Throughput maximization for UAV-assisted wireless powered D2D communication networks with a hybrid time division duplex/frequency division duplex scheme

Abstract

In this paper, we investigate a low-altitude unmanned aerial vehicle (UAV)-assisted wireless power- ed device-to-device (D2D) network with the “harvest-transmit-store” model, where a UAV broadcasts energy to all D2D transmitters and the transmitters then store or use the energy for transmission information. In the proposed model, the UAV does not transmit energy in the transmission information time, resulting in a limited charging time. In addition, the interference may be caused by communicating information of an arbitrary number of D2D pairs simultaneously. In this paper, we propose a hybrid time division duplex/frequency division duplex communication scheme, which performs the power transfer and information transmission simultaneously in the transmission information time. The scheme cancels the cochannel interference by using the time division duplex scheme and guarantees the charging time by using the frequency division duplex. We consider optimizing the throughput based on the scheme via joint power control, spectrum allocation, and scheduling time. We propose a two-step algorithm to solve the formulated nonconvex problem. The simulation results show that the proposed algorithm improves the average throughput compared to the other schemes.

Appendices

Appendix A

A concave programming problem needs to satisfy two conditions: all the constraints are linear conditions and the objective function is a concave function. All the constraints are linear in problem [M2]. We only need to prove that the objective function is a concave function. Let $f(\varvec{a},\varvec{t})$ denote the objective function, where a and t are variable vectors that contain the variables $a_{ij}$ and $t_{ij}$ respectively, $\forall j\in \{1,2,,N\}$ and $\forall i\in \{2,,T\}$. To solve the first derivative of the variables $a_{ij}$ and $t_{ij}$, we use

$$\begin{aligned} \frac{{\partial f(\varvec{a},\varvec{t})}}{{\partial {a_{ij}}}} = \frac{{{h_j}{t_{ij}}W{{\hat{\mu }}_i}}}{{{a_{ij}}{h_j}ln2 + {t_{ij}}W{{\hat{\mu }}_i}\delta ln2}} \end{aligned}$$

(15)

$$\begin{aligned} \frac{{\partial f({{\varvec{a}}},{{\varvec{t}}})}}{{\partial {t_{ij}}}} = \frac{{W{{\hat{\mu }}_i}( - {a_{ij}}{h_j} + ({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i}))}}{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})ln2}} \nonumber \\ *ln[1 + \frac{{{a_{ij}}{h_j}}}{{{t_{ij}}W{{\hat{\mu }}_i}\delta }}] \end{aligned}$$

(16)

From Eqs. (15) and (16), $\frac{{{\partial ^2}f(\varvec{a},\varvec{t})}}{{\partial {a_{ij}}\partial {a_{zk}}}}$ and $\frac{{{\partial ^2}f(\varvec{a},\varvec{t})}}{{\partial {a_{ij}}\partial {t_{zk}}}}$ are zero as $i \ne z$ or $j\ne k$. When $i=z$ and $j=k$, we can obtain the second-order partial derivative of the function $f(\varvec{a},\varvec{t})$ as follows:

$$\begin{aligned} \frac{{{\partial ^2}f(\varvec{a},\varvec{t})}}{{\partial {a_{ij}}^2}} = - \frac{{{h_j}^2{t_{ij}}W{{\hat{\mu }}_i}}}{{ln2{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})}^2}}} \end{aligned}$$

(17)

$$\begin{aligned} \frac{{{\partial ^2}f(\varvec{a},\varvec{t})}}{{\partial {t_{ij}}^2}} = - \frac{{{a_{ij}}^2{h_j}^2W{{\hat{\mu }}_i}}}{{ln2{t_{ij}}{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})}^2}}} \end{aligned}$$

(18)

$$\begin{aligned} \frac{{{\partial ^2}f(\varvec{a},\varvec{t})}}{{\partial {t_{ij}}\partial {a_{ij}}}} = \frac{{{a_{ij}}{h_j}^2W{{\hat{\mu }}_i}}}{{ln2{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})}^2}}} \end{aligned}$$

(19)

We use a Hessian matrix to analyze the objective function. Let ${{H}}({a_{ij}},{t_{ij}})$ represent the Hessian matrix corresponding to the variables $a_{ij}$ and $t_{ij}$. From Eqs. (17)–(19), we have

$$\begin{aligned}&{{H}}({a_{ij}},{t_{ij}}) = \nonumber \\&\left[ {\begin{array}{*{20}{c}} { - \frac{{{h_j}^2{t_{ij}}W{{\hat{\mu }}_i}}}{{ln2{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})}^2}}}}&{}{\frac{{{a_{ij}}{h_j}^2W{{\hat{\mu }}_i}}}{{ln2{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})}^2}}}}\\ {\frac{{{a_{ij}}{h_j}^2W{{\hat{\mu }}_i}}}{{ln2{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})}^2}}}}&{}{ - \frac{{{a_{ij}}^2{h_j}^2W{{\hat{\mu }}_i}}}{{ln2{t_{ij}}{{({a_{ij}}{h_j} + {t_{ij}}W\delta {{\hat{\mu }}_i})}^2}}}} \end{array}} \right] \end{aligned}$$

(20)

We use ${{{H}}^{uv}}({a_{ij}},{t_{ij}})$ to represent the value in row u and column v of matrix $H(a_{ij},t_{ij})$. Since all variables $a_{ij}$ and $t_{ij}$ are nonnegative variables in Eq. (20), ${{{H}}^{11}}({a_{ij}},{t_{ij}})$ is nonpositive. Determining $det(H(a_{ij},t_{ij}))$ of the matrix yields

$$\begin{aligned} det(H(a_{ij},t_{ij}))=0 \end{aligned}$$

(21)

From Eqs. (20) and (21), $H(a_{ij},t_{ij})$ is negative semi-definite. We use H to represent the Hessian matrix of the objective function, which can be expressed as Eq. (22).

$$\begin{aligned}{H} = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{H}({a_{11}},{t_{11}}),}&{}{{H}\left( {{a_{11}},{t_{12}}} \right) = 0,}\\ {0,}&{}{{H}\left( {{a_{12}},{t_{12}}} \right) ,} \end{array}}&{}{\begin{array}{*{20}{c}} \ldots &{}{{{H}}\left( {{a_{11}},{t_{NT}}} \right) = 0}\\ {0,}&{} \ldots \end{array}}\\ {\begin{array}{*{20}{c}} \vdots &{}{ \ddots }\\ 0&{}{ \ldots } \end{array}}&{}{\begin{array}{*{20}{c}} \ddots &{} \vdots \\ \ldots &{}{{{H}}\left( {{a_{NT}},{t_{NT}}} \right) } \end{array}} \end{array}} \right) \end{aligned}$$

(22)

From Eq. (22), the matrix H is negative semi-definite. Therefore, $f(\varvec{a},\varvec{t})$ is a concave function. Problem [M2] is a concave programming problem.

Appendix B

The constraint is a linear condition in problem [M5]. Therefore, we only need to prove that the objective function is a concave function. We use $f(\varvec{\mu })$ to denote the objective function and obtain the first derivative of the function as follows:

$$\begin{aligned} \frac{\partial }{{\partial {\mu _i}}}f( \varvec{\mu } ) = \mathop \sum \limits _{j = 1}^N \left( - \frac{{W{{\hat{t}}_{ij}}{{\hat{P}}_{ij}}{h_j}}}{{( {{{\hat{P}}_{ij}}{h_j} + {\mu _i}W\delta })\ln 2}} + W{{\hat{t}}_{ij}}\log \left( {1 + \frac{{{{\hat{P}}_{ij}}{h_j}}}{{{\mu _i}W\delta }}}\right) \right) \end{aligned}$$

(23)

Based on the Eq. (23), we find the second order of the function $f(\varvec{\mu })$ and obtain

$$\begin{aligned} \frac{{{\partial ^2}f(\varvec{\mu } )}}{{\partial {\mu _i}\partial {\mu _j}}} = 0, i \ne j, \end{aligned}$$

(24)

$$\begin{aligned} \frac{{{\partial ^2}f(\varvec{\mu } )}}{{\partial {\mu _i}\partial {\mu _j}}} = - \mathop \sum \limits _{j = 1}^N \frac{{{{\hat{t}}_{ij}}{{\hat{P}}_{ij}}^2{h_j}^2}}{{{\mu _i}^2W{{(1 + \frac{{{{\hat{P}}_{ij}}{h_j}}}{{{\mu _i}W\delta }})}^2}{\delta ^2}ln2}}, i = j. \end{aligned}$$

(25)

From Eqs. (24) and (25), the Hessian matrix of the function $f(\varvec{\mu })$ can be expressed as Eq. (26).

$$\begin{aligned} \mathrm{{H}} = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - \mathop \sum \limits _{j = 1}^N \frac{{{{\hat{t}}_{1j}}{{\hat{P}}_{1j}}^2{h_j}^2}}{{{\mu _1}^2W{{(1 + \frac{{{{\hat{P}}_{ij}}{h_j}}}{{{\mu _1}W\delta }})}^2}{\delta ^2}ln2}},}&{}{0,}\\ {0,}&{}{ - \mathop \sum \limits _{j = 1}^N \frac{{{{\hat{t}}_{2j}}{{\hat{P}}_{2j}}^2{h_j}^2}}{{{\mu _2}^2W{{(1 + \frac{{{{\hat{P}}_{2j}}{h_j}}}{{{\mu _2}W\delta }})}^2}{\delta ^2}ln2}},} \end{array}}&{}{\begin{array}{*{20}{c}} \ldots &{}{0}\\ {0,}&{} \ldots \end{array}}\\ {\begin{array}{*{20}{c}} \vdots &{}{ \ddots }\\ 0&{}{ \ldots } \end{array}}&{}{\begin{array}{*{20}{c}} \ddots &{} \vdots \\ \ldots &{}{ - \mathop \sum \limits _{j = 1}^N \frac{{{{\hat{t}}_{Tj}}{{\hat{P}}_{Tj}}^2{h_j}^2}}{{{\mu _T}^2W{{(1 + \frac{{{{\hat{P}}_{Tj}}{h_j}}}{{{\mu _T}W\delta }})}^2}{\delta ^2}ln2}}} \end{array}} \end{array}} \right) . \end{aligned}$$

(26)

It can be seen that the matrix is negative semi-definite from Eq. (26). Thus, problem [M5] is a concave programming problem.