Wireless Ad Hoc and Sensor Networks

Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 215for every i = 1, 2, 3, …, n. The lower threshold value for all links can be takenequal to γ for convenience, reflecting a certain QoS the link has to maintainto operate properly. An upper SIR limit is also set, to decrease the interferencedue to its transmitter power at other receiver nodes. In the literature, severalDPC schemes have been proposed. The most recent DPC work includesBambos et al. (2000), CSOPC by Jantti and Kim (2000), SSCD and optimalby Jagannathan et al. (2002), Dontula and Jagannathan (2004), respectively,and they are discussed in the previous section. This algorithm is given next.5.4.2.1 Error Dynamics in SIRIn the previous DPC schemes presented in Section 5.2 and Section 5.3,only path loss uncertainty is considered. As a result, during fading conditions,high value of outage probability, as illustrated using simulations,is observed. The work, presented in this section, is aimed at demonstratingthe performance in the presence of several channel uncertainties.In the time domain, however, the channel is time varying when channeluncertainties are considered and therefore gij( t)is not a constant. In Leeand Park (2002), a new DPC algorithm is presented where gii( t)is treatedas a time-varying function due to Rayleigh fading by assuming that theinterference Ii( t)is held constant. Because this is a strong assumption, inthis paper, a novel DPC scheme is presented (Jagannathan et al. 2006)where gii( t)and the interference Ii( t)are time varying, and channel uncertaintiesare considered for all the mobile users. This relaxes the assumptionof other works where both gij( t)and Pt j( ) are held at a fixed value.Considering SIR from Equation 5.47 where the power attenuationgij( t)is taken to follow the time-varying nature of the channel and differentiatingEquation 5.47 to get( gii( t) Pi ( t)) ′ I( t) − ( gii( t) Pi( t)) I(t)′Ri()t ′ =I () ti 2(5.49)where Ri( t)′ is the derivative of Ri( t), and Ii( t)′ is the derivative of Ii( t).To transform the differential equation into the discrete timexl ( + 1) −xl(domain, x′() t is expressed using Euler’s formula as ) , where T is theTsampling interval. Equation 5.49 can be expressed in discrete time as( gii( l) Pi ( l)) ′ I( l) − ( gii( l) Pi( l)) I(l)′Ri()l ′ =2I () li⎡1⎛⎞′⎤⎢=() ( g ′iiI l( l ) Pi ( l )) I ( l ) + ( gii( l )⎥P2 ⎢′i ()) l Ii() l − ( gii() l Pi()) l ∑ gij() l Pj() l + η i () t⎥i ⎢⎝⎜j≠i⎠⎟⎥⎣⎦(5.50)