Wireless Ad Hoc and Sensor Networks

Adaptive and Probabilistic Power Control Scheme 469domain, where T is the sampling interval. Equation 10.12 can be transformedinto the discrete time domain asRi−rd( l+ 1) −Ri−rd( l) = g ii−rd ⋅ P i( l+ 1)TI () l Ti−g −Iii rd2i⋅ Pl i()⋅() l T⎛[ gij( l+ 1) − gij( l)] Pj( l)⎞⎜⎟⎜j i ⎝+gij()[ l Pj ( l+ ) −Plj()]⎟⎠≠∑1(10.13)After the transformation, Equation 10.13 can be expressed aswhereandR ( l+ 1) = α ( l) R () l + βv()li−rd i i−rd i i≠∑∆g () l P() l + ∆Plg () () lij j j ijj iα i () l = 1−Ii()lβ i= g ii − rdv() l =P( l+ 1) I () li i i(10.14)(10.15)(10.16)(10.17)With the inclusion of noise, Equation 10.14 is written asR ( l+ 1) = α ( l) R () l + βv() l +r() l ω () li−rd i i−rd i i i i(10.18)where w(l) is the zero mean stationary stochastic channel noise with r i (l)being its coefficient.From Equation 10.18, we can obtain the SNR at time instant l + 1 as afunction of channel variation from time instant l to l + 1. The difficulty indesigning the DAPC is that channel variation is not known beforehand.Therefore, a must be estimated for calculating the feedback control. Now,define; y( k ) =R ( k), then Equation 10.18 can be expressed asii−rdy( l+ 1) = α ( l) y() l + βv() l +r() l ω () li i i i i i i(10.19)As a i , r i are unknown, Equation 10.19 can be transformed intoyi()lyi( l+ 1 ) = [ α ⎡ ⎤i( l) ri( l)]⎢ ⎥ ivi⎣⎢ωi()l ⎦⎥ +Tβ () l = θi () l ψi() l + βivi()l(10.20)