Joint design of physical and MAC layer by applying the constellation rearrangement technique in cooperative multi-hop networks

Abstract

Cooperative diversity schemes can be seen as an extension of spatial diversity systems, where distributed antennas are placed in relay nodes distributed in space as compared to a single multi-antenna source or receiver in conventional spatial diversity systems. Such cooperative communication systems improve the diversity gain significantly. In this paper a cross layer optimization scheme, based on cooperative diversity along with constellation rearrangement, is proposed for minimizing energy toll and enhancing the network longevity. By utilizing a cross-layer cooperative strategy, distributive algorithms are proposed for the dependability restriction multi-hop networks. We demonstrate through simulations that the proposed cross-layer cooperative strategies along with constellation rearrangement achieve considerable energy savings and extend the network longevity significantly. Finally the proposed scheme is evaluated through NS2 simulations in terms of throughput and delay.

Appendix

1.1 Proof of (40)

Consider Specified block in Fig. 2; the probability that the source m transmits only, denoted by $\Pr (\psi )$, is computed as

$$\begin{aligned} \Pr (\psi ) & = 1 - \Pr \left( {I_{m,n} \le R^{CT} } \right) + \Pr \left( {I_{m,n} \le R^{CT} } \right)\Pr \left( {I_{m,r} \le R^{CT} } \right) \\ & = 1 - \exp \left( { - h_{CT} d_{mn}^{\gamma } } \right) + \exp \left( { - h_{CT} \left( {d_{mr}^{\gamma } + d_{mn}^{\gamma } } \right)} \right) \\ & = 1 - \exp \left( { - h_{CT} d_{mn}^{\gamma } } \right)\left[ {1 - \exp \left( { - h_{CT} (d_{mr}^{\gamma } )} \right)} \right]. \\ \end{aligned}$$

(63)

where the term $1 - \Pr \left( {I_{m,n} \le R^{CT} } \right)$ denotes the event when the sender-receiver channel is not in failure, while the other term corresponds to the event when both the sender receiver and the sender-relay channels are in failure. The probability that the relay collaborates with the source is as follow:

$$\overline{\Pr (\psi )} \ge 1 - \Pr (\psi ).$$

(64)

Thus, the average transmission rate of the cooperative transmission mode can be computed as

$$R_{av} = R^{CT} \cdot \Pr (\psi ) + \frac{{R^{CT} }}{2} \cdot \overline{\Pr (\psi )} = \frac{{R^{CT} }}{2}\left( {1 - \Pr (\psi )} \right).$$

(65)

where $R^{CT}$ denotes the transmission rate if the sender is sending alone in one time slot, and $R^{CT} /2$ denotes the transmission rate if the relay collaborates with the sender in the following time slot. Since

$$\begin{aligned} \exp \left( { - h_{CT} d_{mn}^{\gamma } } \right) \le 1\,, \hfill \\ 1 - \exp \left( { - h_{CT} d_{mr}^{\gamma } } \right) \le h_{CT} d_{mr}^{\gamma } . \hfill \\ \end{aligned}$$

(66)

Thus

$$\Pr (\psi ) \ge 1 - h_{CT} d_{mr}^{\gamma } .$$

(67)

Using (39)

$$h_{CT} = \frac{{z_{CT} }}{{P_{m}^{CT} }} = \left( {\log QOS_{{}}^{ - 1} } \right)^{ - 1/2} \cdot \frac{1}{{\left[ {\sum\nolimits_{m = 1}^{L} {d_{eq,m}^{1/3} } } \right]^{1/2} d_{eq,m}^{1/3} }}.$$

(68)

There is

$$\Pr (\psi ) \ge 1 - \left( {\log QOS_{{}}^{ - 1} } \right)^{ - 1/2} \cdot \frac{1}{{\left[ {\sum\nolimits_{m = 1}^{L} {x_{m}^{1/3} } } \right]^{1/2} x_{m}^{1/3} }},$$

(69)

where

$$x_{m} = \frac{{d_{eq,m} }}{{d_{mr}^{2\gamma } }} = \frac{{d_{mn}^{\gamma } \left( {d_{mr}^{\gamma } + d_{rn}^{\gamma } } \right)}}{{d_{mr}^{2\gamma } }} \ge 1.$$

(70)

Thus, we have

$$\Pr (\psi ) \ge 1 - \sqrt {\frac{{\log QOS^{ - 1} }}{L}} .$$

(71)

where L is the number of hops of the route. Thus

$$R^{CT} = \frac{{2R_{av} }}{1 + \Pr (\psi )} \le \frac{{2R_{failure} }}{{2 - \sqrt {\frac{{\log QOS^{ - 1} }}{L}} }}.$$

(72)

So to ensure the fulfillment of desired total transmission rate $R^{CT}$ should be at least

$$R^{CT} = \frac{{2R_{failure} }}{{2 - \sqrt {\frac{{\log QOS^{ - 1} }}{L}} }}.$$

(73)

Consider the consumed power of the cooperative block.

$$P_{cb}^{CT} = P^{CT} \cdot \Pr (\psi ) + 2P^{CT} \cdot \overline{\Pr (\psi )} = P^{CT} \cdot \left( {2 - \Pr (\psi )} \right).$$

(74)

Using (39), the total transmission power of the route is

$$\begin{aligned} P_{tot}^{CT} & = \sum\limits_{m = 1}^{L} {P_{cb,m}^{CT} } \le \left( {1 + \sqrt {\frac{{\log QOS_{{}}^{ - 1} }}{L}} } \right)\sum\limits_{m = 1}^{L} {P_{m}^{CT} } \\ & = \left( {1 + \sqrt {\frac{{\log QOS_{{}}^{ - 1} }}{L}} } \right)(2^{{R^{CT} }} - 1)N_{0} (\log QOS^{ - 1} )^{ - 1/2} \cdot \left[ {\sum\limits_{m = 1}^{L} {d_{eq,m}^{1/3} } } \right]^{3/2} . \\ \end{aligned}$$

(75)

For the multiple-relay cooperation mode applying (29), there is

$$P_{m}^{CT} = \left( {2^{{R^{CT} }} - 1} \right)N_{0} \left( {\log QOS^{ - 1} } \right)^{ - 1/(R + 1)} \cdot \left[ {\sum\limits_{m = 1}^{L} {d_{eq,m}^{1/(R + 2)} } } \right]^{1/(R + 1)} \cdot d_{eq,m}^{1/(R + 2)} ,$$

(76)

and the total transmission power is

$$P_{tot}^{CT} = \left( {2^{{R^{CT} }} - 1} \right)\,N_{0} \,.\,\left( {\log QOS_{{}}^{ - 1} } \right)^{ - 1/(R + 1)} \,.\,\left[ {\sum\limits_{m = 1}^{L} {d_{eq,m}^{1/(R + 2)} } } \right]^{(R + 2)/(R + 1)} + \left( {2^{{R^{CT} }} - 1} \right)\,N_{0} \,\sum\limits_{m = 1}^{L} {d_{mn}^{\gamma } } .$$

(77)

Equations (39) and (76) show that the value of $R^{CT}$ has a considerable effect on the transmission power. To minimize total transmission power, a minimum $R^{CT}$ should be used to fulfill the desired total transmission data rate in (1).