Energy efficient joint user scheduling and transmit beamforming in downlink DAS

Abstract

This paper studies the energy efficient joint user scheduling and transmit beamforming in the downlink of a single-cell distributed antenna system. Due to the distributed nature of antenna units, traditional power consumption model cannot be applied without the backhauling power taken into account. Therefore, under the constraints of the minimum rate requirement of each user and the transmit power budget of each distributed antenna unit, we maximize the system energy efficiency through making a tradeoff between the transmitting power and the backhauling power. By employing the parametric equivalence method to deal with the fractional objective and the $\ell _1$-norm relaxation method to cope with the user scheduling problem, the energy efficiency maximization is turned into a standard difference of convex program and then solved via the successive convex approximation method. Finally, the optimal sparse transmit beamforming vectors are obtained during the weighting factor iteration process, with the aim of minimizing the total power consumption while maintaining the achieved rate of each user. Extensive simulations are conducted to demonstrate the effectiveness of proposed scheme. Simulation results show that energy efficiency can benefit from not only the user scheduling but also the transmit beamforming optimization.

Appendices

Appendix A Proof of Proposition 1

With parameter $\lambda$ and initialized $\overline{{\mathbf{W}}}$, let $\widetilde{\varGamma }_{\lambda }({\mathbf{W}},\overline{{\mathbf{W}}})$ denote the value of (18a) when the beam matrix is ${\mathbf{W}}$, which is shown as below

$$\begin{aligned} \widetilde{\varGamma }_{\lambda }({\mathbf{W}},\overline{{\mathbf{W}}})=&\sum _{k=1}^{K}B\left[ f_k({\mathbf{W}})-\left( g_k(\overline{{\mathbf{W}}})+{\text{vec}}(\nabla g_k(\overline{{\mathbf{W}}}))^T{\text{vec}}({\mathbf{W}}-\overline{{\mathbf{W}}})\right) \right] \nonumber \\&-\lambda \varepsilon \sum \limits _{k=1}^{K}{\text{Tr}}({\mathbf{W}}_{k})- \lambda NP_{c}-\lambda P_{sp}. \end{aligned}$$

(23)

Since $g_k({\mathbf{W}})$ is a concave function, we have

$$\begin{aligned} g_k(\overline{{\mathbf{W}}})+{\text{vec}}(\nabla g_k(\overline{{\mathbf{W}}})^T{\text{vec}}({\mathbf{W}}-\overline{{\mathbf{W}}})\ge g_k({\mathbf{W}}), \ \forall k. \end{aligned}$$

(24)

At the mth SCA iteration, with initialized $\overline{{\mathbf{W}}}[m]$, we obtain the optimal objective of problem (18) as follows:

$$\begin{aligned}&\widetilde{\varGamma }_{\lambda }({\mathbf{W}}^*[m],\overline{{\mathbf{W}}}[m]) \nonumber \\&\quad = \sum \limits _{k=1}^{K}B\left[ f_k({\mathbf{W}}^*[m])-\left( g_k(\overline{{\mathbf{W}}}[m])+{\text{vec}}(\nabla g_k(\overline{{\mathbf{W}}}[m]))^T{\text{vec}}({\mathbf{W}}^*[m]-\overline{{\mathbf{W}}}[m])\right) \right] \nonumber \\&\quad - \lambda \varepsilon \sum \limits _{k=1}^{K}{\text{Tr}}({\mathbf{W}}_{k}^*[m])- \lambda NP_{c}-\lambda P_{sp} \nonumber \\&\quad \le \sum \limits _{k=1}^{K}B\left( f_k({\mathbf{W}}^*[m])-g_k({\mathbf{W}}^*[m])\right) -\lambda \varepsilon \sum \limits _{k=1}^{K}{\text{Tr}}({\mathbf{W}}_{k}^*[m])- \lambda NP_{c}-\lambda P_{sp} \nonumber \\&\quad = \varGamma _{\lambda }({\mathbf{W}}^*[m]), \end{aligned}$$

(25)

where $\varGamma _{\lambda }({\mathbf{W}}^*[m])$ is the value of objective function (15a) with the beam matrix ${\mathbf{W}}^*[m]$.

According to the SCA iteration: $\overline{{\mathbf{W}}}_k[m+1]={\mathbf{W}}_k^*[m]$, we have

$$\begin{aligned}&\widetilde{\varGamma }_{\lambda }({\mathbf{W}}^*[m],\overline{{\mathbf{W}}}[m+1]) \nonumber \\&\quad = \sum \limits _{k=1}^{K}B\left[ f_k({\mathbf{W}}^*[m])-\left( g_k(\overline{{\mathbf{W}}}[m+1])+{\text{vec}}(\nabla g_k(\overline{{\mathbf{W}}}[m+1]))^T{\text{vec}}({\mathbf{W}}^*[m]-\overline{{\mathbf{W}}}[m+1])\right) \right] \nonumber \\&\quad -\lambda \varepsilon \sum \limits _{k=1}^{K}{\text{Tr}}({\mathbf{W}}_{k}^*[m])- \lambda NP_{c}-\lambda P_{sp} \nonumber \\&\quad = \sum \limits _{k=1}^{K}B\left( f_k({\mathbf{W}}^*[m])-g_k({\mathbf{W}}^*[m])\right) -\lambda \varepsilon \sum \limits _{k=1}^{K}{\text{Tr}}({\mathbf{W}}_{k}^*[m])- \lambda NP_{c}-\lambda P_{sp} \nonumber \\&\quad = \varGamma _{\lambda }({\mathbf{W}}^*[m]). \end{aligned}$$

(26)

Since ${\mathbf{W}}^*[m+1]$ is the optimal solution of problem (18) with initialized $\overline{{\mathbf{W}}}[m+1]$, we have

$$\begin{aligned} \widetilde{\varGamma }_{\lambda }({\mathbf{W}}^*[m],\overline{{\mathbf{W}}}[m+1])\le \widetilde{\varGamma }_{\lambda }({\mathbf{W}}^*[m+1],\overline{{\mathbf{W}}}[m+1]). \end{aligned}$$

(27)

Finally, combining (25), (26), and (27), we obtain

$$\begin{aligned} \widetilde{\varGamma }_{\lambda }({\mathbf{W}}^*[m],\overline{{\mathbf{W}}}[m])\le \widetilde{\varGamma }_{\lambda }({\mathbf{W}}^*[m+1],\overline{{\mathbf{W}}}[m+1]). \end{aligned}$$

(28)

For simplicity, let $\widetilde{\varGamma }_{\lambda }^*[m]=\widetilde{\varGamma }_{\lambda }({\mathbf{W}}^*[m],\overline{{\mathbf{W}}}[m])$. Thus, with the iteration in (19), the optimal objective sequence $\{\widetilde{\varGamma }_{\lambda }^*[m]\}$ is monotonically increased. Moreover, due to the power constraint of each DAU, $\widetilde{\varGamma }^*_{\lambda }[m]$ is always upper-bounded. Then, we have proved the convergence of the SCA iteration. Furthermore, since the original objective function in (15a) is neither convex nor concave, the convex approximation based on the first-order Taylor series approximation and the SCA iteration can only obtain a local optimum of problem (15).

Appendix B Proof of Proposition 2

The Lagrangian of the problem (18) is

$$\begin{aligned} {\mathcal{L}}({\mathbf{W}},\mathbf{\alpha },\mathbf{\beta })=&\sum \limits _{k=1}^{K}B( f_k({\mathbf{W}})-\widetilde{g}_k({\mathbf{W}}))-\lambda \varepsilon \sum \limits _{k=1}^{K}{\text{Tr}}({\mathbf{W}}_{k})- \lambda NP_{c}-\lambda P_{sp} \nonumber \\&+\sum \limits _{k=1}^{K}\alpha _k\left[{\text{Tr}}\left( {\mathbf{W}}_k{\mathbf{H}}_k\right) - \gamma _k^0\left( \sum _{j=1,j\ne k}^{K}{\text{Tr}}\left( {\mathbf{W}}_j{\mathbf{H}}_k\right) +\sigma ^2 \right) \right] \nonumber \\&+\sum _{n=1}^{N}\beta _n\left[ P_n^{\max }-\sum \limits _{k=1}^{K}{\text{Tr}}({\mathbf{A}}_n\circ {\mathbf{W}}_k)\right] +\sum \limits _{k=1}^{K} {\text{Tr}}({\mathbf{W}}_k{\mathbf{Z}}_k), \end{aligned}$$

(29)

where $\mathbf{\alpha }=[\alpha _1,\ldots ,\alpha _K]^T$ and $\mathbf{\beta }=[\beta _1,\ldots ,\beta _N]^T$ are the dual vectors for the SINR constraints and the power restrictions, respectively. ${\mathbf{Z}}_k$ is an additional slack matrix.

The partial derivative of ${\mathcal {L}}({\mathbf{W}},\mathbf{\alpha },\mathbf{\beta })$ over ${\mathbf{W}}_i$ is

$$\begin{aligned} \frac{\partial {\mathcal {L}}({\mathbf{W}},\mathbf{\alpha },\mathbf{\beta })}{\partial {\mathbf{W}}_i}= \sum \limits _{k=1}^{K}B\left( \frac{\partial f_k({\mathbf{W}})}{\partial {\mathbf{W}}_i}-\frac{\partial \widetilde{g}_k({\mathbf{W}})}{\partial {\mathbf{W}}_i}\right) -\lambda \varepsilon {\mathbf{I}}+ \alpha _i{\mathbf{H}}_i-\sum \limits _{k=1,k\ne i}^{K}\alpha _k\gamma _k^0{\mathbf{H}}_k -\sum \limits _{n=1}^{N}\beta _n{\mathbf{A}}_n+{\mathbf{Z}}_i, \end{aligned}$$

(30)

According to (12) and (17),

$$\begin{aligned} \frac{\partial {f}_k({\mathbf{W}})}{\partial {\mathbf{W}}_i}=&\left\{ \begin{aligned}&\frac{1}{\text{ln}2}\frac{{\mathbf{H}}_i}{\sum _{j=1}^{K}{\text{Tr}}\left( {\mathbf{W}}_j{\mathbf{H}}_i\right) +\sigma ^2}+\lambda \frac{P_{bh}}{C_{bh}}{\text{log}}_2\left( \sum _{j=1,j\ne k}^{K}{\text{Tr}}\left( {\mathbf{W}}_j{\mathbf{H}}_i\right) +\sigma ^2\right) {\mathbf{I}},\ \text{if}\ i=k, \\&\frac{1}{{\text{ln}}2}\frac{{\mathbf{H}}_k}{\sum _{j=1}^{K}{\text{Tr}}\left( {\mathbf{W}}_j{\mathbf{H}}_k\right) +\sigma ^2}+\frac{\lambda }{{\text{ln}}2}\frac{P_{bh}}{C_{bh}}{\text{Tr}}({\mathbf{W}}_k)\frac{{\mathbf{H}}_k}{\sum _{j=1,j\ne k}^{K}\text{Tr}\left( {\mathbf{W}}_j{\mathbf{H}}_k\right) +\sigma ^2}, \ \text{otherwise}. \end{aligned} \right. \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial \widetilde{g}_k({\mathbf{W}})}{\partial {\mathbf{W}}_i}=&\frac{\partial g_k(\overline{{\mathbf{W}}})}{\partial {\mathbf{W}}_i} \nonumber \\ =&\left\{ \begin{aligned}&\frac{\lambda }{\text{ln}2}\frac{P_{bh}}{C_{bh}}{\text{Tr}}(\overline{{\mathbf{W}}}_i)\frac{{\mathbf{H}}_i}{\sum _{j=1}^{K}{\text{Tr}}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_i\right) +\sigma ^2}+ \lambda \frac{P_{bh}}{C_{bh}} \text{log}_2\left( \sum _{j=1}^{K}{\text{Tr}}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_i\right) +\sigma ^2\right) {\mathbf{I}}, \ {\text{if}} \ i=k, \\&\frac{\lambda }{{\text{ln}}2}\frac{P_{bh}}{C_{bh}}{\text{Tr}}(\overline{{\mathbf{W}}}_k)\frac{{\mathbf{H}}_k}{\sum _{j=1}^{K}\text{Tr}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_k\right) +\sigma ^2}+\frac{1}{\text{ln}2}\frac{{\mathbf{H}}_k}{\sum _{j=1,j\ne k}^{K}\text{Tr}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_k\right) +\sigma ^2}, \ \text{otherwise}. \end{aligned}\right. \end{aligned}$$

(32)

Substituting (31) and (32) into (30),

$$\begin{aligned}&\frac{\partial {\mathcal {L}}({\mathbf{W}},\mathbf{\alpha },\mathbf{\beta })}{\partial {\mathbf{W}}_i} \nonumber \\&\quad = B\left[ \frac{1}{{\text{ln}}2}\frac{{\mathbf{H}}_i}{\sum _{j=1}^{K}{\text{Tr}}\left( {\mathbf{W}}_j{\mathbf{H}}_i\right) +\sigma ^2}+\lambda \frac{P_{bh}}{C_{bh}}\text{log}_2\left( \sum _{j=1,j\ne k}^{K}{\text{Tr}}\left( {\mathbf{W}}_j{\mathbf{H}}_i\right) +\sigma ^2\right) {\mathbf{I}}\right] \nonumber \\&\quad -B\left[ \frac{\lambda }{\text{ln}2}\frac{P_{bh}}{C_{bh}}{\text{Tr}}(\overline{{\mathbf{W}}}_i)\frac{{\mathbf{H}}_i}{\sum _{j=1}^{K}{\text{Tr}}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_i\right) +\sigma ^2}+ \lambda \frac{P_{bh}}{C_{bh}} {\text{log}}_2\left( \sum _{j=1}^{K}{\text{Tr}}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_i\right) +\sigma ^2\right) {\mathbf{I}}\right] \nonumber \\&\quad +\sum \limits _{k=1,k\ne i}^{K}B\left[ \frac{1}{{\text{ln}}2}\frac{{\mathbf{H}}_k}{\sum _{j=1}^{K}{\text{Tr}}\left( {\mathbf{W}}_j{\mathbf{H}}_k\right) +\sigma ^2}+\frac{\lambda }{{\text{ln}}2}\frac{P_{bh}}{C_{bh}}{\text{Tr}}({\mathbf{W}}_k)\frac{{\mathbf{H}}_k}{\sum _{j=1,j\ne k}^{K}\text{Tr}\left( {\mathbf{W}}_j{\mathbf{H}}_k\right) +\sigma ^2}\right] \nonumber \\&\quad -\sum \limits _{k=1,k\ne i}^{K}B\left[ \frac{\lambda }{\text{ln}2}\frac{P_{bh}}{C_{bh}}\text{Tr}(\overline{{\mathbf{W}}}_k)\frac{{\mathbf{H}}_k}{\sum _{j=1}^{K}\text{Tr}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_k\right) +\sigma ^2}+\frac{1}{{\text{ln}}2}\frac{{\mathbf{H}}_k}{\sum _{j=1,j\ne k}^{K}\text{Tr}\left( \overline{{\mathbf{W}}}_j{\mathbf{H}}_k\right) +\sigma ^2}\right] \nonumber \\&\quad -\lambda \varepsilon {\mathbf{I}}+ \alpha _i{\mathbf{H}}_i-\sum \limits _{k=1,k\ne i}^{K}\alpha _k\gamma _k^0{\mathbf{H}}_k -\sum \limits _{n=1}^{N}\beta _n{\mathbf{A}}_n+{\mathbf{Z}}_i, \end{aligned}$$

(33)

According to the KKT conditions [38], we have

$$\begin{aligned}&\frac{\partial {\mathcal {L}}(\mathbf{W},\mathbf{\alpha },\mathbf{\beta })}{\partial \mathbf{W}_i^{\star } }=\mathbf{0}, \end{aligned}$$

(34)

$$\begin{aligned}&\mathbf{W}_i^{\star }\mathbf{Z}_i^{\star }=\mathbf{0} , \end{aligned}$$

(35)

where ${\mathbf{W}}_i^{\star }$ and ${\mathbf{Z}}_i^{\star }$ are the converged optimal solution of problem (18) after the SCA iteration. Moreover, due to the convergence of the SCA iteration, we can approximate ${\mathbf{W}}_k^{\star }=\overline{{\mathbf{W}}}_k$, $\forall k$. Then, from (34), we deduce that ${\mathbf{Z}}_i^{\star }={\mathbf{S}}_i^{\star }-{\mathbf{V}}_i^{\star }$, where

$$\begin{aligned}&{\mathbf{S}}_i^{\star }= \sum _{k=1,k\ne i}^{K}\frac{B}{\text{ln}2}\left( 1-\frac{\lambda P_{bh}}{C_{bh}}{\text{Tr}}({\mathbf{W}}_k^{\star })\right) {\mathbf{H}}_k\left[ \frac{1}{\sum _{j=1,j\ne k}^{K}{\text{Tr}}({\mathbf{W}}_j^{\star }{\mathbf{H}}_k)+\sigma ^2}-\frac{1}{\sum _{j=1}^{K}{\text{Tr}}({\mathbf{W}}_j^{\star }{\mathbf{H}}_k)+\sigma ^2}\right] \nonumber \\&\qquad + \frac{B\lambda P_{bh}}{C_{bh}}\left[ \text{log}_2\left( \sum _{j=1}^{K}{\text{Tr}}({\mathbf{W}}_j^{\star }{\mathbf{H}}_i)+\sigma ^2\right) -{\text{log}}_2\left( \sum _{j=1,j\ne i}^{K}{\text{Tr}}({\mathbf{W}}_j^{\star }{\mathbf{H}}_i)+\sigma ^2\right) \right] {\mathbf{I}} \nonumber \\&\qquad +\lambda \varepsilon {\mathbf{I}}+\sum \limits _{k=1,k\ne i}^{K}\alpha _k^{\star }\gamma _k^0{\mathbf{H}}_k +\sum \limits _{n=1}^{N}\beta _n^{\star }{\mathbf{A}}_n, \end{aligned}$$

(36)

$$\begin{aligned}&{\mathbf{V}}_i^{\star }= \frac{B}{{\text{ln}}2}\left( 1-\frac{\lambda P_{bh}}{C_{bh}}\text{Tr}({\mathbf{W}}_i^{\star })\right) \frac{{\mathbf{H}}_i}{\sum _{j=1}^{K}{\text{Tr}}({\mathbf{W}}_j^{\star }{\mathbf{H}}_i)+\sigma ^2}+\alpha _i^{\star }{\mathbf{H}}_i. \end{aligned}$$

(37)

It can be observed that, if $1-\frac{\lambda P_{bh}}{C_{bh}}\text{Tr}({\mathbf{W}}_k^{\star })\ge 0 \ (\forall k)$, then the first part of ${\mathbf{S}}_i^{\star }$ is semi-definite. Moreover, since the Lagrangian multipliers are non-negative values, we know that ${\mathbf{S}}_i^{\star }$ is a definite matrix with a rank of $\sum _{n=1}^{N}M_n$. Also, the first part of ${\mathbf{V}}_i^{\star }$ is semi-definite, and ${\mathbf{V}}_i^{\star }$ is a matrix with rank one. According to (35), we know that $\text{rank}({\mathbf{W}}_i^{\star })\le \sum _{n=1}^{N}M_n-\text{rank}({\mathbf{Z}}_i^{\star })$. Since $\text{rank}({\mathbf{Z}}_i^{\star })\ge \text{rank}({\mathbf{S}}_i^{\star })-\text{rank}({\mathbf{V}}_i^{\star })$, we obtain $\text{rank}({\mathbf{W}}_i^{\star })\le 1$, $\forall i$. Obviously, $\text{rank}({\mathbf{W}}_i^{\star })=0$ is not the optimal solution of problem (15). Finally, one can deduce that if for any user $k=1,\ldots ,K$, there exists $1-\frac{\lambda P_{bh}}{C_{bh}}\text{Tr}({\mathbf{W}}_k^{\star })\ge 0$, then $\text{rank}({\mathbf{W}}_k^{\star })=1$.