of σh 2, and the tr channels are assumed to be independent. Also we assume a quasi-static Rayleigh fading channel which is constant over a block of n time slots and independent from block to block. Then if the power per input symbol transmitted from each transmit antenna is p s /t, the received signal which is a n r matrix Y will be Y = p s /tch + N, (1.8) where N C n r is a matrix of i.i.d. complex Gaussian random variables with zero mean, and the variance of σ 2, representing receiver noise. To simplify the analysis, σ 2 h = 1 is assumed and therefore the signal-to-noise ratio per receive antenna is defined as ρ = p s /σ 2. The channel is known at the receiver but not at the transmitter. The goal is to design the matrix C to achieve full diversity and multiplexing gain.

21 Chapter 1:Introduction Design Criteria Space-time codes design criteria are derived based on maximum likelihood (ML) detection in [2, 14]. The analysis is based on pairwise error probability [11]. For a given channel matrix H, the probability that a ML receiver decides erroneously in favor of the code matrix C j when the code matrix C i is transmitted will be P (C i C j H) = P ( Y C j H 2 Y C i H 2 ) A = Q( 2 (C i,c j )ρ ) exp( A 2 (C 2 i, C j )ρ/4), (1.9) where A 2 (C i, C j ) = (C i C j )H 2. Equation (1.9) needs to be averaged over the channel distribution. An upper bound on the average probability of error in the case of Rayleigh fading channel is obtained in [2] as follows, ( l ) r P (C i C j ) λ i (ρ/4) lr for C i C j, (1.10) i=1 where l is the minimum rank of the difference matrix, D ij = (C i C j ), over different possible code matrices C i C j, and λ i are nonzero eigenvalues of the matrix Λ = D ij D H ij. This results in the following design criteria for space-time codes: Rank Criterion (diversity gain): The achieved transmit diversity at the receiver is the minimum rank of the difference matrix, D ij, over all possible code matrices C i C j. A full diversity code is obtained if l = t. Determinant Criterion (coding gain): The coding gain, g is defined as g = min C i C j ( l ) λ i. (1.11) i=1 Space-time codes are designed to maximize both diversity and coding gain.

22 Chapter 1:Introduction Space-time Coding Schemes In this section, we briefly review a few well known space-time coding (STC) schemes so that we may refer to them later in the thesis. Space-Time Block Coding (STBC): In [1] a STC scheme is proposed for two transmit antennas. The input symbols (s i ) are divided into groups of two symbols each. Then the STC matrix is generated as follows: C = s 1 s 2 s 2 s 1 (1.12) It is shown that because of the orthogonal structure of this code, i.e. CC H = 2 s 2 i I 2 (1.13) i=1 (where I k is a k k identity matrix), ML detection simplifies to a linear processing at the receiver [1]. Also it can be easily shown that this code has full diversity (the difference matrix is full rank). This scheme is later generalized in [3] to an arbitrary number of antennas. For t transmit antennas, the input symbols are divided into groups of t symbols each and then the STC matrix is generated as an orthogonal matrix, i.e. CC H = t i=1 s2 i I t. Here, we summarize some important properties of STBC: STBC are full diversity codes. Real orthogonal codes with rate R = 1 can be designed for any number of transmit antennas. A complex orthogonal design with rate R = 1 exists if and only if t = 2 (see (1.12)). There are also complex orthogonal designs for t = 3, and t = 4 but with a rate R = 3/4.

23 Chapter 1:Introduction 11 ML detection requires only linear processing at the receiver. BLAST Codes: BLAST stands for Bell Laboratories Layered Space-Time. This architecture breaks the data stream into t sub-streams that are transmitted simultaneously from t antennas. Hence, there is no built-in spatial transmit diversity. This scheme is implemented as Diagonal-BLAST (D-BLAST) [4], Vertical-BLAST (V- BLAST) [5] and Turbo-BLAST (T-BLAST) [15]. In particular BLAST is designed to provide very high data rate communications over wireless flat-fading channels. A typical example for V-BLAST when we have two transmit and two receive antennas is C = s 1 s 2 s 3 s 4. (1.14) Comparing this with STBC (1.12), one can see that in BLAST scheme, transmit diversity is not provided for the system. However, two symbols are transmitted per channel use which is twice the rate of STBC in terms of channel symbols per channel use. At the receiver, successive nulling and cancelling is applied. The interference from an already-detected symbol is subtracted out from the received signal before the next symbol is detected. Each symbol is detected based on a zero forcing method [16]. Therefore, it is necessary to have r t. The order in which the symbols are detected affects the overall performance of the algorithm. The best-first cancellation scheme is widely known within the multiuser detection community [17]. This can also be applied at the receiver for BLAST. Based on this scheme, the symbols are ordered based on their received signal-to-interference-plus-noise ratios (SINRs). Then the symbols with higher SINRs are detected first. Because of the particular structure of BLAST, it can be easily seen that the symbols are automat-

24 Chapter 1:Introduction 12 ically received with different SINRs. Here, we summarize some important properties of BLAST: Spatial diversity is not provided at the transmitter. BLAST can be designed for any number of transmit/receive antennas as long as the number of receive antennas is equal or greater than the number of transmit antennas r t. BLAST is a full rate code, R = t. Linear Dispersion (LD) Codes: In this scheme which is proposed in [6], the data stream is broken into Q sub-streams, s q = α q + jβ q, q = 1,..., Q, that are transmitted over space and time as indicated by the codeword matrix C = Q (α q A q + jβ q B q ). (1.15) q=1 The performance of LD codes is dependent on Q, {A q }, and {B q }. The LD codes in [6] were designed to maximize the mutual information between the transmit and receive signals. Note that, in a MIMO single-user channel, if the channel is known at the receiver, the resulting channel capacity is [4, 18]: C(ρ, t, r) = { [ max E log det (I r + ρ )]} R s,t r(r s )=t t HR sh H, (1.16) where the expectation is taken over the distribution of the random matrix H, and R s is the covariance matrix of the input signal. If the channel matrix H C r t is a matrix of i.i.d. complex Gaussian random variables with zero mean, and the variance of σ 2 h = 1, the optimal covariance matrix when H is unknown to the transmitter is R s = I t, and (1.16) becomes { [ C(ρ, t, r) = E log det (I r + ρ )]} t HHH. (1.17)

25 Chapter 1:Introduction 13 By substituting (1.15) in (1.8) we get Y = Q p s /t (α q A q + jβ q B q )H + N. (1.18) q=1 By decomposing the matrices in (1.18) into their real and imaginary parts and then collecting the real and imaginary parts of the received signal in the vector y, the equation (1.18) is re-formulated as [6] y = p s /th g x + ν, (1.19) where H g R 2nr 2Q is a modified channel matrix which is a function of real and imaginary components of A q and B q as well as the original channel gains. y, ν R 2nr 1, and x R 2Q 1 is a vector of real and imaginary parts of transmitted symbols (α q, β q ). Therefore the LD codes are linear in the variables α q, β q and the same detection algorithm as explained for BLAST scheme can be applied. Without loss of generality, s 1,..., s Q are assumed to be unit-variance and uncorrelated. Then E[T r(cc H )] = nt and therefore Q (T r(a H q A q ) + T r(b H q B q )) = 2nt. (1.20) q=1 To design LD codes, first of all Q = n. min(t, r) is chosen. As mentioned earlier the LD codes are designed to maximize the mutual information between the transmit and receive signals. Therefore to choose {A q, B q }, the following optimization problem has to be solved C LD (ρ, n, t, r) = max A q,b q,q=1,...,q subject to one of the following constraints 1 { [ ( 2n E log det I 2nr + ρ )]} t H gh T g, (1.21) 1. Q q=1 (T rah q A q + T rb H q B q ) = 2nt

26 Chapter 1:Introduction T ra H q A q = T rb H q B q = nt, q = 1,..., Q Q 3. A H q A q = B H q B q = n Q I t, q = 1,..., Q The first constraint is the power constraint that ensures E [ T r(cc H ) ] = nt. The second constraint is to make sure that all the data symbols are transmitted with the same power. The third constraint is to transmit all the data symbols with equal energy in all spatial and temporal directions. Here, we summarize some important properties of LD codes: Full diversity is not guaranteed but the codes are shown to provide good performance with respect to the probability of error [6]. The optimization problem in (1.21) is neither convex nor concave. Therefore the optimization problem may lead to a local optimum. The solution (A q, B q ) is not unique. LD code is a full rate code, R = Q/n which if r t results in R = t. TAST Codes: TAST stands for Threaded Algebraic Space-Time. This scheme which is proposed in [7,19,20], uses a threaded structure and algebraic number theoretic tools to design full diversity codes. The codes are directly optimized based on the rank criterion (diversity gain) and determinant criterion (coding gain) (see 1.2.2). The problem of space-time diversity gain is related to algebraic number theory, and the coding gain optimization is related to the theory of simultaneous Diophantine approximation in the geometry of numbers. The coding gain optimization is found to be equivalent to finding irrational numbers, the furthest from any simultaneous rational approximations. Applying a ML detection at the receiver, these codes achieve full diversity while the coding gain is optimized as well.

27 Chapter 1:Introduction 15 For comparison, a design of TAST codes for two transmit/receive antennas is given in the following C φ = s 1 + φs 2 θ(s 3 + φs 4 ) θ(s 3 φs 4 ) s 1 φs 2, (1.22) where θ 2 = φ, and φ = e j/2. It can be seen that the rate of this code is R = 2. Also based on the rank criterion, it can be easily shown that this code achieves full diversity. φ is chosen to maximize the coding gain. Other details for designing these codes are omitted here. The comprehensive explanation is provided in [7, 19, 20]. Here we summarize some key points of TAST codes. TAST codes can be designed for any number of transmit antennas TAST codes are full diversity, full rate (R = t) codes Optimal detection (ML) is required to achieve full diversity Space-Time Coding in a Multiuser System Designing space-time codes for single-user systems is very well understood. However, there has not been an extensive work towards space-time code design for multiuser applications. In fact, splitting of the channel resources among independent users either in the form of multiple access (uplink) or broadcasting (downlink) is often considered a straightforward task involving the concatenation of a multiple access scheme such as CDMA with the space-time (ST) processor [21 23]. For instance, each user can be assigned an orthogonal spreading code, which is used to spread the symbols at the output of a space-time encoder. For that matter, the channel symbols can be generated using

28 Chapter 1:Introduction 16 orthogonal space-time block codes, or any other STC designed for a single-user system. With flat fading synchronous channels, as seen on the downlink, a de-spreading front-end at each receiver results in a single-user channel without multi-access interference (MAI). Note that, the maximum number of active users is only equal to the processing gain (bandwidth expansion) of the system, regardless of the number of space-time dimensions (n time slots, t transmit antennas) used in the STC part of the system. Furthermore, all the constraints and complexities of the applied STC scheme carry over to the multiuser case. As we discussed before, for some STC designs, such as orthogonal space-time block codes, it is simply impossible to allow certain antenna configurations; for others, such as linear dispersion codes, it is necessary to maximize an objective function for a given t and r. In applying the BLAST scheme to a multiuser system, one can use the same spreading code to spread each of the sub-streams. Since the same code is used to spread the sub-streams, the spreading does not aid the receiver in distinguishing among them. As an alternative, different spreading codes could be used for the sub-streams which are transmitted simultaneously from different transmit antennas. In this case, the sub-streams can be separated by their spatial characteristics and their codes. In either case, we can either transmit multiple sub-streams to each user or transmit one sub-stream per user. In addition, different spreading codes can be used to transmit the same sub-stream from different antennas to achieve transmit diversity. In this case, a different spreading code is used on each antenna to distinguish the sub-streams [22]. Although applying different spreading codes over different antennas improves the performance significantly, but as we will show later, it decreases the spectral efficiency. In Chapter 2, we will propose a space-time spreading scheme which is designed for multiuser downlink channel and then we compare the spectral efficiency of the proposed scheme with other STC schemes presented for CDMA systems.

29 Chapter 1:Introduction Precoding The main difficulty in MIMO channels is the separation of the data streams which are sent in parallel. In the context of the multiple access channel, this task is called multiuser detection. In this section we discuss precoding or pre-equalization of the transmitted signals for MIMO systems. This type of processing at the transmitter requires the channel state information (CSI) at the transmitter. In order to be able to obtain CSI at the transmitter, the channel should be fixed (non-mobile) or approximately constant over a reasonably large time period. If CSI is available at the transmitter, the transmitted symbols, either for a single-user or for multiple users, can be partially separated by means of pre-equalization at the transmitter. In this section, we give a brief overview of precoding schemes for single-user and multiuser systems MIMO Single-user Systems A MIMO channel can be described by a very basic model as y = Hx + ν, where x, y are the transmit and receive signal vector respectively, ν represents the receive noise, and H is the r t MIMO channel. In a zero-forcing receiver, the transmit data signals are detected by multiplying the received signal vector by the pseudo inverse of the channel matrix ˆx = (H H H) 1 H H y, if t r. (1.23) For this, the number of receive antennas should be greater than or equal to the number of transmit antennas. It is well known that zero-forcing equalization suffers from noise enhancement. To overcome this deficiency, decision-feedback equalization (DFE) can be applied at the receiver [24]. In DFE, the symbols are detected sequentially. After each symbol is detected, it is cancelled out before the next symbol is detected, therefore DFE

30 Chapter 1:Introduction 18 suffers from error propagation. The structure of DFE is shown in Figure 1.3. The matrix B is a lower triangular matrix representing the decision feedback operation, and matrix F is the feedforward matrix. For the above methods, the CSI is required only at the receiver. By assuming perfect CSI at the transmitter, the interference between the transmitted symbols can be completely avoided at the receiver by multiplying the transmit signal by the pseudo inverse of the channel, which means transmitting x = H H (HH H ) 1 s, if r t, (1.24) rather than transmitting the data vector s. In this linear pre-equalization, instead of enhancing the noise, the average transmit power is increased. Also the number of transmit antennas should be equal or greater than the number of receive antennas. The equalization can also be split among transmitter and receiver. A popular strategy is based on the singular value decomposition (SVD) of the channel matrix. The channel can be written as H = UΣV H, where U, V are unitary matrices and Σ is diagonal. By multiplying the data signal by V at the transmitter, and then applying U H at the receiver, the channel is diagonalized [25]. In this scheme, neither transmit power is increased, nor channel noise is enhanced. The above schemes are considered as linear pre-equalization. noise x H y F ˆx B Figure 1.3: Matrix DFE As mentioned before, the DFE is a non-linear equalizer. With perfect channel knowl-

31 Chapter 1:Introduction 19 edge at the transmitter, the feedback part of the DFE can be transferred to the transmitter which results in a non-linear precoding scheme known as Tomlinson-Harashima precoding (THP). The performance of DFE and THP are the same but since THP is applied at the transmitter, error propagation is avoided [26]. The calculation for the feedforward and feedback filter is as follows. We begin by applying a QL factorization over the channel matrix such that H = Q H S where Q is a unitary matrix and S is a lower triangular matrix. This can be obtained through a Cholesky factorization of H H H because H H H = S H S [26]. Now, we define C = VS where V is a diagonal matrix with the elements equal to the inverse of the diagonal elements of the matrix S so that C becomes a unit-diagonal lower triangular matrix. It can be easily verified that the feedback matrix at the transmitter and the feedforward filter at the receiver should be calculated as B = C I, and F = VQ respectively. Therefore at the transmitter the symbols a i, i = 1,..., K are generated successively from the original data x i 1 a i = x i B(i, l)a l, i = 1,..., K (1.25) l=1 where x i is the ith element of x and B(i, l) is the element in the ith row and the lth column of the matrix B. This strategy will significantly increase the transmit power, therefore the symbols are modulo reduced into the boundary region of the used signal constellation. Mathematically, the integers are added to real and imaginary parts of a i to bound the transmit signals to the constellation region (see [27] and Chapter 3 for more details). Because of this modulo operation, THP is considered as a non-linear precoding. As is shown in [27] the transmit power is still slightly increased, but the scheme outperforms linear pre-coding schemes in the sense of error probability.

32 Chapter 1:Introduction MIMO Multiuser Systems A multiuser downlink channel can be also modelled as y = Hx+ν, while H is the overall downlink channel matrix, and y includes the received signals for all users. However, since the receivers are not collaborating, joint processing of the vector y is not possible, and consequently the schemes proposed for single-user systems may not be applicable. For instance the SVD over the known channel matrix, as explained in the last section, cannot be applied. Also, in THP implementation although the feedback part is moved to the transmitter but still the feedforward filter requires a joint processing of the received signals. However, the THP can be modified to be suitable for a multiuser channel. In fact the feedforward filter is also transferred to the transmitter. The calculation for feedforward and feedback filter for this new structure is as follows. A QR factorization is applied over H H such that H H = QR and H = SQ H where Q is a unitary matrix and S = R H is a lower triangular matrix. By defining C = VS where V is a diagonal matrix with the elements equal to the inverse of the diagonal elements of the matrix S so that C becomes a unit-diagonal lower triangular matrix, the feedforward and the feedback matrices are F = Q, and B = C I respectively. The output signals of the Tomlinson- Harashima precoder are now applied to the feedforward filter before transmitting through the downlink channel. As a result the received signal is equal to HQC 1 = V 1 which is a diagonal matrix, and therefore a joint processing is not required at the receiver. In this scheme the number of transmit antennas has to be equal to or greater than the total number of receive antennas which is a restrictive condition over the number of users in the system or the number of receive antennas at each user. In Chapter 3 we have designed a non-linear precoding scheme based on THP which is valid for any number of transmit/receive antennas. Also, in [8] the authors show that the broadcast channel sum capacity is achieved

33 Chapter 1:Introduction 21 using a precoder with the structure of a DFE that decomposes the broadcast channel into a series of single-user channels with interference pre-subtracted at the transmitter. The proposed precoder is a generalization of the Tomlinson-Harashima precoder. 1.4 Overview of the Thesis The focus of this thesis is on the precoding and beamforming design for the multiuser downlink channel (Figure 1.2) when multiple antennas are employed at both transmitter and receiver sides. We address two scenarios: No channel state information is available at the transmitter (NCSIT) Perfect channel state information is available at the transmitter (CSIT) The design problem with NCSIT is addressed in Chapter 2. The channel is assumed to be flat fading. A space-time spreading matrix is proposed for each user, rather than a temporal spreading code vector as is usual in code-division-multiple-access (CDMA) systems. The spreading matrices are designed to provide full spatial diversity at each receiver while the multiplexing gain is maximized as well. The bandwidth expansion, for a given number of users, is then reduced by a factor of min(t, r), while full spatial diversity is provided for each user, where t is the number of transmit antennas at the base station, and r is the minimum number of receive antennas at user stations. In the downlink since the receivers are portable end-user devices, we are concerned with the complexity at the receiver. Therefore, suboptimal detectors are preferred over optimal detectors (maximum likelihood detection). We have a two-stage interference canceller (IC) applied at each receiver. A power allocation scheme is then suggested to improve the performance of IC towards achieving full diversity.

34 Chapter 1:Introduction 22 The design problem with CSIT is studied in two parts. In the first part which is addressed in Chapter 3, we have a per-user power constraint. In the second part which is addressed in Chapter 4, the design goal is to minimize the total transmitted power in the downlink, while signal-to-interference-noise ratio (SINR) requirements are to be satisfied at each receiver. In the following we explain these two parts in more detail. As mentioned before since we do not have collaboration between the receivers in the downlink BC channel and also low complexity receivers are preferred at mobile stations, our goal is to transfer the processing load from the receivers to the transmitter. It is very well known that assuming perfect channel knowledge at the transmitter, complexity can be moved from the receivers to the transmitter without loss of performance [10], [8]. We also know that the boundary of the capacity region of the broadcast (BC) channel is attained with channel knowledge at the transmitter, and using it for successive dirty paper coding (DPC) [28]. Dirty paper coding is a technique that can be seen as interference pre-cancellation at the transmitter. In this work, assuming perfect channel knowledge at the transmitter, a successive interference pre-subtraction is applied via a matrix version of Tomlinson-Harashima Precoding (THP). In Chapter 3, the multiuser MMSE beamforming is combined with THP to minimize the mean squared error between transmit and receive data streams. The receive beam vectors are obtained with the MMSE criterion, and the transmit beam vectors are obtained through an eigen-value-decomposition scheme. In fact since interference precancellation is applied at the BS, the single user algorithms are applicable over individual single user channels. The proposed scheme is extended to design the beam vectors over the time domain as well. In Chapter 4, the same interference pre-cancellation is applied at the transmitter. However, since the goal is to minimize the total transmit power, the design problem

35 Chapter 1:Introduction 23 is more complicated. We have shown that transmit beamforming is much more complicated than receive beamforming when the total transmit power is to be minimized. We have proposed an iterative algorithm for designing the one transmitter and multiple receivers. An uplink-downlink SINR duality result is proved and used, which computes MMSE beamforming receivers for the virtual uplink and the downlink in turn. Initialization is provided by the eigen-value-decomposition scheme explained in Chapter 3. This algorithm is applicable to design space-time beam vectors as well. In the above proposed algorithms, there is no limitation on the number of transmit/receive antennas. In Chapter 5, the proposed designs in Chapters 2, 3, 4 are extended to perform precoding and beamforming in the MIMO multiuser frequency selective fading channel when orthogonal frequency-division multiplexing (OFDM) is applied. In frequency selective MIMO channels, there is an additional source of diversity, frequency diversity, due to the existence of multiple propagation paths between each transmit and receive antenna pair. In MIMO/OFDM systems, the channel frequency diversity can be also exploited through the proper design of space-frequency codes. In this chapter, first without any knowledge of the channel at the transmitter a multiple access scheme is proposed for the downlink in a MIMO/OFDM system. The space-frequency codes are designed to exploit the space and frequency diversity. Then assuming perfect channel knowledge at the transmitter, the precoding and beamforming design is performed over space and frequency. It is shown that the optimization algorithm benefits from cooperation among the processing at different frequency bins. We conclude the thesis in Chapter 6 where we summarize the contributions of this work and suggest some directions for future work.