Wireless Ad Hoc and Sensor Networks

214 Wireless Ad Hoc and Sensor Networkswhere x is a random variable, and σ 2 is known as the fading envelope ofthe Rayleigh distribution.Because the channel uncertainties can distort the transmitted signals,therefore, the effect of these uncertainties is represented via a channel loss(gain) factor, typically multiples of the transmitter power. Then, the channelgain or loss, g, can be expressed as (Canchi and Akaiwa 1999)−n. ζg= f( d, n, X, ζ)= d ⋅ ⋅X10 01 2(5.46)where d −nis the effect of path loss, 10 01 . ζ corresponds to the effect ofshadowing. For Rayleigh fading, it is typical to model the power attenuationas X 2 , where X is a random variable with Rayleigh distribution.Typically, the channel gain, g, is a function of time.5.4.2 Distributed Power Controller Scheme DevelopmentThe goal of transmitter power control now is to maintain a required SIRthreshold for each network link while the transmitter power is adjustedso that the least possible power is consumed in the presence of channeluncertainties. Suppose there are N∈ Z + links in the network. Let g ij be thepower loss (gain) from the transmitter of the jth link to the receiver of theith link. It involves the free space loss, multipath fading, shadowing, andother radio wave propagation effects, as well as the spreading/processinggain of CDMA transmissions. The power attenuation is considered tofollow the relationship given in Equation 5.46. In the presence of suchuncertainties, our objective is to present a novel DPC and to compare itsperformance with others.The channel uncertainties will appear in the power loss (gain) coefficientof all transmitter–receiver pairs. Calculation of SIR, Ri( t),at the receiverof ith link at the time instant t (Jagannathan et al. 2006), is given bygii() t Pi() tRi()t = =I () ti∑j≠ig () t P()tiiig () t P(t) + η ( t)ijji(5.47)where i , j ∈{ 123… , , , , n},Ii( t) is the interference, Pt i( ) is the link’s transmitterpower, Pt j( ) are the transmitter powers of all other nodes, andη i () t > 0 is the thermal noise at its receiver node. For each link i, there isa lower SIR threshold γ i . Therefore, we requireγ ≤R () t ≤ γ*i i i(5.48)