Optimal power allocation for a wireless cooperative network with UAV

Problem description

The goal of power allocation is to minimize the outage probability under a certain total transmit power P_T. Assume that the maximum transmission power of the source node is P_max1, and the maximum transmission power of the UAV is P _max2. The total transmit power P_T is generally the maximum power allowed for a given data packet to be transmitted from the source node to the destination node, while P_max1 and P_max2 correspond to the maximum power that the source node and UAV can provide, respectively (0.5 P_T<P_max1, P_max2 ≤P_T), then the constrained optimization problem can be expressed as: (12)${\displaystyle \begin{array}{c}\hfill \underset{P_S,P_U}{min}{P}_{out}\hfill \\ \hfill s.t.\left\{\begin{array}{c}\hfill P_S+P_U=P_T\hfill \\ \hfill P_S\le P_{max}\prime P_U\le P_{max2}\hfill \end{array}\right.\hfill \end{array}}$

In the above formula, the objective function is the convex function of the variables P_S and P_U, whether there is a diversity gain or no diversity gain. The constraint function is the linear function of the optimization variables P_S and P_U. Therefore, this is a convex optimization problem. There exists a globally optimal solution, which can be solved by Lagrange Multiplier Method.

Without diversity

The system outage probability without diversity is shown in Eq. (9), then the optimization problem can be expressed as: (13)${\displaystyle \begin{array}{c}\underset{P_S,P_U}{min}{P}_{{}_{out}}^{\mathrm{DF}}=1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}\hfill \\ s.t.\left\{\begin{array}{c}{P}_S+P_U=P_T\hfill \\ P_S\le P_{max},P_U\le P_{max2}\hfill \end{array}\right.\hfill \end{array}}$

In the formula above, minimizing the objective function is equivalent to maximizing the expression $-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}$, then the optimization problem is simplified to be (14)${\displaystyle \begin{array}{c}\hfill \underset{P_S,P_U}{max}\left(-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}\right)\hfill \\ \hfill s.t.\left\{\begin{array}{c}\hfill P_S+P_U=P_T\hfill \\ \hfill P_S\le P_{max},P_U\le P_{max2}\hfill \end{array}\right.\hfill \end{array}}$

To solve the problem, the Lagrange Multiplier Method is picked. If the second restriction is neglected, the objective function is constructed as: (15)${\displaystyle F_1\left(P_S,P_U,\eta \right)=-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}+\eta \left(P_S+P_U-P_T\right)}$

According to the necessary for the existence of extreme values, it can be obtained (16)${\displaystyle \left\{\begin{array}{c}\hfill P_S^{\ast }=\frac{\sqrt{G_2}{P}_T}{\sqrt{G_1}+\sqrt{G_2}}\hfill \\ \hfill P_U^{\ast }=\frac{\sqrt{G_1}{P}_T}{\sqrt{G_1}+\sqrt{G_2}}\hfill \end{array}\right.}$

As the value of the power given in Eq. (16) may be out of range, the second constraint in Eq. (14) is combined to determine the optimal power. Moreover, if $P_S^{\ast }$ is out of bounds, the optimal power allocation can be obtained as (17)${\displaystyle \left\{\begin{array}{c}\hfill P_S^{\ast }=P_{max}\hfill \\ \hfill P_U^{\ast }=P_T-P_{max1}\hfill \end{array}\right.}$

Conversely, if $P_U^{\ast }$ crosses the boundary, the optimal power can be derived by (18)${\displaystyle \left\{\begin{array}{c}\hfill P_U^{\ast }=P_{max}\hfill \\ \hfill P_S^{\ast }=P_T-P_{max}\hfill \end{array}\right.}$

With diversity

Since the system outage probability with diversity gain is given in Eq. (10), the optimization problem, in this case, can be built as: (19)${\displaystyle \begin{array}{c}\hfill \underset{P_S,P_U}{min}{P}_{out2}=\left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}\right)\times \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}\right)\hfill \\ \hfill s.t.\left\{\begin{array}{c}\hfill P_S+P_U=P_T\hfill \\ \hfill P_S\le P_{max}\prime P_U\le P_{max2}\hfill \end{array}\right.\hfill \end{array}}$

Similarly, using Lagrange Method and ignoring the second constraint in Eq. (19), then the simplified optimization function is: (20)${\displaystyle F_2\left(P_S,P_U,\mu \right)=1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}+e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}+\mu \left(P_S+P_U-P_T\right)}$

The equations that the extreme values should be satisfied are as follows, (21)${\displaystyle \left\{\begin{array}{c}\hfill -\frac{{\gamma }_0}{G_0{P}_S^2}{e}^{-\frac{{\gamma }_0}{G_0{P}_S}}-\frac{{\gamma }_0}{G_1{P}_S^2}{e}^{-\frac{{\gamma }_0}{G_1{P}_S}}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}+e^{-\frac{{\gamma }_0}{G_0{P}_S}}{e}^{-\frac{{\gamma }_0}{G_1{P}_S}}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}\left(\frac{{\gamma }_0}{G_0{P}_S^2}+\frac{{\gamma }_0}{G_1{P}_S^2}\right)+\mu =0\hfill \\ \hfill -\frac{{\gamma }_0}{G_2{P}_U^2}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}{e}^{-\frac{{\gamma }_0}{G_1{P}_S}}+\frac{{\gamma }_0}{G_2{P}_U^2}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}{e}^{-\frac{{\gamma }_0}{G_1{P}_S}}{e}^{-\frac{{\gamma }_0}{G_0{P}_S}}+\mu =0\hfill \\ \hfill P_S+P_U-P_T=0\hfill \end{array}\right.}$

The transmission power of the source node and the UAV can be obtained by solving the formula above and the expression that the optimal values need to satisfy is (22)${\displaystyle \left\{\begin{array}{c}\hfill P_S^{\ast }=\sqrt{{\gamma }_0\left(\frac{1}{G_0}+\frac{1}{G_1}\right){\left(\frac{{\gamma }_0}{G_0{P}_S^2}{e}^{\frac{{\gamma }_0}{G_1{P}_S}}{e}^{\frac{{\gamma }_0}{G_2{P}_U}}+\frac{{\gamma }_0}{G_1{P}_S^2}{e}^{\frac{{\gamma }_0}{G_0{P}_S}}-\frac{{\gamma }_0}{G_2{P}_U^2}{e}^{\frac{{\gamma }_0}{G_0{P}_S}}-\frac{{\gamma }_0}{G_2{P}_U^2}\right)}^{-1}}\hfill \\ \hfill P_U^{\ast }=P_T-P_S^{\ast }\hfill \end{array}\right.}$ where $P_S^{\ast }$ and $P_U^{\ast }$ is the optimal allocation power.

Although the equation in (22) is not a closed-form solution about $P_S^{\ast }$ and $P_U^{\ast }$, its optimal solution can be obtained by successive approximation. It should be noted that, if $P_S^{\ast }$>P_max1 during the iteration process appears, the iteration is aborted to ensure the KKT conditions. Thus, the optimal value of $P_S^{\ast }$ is P_max1, and the optimal power allocation is: (23)${\displaystyle \left\{\begin{array}{c}\hfill P_S^{\ast }=P_{max}\hfill \\ \hfill P_U^{\ast }=P_T-P_{max}\hfill \end{array}\right.}$

Similarly, if the power value of the UAV exceeds P_max2 in the iterative process, it is reduced to P_max2, and the reduced power is allocated to the source node. Therefore, the optimal power distribution is given by: (24)${\displaystyle \left\{\begin{array}{c}\hfill P_U^{\ast }=P_{max2}\hfill \\ \hfill P_S^{\ast }=P_T-P_{max2}\hfill \end{array}\right.}$

Problem description

The target of power allocation is to minimize the total power through power allocation under the condition that the outage probability at the destination node guarantees a certain threshold condition.

It is supposed that the threshold of outage probability is ρ₀, then the constrained optimization problem can be expressed as: (25)${\displaystyle \begin{array}{c}\hfill \underset{P_S,P_U}{min}{P}_S+P_U\hfill \\ \hfill s.t.P_{out}\le {\rho }_0\hfill \end{array}}$

Because the constraints are non-convex, the problem above is not a convex optimization. It can be solved by the Lagrange multiplier method.

Without diversity

Substituting Eq. (9) into the expression above, the constrained minimization problem for a system without diversity can be obtained as (26)${\displaystyle \begin{array}{c}\hfill \underset{P_S,P_U}{min}{P}_S+P_U\hfill \\ \hfill s.t.1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}-{\rho }_0\le 0\hfill \end{array}}$

The Lagrange multiplier method is used for optimization, and the constructor is: (27)${\displaystyle L\left(P_S,P_U,\theta \right)=P_S+P_U+\theta \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}-{\rho }_0\right)}$

The optimal value satisfies the equation as follows: (28)${\displaystyle \left\{\begin{array}{c}\hfill \frac{\partial L\left(P_S,P_U,\theta \right)}{\partial P_S}=1+\mu \left(-\frac{{\gamma }_0}{G_1{P}_S^2}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}{e}^{-\frac{{\gamma }_0}{G_1{P}_S}}\right)=0\hfill \\ \hfill \frac{\partial L\left(P_S,P_U,\theta \right)}{\partial P_U}=1+\mu \left(-\frac{{\gamma }_0}{G_2{P}_U^2}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}{e}^{-\frac{{\gamma }_0}{G_1{P}_S}}\right)=0\hfill \\ \hfill \frac{\partial L\left(P_S,P_U,\theta \right)}{\partial \mu }=1-e^{-\frac{{\gamma }_0}{G_2{P}_U}}{e}^{-\frac{{\gamma }_0}{G_1{P}_S}}-{\rho }_0=0\hfill \end{array}\right.}$

From the first two formulas of the equations above, the relationship between P_S and P_U can be obtained as P_S=(G ₂/G ₁) ^1/2P_U, and the optimal power can be expressed as (29)${\displaystyle \left\{\begin{array}{c}{P}_S=-\frac{{\gamma }_0}{ln\left(1-{\rho }_0\right)}\left(\frac{1}{\sqrt{G_1{G}_2}}+\frac{1}{G_1}\right)\hfill \\ P_U=-\frac{{\gamma }_0}{ln\left(1-\rho tensor\ast \left[0^{}\right]\right)}\left(\frac{1}{\sqrt{G_1{G}_2}}+\frac{1}{G_2}\right)\hfill \end{array}\right.}$

With diversity

By substituting Eq. (10) into (25), the constrained minimization problem for a system with diversity can be expressed as (30)${\displaystyle \begin{array}{c}\hfill \underset{P_S,P_U}{min}{P}_S+P_U\hfill \\ \hfill s.t.\left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}\right)\times \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}\right)-{\rho }_0\le 0\hfill \end{array}}$

Use Lagrange Multiplier Method to optimize, the constructor is: (31)${\displaystyle L\left(P_S,P_U,\omega \right)=P_S+P_U+\omega \left[\left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}\right)\times \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}\right)-{\rho }_0\right]}$

According to the necessary existence conditions of extreme values: (32)${\displaystyle \left\{\begin{array}{c}\hfill \frac{\partial L\left(P_S,P_U,\omega \right)}{\partial P_S}=1+\mu \frac{\partial }{\partial P_S}\left[\left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}\right)\times \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}\right)\right]=0\hfill \\ \hfill \frac{\partial L\left(P_S,P_U,\omega \right)}{\partial P_U}=1+\mu \frac{\partial }{\partial P_U}\left[\left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}\right)\times \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}\right)\right]=0\hfill \\ \hfill \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SR}}}}{e}^{-\frac{{\gamma }_0}{\overline{{\gamma }_{RD}}}}\right)\times \left(1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}\right)-{\rho }_0=0\hfill \end{array}\right.}$

From the first two constraints above, the relationship between P_S and P_U can be obtained as (33)${\displaystyle \frac{1}{G_0{P}_S^2}{e}^{-\frac{{\gamma }_0}{G_0{P}_S}}\left(e^{-\frac{{\gamma }_0}{G_1{P}_S}}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}-1\right)=e^{-\frac{{\gamma }_0}{G_1{P}_S}}{e}^{-\frac{{\gamma }_0}{G_2{P}_U}}\left(\frac{1}{G_2{P}_U^2}-\frac{1}{G_1{P}_S^2}\right)\left(e^{-\frac{{\gamma }_0}{G_0{P}_S}}-1\right)}$

By simultaneous the third formula in (32) and the formula in (33), we can obtain the optimal power.

Considering the computation complexity is high, we tried to use the relationship between P_S and P_U to give a relatively simple solution. First of all, as the constraint in Eq. (33) is valid, we could take the equal sign. When the feasible region of P_S needs to meet certain conditions, P_U can be expressed as a function of P_S as (34)${\displaystyle P_U=\frac{-{\gamma }_0}{G_2\left[\frac{{\gamma }_0}{G_1{P}_S}+ln\left(1-\frac{{\rho }_0}{1-e^{-\frac{{\gamma }_0}{G_0{P}_S}}}\right)\right]}}$

On the one hand, as the entire cooperation process only works when the signal-to-noise ratio of the direct channel is greater than the threshold ρ₀, there is (35)${\displaystyle 1-e^{-\frac{{\gamma }_0}{\overline{{\gamma }_{SD}}}}>{\rho }_0}$

Solving the formula above, we can get P_S < P_th1, and P_th1= −γ₀/ln(1 −ρ₀)G ₀. On the other hand, the transmission power of the UAV is non-negative, then it can be derived according to the (32), as (36)${\displaystyle \left(1-e^{-{\gamma }_0/G_1}\right)\left(1-e^{-{\gamma }_0/G_0}\right)<{\rho }_0}$

The values which satisfy P_S<P_th2 can be obtained by numerical methods according to the Eq. (34). Thus, the two-dimensional problem in the Eq. (30) can be simplified to a one-dimensional optimization problem according to Eqs. (34), (35) and (36). The problem can be remodeled as (37)${\displaystyle \begin{array}{c}\underset{P_S,P_U}{min}\stackrel{\sim }{f}\left(P_S\right)=P_S+\frac{-{\gamma }_0}{G_2\left[\frac{{\gamma }_0}{G_1{P}_S}+ln\left(1-\frac{{\rho }_0}{1-e^{-\frac{{\gamma }_0}{G_0{P}_S}}}\right)\right]}\hfill \\ s.t.P_{th1}<P_S<P_{th2}\hfill \end{array}}$

To obtain the minimum total power when the system satisfies the transmission conditions, the derivative of $\bar{\tilde{f}}\left(P_S\right)$ needs to be obtained, and the equation that optimal $P_S^{\ast }$ needs to satisfy can be expressed as (38)${\displaystyle \left[{\gamma }_0{q}_1\left(P_S^{\ast }\right)\right]/\left[G_2{q}_2\left(P_S^{\ast }\right)\right]=-1}$ where, $q_1\left(P_S^{\ast }\right)=\frac{-{\gamma }_0}{{\left(P_S^{\ast }\right)}^2}\left[\frac{{\rho }_0{e}^{-\frac{{\gamma }_0}{G_0}}}{G_0\left(1-e^{-\frac{{\gamma }_0}{G_0}}\right)\left(1-e^{-\frac{{\gamma }_0}{G_0}}-{\rho }_0\right)}+\frac{1}{G_1}\right]$ and $q_2\left(P_S^{\ast }\right)={\left[\frac{{\gamma }_0}{G_1}+ln\left(1-\frac{{\rho }_0}{1-e^{-\frac{{\gamma }_0}{G_0}}}\right)\right]}^2$.

It should be noted that P_U is a monotonically decreasing function of P_S within the feasible range. Therefore, the optimal transmit power $P_S^{\ast }$ of the source node can be easily obtained by numerical methods if an initial value of P_S is given. Then the optimal transmission power of the UAV can be obtained by substituting $P_S^{\ast }$ into the Eq. (34).

Simulation

The theoretical derivation in the previous is simulated in this section, and the correctness of the theory is verified. Moreover, the influence of the signal-to-noise ratio threshold on the optimal power allocation strategy is further discussed under different diversity reception modes. The simulation parameters are set as follows: the corresponding average power gains are given as G ₀ = 1, G ₁ = 5, G ₂ = 0.5, respectively and threshold of the signal-to-noise ratio is set as γ₀ = 5dB.

Figure 3 shows the optimal power allocation strategy for a network with diversity when the signal-to-noise ratio at the destination node is no less than the preset threshold.

We simulate the third equation in Eqs. (32) and (33) (the curve and virtual real line in Fig. 3 respectively) and find the intersection of the two lines which is the value of optimal power. To reduce the computational complexity, the optimal power can also be obtained by simultaneous Eqs. (34) and (37), as shown by the black dot in the figure. We find that the same optimal power can be obtained by the two methods, which verifies the correctness of the theoretical derivation.

The optimal transmission power for both with/without diversity is depicted in Figs. 4 and 5, which vary with the outage probability threshold, respectively. The simulation value range of the outage probability threshold is from 10 ⁻³ to 0.1. It can be seen that the optimal transmission power required by the network is significantly reduced since the outage probability constraint is relaxed. In addition, the optimal power with diversity is significantly less than it without diversity for the same outage probability threshold. This is because there is an extra direct link from the source node to the destination node in the diversity system, which strengthens the SNR at the destination node.