COOPERACIÓN AL DESARROLLO, HAMBRE Y POBREZA RURAL

constraint is imposed [34].

9.4 Opportunistic Amplify and Forward

Like for ODF, in OAF, the relay is used whenever it allows, according to the goal, for capacity improvements or power saving w.r.t. direct transmission. To be more precise, the source node sends data to the destination node according to two modes, DT or AF. In DT, S transmits to D occupying all the time frame Tf. In AF mode [86, chapter 5], S transmits its data to both R and D during the first half of the frame, then in the second half, R amplifies and forwards the same data to D while the source is silent. Finally, D decodes the message combining the data received in both time slots from both the source and the relay. Fig. 9.2 shows the DT and AF modes, and the corresponding time slot allocation. The capacity of OAF can be then computed as

COAF = max {CDT, CAF} , (9.23) where CDT is given by (9.2). To compute the capacity of AF, namely CAF, we make the following assumptions. For all the network nodes, at the physical layer, we assume multi-carrier modulation with M sub-channels, and the noise at both the relay and destination nodes is considered Gaussian. The frame has normalized duration Tf = 1. The relay amplifies the signal received in sub-channel k by the quantity

g(k)₌ v u u u t P_R,AF(k) P_S,AF(k)  H (k) S,R    2 + P_w,R(k) , (9.24)

to assure that the relay transmits the power P_R,AF(k) in sub-channel k during the second half of the time frame. Finally, we assume the receiver to adopt maximal ratio combining for the data received from the source and the relay. Therefore, assuming Gaussian input signals, the AF capacity can be written

Chapter 9 - Opportunistic Relaying over PLC Networks as [80, 82, 86] CAF(PS,AF, PR,AF) = (9.25) 1 2M T X k∈Kon log2 1 + PS,AF(k) η (k) S,D+ P_S,AF(k) η(k)_S,RP_R,AF(k) η(k)_R,D 1 + P_S,AF(k) η_S,R(k) + P_R,AF(k) η(k)_R,D ! .

From (9.25), we note that the second and the third arguments of the log function respectively denote the SNR obtained with the direct link, and the one obtained with the relay. Furthermore, the term 1/2 accounts for the slot duration.

In Section 9.4.1, we will deal with the power allocation to maximize the OAF capacity. Then, in Section 9.4.2, we will focus on the power saving problem.

9.4.1 Capacity Improvements with OAF

Looking at (9.23), we see that the use of OAF can bring capacity improvements w.r.t. the DT. To maximize the capacity of OAF we need to optimally allocate the power for both DT and AF modes.

As explained in Section 9.3.1, assuming that the network nodes have to sat- isfy a PSD mask constraint, the sub-channel power allocation that maximizes the DT capacity corresponds to the one given by the same PSD constraint, namely, the DT capacity is maximized when P_S,DT(k) = P , ∀ k ∈ Kon.

To maximize the AF capacity we need to solve the following optimization problem

min {−CAF(PS,AF, PR,AF)}

s.t. 0 ≤ P_S,AF(k) ≤ P , (9.26)

0 ≤ P_R,AF(k) ≤ P , ∀ k ∈ Kon.

Looking at problem (9.26), we notice that the Hessian associated to the ob- jective function is not symmetric. Therefore, we can not use the Sylvester’s criterion [85], or check the sign of the eigenvalues to see whether the objective

9.4 - Opportunistic Amplify and Forward

function is convex.

To compute the solution of (9.26) we notice that the objective function is sum of monotonic increasing functions of both the power at the source node and the power at the relay node, further, since we have a constraint on the PSD, we can assert that the optimal power allocation is equal to that given by the same PSD.

Another way to compute the optimal solution of (9.26) is to exploit the notion of invex functions2 _{[87]. This is because in 1985, Martin [88] proved} the following proposition.

Proposition 1. Besides convex functions, a broader class of functions in which KKT conditions guarantee global optimality are the so called invex func- tions. So if equality constraints are affine functions, inequality constraints and the objective function are continuously differentiable invex functions then KKT conditions are sufficient for global optimality.

Therefore, if we show that the objective and the inequality constraint func- tions of (9.26) are invex, we can find the optimal power distribution just solv- ing the KKT conditions. The definition of an invex function follows. Let us consider a general minimization problem with inequality constraint

min

x∈S f (x), s.t. g(x) ≤ 0, (9.27)

where x ∈ Rn_{, f ∈ R, g ∈ R}m_{, f and g are differentiable on a open set} containing the feasible set S.

With respect to problem (9.27), f (x) and g(x) are Invex at u ∈ S w.r.t. a common function γ(x, u) ∈ Rn_{, if ∀ x ∈ S}

f (x) − f (u) ≥γ(x, u)T_{∇f (u), (9.28)} gi(x) − gi(u) ≥γ(x, u)T∇gi(u), i = 1, . . . , m. (9.29) In (9.28), ∇ denotes the gradient operator. A slight generalization of the

2

Note that the proposed procedure can be applied to the more general case of a constraint on the total transmitted power.

Chapter 9 - Opportunistic Relaying over PLC Networks

previous definition is Type I Invexity, which, with reference to problem (9.27), is defined as

f (x) − f (u) ≥γ(x, u)T_{∇f (u), (9.30)} −gi(u) ≥γ(x, u)T∇gi(u), i = 1, . . . , m. (9.31) It is interesting to note that if a function is Tipe I invex then it is also invex. Now, let us consider the following theorem that is given in [89].

Theorem 1. Suppose that problem (9.27) has a minimal solution at u ∈ S. Suppose also that there is a point x ∈ S such that gi(x) < 0 for some i ∈ I = {i : gi(u) = 0}. If the Kuhn-Tucker conditions apply at u and all the Kuhn- Tucker multipliers y ∈ Rm _{are bounded, then the active constraint functions} and the objective function are Type I invex w.r.t. a common nontrivial γ at u.

The previous theorem allows us to affirm that if the KKT conditions applied to problem (9.26) give a solution for which the constraints are satisfied and the multipliers are bounded, then the objective function of (9.26) is Type I invex. Note that the inequality constraints in (9.26) are affine fuctions and thus they are also Type I invex.

In Appendix 11.8, we prove that applying the KKT conditions to problem (9.26), we obtain a solution that satisfies the hypothesis of Theorem 1, and thus the objective function of (9.26) is Type 1 invex. More in detail, we have found that the sub-channel power allocation solution of the KKT conditions for problem (9.26) corresponds to the one given by the PSD constraint for both the source and the relay node, i.e., P_x,AF(k) = P , with x ∈ {S, R}, and k ∈ Kon. Now, exploiting Proposition 1, we can state that this solution corresponds to the optimal sub-channel power allocation that maximizes the AF capacity under a PSD mask constraint. Therefore, to compute the OAF capacity when the system is subject to a PSD constraint, we can simply compute the DT and the AF capacities obtained setting the powers to P , and then we can choose the mode that gives the highest capacity (9.23).