INTERFERENCE AVOIDANCE FOR WIRELESS SYSTEMS WITH MULTIPLE RECEIVERS

Transcription

1 INTERFERENCE AVOIDANCE FOR WIRELESS SYSTEMS WITH MULTIPLE RECEIVERS BY OTILIA POPESCU A dissertation submitted to the Graduate School New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Electrical and Computer Engineering Written under the direction of Professor Christopher Rose and approved by New Brunswick, New Jersey May, 2004

2 c 2004 Otilia Popescu ALL RIGHTS RESERVED

3 ABSTRACT OF THE DISSERTATION Interference Avoidance for Wireless Systems with Multiple Receivers by Otilia Popescu Dissertation Director: Professor Christopher Rose Interference avoidance is a class of adaptive modulation techniques in which, with feedback from the receiver, a transmitting radio is instructed to vary its waveform so as to maximize the signalto-interference plus noise-ratio. For an ensemble of users connected to a single receiver interference avoidance methods have been thoroughly analyzed. This thesis provides insight into application of interference avoidance methods to systems with multiple transmitters and receivers, which consist of a collection of base stations and associated users that share the same bandwidth for communication. After a brief review of interference avoidance for single base station systems, we present a codeword adaptation algorithm for collaborative multi-base wireless systems based on greedy interference avoidance. The system is modeled with multiple inputs and multiple outputs and information is transmitted using multicode CDMA. We continue with the collaborative scenario and derive bounds on sum capacity and total squared correlation in the case of flat channels between users and bases. We also investigate structural properties which must be satisfied by user transmit covariance matrices at the optimal sum capacity/tsc point, and show that for multi-base systems maximizing sum capacity and minimizing TSC are in general not equivalent problems. Application of greedy interference avoidance in a non-collaborative scenario is presented next, starting with a simple system consisting of two bases, each base having only one user associated ii

4 with it. Fixed points of the greedy interference avoidance algorithm in this case correspond to a simultaneous water filling solution which is also a Nash equilibrium for the system, and in general is not unique. A detailed analysis of simultaneous water filling for mutually interfering systems, as is the case with non-collaborative multi-base systems, is also included in the thesis. We show that water filling alone does not generally result in optimum resource sharing and in some cases is a poor solution. For weakly interfering systems we show that the simultaneous water filling solution is unique, but this solution is sub-optimal since usually simple separation of users in signal space offers much better performance. We also present a distributed algorithm which iteratively moves users toward separation in the signal space. For strongly interfering systems the simultaneous water filling solution implies a set of fixed points, among which the information theoretic capacities of users vary widely, and in this case we propose a procedure to move the system from a possible sub-optimal point to a better simultaneous water filling fixed point, eventually to the optimal point. iii

5 Acknowledgements When something comes to an end you look back and think of all the people that have been with you and helped you, and you feel indebted to all of them. Same with my doctoral studies at Rutgers University, a wonderful time of my life. First I would like to thank my advisor, Professor Christopher Rose, without whose guidance I could not have written this thesis. He opened my way to a new field, that of wireless communications, and guided my first steps into this new world. I am grateful to him for his confidence in my abilities, for his permanent support, and for all those moments when with a smiling face he encouraged me and never let me down. The other members of my doctoral committee are also gratefully acknowledged: Professors Zoran Gajić, Narayan Mandayam, and Roy Yates, of WINLAB, and Professor Şennur Ulukuş of University of Maryland, who served as the external examiner in the committee. Special thanks are due to Dr. Joseph Wilder, of the Center for Advanced Information Processing (CAIP) for his guidance and support in the early stages of my doctoral program at Rutgers. I owe Chris the honor of making me a part of the WINLAB team, but I would like to thank everybody in this wonderful team for making WINLAB such a great place for research and study. I feel fortunate for having had the chance to learn from the excellent researchers and teachers at WINLAB. I would also like to thank the technical and administrative staff at WINLAB: Ivan Seskar, Kevin Wine, Melissa Gelfman, Noreen DeCarlo, Elaine Connors, and Jeanne Sullivan. Many other people helped me greatly along the way, and I would also like to thank Barbara Klimkiewicz, Lynn Ruggiero and Dianne Coslit, in the ECE Department, and Urmi Otiv in the Center for International Faculty and Student Services at Rutgers. Barbara was especially close to me during difficult times, and I thank her for her warmth and support. Finally, I would like to express my gratitude to my family. I will never find the best words to thank my parents. In all my hard moments I was thinking of my mother, and how she always told iv

6 me that since I was a little girl I dreamed to be a researcher and think of something that nobody else did before. She did everything she could to help me pursue my dream. I am deeply grateful to my father for his constant support throughout the years, for the trust he always had in all my decisions, and for dreaming with me how wonderful the world can be. I will never forget his words when he called me a runner in a long race, telling me to never get discouraged by a small failure, to always stand up and look forward for the next target. From where they are, I know they both look smiling now to me. I have special thanks to my husband and life long best friend Dimitrie, for standing by me all this time. I hope I can help him as much as he helps me. Last, but not least, my thoughts turn to our son, Gabriel. I started my studies at Rutgers in the seventh month of pregnancy, and all these years I shared my time between my son and my studies. He changed our life in a way I never thought before, and I thank him for this. Now, at six years old, he dreams to be an astronaut, a diver, a scientist, a doctor and an inventor, to build machines that nobody else was ever able to build before, reminding me of my own little girl dreams. I thank you all, and hope I made you proud of my accomplishments. This research was supported by the National Science Foundation under grant CCR , and by the New Jersey Commission on Science and Technology under the New Jersey Center for Wireless Technologies grant v

7 Dedication to GABRIEL, wishing him to be lucky to fulfill his dreams vi

8 Table of Contents Abstract ii Acknowledgements iv Dedication vi List of Figures x 1. Introduction Interference Avoidance: A Brief Review of Single Base Results Greedy Interference Avoidance and The Eigen-Algorithm Greedy Interference Avoidance for Multiaccess Vector Channels Thesis Road Map Greedy SINR Maximization in Collaborative Multi-Base Systems Problem Statement Greedy SINR Maximization Through Distributed Codeword Adaptation Fixed Point Properties of the Greedy SINR Maximization Algorithm Making the Connection with Sum Capacity Maximization Numerical Results and Discussion Full Codeword Complement (N CodewordsPerUser) One Codeword Per User Chapter Summary Sum Capacity and TSC Bounds in Collaborative Multi-Base Systems Problem Statement Bounds on Sum Capacity vii

9 3.3. Bounds on GTSC Incorporating Carrier Phase Delays Chapter Summary A. Proof of Theorem A.1. The J = 2 case A.2. The Recursive Extension for J> B. Proof of Theorem Greedy Interference Avoidance in Non-Collaborative Multi-Base Systems The Greedy Interference Avoidance Algorithm Fixed Points and Social Optimum: Nash Equilibrium Chapter Summary Analyzing the Water Filling Distribution for Multiple Bases Simultaneous Water Filling Structures Complete Overlap Between Users Incomplete Overlap Between Users The case of k 1,k 2 <N The case of k 1 = N and k 2 <N No Overlap Between Users The Simultaneous Water Filling Region More Than Two User-Base Pairs Chapter Summary Weakly Interfering Systems: Simultaneous Water Filling Versus Signal Space Partitioning Simultaneous Water Filling: An Inefficient Nash Equilibrium The Social Optimum for a Multiple User/Base System A Distributed Splitting Algorithm Chapter Summary viii

10 7. Improving Performance for Strongly Interfering Systems Strong Interference and Simultaneous Water Filling A Procedure for Performance Improvement The Bouncing Region Chapter Summary Conclusions and Future Work Thesis Summary Future Directions References Vita ix

11 List of Figures 1.1. Multicode CDMA approach for sending frames of information. Each symbol in user l s frame is assigned a distinct signature waveform and the transmitted signal is a superposition of all signatures scaled by their corresponding information symbols A multibase system with B receiving bases and L transmitting locations, each location k using M k signatures. Triangles denote receivers and circles denote transmitters/users A system with two transmitters and two receivers Distribution of Eigenvalues of Covariance Matrices R 1 and R 2 Corresponding to Simultaneous Water Filling for a Two User-Base System with N =6,g 1 =0.2311, g 2 =0.6068, and η 0 = Distribution of Eigenvalues of Covariance Matrices R 1 and R 2 Corresponding to Simultaneous Water Filling for a Two User-Base System with N =6,g 1 =1.7643, g 2 =4.0658, and η 0 = Distribution of Eigenvalues of Covariance Matrices R 1, R 2,andR 3 Corresponding to Simultaneous Water Filling for a Three User-Base System Water Filling Distribution of Power From a Single User Perspective Simultaneous Water Filling Distribution of Power for Two Users Complete overlap between users Incomplete overlap between users Incomplete Overlap Between Users: The Nesting Case Complete separation between users Simultaneous Water Filling Region as a Function of Gain Simultaneous Water Filling Versus User Separation for g 1 g 2 < Simultaneous Water Filling For a System with Three User-Base Pairs x

12 5.10. Simultaneous Water Filling with Three Users Separated in the Signal Space Capacity variations as a function of user subspace width Water-filling and separation collective capacity as a function of gain and noise level for a symmetric system with M = 2 users Water-filling and separation collective capacity as a function of gain and noise level for a symmetric system with M = 5 users Water-filling and separation collective capacity as a function of gain and noise level for a symmetric system with M = 100 users The splitting algorithm for M = 3 users. In step k = 1, users 1, 2 and 3 each retreat from half the signal space which half is arbitrary but shown simply-connected for clarity. In step k = 2, users 1 and 2 then retreat from one third of their current occupancy which in this case is only a portion of that which they share with user 3. User 3 then retreats from the dimensions which overlap users 1 and 2 to fully complete the process The strong interference case. A user is closer to the base to which it produces interference, than to the base to which its transmission is intended The variation of r max in equation (7.6) as a function of number of users and noise level for P = N = 100. Peak value for η 0 =0.5, 1, 5, 10 corresponds to n =2, 4, 20, User 1 s attack for dimension k shared with user User 1 s attack for dimension k occupied by user 2 only Competing for maximum signal space occupancy Capacity variations during the learning game for g =2,P = N = 100, η 0 = Signal space partition for two orthogonal users Comparing the capacity performances between collaborative and non-collaborative scenarios xi

13 1 Chapter 1 Introduction Recent developments in telecommunications, in particular the emergence of software radios [1, 2, 28, 29, 50, 51], and the release of unlicensed spectrum for the Unlicensed National Information Infrastructure (U-NII) [10], have opened new research directions in the area of wireless systems. The versatility of transmitters and receivers implied by software radios, which allow adaptation of modulation and demodulation methods for transmitters and receivers in a wireless system, has generated increased interest in adaptive modulation techniques, in particular in the area of codeword optimization for CDMA systems [12,18,23,25,56 60]. In addition, operation in unlicensed environments where mutually interfering independent systems will have to coexist with no central control or coordination, poses additional challenges from a spectrum management perspective in order to achieve efficient use of the available shared bandwidth. This thesis provides insight into application of interference avoidance methods to systems with multiple transmitters and receivers, which consist of a collection of base stations and associated users that share the same bandwidth for communication. Interference avoidance is a class of adaptive modulation techniques in which, with feedback from the receiver, a transmitting radio is instructed to vary its waveform so as to maximize the signalto-interference plus noise-ratio (SINR). For an ensemble of users connected to a single receiver, which corresponds to the uplink of a single base wireless system, interference avoidance methods have been thoroughly analyzed [31, 38, 45, 46], and it has been shown that they lead to optimized use of the shared medium, which can be reached in a distributed fashion, through individual greedy optimization by each user. From an abstract signal space perspective, the socially optimal solution for single base systems corresponds to a simultaneous water filling distribution of user powers [63, 64], in which each user performs a water filling of its corresponding signal space while regarding all other users in the

14 2 system as noise. A simultaneous water filling solution can be achieved for multiple base systems as well, either by application of interference avoidance, or through alternative methods like iterative water filling [63,64]. The thesis provides an in-depth analysis of simultaneous water filling for multi base systems, and shows that, unlike single base systems, this does not always correspond to a socially optimal solution. 1.1 Interference Avoidance: A Brief Review of Single Base Results Interference avoidance methods allow users in a CDMA system to adapt their codewords, or signature sequences, to achieve better performance. The main criterion used in the codeword adaptation process is maximization of the signal-to-interference plus noise-ratio. Interference avoidance was originally introduced in the context of MMSE receiver filters [55, 56], but we concentrate our attention on greedy interference avoidance which uses matched filter receivers and was introduced in [45, 46] and explored in more detail in [31] Greedy Interference Avoidance and The Eigen-Algorithm In order to review the greedy interference avoidance procedure we consider the uplink of a synchronous CDMA system represented in an arbitrary N-dimensional signal space, in which each user l is assigned a unit norm N-dimensional codeword s l, to convey its information symbol b l. The received signal vector at the base station receiver is L r = b l s l + n = Sb + n (1.1) l=1 where S is the N L codeword matrix having the user codewords s l as columns, b =[b 1...b L ] is the vector containing the information symbols sent by users, and n is the additive noise vector that corrupts the received signal. The covariance matrix of the received signal is R = E[rr ]=SS + W (1.2) Assuming simple matched filters at the receiver for all users, the signal-to-interference plus noise-ratio (SINR) for user l is γ l = 1 s l R ls l (1.3)

15 3 with R l = R s l s l being the covariance matrix of the interference-plus-noise seen by user l. In this framework, greedy interference avoidance is defined by replacement of user l s codeword s l with the minimum eigenvector of R l. This procedure is referred to as greedy interference avoidance since by replacing its current codeword with the minimum eigenvector of the interferenceplus-noise correlation matrix, user l avoids interference by placing its transmitted energy in that region of the signal space with minimum interference-plus-noise energy and greedily maximizes SINR without looking at the potentially negative effects this may have on other users in the system. In addition to maximizing user l s SINR, greedy interference avoidance also monotonically increases sum capacity [31, 34, 38] C s = 1 2 log (det R) 1 log(det W) (1.4) 2 Sequential application by all users of this greedy SINR maximization procedure defines the eigenalgorithm for interference avoidance [38, 46], formally stated below: The Eigen-Algorithm 1. Start with a randomly chosen codeword ensemble specified by the codeword matrix S 2. For each user l =1...L replace user l codeword s l with the minimum eigenvector of the autocorrelation matrix of the corresponding interference-plus-noise process R l 3. Repeat step 2 until a fixed point is reached. Numerically, a fixed point of the eigen-algorithm is defined with respect to a stopping criterion. That is, we say that a fixed point is reached when the difference between two consecutive values of the stopping criterion is within a specified tolerance ɛ. The stopping criterion can be an individual one, like the codeword SINR or the Euclidian distance between codewords and their corresponding replacements, or a global one like sum capacity. We note that in the case of individual stopping criteria all values corresponding to all codewords must be within specified tolerance for the algorithm to stop. Mathematically, convergence of the eigen-algorithm to a fixed point is ensured by the

16 4 fact that the algorithm monotonically increases sum capacity, and that sum capacity is upper bounded. While this does not necessarily imply that the fixed point is unique, and theoretically many fixed points of the eigen-algorithm are possible, empirical evidence [46] has shown that when starting with randomly chosen codewords this fixed point is the optimal point where sum capacity C s is maximized. A thorough theoretical analysis of eigen-algorithm fixed point properties can be found in [45] along with a procedure to escape any suboptimal fixed points. Thus, the eigen-algorithm (with escape modifications) always converges to the optimal fixed point where the resulting codeword ensemble maximizes sum capacity. As a necessary by-product, the aggregate power distribution corresponds to a water filling distribution over those dimensions of the signal space with minimum noise energy. We emphasize that the water filling solution and the implied maximization of sum capacity are emergent properties of greedy interference avoidance, as individual users do not directly attempt to maximize sum capacity through an individual or ensemble water filling scheme, but rather, they greedily maximize the SINR of their own codeword. In fact, individual water filling schemes over the whole signal space are impossible in this framework since each user s transmit covariance matrix X l = s l s l is of rank one and cannot possibly span the N-dimensional signal space. We also note that the aggregate water filling of the signal space implies that all users achieve uniform maximum SINR [46], and that matched filters are optimal linear receivers in this case [58, 59] Greedy Interference Avoidance for Multiaccess Vector Channels Application of greedy interference avoidance can be extended to general multiaccess vector channels for which the received signal at the base station receiver is expressed (see [33, 38, 64]) L r = H l x l + n (1.5) l=1 where x l of dimension N l is the input vector corresponding to user l, l =1,...,L, r of dimension N is the received vector at the common receiver corrupted by additive noise vector n of the same dimension, and H l is the N N l channel matrix corresponding to user l. It is assumed that N N l, l =1,...,L. This is a general approach to a multiuser communication system in which different users reside in different signal subspaces, with possibly different dimensions and potential

17 5 User l symbols (frame) Serial to Parallel ( l ) b1 ( l ) b i x x ( l ) s 1 ( l) s i + x = l ( l ) ( l ) bi s i i Σ ( l) b Ml x ( l) s M l Figure 1.1: Multicode CDMA approach for sending frames of information. Each symbol in user l s frame is assigned a distinct signature waveform and the transmitted signal is a superposition of all signatures scaled by their corresponding information symbols. overlap between them, but all of which are subspaces of the receiver signal space. We note that each user s signal space as well as the receiver signal space are of finite dimension implied by a finite signaling interval T, and finite bandwidths W l for each user l, respectively W for the receiver (which includes all W l s corresponding to all users) [24]. We also note that for memoryless channels the channel matrix H l merely relates the bases of user l signal space and receiver signal space, but a similar model applies to channels with memory in which case the channel matrix H l also incorporates channel attenuation and multipath [34, 44, 64]. In this signal space setting we assume that during the signaling interval of duration T each user l sends a frame of data using a multicode CDMA approach wherein each symbol is transmitted using a distinct signature waveform which spans the frame. This scenario is depicted in Figure 1.1. That is, the sequence of information symbols b l =[b (l) 1...b(l) M l ] is transmitted as a linear superposition of distinct, unit-energy waveforms s (l) m (t) x l (t) = M l b (l) m s(l) m m=1 as if each symbol in the frame corresponded to a distinct virtual user. (t) (1.6) In the N l -dimensional signal space corresponding to user l, each waveform can be represented as an N l -dimensional vector, thus the input vector x l corresponding to user l is equivalent to a

18 6 linear superposition of unit norm codeword column vectors s (l) m scaled by the corresponding b (l) m. That is, each user uses an N l M l codeword matrix S l S l = s (l) 1 s (l) 2 s (l) M l so that (1.7) Therefore, the received signal can be rewritten as r = x l = S l b l (1.8) L H l S l b l + n (1.9) l=1 Note that under the assumption that M l N l,then l N l transmit covariance matrix of user l, X l = E[x l x l ]=S ls l, has full rank and spans user l s signal space. Extension of the greedy interference avoidance procedure to this general multiaccess vector channel setting is presented in [31,33,38]. The procedure starts by separating the interference-plusnoise seen by a given user k and rewriting the received signal in equation (1.9) from the perspective of user k as r = H k S k b k + L H l S l b l + n l=1,l k }{{} z k =interference + noise The covariance matrix of the interference-plus-noise seen by user k Z k = E[z k z k ]= L l=1,l k = H k S k b k + z k (1.10) H l S l S l H l + W (1.11) is then used to define a whitening transformation T k of the interference-plus-noise seen by user k. The equivalent problem in which user k sees white interference-plus-noise is then projected onto the user k signal space using the singular value decomposition (SVD) [53, p. 442] of user k s transformed channel matrix. This reduces the problem to an equivalent one given by an equation identical in form to equation (1.1) which allows straightforward application of greedy interference avoidance. In this case numerous interference avoidance algorithms can be formulated based on repeated application of the greedy interference avoidance procedure presented in the previous section. These

19 7 are defined by the various ways in which user codewords are selected for replacement. For example, one algorithm could be defined by replacement at a given step of one codeword of a given user, followed by replacement of a randomly selected codeword of a randomly selected user. Alternatively, at a given step of the algorithm, one could replace the codeword with the lowest SINR over all codewords and users. Or one could replace the codeword which will yield the maximum increase in sum capacity. We note that, since greedy interference avoidance monotonically increases sum capacity, all these procedures are guaranteed to reach a fixed point. Furthermore, when users have at least as many codewords as signal space dimensions, this fixed point reached by interference avoidance invariably corresponds to a simultaneous water filling solution for all users [31 33, 38] and corresponds to an ensemble of codewords which maximizes sum capacity [63, 64]. We conclude our review of interference avoidance by noting that greedy interference avoidance is a versatile tool for codeword optimization in CDMA systems, and can be applied for various scenarios. Among these we mention codeword optimization in the uplink of a CDMA system with non-ideal (dispersive) channels, multiuser systems with multiple inputs and outputs (MIMO), and asynchronous CDMA systems [34 37]. 1.2 Thesis Road Map For each chapter, a detailed introduction, literature review about the specific problem, and a statement of the problem are presented in the begining in an attempt to make chapters more independently readable. In Chapter 2 we present a codeword adaptation algorithm for collaborative multi-base wireless systems. The system is modeled with multiple inputs and multiple outputs (MIMO) in which information is transmitted using multicode CDMA, and codewords are adapted based on greedy maximization of the SINR. The procedure monotonically increases sum capacity, and repeated iteratively for all codewords in the system converges to a fixed point. Fixed-point properties and a connection with sum capacity maximization, along with a discussion of simulations that corroborate the basic analytic results, are also included in Chapter 2. In Chapter 3 we continue with the collaborative scenario and derive bounds on sum capacity and

20 8 total squared correlation in the case of flat channels between users and bases. We also investigate structural properties which must be satisfied by user transmit covariance matrices at the optimal sum capacity/tsc point, and show that for multi-base systems maximizing sum capacity and minimizing TSC are in general not equivalent problems. A new mathematical result on determinant maximization for a special class of block partitioned matrices is also presented in Chapter 3. Application of greedy interference avoidance in a non-collaborative scenario is presented next in Chapter 4. We start with a very simple system consisting of two bases, each base having only one user associated with it, for which we apply greedy interference to adaptation of user codewords. Fixed points of the greedy interference avoidance algorithm correspond to a simultaneous water filling solution which is also a Nash equilibrium for the system, and in general is not unique. A detailed analysis of simultaneous water filling is presented in Chapter 5 where we seek to analytically understand properties of different fixed points that correspond to a simultaneous water filling distribution. We show that water filling alone does not generally result in optimum resource sharing and in some cases is a poor solution. In Chapter 6 we continue our analysis of a non-collaborative system and focus on the weak mutual interference case for which the simultaneous water filling solution is unique. We show that when users at other bases are treated as Gaussian noise, simple separation of users in signal space usually offers much better performance than simultaneous water filling, and present a distributed algorithm which iteratively moves users toward greater separation in the signal space. In Chapter 7 we follow the analysis of the strong interference case for which the simultaneous water filling solution implies not an unique fixed point but a set of fixed points, among which the information theoretic capacities of users vary widely. We propose a dynamic game to move the system from a possible suboptimal point to a better simultaneous water filling fixed point, eventually to the optimal point. The results in this thesis have been presented in part at previous conferences. The work on interference avoidance and in a collaborative scenario in Chapters 2 and 3 was presented in part at the 39 th Annual Allerton Conference on Communication, Control, and Computing [39, 40]. Maximizing the determinant for a special class of block-partitioned matrices is a new mathematical result that will appear in [42]. The analysis of simultaneous water filling for non-collaborative

21 9 systems in Chapter 5 was presented at the 2003 IEEE Global Telecommunications Conference GLOBECOM 03 [41], and the strong interference case was presented at the 38 th Conference on Information Sciences and Systems CISS 04 [43].

22 10 Chapter 2 Greedy SINR Maximization in Collaborative Multi-Base Systems We consider a wireless communication system with multiple transmitters and receivers geographically distributed over some area as described schematically in Figure 2.1. We assume that the available spectrum is shared by all users and bases, as is the case in unlicensed bands, and make the simplifying assumption that receivers are allowed to share information using a sufficiently high speed backbone. As a specific example, imagine an abstraction of WiFi/ access points which can share information over a high speed landline to do joint decoding. This implies that the system under consideration can be regarded as a system with multiple inputs and multiple outputs (MIMO), unlike the usual cellular scenario in which users are assigned to bases based on a quality of service criterion (like the signal-to-interference-plus-noise-ratio SINR for example). However, unlike most MIMO systems where individual antenna elements are assumed co-located, here they are geographically dispersed and cannot share power budgets. We note that in a usual cellular system, when no cooperation among users/bases is assumed, the problem of decoding one user at its associated base station under interference generated by all the other users in the system is an instance of the interference channel [13, Ch. 14], and is still a mostly open research problem. The collaborative scenario considered here provides upper bounds on various measures of interest as one can do no better than jointly decode, and has been treated by researchers in previous work dealing with systems with multiple transmitters and receivers [17, 20, 54]. While unusual in the context of current multiple base cellular communications systems where, in general, bases do not share information, the availability of relatively low-cost high-speed terrestrial links makes collaboration practicable in future generation wireless systems. We present a codeword adaptation algorithm for collaborative multi-base wireless systems based on greedy SINR maximization. A multicode CDMA approach is used for transmitting information, and the algorithm is based on selfish optimization of individual SINRs. We show that, in addition

23 11 S NxM 1 1 B 2 S NxM L L B 1 NxM 4 S 4 B B S NxM 2 2 NxM 3 S 3 Figure 2.1: A multibase system with B receiving bases and L transmitting locations, each location kusingm k signatures. Triangles denote receivers and circles denote transmitters/users. to maximizing SINRs at each step, the algorithm monotonically increases sum capacity, which is a global criterion. Fixed-point properties of the proposed algorithm are investigated, and the connection with sum capacity maximization is made. We note that the optimal fixed point of the algorithm corresponds to maximum sum capacity and optimal codeword ensembles satisfy a simultaneous water filling solution [64]. However, the algorithm is not a water filling procedure, but a codeword adaptation one, and in the most general scenario replacement of one codeword of one user is followed by replacement of another codeword of a different user. In fact, when the number of codewords assigned to users is such that the transmit covariance matrices do not have full rank, then water filling schemes may not even be applicable, since maximization of sum capacity under trace and rank constraints on user transmit covariance matrices is no longer a convex optimization problem and does not enjoy global convergence properties [61]. We have also performed simulations to corroborate our analytical results. 2.1 Problem Statement The system model is depicted in Figure 2.1 and consists of B base stations that are situated in a given geographical area, and L users transmitting from various locations within the same region. We assume a common signal space representation of dimension N for all users/bases implied by

24 12 finite bandwidth and finite signaling interval constraints [24]. In this signal space setting we assume that during each signaling interval of duration T users transmit frames of data using a multicode CDMA approach in which each symbol in a given user s frame is assigned a distinct signature (codeword), and the transmitted signal is a superposition of all the codewords scaled by their corresponding information symbols, as described schematically in Figure 1.1. Thus, each user l at a given location transmits the signal where x l = M l m=1 S l = s (l) m b (l) m = S l b l, l =1,...,L (2.1) s (l) 1 s (l) m s (l) M l (2.2) is the N M l codeword matrix corresponding to user l, andb l =[b (l) 1...b(l) M l ] is the vector containing the information symbols transmitted by user l that are assumed uncorrelated, with zero mean and unit variance. We note that the model is general and allows coexistence of users with different data rates, since the number of symbols in a frame transmitted during a signaling interval of duration T may not be the same for all users. All signature sequences are assumed to have unit energy, s (l) m =1, m =1,...,M l,l =1,...,L, and user transmit power, which is the same for all symbols in the frame and should appear as a scalar multiplying the codeword matrix, is incorporated in the N N gain matrix G lj which characterizes the vector channel between user l and base station j. In general, this gain matrix incorporates channel attenuation and multipath [64]. The received signal at base station j is L r j = G lj S l b l + w j j =1,...,B (2.3) l=1 where w j is an additive Gaussian noise term with covariance matrix W j. Assuming collaboration, the information received at all base stations is pooled, forming a BN-dimensional received vector r 1 L G l1 w 1. =. S l b l +. (2.4) r B } {{ } r l=1 G lb } {{ } G l w B } {{ } w

25 13 with correlation matrix L R = E[rr ]= R(l)+W (2.5) where matrix R(l) represents user l s contribution to R and is expressed in terms of its codeword and corresponding gain matrices as l=1 R(l) =G l S l S l G l (2.6) and W is the covariance matrix of the resulting noise vector w. Thuswehave L R = G l S l S l G l + W (2.7) l=1 Under Gaussian signaling and noise assumptions, sum capacity for the multi-base system with collaboration in equation (2.4) is given by [20] Csum = 1 2 log R 1 log W (2.8) 2 which is identical to the sum capacity of a MIMO system described by a similar equation [64]. Our objective is to define a codeword adaptation algorithm which iteratively updates codewords of all users in the system until an optimal ensemble of codewords is obtained. While our algorithm is greedy and based on maximization of the SINR for individual codewords, it also monotonically increases sum capacity, and in so doing leads the system toward the socially optimal ensemble corresponding to maximum sum capacity. 2.2 Greedy SINR Maximization Through Distributed Codeword Adaptation For the MIMO system in equation (2.4) we assume linear filtering at the receiver, and would like to adjust codewords in the system, one at a time, so that their SINR is maximized. This is a two-step process, in which we first derive the expression of the linear filter which yields maximum SINR for a given codeword, and then look to replace the codeword with a new one which will increase the SINR. A similar approach was used for a single base system in [56]. However, the algorithm in [56] updates codewords by replacing them with the current maximum SINR filter, as opposed to choosing a codeword and a filter which absolutely maximize the SINR as we are doing here.

26 14 We simplify our notation and express the covariance matrix in equation (2.7) in terms of individual codewords s i, rather than user codeword matrices, as R = M M G i s }{{} i s i G i + W = y i yi + W (2.9) y i i=1 i=1 where M = L l=1 M l is the total number of codewords in the ensemble, and the gain matrix G i will be equal for all the codewords of a given user under the multicode assumption. This is because in the most general scenario one may replace one codeword of a given user followed by replacement of one codeword of a different user. Thus, it is not the user index which is relevant in the update process, but the codeword index in the ensemble. We denote by c i the NB-dimensional linear filter associated with the received vector y i = G i s i which implies that the SINR for codeword s i is γ i = (c i y i) 2 M (c i y k ) 2 + E[(c i n) 2 ] k i = c i c i y iyi c i [ R y i yi ] c i = c i y i y i c i c i R ic i (2.10) where R i = R y i y i. Since R i is positive definite 1 we can consider an eigen-decomposition R i = Φ i Λ i Φ i, and define a new vector z i = Λ 1/2 i Φ i c i, such that c i = ΦΛ 1/2 i z i. This implies that the SINR for codeword s i can be rewritten as γ i = z i Λ 1/2 i Φ i y iyi Φ iλ 1/2 i z i z i z (2.11) i which represents the Rayleigh quotient of a rank one matrix and is maximized when z i = Λ 1/2 i Φ i y i. Thus, the SINR maximizing linear filter c i is which is an MMSE type filter [26] and for which the SINR value is c i = R 1 i y i = R 1 i G i s i (2.12) γ i = yi R 1 i y i = s i G i R 1 i G i s i (2.13) We now note that γ i is maximized when s i is the eigenvector x i corresponding to the maximum eigenvalue (maximum eigenvector for short) of G i R 1 i G i. Thus, replacement of codeword s i by 1 R i contains the noise covariance matrix W which is positive definite; otherwise, the capacity will be infinite.

27 15 x i maximizes the SINR of codeword i which becomes γ i = x i G i R 1 i G i x i γ i (2.14) Applying this procedure iteratively for all codewords in the ensemble defines a codeword adaptation algorithm which is formally stated below: Greedy SINR Maximization Algorithm 1. Initialize codewords {s i } and gain matrices G i 2. For each codeword in the ensemble i =1,...,M do Replace codeword s i with the maximum eigenvector of the matrix G i R 1 i G i 3. Repeat step 2 until a fixed point is reached. Numerically, a fixed point of the algorithm is defined with respect to a stopping criterion. That is, we say that a fixed point is reached when the difference between two consecutive values of the stopping criterion is within a specified tolerance ɛ. The stopping criterion can be an individual one, like the codeword SINR or the Euclidian distance between codewords and their corresponding replacements, or a global one like sum capacity. We note that in the case of individual stopping criteria all values corresponding to all codewords must be within the specified tolerance for the algorithm to stop. Mathematically, convergence of the greedy SINR maximization algorithm to a fixed point is ensured by the fact that the algorithm monotonically increases sum capacity, and that sum capacity is upper bounded. This does not necessarily imply that the fixed point is unique, and theoretically many fixed points of this algorithm are possible. However, extensive simulations have shown that the algorithm has always reached the maximum sum capacity point when starting with randomly selected codewords, though we were not able to prove this result in general. In order to see that the proposed greedy SINR maximization procedure monotonically increases sum capacity, we consider the determinant of R in the sum capacity expression in equation (2.8) which we write in terms of codeword s i as M R = G k s k s k G k + W + G is i s i G i k i = R i + G i s i s i G i (2.15)

28 16 Since R i is invertible it can be factored out [ R = R i I BN + R 1 G i s i s i G i i ] (2.16) which implies that R can also be written as R = R i I BN + R 1 G i s i s i G i i ( ) = R i 1+s i G i R 1 i G i s i (2.17) where the last equality follows from I k + AB = I m + BA, A M k m B M m k (2.18) From equation (2.14) we obtain R = R i (1 + γ i ) (2.19) which shows that each iteration monotonically increases R. This in turn implies a monotonic increase in sum capacity, and because sum capacity is upper bounded, this proves our claim that the greedy SINR maximization algorithm will reach a fixed point. Properties of this fixed point as well as a connection to maximizing sum capacity are presented in the following sections. To conclude, we note that the formal statement of the greedy SINR maximization algorithm does not impose a particular order on codeword adaptation, and in the most general case, replacement of one codeword of one user is followed by replacement of one codeword of a different user. We also note that the proposed SINR maximization algorithm is a greedy interference avoidance algorithm [46] since at each step it greedily maximizes the SINR of an individual codeword without paying attention to the consequences that this action may have on the ensemble of codewords. 2.3 Fixed Point Properties of the Greedy SINR Maximization Algorithm Let {λ i j } be the set of eigenvalues for the matrix G i R 1 i G i in decreasing order, λ i 1 λi 2... λi N, with {x i j } the corresponding eigenvectors. At a fixed point of the greedy SINR maximization algorithm any further change in user codewords brings no improvement in the SINR values, which will be equal to the maximum eigenvalues γ i = λ i 1. The codewords will be the maximum eigenvectors of G i R 1 i G i,thatiss i = x i 1,andwecanwrite G i R 1 i G i s i = λ i 1s i = γ i s i (2.20)

29 17 In addition, any eigenvector of G i R 1 i G i is also an eigenvector of G i R 1 G i.inordertoseethis we start by writing and using the matrix inversion lemma we get R 1 = R 1 i R = R i + G i s i s i G i = R i + G i x i 1(x i 1) G i (2.21) ( ) 1 R 1 i G i x i 1 1+(x i 1 ) G i R 1 i G i x i 1 (x i 1 ) G i R 1 i (2.22) G i R 1 G i x i j = λ i jx i j γ2 i 1+γ i x i 1(x i 1) x i j = where δ ij is the Kronecker delta operator which is 1 for i = j and 0 for i j. ( ) λ i j (λi j )2 1+λ i δ 1j x i j (2.23) j Thus, at a fixed point, the matrices G i R 1 i G i and G i R 1 G i share the same set of eigenvectors. In addition, they have the same eigenvalues with the exception of the one that corresponds to s i, for which we have G i R 1 i G i s i = γ i s i and G i R 1 G i s i = γ i 1+γ i s i (2.24) We now consider all codewords of a given user l, which share the same gain matrix G l. If, at a fixed point of the algorithm, they have different SINRs, then they must be orthogonal since they are eigenvectors of the same matrix G l R 1 G l corresponding to different eigenvalues. More precisely, if s (l) m and s (l) n are two distinct codewords corresponding to user l but with different SINRs γ (l) m γ n (l),thens (l) m s (l) n. Alternatively, when all codewords of a given user have the same SINR at a fixed point, then user l s codeword matrix satisfies G l R 1 G l S l = γ l 1+γ l S l (2.25) and γ l /(1 + γ l ) is the maximum eigenvalue of G l R 1 G l. To conclude this section we note that empirically, we have always observed convergence of the algorithm from random initial points to a fixed point in which all codewords of a given user have the same SINR. 2.4 Making the Connection with Sum Capacity Maximization We have seen in section 2.2 that the greedy SINR maximization algorithm for codeword adaptation monotonically increases sum capacity. We have also noted in section 2.2 that the algorithm is

30 18 a greedy interference avoidance algorithm. Putting this observation together with the fact that previously proposed interference avoidance algorithms converge to a codeword ensemble that maximizes sum capacity [4, 33, 45, 46] makes us suspect that our proposed greedy SINR maximization algorithm for collaborative multibase systems also yields sum capacity maximizing codeword ensembles. Thus, in this section we examine some properties of codeword ensembles which maximize sum capacity and relate them to the fixed point properties of the proposed algorithm. Maximization of sum capacity for a general multiaccess vector channel was solved in [64] as a spectral optimization problem where it has been shown that optimal user transmit covariance matrices X l can be obtained as a solution of the following convex optimization problem max X l Csum subject to Trace [X l ]=const.,l=1,...,l (2.26) If we assume that the noise is stationary with fixed covariance matrix W, then equation (2.8) shows that maximizing sum capacity is equivalent to maximizing R. According to [64] this implies a simultaneous water filling distribution for all users, for which eigenvectors of the interference plus noise covariance matrix seen by a given user l align with those of its transmit covariance matrix X l whose eigenvalues satisfy a water filling distribution [13, p. 253]. We note that optimal transmit covariance matrices may be obtained through an iterative water filling procedure [64], and then subsequently used to construct optimal codeword ensembles that maximize sum capacity using established algorithms [57, 58]. When additional rank constraints are imposed on user transmit covariance matrices, these can no longer be obtained by solving a convex optimization problem and using the iterative water filling procedure. Maximizing sum capacity subject to trace and rank constraints on user transmit covariance matrices is currently an open problem [61]. In our formulation, user transmit covariance matrices are expressed in terms of codeword matrices as X l = S l S l, and when users are assigned at least M l = N codewords for transmission, then S l S l is not rank constrained and may span the entire signal space. In this case, optimal S ls l must satisfy a simultaneous water filling distribution as described in the previous paragraph. When users are assigned M l <N codewords for transmission, then S l S l is rank constrained. However, from the perspective of the simultaneous water filling solution [64] at the maximum sum capacity point, R will be maximized when matrices S l S l satisfy a water filling distribution on a lower dimension eigensubspace containing the smallest eigenvalues of the interference plus noise covariance matrix