CHAPTER 7 Non Cooperative Games Theory based uniroma1 it

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7 1 Related Works and Proposed Contributions, Power control and spatial signal shaping are central issues for the optimized. design of MAI limited ad hoc networks In fact in these networks the informa. tion throughput measured in bits slot conveyed by each link depends not only. on the power allocation and signal shaping performed by each transmitter but. also on the power allocations and signal shapings of all other transmitters active. over the network Thus the optimized design of the overall network involves a. performance tradeo among all active transmitters Such tradeo is the subject. of the present work Speci cally the proposed power control and signal shaping. algorithms aim to maximize the information throughput conveyed by each link. active over the network and are based on the modelling of the ad hoc network. as a noncooperative strategic Game, The Game Theory point of view has been adopted in several recent contributions. dealing with the power control problem for wireless networks 1 4 5 However all. these works focus on scenarios characterized by single antenna terminals and then. fully neglect the spatial dimension of the system On the contrary in emerging. next generation ad hoc networks built up by Multi Antenna transceivers the. spatial dimension of the overall system is crucial and it must be explicitly taken. into account in order to optimized the network throughput. The main result of this Chapter is that under suitable conditions the Multi. Antenna MAI channel Game has a unique Nash equilibrium under both Best. E ort and contracted QoS access policies This result leads to iterative fully. scalable power control and signal shaping algorithms able to achieve the equilib. rium point in a fully distributed and asynchronous way see the Proposition 3 of. sub Paragraph 7 6 2 Speci cally the presented power control and signal shaping. algorithms exhibit the following advantages over conventional centralized orthog. onal access methods as TDMA FDMA CDMA, The proposed algorithms are fully scalable and may be implemented without. any centralized controller, The proposed algorithms are competitively optimal and then they strike an. optimized balance between maximizing each user s own information through. put and minimizing its induced interference e ect For achieving this task. the spatial dimension of the underlying Multi Antenna channels is explicitly. The proposed algorithms allow to implement Best E ort and Contracted. QoS access policies and may account for multiple QoS classes. Several numerical results support the conclusion that the proposed distributed. algorithms outperform conventional centralized ones as TDMA FDMA CDMA. in terms of conveyed information throughput specially in networking scenar. ios a ected by strong MAI,7 1 1 Organization of the Chapter.
The remainder of this paper is organized as follows After the system modeling. of Paragraph 7 2 and 7 3 deals with the evaluation of the conveyed information. throughput in networking environments a ected by MAI Paragraph 7 4 the opti. mized power allocation over the transmit antennas is presented for the simple case. of a single transmit receive pair impaired by static e g no time variant MAI. Thus after shortly reviewing in Paragraph 7 5 the MAI model for ad hoc net. works proposed in 16 Paragraph 7 6 formalizes the concept of Nash Equilibrium. while Paragraph 7 7 presents iterative and fully decentralized algorithms for achiev. ing the competitively optimal e g throughput maximizing power allocation and. signal shaping of all transmit receive pairs active over the considered network. Analytical conditions for the convergence of the operating point of the overall net. work towards a stable state e g the Nash equilibrium of the underlying Game. are also provided in Paragraph 7 8 and proved in the nal Appendices Actual. e ectiveness of the proposed power allocation and signal shaping Game is numer. ically tested in Paragraph 7 8 where some nal conclusions are also drawn. 7 2 The System Modeling, The considered application scenario models emerging wireless ad hoc net. works 9 where a large number of uncoordinated transmit receive nodes simul. taneously attempt to communicate over a limited size hot spot cell and then give. arise to MAI The complex base band equivalent point to point radio channel. linking a transmitter node Tx to the corresponding receiving node Rx is sketched. in Fig 7 1, Simply stated it is composed by a transmit unit equipped with t 1 antennas. communicating to a receive unit equipped with r 1 antennas via a Multiple. Input Multiple Output MIMO radio channel impaired by both slow variant at. Rayleigh fading and additive MAI induced by adjacent transmit nodes active over. the same hot spot cell The path gain hji from the transmit antenna i to the. receive one j may be modelled as a complex zero mean unit variance proper com. plex random variable r v 2 3 11 and for su ciently spaced apart antennas. overall path gains hji C1 1 j r 1 i t may be considered mutually. Multiple Access Interference,Space Time 2 2 Demodulator Detected. Message Encoder and channel Message,Tx Modulator estimator and Rx. decoder with r,antennas M,MIMO FORWARD CHANNEL H,Kd FEEDBACK LINK.
Figure 7 1 Multi Antenna system equipped with imperfect forward channel es. timates H and impaired by MAI with spatial covariance matrix Kd. uncorrelated Furthermore for low mobility applications as those serving users no. madic over hot spot cells the path gains hji may be also assumed time invariant. over T 1 signalling periods after which they change to new statistically inde. pendent values held for another T signalling periods and so on The resulting. block fading model well captures the main features of several frequency hopping. or packet based interleaved 4G systems where each transmitted packet is detected. independently of any other 2 3 11 16 About the MAI a ecting the link of Fig 7 1. its statistics mainly depend on the network topology 2 3 11 and in the applica. tion scenario here considered it is reasonable to assume these last constant over. at least an overall packet 16 However since both path gains hji and MAI. statistics may change from a packet to another it is assumed that Tx and Rx in. Fig 7 1 are not aware of them at the beginning of each transmitted packet Hence. according to Fig 7 2 it is assumed that the coded and modulated streams radiated. by the transmit antennas are split into packets composed by T 1 slots where. the rst TL 0 slots are used by the receiver for learning the MAI statistics. the second Ttr 0 slots are employed for estimating the path gains hji of the. forward MIMO channel and the last Tpay T Ttr TL slots convey payload. TL learning Ttr training Tpay payload,Figure 7 2 The packet structure T TL Ttr Tpay. 7 2 1 The Learning Phase, As described in the previous Chapter during the learning phase see Fig 7 2. no signals are radiated by the transmitter of Fig 7 1 so to allow the corresponding. receiver to learn the statistics of the impairing MAI Speci cally the r dimensional. complex column vector y n y 1 n y r n T collecting the outputs of the r. receive antennas over the n th slot of the learning phase may be modeled as 10 11. y n d n v n w n 1 n TL 7 1, where y n is the superposition of two mutually independent components w n. w 1 n w r n T and v n v 1 n v r n T Since the rst component accounts. for the receiver thermal noise then w n Cr 1 n TL may be modeled. as a zero mean proper complex spatially and temporally white Gaussian sequence. with covariance matrix equal to,E w n w m N0 Ir m n 7 2. where N0 watt Hz is the level of the receiving thermal noise The component. v n in 7 1 accounts for the MAI induced by multiple co located transmit. nodes active over the same hot spot cell and then it may be adequate to model. v n Cr as a zero mean temporally white spatially colored proper Gaussian. sequence 12 16 whose covariance matrix,Kv E v n v n.
remains constant over time intervals at least equal to the duration of an overall. packet see Fig 7 2 However Kv may change from a packet to another so that. it is reasonable to assume that both the transmit and receive nodes in Fig 7 1 are. not aware of the covariance matrix of the overall disturbance. Kd E y n y n Kv N0 Ir 7 4, at the beginning of each transmitted packet However since the received signal. y n in 1 equates the MAI one d n during the learning phase the Law of Large. Numbers guarantees that an unbiased and consistent e g asymptotically exact. estimate Kd of the a priori unknown covariance matrix Kd may evaluated via. the following sample average,Kd y n y n 7 5, As pointed out by the analysis reported in 11 the e ects of possible mismatches. between actual Kd and the estimated one Kd are no so critical so that in the. following Kd Kd see 11 for more details on this topic. 7 2 2 The Training Phase, Thus during the training phase the Tx transmit node of Fig 7 1 is able to. xi n C1 TL, perform the optimized shaping of the deterministic pilot streams. 1 n TL Ttr 1 i t to be used for estimating the a priori unknown. r t path gains hji of the MIMO forward channel of Fig 7 1 In particular. yj n C1 TL 1 n TL Ttr 1 j r measured at,the sampled signals.
the output of j th receive antenna during the training phase may be modelled as. y j n i n d j n TL 1 n TL Ttr 1 j r,where the corresponding overall disturbance. d j n v j n w,j n TL 1 n TL Ttr 1 j r 7 7, is independent from the path gains hji and exhibits the same statistics previously. reported in 7 4 for the learning phase Thus after assuming the usual power. constraint 2,xi n 2 P TL 1 n TL Ttr, on the average power P radiated by the transmit antennas over each slot of the. training phase the resulting signal to interference plus noise ratio SINR. sured at the output of j th receive antenna equates see eqs 7 7 7 8. j P N0 cjj 1 j r, where N0 cjj is the j th diagonal entry of the MAI matrix Kd in 7 4 Therefore. as in 2 the Ttr r complex samples gathered at the outputs of the r receive. antennas during the overall training phase may be organized into the Ttr r. y y given by 10 11,observed matrix Y 1 r, x 1 x t is the Ttr t matrix composed by the deterministic radi.
ated pilot symbols H h1 hr is the t r complex matrix composed by the. path gains hji to be estimated and the Ttr r matrix D collects. the disturbance samples d j n in 7 7 experienced during the training phase. in 7 10 must satisfy the, Obviously from 7 8 it follows that the pilot matrix X. following power constraint,T ra X tTtr P 7 11, As detailed in 11 the trained observations Y in 7 10 are employed by the. receive node of Fig 7 1 for computing the Minimum Mean Square Error MMSE. of the MIMO channel matrix H In turn at step,matrix estimate H E H Y. n TL Ttr e g at the end of the training phase this estimate H is communicated. back to the transmitter via the ideal feedback link of Fig 7 1. 7 2 3 Some consideration about the optimized training. A detailed analysis about the structure and performance of the estimator. computing the set h ji E hji Y 1 j r 1 i t of the r t MMSE. channel estimates of the MIMO forward channel H in 7 10 has been recently. presented in 11 so that thereinafter the sub Paragraph summarizes some basic. results exploited in the next Paragraphs In particular in 11 is proved that. under the power constraint 7 11 the pilot matrix X minimizing the total average. squared estimation error,tot E h ji hji 2 E ji 2,j 1 i 1 j 1 i 1. must satisfy the following relationship,d X X aIrt 7 11 a.
a T ra K 1, Furthermore when X meets 7 11 a the resulting MMSE channel estimator errors. ji h ji hji 1 j r 1 i t are mutually uncorrelated zero mean proper. complex equi distributed Gaussian r v s sharing the following variance 11. 2 E ji 2 E h ji hji 2 1 a t 1 7 11 c, In the following it is assumed eq 7 11 a met so to evaluate the performance of. the considered system of Fig 7 1 on the basis of eq 7 11 c Additional details. about the overall topic of the optimized MMSE MIMO channel estimates as well. as considerations about the system sensitivity to noisy e g no ideal feedback. link may be found in 14,7 2 4 The Payload Phase, Afterwards on the basis of the available Kd and H matrices and actual. packet M to be transmitted the transmit node of Fig 7 1 suitable shapes the signal. streams i n C1 TL Ttr 1 n T 1 i t to be radiated during the. payload phase The corresponding sampled signals yj n C1 TL Ttr 1. n T 1 j r measured at the outputs of the receive antennas may be. modelled as 10 11,yj n hji i n dj n TL Ttr 1 n T 1 j r 7 12. where the sequences dj n vj n wj n 1 j r account for the overall. disturbances e g MAI plus thermal noise experienced during the payload phase. They still exhibit the same statistics previously detailed in 7 4 and may be also. assumed independent from both path gains hji and payload streams i 10. Therefore after assuming that the transmitted streams meet the usual power. constraint 2,E i n 2 P TL Ttr 1 n T 7 13, the resulting SINR j measured at the output of the j th receive antenna during.
the payload phase equates see eqs 7 12 7 13,j P N0 cjj 1 j r 7 14. Furthermore from 7 12 it is also deduced that the r 1 column vector y n. y1 n yr n T collecting the outputs of the r receive antennas over the n th pay. load slot is linked to the t 1 column vector n 1 n t n T of the. algorithms exhibit the following advantages over conventional centralized orthog onal access methods as TDMA FDMA CDMA The proposed algorithms are fully scalable and may be implemented without any centralized controller The proposed algorithms are competitively optimal and then they strike an optimized balance between maximizing each user s own information through

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