Wireless Ad Hoc and Sensor Networks

Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 217with the inclusion of noise, Equation 5.52 is written asR( l+ 1) = α ( l) R() l + β () l v() l + r() l ω () li i i i i i i(5.57)where ω( l)is the zero mean stationary stochastic channel noise with rl i( ) asits coefficient.The SIR of each link at time instant l is obtained using Equation 5.57.Carefully observing Equation 5.57, it is clear that the SIR at the timeinstant l + 1 is a function of channel variation from time instant l to l + 1.The channel variation is not known beforehand, and this makes the DPCscheme development difficult and challenging. Because α is not known,it has to be estimated for DPC development. As indicated earlier, all theavailable schemes so far ignore the channel variations during DPC development,and therefore, they render unsatisfactory performance.Now define y() l = R (), l then Equation 5.57 can be expressed asiiy( l+ 1) = α ( l) y() l + β () l v() l + r() l ω () li i i i i i i(5.58)The DPC development is given in two scenarios.CASE 1 α i , β i , and r i are known. In this scenario, one can select feedbackas−1v() l = β ()[ l −α () l y() l − r() l ω () l + γ + k ei i i i i i v i()] l(5.59)where the error in SIR is defined as e() l = R() l −γ . This implies thatiie( l+ 1) = k e( l)iv i(5.60)Appropriately selecting k v by placing the eigenvalues within a unitcircle, it is easy to show that the closed-loop SIR system is asymptoticallystable in the mean or asymptotically stable, lim Ee { i( l)} = 0.This rendersl→∞that yi( l) → γ .CASE 2 α i , β i , and r i are unknown. In this scenario, Equation 5.58 canbe expressed as⎡ yi()l ⎤yi( l+ 1) = [ αi( l) ri( l)]⎢ ⎥ + βi()lvi()l⎣ωi()l ⎦(5.61)= θ () lψ () l + β () l v()li T i i i