Direct transmission vs. multihop. Consider the scenario depicted in Figure 10.5. The N1 nodes are placed in

Direct transmission vs. multihop.

Consider the scenario depicted in Figure 10.5. The Nþ1 nodes are placed in a straight line. All adjacent nodes are spatially separated by distance r. The radios on the nodes consume E J per bit when turned on, in either transmit or receive mode. The radio front-end consumes  J(bitmx) to amplify the signal, to arrive at a receiver 1m away, having a desired SNR of s1 (where x is the order of power drop-off with distance). For both second order and fourth order power drop-offs:

(a) Derive the energy equations for the direct transmission andmultihop scenarios.

(b) What is the condition for which direct transmission consumes less energy per bit than multihop transmissions?

(c) Plot the feasible region for direct communications in the r–N plane and comment on the consequences. For your plot, assume the low-power radios, operating at 2.5 V and 30 mA, with a power efficiency of approximately 5%

have E¼7.9
107 J per bit and ¼7.3 nJ /(bitm4). These values are valid for 2-FSK non-coherent reception, with a noise power ofNp¼67 dBm, at a symbol rate of 1/Tb¼10 kb/s and a bit error rate of 103.

You may disregard the receiver energy needed at the destination because it is fixed regardless of the number of intervening relays.