# The relationship between the transmitted and the received power of an RF signal follows the…

The relationship between the transmitted and the received power of an RF signal follows the inverse-square law shown in Equation (1.5), that is, power density and distance have a quadratic relationship. This can be used to justify multi-hop communication (instead of single-hop), that is, energy can be preserved by transmitting packets over multiple hops at lower transmission power. Assume that a packet p must be sent from a sender A to a receiver B. The energy necessary to directly transmit the packet can be expressed as the simplified formula E AB = d(A, B) 2 + c, where d(x, y) (or simply d

»The relationship between the transmitted and the received power of an RF signal follows the inverse-square law shown in Equation (1.5), that is, power density and distance have a quadratic relationship. This can be used to justify multi-hop communication (instead of single-hop), that is, energy can be preserved by transmitting packets over multiple hops at lower transmission power. Assume that a packet p must be sent from a sender A to a receiver B. The energy necessary to directly transmit the packet can be expressed as the simplified formula E AB = d(A, B) 2 + c, where d(x, y) (or simply d in the remainder of this question) is the distance between two nodes x and y and c is a constant energy cost. Assume that you can turn this single-hop scenario into a multi-hop scenario by placing any number of equidistant relay nodes between A and B. (a) Derive a formula to compute the required energy as a function of d and n, where n is the number of relay nodes (that is, n = 0 for the single-hop case). (b) What is the optimal number of relay nodes to send p with the minimum amount of energy required and how much energy is consumed in this optimal case for a distance d(A, B) = 10 and (i) c = 10 and (ii) c = 5?

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