Wireless Ad Hoc and Sensor Networks

Adaptive and Probabilistic Power Control Scheme 465Merging Equation 10.6 and Equation 10.7, we can calculate the actualdetection range r actual in terms of R rd asractualRrd= r⎛ d⎝ ⎜ ⎞R⎟required ⎠1/4q(10.8)For analysis purposes, we assume that any tag within such a range canbe successfully detected by the reader on account of BER specification. Ifa reader is completely isolated, meaning no interference, a maximumrange r max can be achieved by using it at the maximum power P max of agiven reader. In a practical application, it is not possible to expect thismaximum range owing to interference. It is important to note fromEquation 10.8 that the detection range and SNR are interchangeable and,therefore, our proposed algorithms target the required SNR. By viablepower control both read rate and coverage can be achieved.By substituting Equation 10.3 and Equation 10.4 into Equation 10.2, theSNR as a time-varying function is given byPbs()t⎛⎞Ri()t = = gii ⋅ Pi () t gij() t Pj() t +Ii()t ∑ ui( t)⎝⎜j≠i⎠⎟(10.9)Note that g ii is constant for a particular reader–tag link if we assumethat the tag is stationary. If the desired range for the reader is defined asr d , which is less than r max , then we can define the SNR for the backscattersignal from a tag placed at a distance r d to a reader aswhereRi−rdPbs−rd()t⎛⎞() t = = gii−rd ⋅Pi () t ⎜ gij() t P j () t ui()t ⎟Ii()t⎝⎜j i⎠⎟≠∑ +(10.10)g(10.11)Equation 10.10 shows the basic relationship between the SNR and theoutput power of all readers through interference experienced at a particularin the network. This relationship is used throughout this chapter toderive the power control algorithms.Krii− rd = 14qd10.2.2 Two-Reader ModelTo better understand the problem, a simple two-reader model is consideredfirst. Two readers i and j spaced D(i, j) apart, each with the desiredrange R i_1 and R j_1 , respectively, are shown in Figure 10.1. Readers must