Energy-efficiency resource allocation of very large multi-user MIMO systems

Abstract

With increasing demand in multimedia applications and high data rate services, energy consumption of wireless devices has become a problem. At the user equipment side, high-level energy consumption brings much inconvenience, especially for mobile terminals that cannot connect an external charger, due to an exponentially increasing gap between the available and required battery capacity. Motivated by this, in this paper we consider uplink energy-efficient resource allocation in very large multi-user MIMO systems. Specifically, both the number of antenna arrays at BS and the transmit data rate at the user are adjusted to maximize the energy efficiency, in which the power consumption accounts for both transmit power and circuit power. We proposed two algorithms. Algorithm1, we demonstrate the existence of a unique globally optimal data rate and the number of antenna arrays by exploiting the properties of objective function, then we develop an iterative algorithm to obtain this optimal solution. Algorithm2, we transform the considered nonconvex optimization problem into a convex optimization problem by exploiting the properties of fractional programming, then we develop an efficient iterative resource allocation algorithm to obtain this optimal solution. Our simulation results did not only show that the the proposed two algorithms converge to the solution within a small number of iterations, but demonstrated also the performances of the proposed two algorithms are close to the optimum. Meanwhile, it also shows that with a given number iterations the performance of proposed algorithm1 is superior to proposed algorithm2 under small p _C . On the contrary, the performance of proposed algorithm2 is superior to proposed algorithm1 under large p _C .

Appendices

Appendix 1

Proof of lemma 1

• i. Denote the upper contour sets of U(r, M) as S _α = { M ≥ K + 1 | U(r,M) ≥ α }.

According to proposition C.9 of [23], U(r, M) is strictly quasi-concave if and only if S _α is strictly convex for any real number α. When α ≤ 0, no points exists on the contour U(r, M) = α. When α > 0, S _α is equivalent to S _{α={ M ≥ K+1 | α P_T} (r,M) + α P _C (M) − R(r) ≤ 0 }. Since P _T (r, M) and P _T (M) are strictly convex in M, S _α is also strictly convex. Hence, we have the strict quasiconcavity of U(r, M)

The partial derivative of U(r, M) with M is

$$\begin{aligned} \frac{\partial U(r,M)}{\partial M} & =\frac{-R(r)[P^{'}_{T}(r,M)+P^{'}_{C}(M)]}{[P_{T}(r,M)+P_{C}(M)]^{2}}\\ & \mathop{=}\limits^{\triangle} \frac{\phi(r,M)}{[P_{T}(r,M)+P_{C}(M)]^{2}} \end{aligned}$$

(14)

where P ^' _T (r, M) is the first partial derivative of P _T (r, M) with respect of M, P ^' _C (M) is the first partial derivative of P _C (M) with respect of M. According to Lemma 1, if M* exists such that $\frac{\partial U(r,M)}{\partial M}|_{M=M^{*}}=0$, it is unique, i.e. There is a M* such that ϕ (r,M) = 0, it is unique. In the following, we investigate the conditions when M* exists.

The derivative of ϕ(r, M) with respect of M is $\phi^{'}(r,M)=-RP_{T}^{\prime\prime}(r,M)<0$. Where $P_{T}^{\prime\prime}(r,M)$ is the second partial derivative of P _T (r, M) with respect of M. Hence, ϕ(r, M) is strictly decreasing. According to the L’Hopital’s rule, it is easy to show that

$$\begin{aligned} \mathop{lim}\limits_{M\rightarrow \infty} \phi(r,M) & = \mathop{lim}\limits_{M\rightarrow \infty}\{ -R(r)[P_{T}^{'}(r,M)+P_{C}^{'}(M)] \}\\ & = \mathop{lim}\limits_{M\rightarrow \infty} \{ \frac{-R(r)[P_{T}^{'} (r,M)+P_{C}^{'}(M)]}{M}M \}\\ & = \mathop{lim}\limits_{M\rightarrow \infty} [\frac{-R(r)P_{T}^{\prime\prime}(r,M)}{1}M]<0 \end{aligned}$$

(15)

Because of the M ≥ K + 1, where K is the number of users, so

$$\begin{aligned} \mathop{lim}\limits_{M\rightarrow K+1} \phi(r,M)& = \mathop{lim}\limits_{M\rightarrow K+1}\{ -R(r)[P_{T}^{'}(r,M)+P_{T}^{'}(M)] \}\\ & = -R(r)[-K(2^{r}-1)+p_{C}]\\ & = R(r)[K(2^{r}-1)-p_{C}] \end{aligned}$$

(16)

• 1. when $R(r)[K(2^{r}-1)-p_{C}]\geq 0, \mathop{lim}\limits_{M\rightarrow K+1}\phi(r,M)\geq 0$. We can see that M* exists and U(r, M) is first strictly increasing and then strictly decreasing in M.

• 2. when $R(r)[K(2^{r}-1)-p_{C}]<0, \mathop{lim}\limits_{M\rightarrow K+1}\phi(r,M)<0$. We can see that U(r, M) is always strictly decreasing in M. Hence, U(r, M) is maximized at M = K + 1.

• ii. The quasi-concavity of U(r, M) with respect of r is the same as the quasi-concavity of U(r, M) with respect of M.

The partial derivative of U(r, M) with r is

$$\begin{aligned} \frac{\partial U(r,M)}{\partial r} & = \frac{R^{'}(r)[P_{T}(r,M)+P_{C}(M)]- R(r)\bar{P_{T}^{'}}(r,M)}{[P_{T}(r,M)+P_{C}(M)]^{2}}\\ & \mathop{=}\limits_{\triangle} \frac{\varphi(r,M)}{[P_{T}(r,M)+P_{C}(M)]^{2}} \end{aligned}$$

(17)

where $\bar{P}_{T}^{'}(r,M)$ is the first partial derivative of P _T (r, M) with respect of r, R ^'(r) is the first partial derivative of R(r) with respect of r. According to Lemma 1, if r* exists such that $\frac{\partial U(r,M)}{\partial r}|_{r=r^{*}}=0$, it is unique, i.e. there is a r* such that $\varphi(r^{*},M)=0$, it is unique. In the following, we investigate the conditions when r* exists.

The derivative of $\varphi(r,M)$ with respect of r is

$$\varphi^{'}(r,M)=-R(r)\bar{P}_{T}^{\prime\prime}(r,M)<0$$

(18)

where $\bar{P}_{T}^{\prime\prime}(r,M)$ is the second partial derivative of P _T (r, M) with respect of r. Hence, $\varphi(r,M)$ is strictly decreasing. According to the L’Hopital’s rule, it is easy to show

$$\begin{aligned} \mathop{lim}\limits_{r\rightarrow\infty}\varphi(r,M) &= \mathop{lim}\limits_{r\rightarrow\infty} \{ R^{'}(r)[P_{T}(r,M)+P_{C}(M)]-R(r) \bar{P}_{T}^{'}(r,M) \}\\ & = \mathop{lim}\limits_{r\rightarrow\infty}\left\{ \frac{R^{'}(r)[P_{T}(r,M)+ P_{C}(M)]-R(r)\bar{P}_{T}^{'}(r,M)}{r}r \right\}\\ & = \mathop{lim}\limits_{r\rightarrow\infty} \left[\frac{-R(r)\bar{P}_{T}^{\prime\prime}(r,M)}{1}r\right]<0 \end{aligned}$$

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Besides

$$\mathop{lim}\limits_{r\rightarrow 0}\varphi(r,M)= \mathop{lim}\limits_{r\rightarrow 0}R^{'}(r)[P_{T}(r,M)+P_{C}(M)]- R(r)\bar{P}_{T}^{'}(r,M) = KP_{C}(M)>0$$

We can see that r* exists and U(r, M) is first strictly increasing and then strictly decreasing in r.

Appendix 2

Proof of the concavity of the problem

Without loss of generality, we define function

$$\begin{aligned} f & =R(r)-q[P_{T}(r,M)+P_{C}(M)]\\ & = Kr - q(K\frac{2^{r}-1}{M-K}+Mp_{C}) \end{aligned}$$

(20)

The Hessian matrix is given by $H(f)= \left( \begin{array}{cc}-qK\frac{2^{r}(ln2)^{2}}{M-K} & qK\frac{2^{r}ln2}{(M-K)^{2}} \\ qK\frac{2^{r}ln2}{(M-K)^{2}} & -2qK\frac{2^{r}-1}{(M-K)^{3}} \\ \end{array}\right)$

So, f is jointly concave w.r.t r and M. Therefore, the objective function is jointly concave w.r.t r and M.

Appendix 3

Proof of algorithm 2 convergence

We follow a similar approach as in [25] for proving to the convergence of algorithm 2. We first introduce two propositions.

Proposition 1

F(q) is strictly monotoinc decreasing, i.e. $F(q^{\prime\prime})<F(q^{'})$ if $q^{'}<q^{\prime\prime}$

Proof

let $\{ r^{\prime\prime}, M^{\prime\prime} \}$ maximize $F(q^{\prime\prime})$, then

$$\begin{aligned} F(q^{\prime\prime})& =\underbrace{max}_{\{M,r\}}R(r)-q^{\prime\prime}[P_{T}(r,M)+ P_{C}(M)]\\ & = R(r^{\prime\prime})-q^{\prime\prime}[P_{T}(r^{\prime\prime},M^{\prime\prime})+ P_{C}(M^{\prime\prime})]\\ & < R(r^{\prime\prime})-q^{'}[P_{T}(r^{\prime\prime},M^{\prime\prime})+ P_{C}(M^{\prime\prime})]\\ & \leq \underbrace{max}_{\{M,r\}} R(r) - q^{'}[P_{T}(r,M)+P_{C}(M)]\\ & =F(q^{'}) \end{aligned}$$

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Proposition 2

let r′, M′ be an arbitrary feasible solution and $q^{'}=\frac{R(r^{'})}{P_{T}(r^{'},M^{'})P_{C}(M^{'})}$, then F(q ^') > 0.

Proof

$$\begin{aligned} F(q^{'})&= \underbrace{max}_{\{M,r\}} R(r)-q^{'}[P_{T}(r,M)+P_{C}(M)]\\ & \geq R(r^{'})-q^{'}[P_{T}(r^{'},M^{'})+P_{C}(M^{'})]\\ & = 0 \end{aligned}$$

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We are now ready to prove the convergence of algorithm 2.

• a. First we shall prove that the energy efficiency q increases in each iteration. Suppose q _k  ≠ q* and q _k+1 ≠ q* represent the energy efficiency of the considered system in iteration k and k + 1, respectively. Theorem 2 and proposition 2 imply F(q _k ) > 0. By definition we have R(r _k ) = q _k+1[P _T (r _k ,M _k ) + P _C (M _k )]. Hence

  $$\begin{aligned} F(q_{k})& =R(r_{k})-q_{k}[P_{T}(r_{k},M_{k})+P_{C}(M_{k})]\\ & = q_{k+1}[P_{T}(r_{k},M_{k})+P_{C}(M_{k})]-q_{k}[P_{T}(r_{k},M_{k})+P_{C}(M_{k})]\\ & = (q_{k+1}-q_{k})[P_{T}(r_{k},M_{k})+P_{C}(M_{k})]\\ & > 0\\ \end{aligned}$$

  (23)

  We have q _k+1 > q _k .

• b. Obviously, the energy efficiency q converges to the optimal q* such that it satisfies the optimality condition in theorem 2, i.e. F(q*) = 0.