**2.3. Basics of game theory**

4 Will-be-set-by-IN-TECH

The unique characteristics of the IR-UWB physical layer provide both challenges and opportunities for the MAC layer design. A concrete challenge is the impossibility of carrier sensing, as practised in narrowband systems, since an IR-UWB signal has no carrier. Great opportunities such as robustness against MUI and multipath fading derive from IR-UWB's high temporal resolution. This makes uncoordinated access to the spectrum possible, provided that the local offered load is low compared with the available system bandwidth. For instance, at moderate pulse rates, the "dead time" between pulses allows several uncoordinated, concurrent transmissions to be time interleaved. As a result, ALOHA emerges as the most straightforward MA approach for low data rate IR-UWB networks [3, 9]. The inherent resilience of IR-UWB to MUI can be further increased if different links employ different pseudo-random THC in order to determine the temporal position of the transmitted pulses. The combination of ALOHA with TH coding leads to Time-Hopping Multiple-Access (THMA) [21], which is the most representative MA scheme for low data rate

However TH codes are not perfectly orthogonal, and even if, due to asynchronicity between sources and multipath fading, THMA is still sensitive to impulsive interference. As for CDMA systems, interference coming from a near-by interferer, the so called *near-far* effect, can be very harmful and must be managed in order to avoid performance degradation. For instance, at the physical (PHY) layer multi-user receivers3 can efficiently address the *near-far* effect at the cost of moderate to high additional hardware [19]. At the link layer, based on the estimation of the wireless link quality, it is possible to adapt the transmission rate to the level of interference experienced at the receiver; the goal can be the improvement of the data rate while satisfying

Within the MAC layer the adaptation of transmission parameters corresponds to a resource allocation task. In the technical literature some approaches on THMA resource allocation can be found, for instance in [18]. A broadly accepted result concerns the optimal power allocation in terms of *proportional fairness*4. Note that in autonomous networks with random topology, *proportional fairness* satisfyingly combines fairness and efficiency and outperforms other known performance metrics such as *max-min fairness* or *max total capacity*. This result indicates that the only necessary power control in a IR-UWB network, whose physical layer is in the linear regime5, is the scheduling function 0 <sup>−</sup> *Pmax*. In other words, each node should either transmit with full power or not transmit at all [18]. Another common finding is the existence of an *exclusion region* around every destination such that the reference source and nodes within this area cannot transmit simultaneously. This is similar to the IEEE 802.11 CSMA/CA strategy. Certainly, the *exclusion region* vanishes granted that the system has infinite bandwidth (*W* → ∞). However, in a practical IR-UWB network the bandwidth is always finite and the size of the *exclusion region* may become a critical issue

It can be concluded that for IR-UWB networks, and when maximizing rates is the design objective, near-far effects should be tackled by combining scheduling and rate adaptation. Particularly, for low power networks the optimal MAC layer design follows an "all at once"

<sup>4</sup> The power allocation that maximises the sum of the log of source rates, which is a concave objective function. <sup>5</sup> When the rate of a link can be approximated by a linear function of the signal-to-interference-and-noise ratio.

forcing the MAC designer to concentrate on the scheduling task.

scheduling while it adapts the transmission rates to interference [18].

<sup>3</sup> Those which can receive on *M* channels concurrently.

IR-UWB networks.

a minimum BER requirement [12].

Game theory has been applied in the recent past to model complex interactions among radio devices that have possibly conflicting interests. For the designer of wireless communication systems game theory is a powerful tool to analyse and predict the behaviour of distributed algorithms and protocols. Respected reference books are [7]. A short overview focusing on the application of game theory in the field of wireless communications can be found in [5].

A resource allocation problem can be naturally modeled as a game, in which the players are the radio devices willing to transmit or receive data. In general, there is an interest conflict since the players have to cope with a limited transmission resource such as power, bandwitdh or pulse load. In order to resolve this conflict they can make certain decisions (or take certain actions) such as changing their transmission parameters. The most familiar game form is the *strategic* form, which models a single-shot, simultaneous interaction among players. It is worth to mention that "simultaneous" is not used here in its strict temporal meaning. It does not imply that players have to choose their actions at the same point of time, but much more that no player is aware of the choice of any other player prior to making his own decision.

<sup>A</sup> *strategic* form game <sup>Γ</sup> �**N**, **<sup>A</sup>**, **<sup>u</sup>**� is defined by the following elements6:


To model asynchronous, continuous interactions as those which characterise resource allocation problems in AN the *strategic* form game model is however insufficient. The author agrees with the opinion advanced in [14] and considers *asynchronous, myopic, repeated* games as the most appropriate game model to analyse those problems. At this point it has to be stressed that the work in [14] provides a key result for the application of game theory in the analysis and modeling of AN, since it formalises the relation between the AN's *steady states* and the game behaviour characterised by the Nash equilibria (NE). Theorem 4.1 in [14] proofs that the *steady states* of an AN modeled by an *asynchronous, myopic, repeated* game with stage game Γ �**N**, **A**, **u**�, where all players are rational and act autonomously, coincide with the Nash equilibria of the stage game Γ �**N**, **A**, **u**�.

Notice that an NE is a strategy combination **a**∗, where no player can improve his utility by individually deviating from its strategy. The existence of an NE for a game significantly

<sup>6</sup> We use bold capital letters to denote sets and bold small letters for vectors; for the sake of simplicity we have omitted the superscript T for transposed vectors

depends on the choice of the utility function; specifically on its mathematical properties. The most common existence result for a game's NE is given by the Glicksberg-Fan-Debreu fixed point theorem [7].

**2.5. Simulation model**

in this chapter and is covered in section 3.

**3. MUI Mitigation by distributed pulse rate control**

and interference conditions.

and rate adaptation.

selected.

The dynamic simulation model has been developed with the discrete event simulation system OMNeT++ [15]. The CH and each SN comprise a PHY layer, a DLC layer and an application layer instance; network and transport layer operation is transparent. For the air interface the superframe structure described in Section 5.4.1 of the IEEE 802.15.4a standard [8] has been

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 57

At the application layer each SN generates data packets according to an exponential distributed packet inter-arrival process. Packets are addressed to the CH; its size has been chosen to be *Lp* = 400 bits. The exponential processes are statistically independent from each other, and a maximum information data rate of 1 Mbps is considered. The DLC layer implementation corresponds with the basic "data transfer model to a coordinator" in the standard IEEE 802.15.4a [see 8, Section 5.5.2.1]. For each DLC packet a packet error rate (PER) is calculated as a function of the received power, interference from concurrent transmissions and thermal noise. At the MAC layer a link adaptation function has been implemented which aims at optimising link/system capacity under several channel and interference conditions. The development and analysis of this function is the main achievement of the work presented

It is assumed that the network has a fixed chip duration, *Tc*, so that all changes at the PHY layer transmission parameters are induced by instructions coming from the MAC/DLC layer. The selected set of PHY layer parameters remains constant for one MAC packet transmission, but can be changed from packet to packet according to the time variant channel

Interference mitigation is a fundamental problem in wireless networks. In CDMA systems, a well-known technique for this is to control the nodes' transmit powers [22]. The work in [17] has shown that for wireless networks in the *linear regime*, and that allow fine-grained rate adaptation, the optimal power allocation is to let nodes either transmit at full power or do not transmit at all. IR-UWB conforms to both attributes, and thus, according to [17], the MAC layer should concentrate on alternative interference mitigation techniques such as scheduling

The term rate adaptation embraces all technical means in a system to adapt the transmission speed (rate) to the current quality of the radio link. In IR-UWB networks, rate control can be achieved by adapting the channel coding rate, the modulation order or the processing gain. In order to adapt these parameters, the link's transmitter must have an estimate of the level of interference at its intended receiver. In autonomous networks, most approaches make use of feedback information from the receiver to the transmitter, for example within ACK packets. This information can take various forms; conventionally, it is a function of the signal-to-noise-plus-interference-ratio (SNIR). However, measuring the SNIR is difficult in practice due to the very low transmit power of UWB signals. Therefore, recent approaches [9] rely on information provided by the channel decoder, namely on BER estimations. This

work follows theses approaches and also considers BER instead of SNIR feedbacks.
