Wireless Ad Hoc and Sensor Networks
Wireless Ad Hoc and Sensor Networks 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)
470 Wireless Ad Hoc and Sensor Networkswhere θi T yi() l() l = [ αi() l ri()]l is a vector of unknown parameters, and ψ i() l = [ ωi()l ]is the regression vector. Now, selecting feedback control for DAPC asv() l = − 1β [ − ˆ θ () l ψ () l + γ + k e ()] li i i i v i(10.21)where ˆ θ i () l is the estimate of θ i () l , the SNR error system is expressed ase( l + 1 ) = k e( l) + θ () l ψ () l − ˆ θ () l ψ () l = k e () l + ̃θ () l ψ () li v i i T i i T i v i i T i(10.22)where ̃θi() l = θi() l − ˆ θi()l is the error in estimation.From Equation 10.22, it is clear that the closed-loop SNR error systemis driven by the channel estimation error. If the channel uncertainties areproperly estimated, then SNR estimation error tends to be zero; therefore,the actual SNR approaches the target value. In the presence of error inestimation, only the boundedness of error in SNR can be shown. Giventhe closed-loop feedback control and error system, we can now advanceto the channel estimation algorithms.Consider now the closed-loop SNR error system with channel estimationerror, e(l), ase( l+ 1) = k e( l) + ̃θ () l ψ () l + ε () li v i i T i(10.23)where e(l) is the error in estimation, which is considered bounded above|()| ε l ≤ ε , with e N a known constant.NTHEOREM 10.3.1Given the DPC scheme above with channel uncertainties, if the feedback for theDPC scheme is selected as in Equation 10.21, then the mean channel estimationerror along with the mean SNR error converges to zero asymptotically, if theparameter updates are taken asˆ θ ( l+ 1 ) = ˆ θ () l + σψ () l e ( l+ 1 ) −Γ I−ψ () l ψ ( l) ˆ θ () li i i i T i T ii(10.24)where e(l) is the error in estimation, which is considered bounded above|()| ε l ≤ εN, e with e N a known constant. Then the mean error in SNR and theestimated parameters are bounded.10.3.2 Selective BackoffIn a dense reader environment, it is inconceivable that all readers are ableto achieve their target SNR together, owing to severe congestion thataffects both read rates and coverage. These readers will eventually reachmaximum power as a result of the adaptive power update. This demandsa time-based yielding strategy, with some readers allowing others toachieve their target SNR.
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470 <strong>Wireless</strong> <strong>Ad</strong> <strong>Hoc</strong> <strong>and</strong> <strong>Sensor</strong> <strong>Networks</strong>where θi T yi() l() l = [ αi() l ri()]l is a vector of unknown parameters, <strong>and</strong> ψ i() l = [ ωi()l ]is the regression vector. Now, selecting feedback control for DAPC asv() l = − 1β [ − ˆ θ () l ψ () l + γ + k e ()] li i i i v i(10.21)where ˆ θ i () l is the estimate of θ i () l , the SNR error system is expressed ase( l + 1 ) = k e( l) + θ () l ψ () l − ˆ θ () l ψ () l = k e () l + ̃θ () l ψ () li v i i T i i T i v i i T i(10.22)where ̃θi() l = θi() l − ˆ θi()l is the error in estimation.From Equation 10.22, it is clear that the closed-loop SNR error systemis driven by the channel estimation error. If the channel uncertainties areproperly estimated, then SNR estimation error tends to be zero; therefore,the actual SNR approaches the target value. In the presence of error inestimation, only the boundedness of error in SNR can be shown. Giventhe closed-loop feedback control <strong>and</strong> error system, we can now advanceto the channel estimation algorithms.Consider now the closed-loop SNR error system with channel estimationerror, e(l), ase( l+ 1) = k e( l) + ̃θ () l ψ () l + ε () li v i i T i(10.23)where e(l) is the error in estimation, which is considered bounded above|()| ε l ≤ ε , with e N a known constant.NTHEOREM 10.3.1Given the DPC scheme above with channel uncertainties, if the feedback for theDPC scheme is selected as in Equation 10.21, then the mean channel estimationerror along with the mean SNR error converges to zero asymptotically, if theparameter updates are taken asˆ θ ( l+ 1 ) = ˆ θ () l + σψ () l e ( l+ 1 ) −Γ I−ψ () l ψ ( l) ˆ θ () li i i i T i T ii(10.24)where e(l) is the error in estimation, which is considered bounded above|()| ε l ≤ εN, e with e N a known constant. Then the mean error in SNR <strong>and</strong> theestimated parameters are bounded.10.3.2 Selective BackoffIn a dense reader environment, it is inconceivable that all readers are ableto achieve their target SNR together, owing to severe congestion thataffects both read rates <strong>and</strong> coverage. These readers will eventually reachmaximum power as a result of the adaptive power update. This dem<strong>and</strong>sa time-based yielding strategy, with some readers allowing others toachieve their target SNR.
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