Wireless Ad Hoc and Sensor Networks

Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 221Consider now the closed-loop SIR error system with channel estimationerror, ε( l),ase( l+ 1) = k e( l) + ̃θ () l ψ () l + ε () li v i i T i(5.73)using the proposed DPC. The DPC scheme, presented in Table 5.4, will beemployed when the channel estimation error is nonzero. In fact, in thefollowing theorem, it is demonstrated that the SIR and channel parameterestimation errors are bounded when the channel estimation error is nonzero.THEOREM 5.4.2Assume the hypothesis as given in Theorem 5.4.1, with the channel uncertainty(path loss, shadowing, and Rayleigh fading) now estimated byˆ θ ( l+ 1 ) = ˆ θ () l + σψ () l e ( l+ 1 ) − I−ψ () l ψ ( l) ˆ θ i () li i i i T i T i(5.74)where ε( l)is the error in estimation that is considered bounded above|()| ε l εN, with a known constant. Then the mean error in SIR and theestimated parameters are bounded provided Equation 5.65 and Equation5.66 hold.PROOF See Jagannathan et al. (2006).≤ ε Ny l l r lTABLE 5.4Distributed Power Controller during Fading Channels: Nonideal Case(Estimation Error Nonzero)SIR system state equationwhereSIR errorFeedback controllerChannel parameter updatePower update⎡ yi()l ⎤( + 1) = [ α ( ) ( )] ⎢ ⎥ + βi()lvi()l⎣ωi()l ⎦i i iy () l = R()li= θ () lψ () l + β () l v () le () l = R()l −γi i iii T i i iv () l = − 1β ()[ l − ˆ θ () lψ () l + γ + k e ()] li i i i i v iˆ θ ( l+ 1 ) = ˆ θ () l + σψ () l e ( l+ 1 ) − | I−ψ ( l ) ψ ()|ˆ l () li i i i T i T ip( l+ 1) = ( v ( l) I ( l) + p( l))i i i iθ iwherek v, γ , σ,andηare design parametersii