A Game-theoretic Approach to Power Management in MIMO-OFDM. Ad Hoc Networks. A Dissertation. Submitted to the Faculty. Drexel University.

Transcription

1 A Game-theoretic Approach to Power Management in MIMO-OFDM Ad Hoc Networks A Dissertation Submitted to the Facuty of Drexe University by Chao Liang in partia fufiment of the requirements for the degree of Doctor of Phiosophy in Eectrica and Computer Engineering September 2006

2 c Copyright September 2006 Chao Liang. This work is icensed under the terms of the Creative Commons Attribution-ShareAike icense. The icense is avaiabe at creativecommons.org/icenses/by-sa/2.5/.

3 ii Dedications For Mom and Dad

4 iii Acknowedgements I fee very fortunate to have met a the peope I have met during the journey to my Ph.D. I have earned a ot from these incredibe individuas. Foremost, my dissertation advisor, Dr. Kapi Dandekar has my sincerest gratitude for his guidance and support throughout the past four years. Dr. Dandekar is a great motivator and inspirer and I have benefited tremendousy from his energy, vision and knowedge. I aso thank him for giving me the freedom to investigate my own interests and for constanty encouraging my academic growth. My thanks aso extend to my Ph.D. candidacy exam committee members, Dr. Jaudeice De Oiveira, Dr. Stanisav Keser, Dr. Ruifeng Zhang and Dr. Herman Gowitzer, and my dissertation defense committee members, Dr. Moshe Kam, Dr. Steve Weber, Dr. Wiiam Regi and Dr. Nagarajan Kandasamy. Thank you a for your vauabe comments and encouragement. I woud ike to thank a my coeagues at Drexe University who have contributed greaty to provide a supportive and coaborative research atmosphere. Chapter 2 can trace back to co-work with Sumant Kawae and the channe measurement section in Chapter 4 woud not have been possibe without working with Nichoas Kirsch. Many thanks to Matt Garfied, John Kountouriotis, Abhishek Khemka, Danie Piazza and Marae Fakhereddin, with whom I have had opportunities to coaborate on various subjects. I woud ike to sincerey thank my parents for their support, encouragement and ove throughout my ife. This dissertation is dedicated to them.

5 iv Tabe of Contents List of Figures... vi Abstract... viii 1. Introduction Introduction to MIMO Introduction to OFDM Power Management in Wireess Communication Systems Outine of Dissertation Notations Power Management in MIMO Ad Hoc Networks - A Game Theoretic Approach Probem Formuation Existing techniques to cacuate system mutua information Independent Water-fiing Mutiuser Water-fiing Gradient Projection Method Game theoretic approach to power contro Game Formuation Utiity Function Design Game Theoretic Approach Game Anaysis Simuation Resuts Summary Power Management in MIMO Ad Hoc Networks - A Soft Shut-down Mechanism Power Contro Game with a Soft Shut-down Mechanism Utiity Function Design Game Formuation Iterative Power Contro with Game Theory Impication of Pricing Factor γ on Power Aocation... 38

6 v 3.2 Game Anaysis Simuation Resuts with a Soft Shutdown Mechanism Summary Power Contro for MIMO-OFDM Ad Hoc Networks Probem Formuation Signa Mode Mutua Information Power Distribution and Subcarrier Assignment Power distribution for a fixed subcarrier aocation A Subcarrier Assignment Scheme Iustrative Exampe Resource Aocation in MIMO-OFDM Ad Hoc Networks - A Game Theoretic Approach Utiity Function and Game Formuation Reation between subcarrier assignment and choice of γ (k) Game Agorithm Game Anaysis Numerica Resuts Based on Channe Measurements Summary Concusion Bibiography... 76

7 vi List of Figures 2.1 Water-fiing power aocation z i = (µ σ 1 i ) Iterative descent for maximizing a utiity function [1] g(c ) Iustration of the game theoretic approach with a mechanism for shutting down inks (A) Situation in which ink 1-2 shoud reduce transmit power (B) Situation in which ink 1-2 shoud be shut off Iustration of Phiadephia downtown simuation. Node numbers are not circed and ink numbers are circed Sum data rate of different methods Energy efficiency of different methods System capacity and power consumption vs η Power aocation vs γ Power aocation vs γ (a specia case) Power aocation vs γ : a numerica exampe Sum data rate of different methods Energy efficiency of different methods System capacity and power consumption vs η Non-overapping subcarrier assignment Ratio of data rate (C (D )/C (N)) Subcarrier aocation of a inks (β = 0.7, SNR=20dB) Subcarrier aocation of a inks (β = 0.8, SNR=20dB) Subcarrier aocation of a inks (β = 0.9, SNR=20dB) Subcarrier aocation of a inks (β = 0.95, SNR=20dB)... 65

8 vii 4.7 Indoor measurement testing environment and node ocations Link data rate using game theoretic technique Link data rate using mutiuser water-fiing Sum data rate of GGT and MUWF Sum data rate versus β in the GGT method Energy efficiency β in the GGT method... 73

9 viii Abstract A Game-theoretic Approach to Power Management in MIMO-OFDM Ad Hoc Networks Chao Liang Advisor: Kapi R. Dandekar, Ph.D. With the increasing demand for wireess services, the efficient use of spectra resources is of great importance. MIMO-OFDM communication systems hod great promise in using radio spectrum efficienty whie power contro wi improve energy efficiency. Existing approaches such as mutiuser water-fiing and gradient projection assign a fixed transmit power to each ink and each transmitter node aocates power among different antennas in order to optimize the ink capacity or sum data rate. If bad channe conditions exist in some communicating inks, these methods are not energy efficient. We propose a new technique for power management and interference reduction based upon a game theoretic approach. Utiity functions are designed and power aocation in each ink is buit into a non-cooperative game. To avoid unnecessary power transmission under poor channe conditions, a mechanism of shutting down inefficient inks is integrated into the game theoretic approach. Two kinds of ink shut-down mechanism are presented in this dissertation. The first one is caed hard shut-down, because once the transmit node decides to shut down, the node wi not resume transmission no matter how the interfering channes change. The other mechanism is caed soft shut-down, in which the transmit power is reated to the pricing factor of that ink and the interference it is exposed to. With this mechanism, the transmit power can change adaptivey in response to the condition of interference. We aso investigate the probem of subcarrier assignment and power distribution among mutipe antennas for point-to-point inks in a network without base stations. A subcarrier assignment scheme is proposed which seects a set of subcarriers for each ink so that high data rate can be achieved and co-channe interference can be mitigated. The power management in a MIMO-OFDM ad hoc network is aso buit into a non-cooperative game

10 ix in which each ink cacuates its optima power aocation vector in order to maximize the net utiity. The designed utiity function faciitates subcarrier assignment schemes by using a tunabe pricing factor, which heps a ink to admit or drop subcarriers in a soft and adaptive fashion.

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12 1 1. Introduction 1.1 Introduction to MIMO Demands for capacity in wireess communications, driven by ceuar mobie, Internet and mutimedia services have been rapidy increasing wordwide. On the other hand, the avaiabe radio spectrum is imited and the communication capacity needs cannot be met without a significant increase in communication spectra efficiency. Significant further advances in spectra efficiency are avaiabe through increasing the number of antennas at both the transmitter and the receiver [13][34]. The benefits of expoiting MIMO can be categorized by the foowing : Array gain Array gain refers to the average increase in the SNR at the receiver that arises from the coherent combining effect of mutipe antennas at the receiver or transmitter or both. The average increase in signa power at the receiver is proportiona to the number of receive antennas. Diversity gain Signa power in a wireess channe fuctuates. When the signa power drops significanty, the channe is said to be in a fade. Diversity is used in wireess channes to combat fading. Utiization of diversity in MIMO channes requires antenna diversity at both receive and transmit side. The diversity order is equa to the product of the number of transmit and receive antennas, if the channe between each transmit-receive antenna pair fades independenty.

13 2 Spatia mutipexing (SM) SM offers a inear (in the number of transmit-receive antenna pairs or min(m r, M t )) increase in the transmission rate for the same bandwidth and with no additiona power consumption. Interference reduction Co-channe interferece arises due to frequency reuse in wireess channes. When mutipe antennas are used, the difference between the spatia signatures of the desired signa and co-channe signas can be expoited to reduce the interference. This operation is done at the receiver side and it requires knowedge of the channe of the desired signa. Interference reduction can aso be done at the transmitter, where the goa is to minimize the interference energy sent towards the co-channe users whie deivering the signa to the desired user. 1.2 Introduction to OFDM Orthogona Frequency Division Mutipexing (OFDM) has become a popuar technique for transmission of signas over wireess channes. OFDM is a muticarrier transmission technique, which divides the avaiabe spectrum into many subcarriers, each one being moduated by a ow-rate data stream. It efficienty uses the spectrum by spacing the channes coser together. This is achieved by making a the subcarriers orthogona to one another, preventing interference between cosey spaced carriers. OFDM converts a frequency-seective channe into a parae coection of frequency fat subchannes. A key advantage of OFDM systems is their inherent capabiity to aow adaptive resource aocation where the eve of moduation, the number of bits oaded and the transmit power in each subcarrier, can be seected to increase the data rate or reduce the required transmit power [6]. For instance, if knowedge of the channe is avaiabe at the transmitter, then the OFDM transmitter can adapt its signaing strategy to match the channe and the idea water-fiing capacity of a frequency-seective channe can be approached [33].

14 3 As it is we known that the MIMO technique utiizes spectrum efficienty and enhances energy efficiency, the hybrid design of MIMO and OFDM further enabes the diversities from spatia, tempora and spectra domains, which can ead to significant improvement in system performance. 1.3 Power Management in Wireess Communication Systems Power contro pays a key roe in improving energy efficiency of wireess communication systems. It is important to identify ways to use ess power whie maintaining a certain quaity of service (QoS). There has been a considerabe amount of research on power management in wireess systems. In [36, 41], power contro agorithms were deveoped for ceuar systems. Power contro has aso been studied with a combination of mutiuser detection, beamforming and adaptive moduation[38, 22]. In [21, 32] adaptive agorithms were deveoped to improve system performance by controing power aocation and data rate. As the use of MIMO technoogy in ad hoc networks grows, MIMO interference systems have attracted a great dea of attention. [40, 2] studied the interactions and capacity dependencies of MIMO interference systems and [37, 26] expored methods for power management and interference avoidance in MIMO systems. In recent years there has been a growing interest in appying game theory to study wireess systems. [29, 16] used game theory to investigate power contro and rate contro for wireess data. Their work studied mobie ceuar networks with transmitter and receiver pairs using Singe-Input-Singe-Output (SISO) antennas. The anaysis was based on the Signa to Interference pus Noise Ratio (SINR), which was a function of the transmit power of each individua transmitter. Since SISO antenna system was used, the controing variabe of each user was its transmit power which was a scaar. Hicks provided a game theory perspective on interference avoidance in [17]. A synchronous interference avoidance (IA) scheme was modeed as a potentia game. In addition, when the IA system s signa environment is eveabe, the noisy best response iteration amost surey converges for a game in which inks independenty choose from a set of

15 4 metrics given in that paper. A game-theoretic approach to study power aocation in MIMO channes was deveoped in [25]. This paper considers the case in which even the channe statistics are not avaiabe at the transmitter, obtaining a robust soution under channe uncertainty by formuating the probem within a game-theoretic framework. The payoff function of the game is the mutua information and the payers are the transmitter and a maicious nature. The probem turns out to be the characterization of the capacity of a compound channe which is mathematicay formuated as a maximin probem. The uniform power aocation is obtained as a robust soution. In practica wireess communication systems, transmitters try to obtain channe state information (CSI). It can be acquired either via a feedback channe or the appication of the channe reciprocity property to previous receive channe measurements when the transmit and receive channes are sufficienty correated. When CSI is avaiabe at the transmitter, the optima power aocation that achieves capacity is we known [9]. In such a case, capacity is achieved by adapting the transmitted signa to the specific channe reaization. To be more specific, the channe matrix is diagonaized and the set of constituent subchannes or eigenmodes is obtained. The optima signaing directions are the eigenmodes of the channe matrix and power is aocated preferaby to those eigenmodes with higher eigenvaues. In this dissertation, we wi assume transmitters instanty know channe conditions due to a no-deay feedback mechanism. With this information, the transmitters find out their power aocations based on water-fiing soutions. These soutions correspond to maximizing channe mutua information by using the maximum transmit power avaiabe. In a wireess ad hoc network with mutipe co-channe inks, the transmitter in a ink not ony sends data to its designated receiver, but aso causes interference to other inks. Some inks may have exceent channe conditions and accordingy support high data rate. However, if the capacity of a ink is more than enough to maintain a certain eve of QoS, reducing the capacity by decreasing transmit power wi mitigate the interference

16 5 sent to other inks, i.e., it is not necessary to transmit the maximum amount of power. In contrast, some inks may have the transmitter and the receiver nodes far apart with poor channe conditions, thus the data rates that these inks can support are ow even the maximum power is transmitted. Those ow-rate inks are not ony useess for data transmission but aso may bring down the data rates of other inks due to generated interference. To counter interference, other inks may aso increase their transmit power, which resuts in ow energy efficiency across the network. To avoid this negative effect, it is advisabe to shut down ow-efficiency inks. In order to accommodate the two scenarios described above, we can assign each ink a utiity function, which has an intrinsic power contro property. By maximizing its utiity function, each ink tries to reach a high data rate without necessariy transmitting the highest power. If even a minimum data rate cannot be supported by a ink, the ink shuts off in order to save power and reduce interference. In a wireess ad hoc network, there is no centra controer to determine the strategy of resource aocation for each ink. Instead, a ink acquires the information about channe state as we as interference, then makes its own decision how to aocate resource. This kind of interaction among wireess inks can be modeed as a non-cooperative game [15], where each ink attempts to sefishy maximize its utiity. If the utiity function is we-designed with the power contro property described before, there wi be impicit coordination among wireess inks so that metrics refecting socia preference such as energy efficiency or sum data rate of the network can be improved. 1.4 Outine of Dissertation In genera terms, this dissertation focuses on power management in MIMO-OFDM ad hoc networks. In Chapter 2, we consider a stationary MIMO ad hoc network, where each transceiver pair is hindered by cochanne interference coming from other transceiver pairs operating in the same frequency band. We investigate optimum signaing for MIMO interference systems with feedback in a reaistic ad hoc network environment and study how power contro improves energy efficiency by using a game theoretic approach. Com-

17 6 putationa eectromagnetic simuations [10] are used to study the effect of interference on a network composed of mutipe, cochanne MIMO inks. A game theoretic approach for power contro is proposed where we construct a non-cooperative power contro game and show how to design a utiity function suitabe for MIMO ad hoc networks. A mechanism of shutting down inefficient inks is incuded in the power management. Asymptotic behaviors of that power contro game are investigated as we. In Chapter 3, we point out that the previous chapter contains a hard ink shut-down mechanism, and the game anaysis based on that hard mechanism is ony appicabe to those viabe inks. In rea situations, the channe condition of a ink may change from time to time and the interference that ink experiences aso fuctuates, so it wi be beneficia to set up a mechanism which aows a ink to adaptivey contro its transmit power. This mechanism is expected to turn off a ink if its data rate is too ow, but aows a ink to transmit if channe condition has improved, therefore it is caed a soft shut-down mechanism. In this chapter, we carefuy design a new utiity function which is the ink data rate minus the scaed transmit power. The second term is considered a price charged for using resource. We wi prove in theory that a ink is actuay shut down if the pricing factor is propery chosen. The power management is again buit into a non-cooperative game and we wi investigate the existence of Nash equiibrium. In Chapter 4, power management in a game theoretic approach is extended to MIMO- OFDM ad hoc networks, where there are mutipe point-to-point wireess inks. Since every ink works in the same frequency band, co-channe interference (CCI) is a key factor to imit the data rate of each ink. Therefore, resource aocation has to be done with serious considerations in interfering inks. OFDM systems provide a degree of freedom to aocate power in each subcarrier. If different inks use a different subset of avaiabe subcarriers, interference experienced by each ink may be mitigated due to the fact that a particuar subcarrier may not be used by a inks. In this chapter, we investigate joint subcarrier assignment and power aocation for MIMO-OFDM ad hoc networks and a subcarrier assignment scheme is proposed. The power management in a MIMO-OFDM

18 7 ad hoc network is aso buit into a non-cooperative game and it is shown that the designed utiity function faciitates subcarrier assignment schemes by using a tunabe pricing factor, which heps a ink to admit or drop subcarriers in a soft and adaptive fashion. 1.5 Notations We wi try to remain consistent with notations throughout this dissertation. However, if inconsistence does occur, the notation shoud be cear from context or we wi define it immediatey. In this dissertation notations are used as foows: Lower case and bod face etters denote vectors, e.g., x, q, whose eements may be scaars, vectors or matrices. Upper case and bod face etters denote matrices or the set of matrices, e.g., H, Q. E{ } denotes expectation, f( ) denotes the gradient of f( ). For a matrix A, A T denotes transpose, A denotes the conjugate transpose, Tr(A) denotes the trace, det(a) denotes the determinant and A 0 denotes that A is positive semidefinite. A n-dimensiona identity matrix is denoted by I n or I in case it is sef-evident. R n + denotes the n dimensiona nonnegative orthant.

19 8 2. Power Management in MIMO Ad Hoc Networks - A Game Theoretic Approach With the increasing demand for wireess services, the efficient use of spectra resources is of great importance. Mutipe Input Mutipe Output (MIMO) communication systems hod great promise in using radio spectrum efficienty [14] whie power contro wi improve energy efficiency. In appications ike wireess ad hoc networks, battery ife is the argest constraint in designing agorithms [7]. Therefore, it is important that power aocation be managed effectivey by identifying ways to use ess power whie maintaining a certain quaity of service (QoS). In this chapter, we consider a stationary MIMO ad hoc network, where each transceiver pair is hindered by cochanne interference coming from other transceiver pairs operating in the same frequency band. It is known that minimizing interference using power contro increases capacity and extends battery ife for ceuar systems [29]. We investigate optimum signaing for MIMO interference systems with feedback in a reaistic ad hoc network environment and study how power contro improves energy efficiency by using a game theoretic approach. We use computationa eectromagnetic simuations [10] to study the effect of interference on a network composed of mutipe, cochanne MIMO inks. These simuations, given a network topoogy and environment, cacuate the received eectromagnetic fieds due to a of the mutipath rays between every transmitter and every receiver. The simuations are performed using the software system FASANT, which has been used as a too in system panning and has been vaidated using urban propagation measurements [5]. This chapter is organized as foows. Section 2.1 introduces the system mode and formuates the optimization probem. Existing techniques from the iterature are aso introduced as a basis for comparison. In Section 2.3, a game theoretic approach for power contro is proposed where we construct a non-cooperative power contro game and show

20 9 how to design a utiity function suitabe for MIMO ad hoc networks. Asymptotic behaviors of that power contro game are investigated as we. Simuation resuts with a methods are given and discussed in Section 2.4. Section 2.5 provides the summary. 2.1 Probem Formuation Consider an ad hoc network with a set of inks denoted by L = {1, 2,..., L}, where each ink undergoes cochanne interference from the other L 1 inks. Each node uses N t transmit antennas and N r receive antennas. The channe between the receive antennas of ink and the transmit antennas of ink j is denoted by H,j C Nr Nt. For a of the L inks, the transmitted signa vector, x C Nt 1 has covariance matrix Q = E{x x } and the receiver array performs independent singe-user detection. The received baseband signa of ink, y C Nr 1, is given by y = H, x + L j=1,j H,j x j + n (2.1) where n C Nr 1 is the noise vector with independent compex Gaussian entries. We aso ca Q a power aocation matrix with the transmit power for ink given by Tr(Q ). The instantaneous data rate of ink is obtained as [9] C (Q 1,..., Q L ) = og 2 det(i + H, Q H, R 1 ) (2.2) where R = I + L j=1,j H,jQ j H,j is the interference-pus-noise matrix of ink. The channe matrix H,j and R are cacuated by our computationa eectromagnetic simuations. In addition, due to an assumed no-deay channe feedback mechanism, the transmitters instanty know channe conditions. Each transmitter adjusts its power aocation in an effort to maximize its data rate. Power adjustment can be done in two ways. In the first technique, for a fixed transmit power of each node, the power is aotted among the mutipe transmit antennas to achieve capacity maximization. The second technique aows power contro for transmitters, i.e.,

21 10 the transmit power for a certain ink, p = Tr(Q ), can be adjusted. Using this power contro, the transmitter can foow two courses of action: it can change the tota power aotted to the ink and it can aso aot this power in different ways among the mutipe antennas of the ink. For an ad hoc network, the mutua information of the L-ink system given a channe matrices H 1,1,..., H L,L is C(Q 1,..., Q L ) = L =1 og 2 det(i + H, Q H, R 1 ) (2.3) For a metric of energy efficiency, we use the ratio of the system capacity over the tota power consumption. This metric corresponds to the amount of achievabe capacity per unit energy. λ = L =1 C L =1 p (2.4) 2.2 Existing techniques to cacuate system mutua information Different agorithms have been deveoped to cacuate system mutua information. The most commony used ones wi be introduced beow Independent Water-fiing This approach is optimum for the non-interference situation, where the transmitter pretends that there is no interference from other inks. For each ink, it is essentiay a singe-user system and the mutua information is given by C(Q) = og 2 det(i + HQH ) (2.5) and the power aocation matrix is subject to Tr(Q) p and Q 0, where p is the maximum transmit power. This singe-ink optimization probem has a we-known water-fiing soution [9]. Let

22 11 H H = VΣV be an eigenvaue decomposition of H H, with V unitary and Σ diagona, then the optimum power aocation is given as Q = V(µI Σ 1 ) + V where µ is caed the water eve and is chosen to satisfy Tr(µI Σ 1 ) + ) = p. This soution indicates that the optima signaing directions are the eigenmodes of the channe matrix and power is aocated preferaby to those eigenmodes with higher eigenvaues. Fig.2.1 iustrates how power is aocated reative to the eigenvaues and the water eve, where σ i s are the diagona eements of Σ and z i s are the diagona eements of (µi Σ 1 ) +. For a network system, if a inks are coordinated such that ony one ink is transmitting at a time, that ink does not undergo interference and therefore its mutua information can be maximized with an independent water-fiing soution. Another interesting scenario occurs in an OFDM-based MIMO system, where we assume perfect frequency synchronization is in pace. If a subcarriers are partitioned into non-overapping subsets and each ink in the network uses a subset of the subcarriers, this system is in fact interference-free as transmission on one frequency does not cause interference to another. Power μ Water Leve z2 z1 z 3 z5 z N t σ 1 σ 2 σ 3 σ σ σ Nt 1 1 σ N t N t 1 N t Subchanne (eigenmode). Figure 2.1: Water-fiing power aocation z i = (µ σ 1 i ) +

23 Mutiuser Water-fiing In a network with mutipe interfering inks, the mutua information of a ink is reated to the noise and interference matrix, which varies with the transmitter correation matrices of the interfering nodes. A change in the power aocation matrix of one ink induces a change in the optimum power aocation matrices of the other co-channe inks. Yu proposed an iterative water-fiing agorithm to compute the optima input distribution of a Gaussian mutipe access channe with mutipe vector inputs and a singe vector output so that the sum data rate of the channe is maximized [40]. In this method, each transmitter is assumed to know its own channe information H, as we as the noise and interference matrix R. It is shown that the sum data rate probem can be broken down into singe-user probems so that the power aocation matrices can be found iterativey. Specificay, at each iteration, transmitter tries to sove the foowing optimization probem. max Q og 2 det(i + H, Q H, R 1 ) s.t. Tr(Q ) p (2.6) Q 0 This probem has a simiar water-fiing soution to the interference-free case. Mathematicay, using the conversion H, R 1 H, = H 1, R 2 R 1 2 H, = (R 1 2 H, ) (R 1 2 H, ) = H H,, = U Σ U (2.7) with U unitary and Σ diagona, ink empoys the power aocation Q = U (µ I Σ 1 ) + U to maximize instantaneous mutua information and µ is chosen so that Tr(µ I Σ 1 ) + ) = p. Comparing the mutiuser water-fiing with the independent water-fiing, we can see that they are essentiay the same if in the former the substitution H, = R 1 2 H, is appied. This operation is caed spatiay whitening transform [11], since it simpifies an interference pus noise MIMO channe into a noise-ony MIMO channe as far as the mutua

24 13 information is concerned. In [40] the iterative water-fiing agorithm aways converges to the sum rate given the output is a singe vector. However, for an ad hoc network, each inks tries to maximize its own mutua inforamtion and the system mutua information is the sum data rate of a inks, where the number of output vectors is the same as the number of inks, thus convergence is not guaranteed. In fact, it is pointed out in [37] that this agorithm does not aways converge in the MIMO ad hoc network scenario Gradient Projection Method In the mutiuser water-fiing method every ink non-cooperativey competes with others so as to achieve the highest data rate individuay. A more compex probem is to maximize C(Q 1,..., Q L ), which is the sum data rate of the network. To achieve this, the transmitters need to cooperate in a certain way when deciding their covariance matrices. In [37], the gradient projection (GP) method [1] which is widey used as an unconstrained steepest descent method, is extended to convex constrained probems in order to sove the sum data rate probem. The main idea is: Each ink tries to maximize C(Q 1,..., Q L ) iterativey. When ink is optimizing the objective function, it assumes the power aocation matrices of a the other inks are fixed, thus ony Q is adjusted. Therefore, for the sake of simpifying notations, we denote the sum data rate with C(Q ). For ink to maximize the function C(Q ), where Q Φ and Φ={Q C Nt Nt Q is positive semidefinite, and Tr(Q) p } is a convex set, we start from an initia feasibe point Q (1) the vaue according to Φ, then update Q (k+1) = Q (k) (k) + α k ( Q Q (k) ) (2.8) where Q (k) = [Q (k) + s k C(Q (k) )] Φ (2.9) Here, 0 < α k 1 is the stepsize, s k > 0 is a scaar, C(Q (k) ) is the gradient of

25 14 C = (1) ( Q ) c1 C > (2) ( Q ) = c2 c1 C > (3) ( Q ) = c3 c2 (2) Q (4) Q... (3) Q (1) Q Figure 2.2: Iterative descent for maximizing a utiity function [1]. C(Q ) at the point Q (k), and [ ] Φ denotes the projection on the convex set Φ. In this method, Q (k) Q (k) is gradient reated, which guarantees that there exists some α k such that C(Q (k+1) ) > C(Q (k) ). Q (k) is updated iterativey such that C(Q ) is increased at each iteration (see Fig.2.2). According to the convergence anaysis in [1], if α k and s k are chosen propery, the GP method aways converges to a stationary point Q, which satisfies Tr(( C( Q )) (Q Q )) 0 for any Q Φ. This method has a much higher computationa compexity as it invoves cacuating gradients and projections, both on matrix variabes. With this method, the transmitters are assumed to cooperate with a centraized contro mechanism which has access to a of the channe state information and the covariance matrices of each user. 2.3 Game theoretic approach to power contro In the existing methods discussed above, each ink uses the maximum power avaiabe to it, thus the power contro is imited to spatia aocation among antenna eements. Athough transmitting the highest power can resut in high data rate, the energy efficiency might be ow. In this section, we wi propose a new technique for interference management

26 15 in MIMO ad hoc networks using a game theoretic approach, in an effort to achieve both good sum data rate and energy efficiency Game Formuation In the context of game theory, in adjusting its transmit power, each transmitter pursues a strategy that aims to maximize its utiity. If we assume the cooperation between wireess inks is not feasibe, the probem can be modeed as a non-cooperative game (NCG), where each ink is ony concerned about its own utiity rather than the system mutua information or system utiity. Modeing this optimization process as an NCG, the three components that are present in the game are the foowing: (1) Set of inks: This is the set that contains a the inks in the network. In the context of our network, we consider that a nodes are capabe of both transmitting and receiving. However a node can act ony as either a transmitter or receiver at a particuar instance of time, thus there is no sharing of a transmitter or a receiver between inks. (2) Set of actions: The set of actions is basicay the changes that a transmitter can make to hep achieve equiibrium in the system. Since this is a non-cooperative game, goba knowedge is not required. The transmitter changes the power aocations to increase the utiity of its ink in the face of changing interference. A practica constraint for a transmitter is that its transmit power cannot be higher than a certain vaue, i.e., Tr(Q ) p. (3) Set of utiity functions: This set provides functiona descriptions of individua preferences. Formay, a utiity function is defined as foows [15]. Definition 1: A function that assigns a numerica vaue to the eements of the action set u(a R 1 ) is a utiity function, if for a x, y A, x is at east as preferred compared to y if and ony if u(x) u(y). Putting together the three components shown above, the non-cooperative power contro game can be expressed as: G = [L, {A }, {u ( )}], where L is the set of inks, A ={Q C Nt Nt Q is positive semidefinite, and Tr(Q) p } is the set of power aocation actions

27 16 and u ( ) is the utiity function of ink. The transmit power of ink is imited as no higher than p, whie A is a convex set. We further denote the outcome of the game at certain time τ k by the power aocation vector q(τ k ) = (Q 1 (τ k ),..., Q L (τ k )). In order to singe out the action of ink, et q (τ k ) denote the vector consisting of eements of q(τ k ) without the th ink. For any ink at time τ k, the transmitter tries to find Q (τ k ) A, such that for any other Q (τ k) A, u (Q (τ k ), q (τ k )) u (Q (τ k), q (τ k )). Each transmitter cacuates its own ink data rate and tries to optimize its utiity function. Nash equiibrium is a we-known soution to a non-cooperative game. It is defined as a strategy profie in which no payer may gain from uniatera deviation from this profie [24]. For the game G, the definition of Nash equiibrium is as foows. Definition 2: A power aocation vector q(τ k ) = (Q 1 (τ k ),..., Q L (τ k )) is defined to a Nash equiibrium point of G = [L, {A }, {u ( )}] if u (Q, q ) u (Q, q ) hods for a L and Q A. Nash equiibrium does not necessariy exist, nor is it unique. It may be reached by each transmitter successivey changing its power aocation in response to the interference environment from the other inks in the network. A other inks take the new aocation into consideration whie cacuating interference and the process is repeated for a inks in the system unti the power aocations for each user converges. The utiity functions used most commony in the appication of game theory to communications are derived from SINR, BER [36, 30] and as in our case, ink data rate Utiity Function Design Utiity is the measure of satisfaction that a ink obtains from using the channe, which is aso caed the payoff of the ink. For a wireess ad hoc network, the utiity function of a particuar ink is reated to the transmit power and the achievabe data rate of that ink, i.e., u = u (p, C ), where p and C are utimatey functions of Q. We

28 17 design a utiity function which takes the foowing structure u (p, C ) = kg(c ) p 1/2 (2.10) g( C ) 1 η 0 0 C 0 C ( bit / sec/ Hz) Figure 2.3: g(c ) where k is a positive constant whose unit is Vots so that the utiity is a dimensioness number and g(c ) = 1 (1 η 0 ) C /C 0 is potted in Fig.2.3. Here C 0 is chosen to be a certain ratio (η 0 ) of the ink s maxima data rate C max,, where C max, is defined to be interference-free water-fiing capacity of ink. It is obvious that if the transmit power for ink is given, maximum capacity occurs when none of the other inks transmit and ink aocates power using water-fiing. Equation(2.10) accommodates the mechanism of power contro as the utiity depends on both power and data rate. If interference is fixed and transmit power increases, ink capacity and corresponding g(c ) does not change at the same rate as p 1/2. When p increases in the neighborhood of zero, g(c ) goes up

29 18 faster than p 1/2 and u increases, thus the transmitter is encouraged to transmit more power so as to maximize utiity. On the other hand, after p goes up to a certain eve but sti increases, g(c ) grows at a ower rate than p 1/2 and u goes down. Therefore, this discourages the transmitter from sending more power which woud decrease utiity. This intrinsic power contro property is usefu for a utiity function in a power contro game. The utiity function (2.10) is not perfect because it wi resut in a degenerate case in which maximum utiity is achieved when a nodes transmit zero power. To avoid that degenerate situation, we modify Eq.(2.10) to be u (Q ) = C k Tr(Q ) 1/2 (1 2(1 η C 0) 0 ) (2.11) From Eq.(2.10) to Eq.(2.11), we changed the coefficient of the exponentia term to be 2. In addition, we switched the variabes from (p, u ) to Q because the former is determined by the atter, as shown by Eq.(2.2) and p = Tr(Q ). The new utiity function (2.11) has properties shown beow. Property 1: u is a monotonicay increasing function of C for a fixed p = Tr(Q ). This can be shown by u C = where β = (1 η 0 ) 1/C 0 (0, 1). k p 1/2 ( 2β C ) n β > 0 (2.12) Property 2: u obeys the aw of diminishing margina utiity for arge C given p is fixed, as shown by u im = C C im C k p 1/2 ( 2β C ) n β = 0 (2.13) Property 3: u is monotonicay decreasing when p increases, for a fixed C s.t. C >

30 19 og β 1 2. u = p k 2p 3/2 Property 4: u tends to zero in the imit that p goes to infinity. (1 2β C ) < 0 (2.14) im u = im p p k p 1/2 (1 2β C ) = 0 (2.15) For any ink in an ad hoc network, power consumption and achieved data rate determine the payoff of that ink. Properties 1 and 2 show that for a fixed transmit power in a ink, the higher the data rate, the greater the payoff. However, the payoff tends to saturate as the data rate grows. In practica data transmission situations, ink capacities are preferred to be high enough so as to maintain a certain degree of QoS. Once this requirement is met, however, the increment of satisfaction provided by additiona ink capacity wi diminish unti utimatey reaching saturation. Property 3 enforces that if the data rate of a ink is high enough and can be guaranteed, transmitting more power is not desirabe because it causes more interference to other inks, thus bringing down the utiity. The asymptotic behavior of transmitting a arge amount of power is described in Property 4 which shows that extremey high transmit power resuts in zero satisfaction. To further investigate the utiity function for an ad hoc network, consider the two situations shown in Fig In both situations, node 1 transmitting to node 2 is the ink of interest with the soid arrow showing desired communication inks and the hatched arrow showing the propagation of interference. Fig. 2.4A iustrates the case that if the capacity of a particuar ink is more than enough to maintain a certain eve of QoS, reducing the capacity by decreasing transmit power wi mitigate the interference sent to other inks. Since each ink tries to maximize its utiity, the transmitter wi not be encouraged to transmit the maximum power. Another case of interest is shown in Fig. 2.4B. The achievabe data rate for a particuar ink (node 1 to node 2) is ow even when the transmitter sends data using maximum power. If such a ow-rate but high power-

31 20 consumption ink exists, it has two major negative effects on the network. First, a ow-rate ink is not ony useess in terms of data transmission but aso it may bring down the data rate of other inks due to generated interference. Second, the tota power consumption of the network is inefficienty increased as other inks may aso transmit more power in order to counter interference. To avoid these negative effects, it is advisabe to shut down ow-efficiency inks. The outcomes of utiity-based power contro with and without such a ink shut-down mechanism wi be shown in Section 2.4 of this chapter (A) (B) Figure 2.4: Iustration of the game theoretic approach with a mechanism for shutting down inks (A) Situation in which ink 1-2 shoud reduce transmit power (B) Situation in which ink 1-2 shoud be shut off Whether to shut down a particuar ink depends on the minimum data rate that is required by the network. For a non-priority network with homogeneous services, such as video-dedicated, each ink has the same requirement for minimum data rate regardess of the topoogy of the network. Therefore, it is possibe to assign a fixed threshod C t which can be used to decide if a particuar ink shoud be shut down. This threshod is determined by the type of service that the network provides as we as the overa channe conditions which reate the QoS eve to the threshod. Specificay, if the wireess channe conditions are good, the network administrator may set a high threshod so that a viabe

32 21 inks wi have high data rates because the minimum requirement is high. However, once a network is buit up, C t does not depend on the channe condition of each individua ink due to the assumption that every ink has the same QoS requirement for a homogeneous network. Choosing the right (C 0, η 0 ) is important in designing the utiity function. To do so, we rewrite the condition in Property 3, i.e., C > og β 1 2, by pugging C 0 = η 0 C max, as C > og β 1 2 = η 0C max, n 1 2 n(1 η 0 ) (2.16) C is the actua data rate of ink, which is greater than C t due to the ink shutting-down mechanism. Therefore, a more strict condition than (2.16) that guarantees Property 3 is C t > η 0 C max, n 1 2 n(1 η 0 ) (2.17) which can be further converted to n(1 η 0 ) η 0 < C max, C t n 1 2 (2.18) (2.18) shows how to choose η 0. It depends on the pre-determined threshod and the interference-free water-fiing capacity of each ink, where the atter is reated to the network topoogy. In this chapter, we assume that each ink in the network can find out the channe condition when other inks are off, thus C max, can be cacuated for every. (2.18) gives the vaue range for η 0. A different η 0 seected together with corresponding C 0 determines a unique utiity function which eads to a different outcome of this power contro game. In Section 2.4, simuation resuts wi show how sensitive the system capacity and power consumption are to η 0. We aso imit the network topoogy to a stationary case and assume that a nodes are powered on simutaneousy. At start-up, each ink cacuates its mutiuser water-fiing capacity. If it is ower than the threshod, the ink is shut down. Since the nodes are stationary, a ow capacity ink which has been shut down cannot be

33 22 turned back on Game Theoretic Approach Based on the above utiity function, we propose an agorithm in which a inks update their power aocation matrices iterativey. Initiay a users agree on a certain capacity threshod C t to keep the ink viabe and start with the same transmit power p 0 which is aotted equay to a antennas. Each ink cacuates its independent water-fiing capacity [9] assuming there is no co-channe interference, and the mutiuser water-fiing capacity [39, 40] when interfering inks are considered. If the mutiuser capacity for ink j is ower than the threshod C t, then that ink is shut down and the power aocation matrix Q j is set to zero. For any viabe ink, the transmitter cacuates the optimum p (0 p p ) such that the utiity function is maximized. This p corresponds to a Q which is determined by maximizing (2.2). The power contro process is performed iterativey unti the utiity for every ink in the network converges. Agorithm 1: Power contro based on game theoretic approach 1 Initiaization: Set k = 0. Each ink cacuates its interference-free water-fiing capacity and mutiuser water-fiing capacity, then decides on if the ink shoud be shut down or not. 2 Update power aocations: Let k = k + 1. For a inks L, given power aocation vector q(τ k 1 ), compute Q (τ k ) = argmax Q A = u (Q, q (τ k 1 )). Repeat step 2 unti the utiity of every ink converges Game Anaysis In the power contro game G = [L, {A }, {u ( )}], the choice of power aocation matrix Q for user impacts not ony its own ink capacity and utiity, but aso those of other inks. Generay speaking, the ink capacity in (2.2) and the utiity in (2.11) are not concave or convex functions of Q 1,..., Q L. Therefore, anaytica investigation of game

34 23 properties without simpified assumptions is prohibitivey difficut. In this section, we study the concavity of the utiity function in the situation when interference for each ink in the ad hoc network is extremey arge, as one coud assume in a dense outdoor network. Theorem 1. A Nash equiibrium exists in the NCG: G = [L, {A }, {u ( )}] when interference is sufficienty arge for each ink. Proof. For any ink L, its data rate is determined by (2.2), which can be rewritten as C = og 2 det(i + Q H, R 1 H, ) = og 2 det(i + Q U Σ U ) (2.19) where the determinant identity det(i + XY) = det(i + YX) is used and H, R 1 H, = U Σ U is an eigenvaue decomposition of H, R 1 H,, with U unitary and Σ = diag(σ 1, σ 2,..., σ Nt ). When the transmitter sends out data with power p, the we-known water-fiing agorithm can be used to obtain the optimum power aocation matrix Q that maximizes (2.19), which is given by Q = U (µi Σ 1 ) + U (2.20) where µ is known as the water eve and is chosen to satisfy Tr(µI Σ 1 ) + = p. (µi Σ 1 ) + is a diagona matrix with rank r where r N t and we denote it by Ω = diag{ω 1, ω 2,..., ω r, 0,..., 0}. Substituting Ω into (2.20) and (2.19) yieds the capacity of ink as C = og 2 det(i + U Ω U U Σ U ) = og 2 det(i + Ω Σ ) r = og 2 (1 + ω i σ i ) (2.21) i=1 Given interference to ink is sufficienty arge, the eigenvaues of H, R 1 H, are diminishingy sma. We further assume a these eigenvaues are cose to each other, i.e.,

35 24 σ i σ 0. Therefore, (2.21) can be approximated as C 1 n 2 r i=1 ω i σ i σ 0 n 2 r i=1 ω i = σ 0 n 2 p (2.22) Pugging the above approximation into (2.11), we obtain a new utiity function as u (p ) = k p 1/2 (1 2β σ 0 n 2 p ) = k p 1/2 (1 2α p ) (2.23) where α = β σ 0 n 2 is cose to 1 because of sufficienty sma σ 0. Given interference seen by ink, the power aocation matrix Q is uniquey determined by p [40], thus the power contro game G can be equivaenty stated as NCG: S = [L, {P }, {u ( )}], where P is the strategy set in terms of transmit power and we denote the outcome of the game by the transmit power vector p = (p 1, p 2,..., p L ). In [29, 15], it has been shown that a Nash equiibrium exists, if for any : (1) P is a nonempty, convex and compact subset of some Eucidean space R L. (2) u (p) is continuous in p and quasi-concave in p. Each ink has a strategy space that is given by 0 p p. For any p (1), p (2) P and 0 < γ < 1, p (3) 1 = γp (1) + (1 γ)p (2) P. Thus the first condition is satisfied. Eq.(2.23) indicates that u is a continuous function in p. Next we wi show that Eq.(2.23) is concave in p. We approach this probem by investigating the second-order derivation of u with respect to p. 2 u p 2 = kp 5 2 [0.75(1 2α p ) + 2α p (1 p n α)p n α] kp 5 2 [0.75(1 2α p ) + 0.5α p ] = kp 5 2 (0.75 α p ) < 0 (2.24) where in the first inequaity x(1 x) 1/4 (x R) is used and in the second one α p is approximated to 1 since α is cose to 1. (2.24) indicates that u is a concave function in p. Because both conditions are sat-

36 25 isfied, there exists a Nash equiibrium for the power contro game G = [L, {A }, {u ( )}] when interference is sufficienty arge 1. Theorem 2. The NCG: S = [L, {P }, {u ( )}] has a unique equiibrium point when interference is sufficienty arge for each ink. Proof. Theorem 4.4 in [12] shows that under certain conditions a game has at most one equiibrium point. In the context of our power contro game, those conditions are formuated as (i) P = {p R, g (p ) = p p 0} is nonempty and g is a continuousy differentiabe concave function in an open set containing P for each = 1,..., L. (ii) There exists a p P to satisfy g ( p ) > 0 for a = 1,..., L. (iii) A payoff functions u are concave in p with fixed vaues of p j (j ) and twice continuousy differentiabe in an open set containing P = P 1... P L. (iv) Game S is diagonay stricty concave on P, i.e., for any p (0) p (1), p (0), p (1) P and for some t 0 (t R L ), the foowing inequaity hods. (p (1) p (0) )h(p (0), t) + (p (0) p (1) )h(p (1), t) > 0 (2.25) where the function h : R L R L is defined as h(p, t) = t 1 u 1 p 1... t L u L p L (2.26) Next we wi show that the game S = [L, {P }, {u ( )}] satisfies a of the conditions above. For (i) and (ii), g (p ) 0 comes from the power constraint and g (p ) is a inear function with respect to p, thus (i) and (ii) are readiy fufied. It has been proven in Theorem 1 that the utiity function u is a concave function of p in case of sufficienty arge interference, thus (iii) is satisfied. The condition (iv) requires that (2.25) hod. To 1 As shown in the proof, sufficienty arge means σ 0 σ 1... σ r and og 2 (1 + ω iσ i) ω iσ 0/ n 2

37 26 show that, we pug (2.26) into (2.25) and rewrite the LHS in an eement-wise format as the foowing LHS = (p (1) p (0) )[h(p (0), t) h(p (1), t)] = (p (1) 1 p (0) 1,..., p(1) = L =1 L p(0) L ) ( t (p (1) p (0) u ) p (0) ( u 1 p (0) 1 ( u L p (0) L u p (1) ) u 1 p (1) 1... u L p (1) L )t 1 )t L (2.27) where transmit power vector p (i) = (p (i) 1,..., p(i) L ) for i = 0, 1 and t = (t 1,..., t L ) 0. Given p (0) p (1), j L, s.t. p (0) j p (1) j. If p (0) j < p (1) j, because of the concavity of u j (p j ), u j / p j is monotonicay decreasing on p j, which yieds u j / p (0) j > u j / p (1) j. Therefore, we have ( t j (p (1) j p (0) uj j ) p (0) j u ) j p (1) > 0 (2.28) j and (2.27) is positive. If p (0) j > p (1) j, simiary we can show (2.27) is positive. Therefore, condition (iv) is satisfied. Since a of the requirements in (i)-(iv) are met, the game S where interference is sufficienty arge has at most one equiibrium point. According to Theorem 1, there exists a Nash equiibrium point in the game S, therefore it is unique. Note that the concavity of the utiity function is a sufficient condition for the uniqueness of Nash equiibrium, but it is not a necessary one. Thus, whie the numerica simuations in the next section may not meet the requirement for sufficienty arge interference for each ink, a unique Nash equiibrium is found for our simuations.

38 Simuation Resuts The ad hoc network is simuated in downtown Phiadephia (shown in Fig. 2.5) with computationa eectromagnetics [10]. The topoogy is static and contains transmit-receive nodes 5-15 (ink 1), 8-11 (ink 2), 14-3 (ink 3), 10-2 (ink 4), 1-6 (ink 5) and 4-9 (ink 6). A of these inks are singe-hop and no node reays information. We compare the sum data rate and energy efficiency under different methods, namey, game theoretic approach with ink shut-down mechanism (GTWS) or without that mechanism (GTWOS), gradient projection (GP) approach and mutiuser water-fiing (MUWF) approach. Since power contro is appied to the game theoretic approach, each individua ink may not transmit the same amount of power even though the initia transmit power of each ink is the same. In order to compare the performance of different methods in a fair way, we fix the tota power consumption of the network, which is determined by the GTWS technique, and divide power among inks. Specificay, we first set each ink to transmit the same amount of power and use GTWS to compute the sum data rate and tota power consumption( 6 =1 p ), then divide the tota power( 6 =1 p ) equay to a 6 inks and compute sum data rate using MUWF and GP method. When appying the GTWOS method, we et each ink start with equa transmit power ( =1 p ) but assign a different maximum power constraint in order to get the same tota power consumption as GTWS and thus make a fair comparison. Fig. 2.6 shows the sum data rate of the network for different agorithms. The SNR is cacuated on the basis of equa transmit power for GP and MUWF methods. We can see that for every SNR the GTWS method achieves the highest sum data rate whie the MUWF method resuts in the owest system capacity 2. The GP method and MUWF method assume that each transmitter sends a constant amount of power no matter how inefficient that particuar ink is, which might cause severe interference to other inks. The same situation coud happen in the GTWOS method as we, where inefficient inks may 2 The convergence point of the GP method depends on the initia condition. However, [37] found out the ergodic mutua information curves are extremey cose to each other and one choice of initia condition is not evidenty better than another. In this dissertation, the initia condition is set to equa power aocation.

39 Figure 2.5: Iustration of Phiadephia downtown simuation. circed and ink numbers are circed. Node numbers are not transmit even higher power because power contro is aowed. For our proposed method, inefficient inks are shut down so as to avoid a waste of power. For instance, in the simuation at SNR = 19.6dB, ink 1 has such a ow capacity that it is unusabe. As a resut, it is shut off. However, the sum capacity is sti higher than the other methods, which indicates that athough shutting off deficient inks reduces the number of users in the network, it improves the data rate of existing users. Among the three approaches without the mechanism of ink shut-down, the GP method resuts in the highest sum data rate. This is reasonabe because its objective function is to maximize the sum data rate whie in the other two methods each ink aims to maximize its own data rate or utiity. GTWOS and MUWF are both game-based agorithms[40][37], in which the sefish utiity function introduces conficts among inks and power aocations at the Nash equiibrium are ess efficient than possibe power aocations acquired through cooperation. Comparing GTWOS and MUWF methods, GTWOS eads to a higher sum