Power bandwidth trade-off for sensory and ad-hoc wireless networks

We look at the power bandwidth trade-off in random sensory and ad-hoc wireless networks with n users and r≤n¹2/ simultaneous source/destination pairs. Under a specific protocol, we show that the minimum power required for maintaining an achievable scaling law of R_sum=Θ(f(n)) for the sum-rate in the network, scales like Θ(f(n)/n¹2/). The required bandwidth B in this case is of order Θ(f(n)/r). It is also proved that the minimum achievable energy per information bit (E_b/N₀)_min for this protocol, scales as Θ(1/n¹2/) and in this case the spectral efficiency is nonzero and is of constant order.