In this section, unconstrained NEE optimization of different schemes with distinct cooperation levels is considered. We formulate the maximizing NEE problem at first, and then three schemes, i.e., IA-GT, ICIC, and MC-JP are taken into account. IA-GT requires only both the CSIT and data of BSs' own cell and performs selfish eigen-beamforming. Hence, non-cooperative power control should be employed in IA-GT. ICIC requires local CSIT and needs no data sharing. Each BS proactively cancel its own interference to other cells in the cooperative cluster and non-cooperative power control is utilized in ICIC. MC-JP requires full data and CSIT sharing and the cooperative cluster can be treated as a"super BS". We consider global zero-forcing beamforming there and cooperative power control is available.

### 3.1 Problem formulation

The problem is formulated in this subsection where NEE is the optimization object. As precoding design is based on eigen-beamforming and zero-forcing beamforming, respectively, as shown above, only the transmit power *P*_{
t,i
}needs to be optimized. The optimization problem can be defined as

\mathbb{P}1:\underset{{\left\{{P}_{t,i}\right\}}_{i=1}^{M}:{P}_{t,i}\ge 0}{max}\mathsf{\text{NEE}}.

(8)

In the above problem, NEE is first addressed as the performance metric to represent the EE of the multi-cell systems. Although NEE have been considered in the uplink multi-cell channels [17], we believe that it is more suitable for the downlink multi-cell systems because of the two reasons as follows. For one thing, maximizing the NEE needs the global information in the cooperative multi-cell system but the users are difficult to get these global information to control their power cooperatively in the uplink systems. For another, battery limitation is important for the users in the uplink channels and the remaining battery energy is always different for each user, and hence, NEE maximizing cannot indicate the EE requirement of each users, respectively. Therefore, designing to maximize the LEE is more suitable for the uplink systems. Things change for the downlink systems. First, backhaul connection among different BSs makes it possible to exchange the CSIT and data information to preform joint optimization, especially CU in the CoMP systems can help the cooperation. Second, different from the battery limitation in the user side, the total power consumption is more important for the BSs, so NEE is provided with practical significance for the downlink cellular networks. Hence, NEE can better externalize the network behavior compared with the previous LEE.

Considering different capability of backhaul connection, limited CSIT and data sharing are also taken into account. Interestingly, maximizing LEE with limited CSIT and data sharing is a sub-optimal choice without extra information exchanging. We discuss these issues later.

### 3.2 Different transmission schemes

The solution of problem \mathbb{P}1 with three different schemes are discussed in this subsection.

#### 3.2.1 Interference aware game theory

IA-GT is a non-cooperative transmission scheme. In this scheme, only the CSIT between the home BS to its dominated user is available for each BS and no data sharing is available. Each BS selfishly determines the precoding vector based on the eigen-beamforming. If the signal for user *i* is denoted as *s*_{
i
}, precoding vector is denoted as **f**_{
i
}, then the transmitted signal at BS *i* is

{x}_{i}={f}_{i}{s}_{i}={h}_{i,i}^{H}\u2215\left|\right|{h}_{i,i}\left|\right|{s}_{i}.

(9)

The SINR of user *i* can be denoted as

\mathsf{\text{SIN}}{\mathsf{\text{R}}}_{i}=\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}+\sum _{j=1,j\ne i}^{M}{P}_{t,j}|{h}_{j,i}{f}_{j}{|}^{2}}.

(10)

In this case, problem \mathbb{P}1 can be rewritten as

\mathbb{P}1:\underset{{\left\{{P}_{t,i}\right\}}_{i=1}^{M}:{P}_{t,i}\ge 0}{max}\frac{\sum _{i=1}^{M}Wlog\left\{1+\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}+\sum _{j=1,j\ne i}^{M}{P}_{t,j}|{h}_{j,i}{f}_{j}{|}^{2}}\right\}}{\sum _{i=1}^{M}{P}_{\mathsf{\text{total}},i}}.

(11)

As data and CSIT sharing is not available in IA-GT, joint optimizing above problem is impractical in IA-GT. A sub-optimal but practical solution is that each user optimize its own transmit power *P*_{
t,i
}as follows excluding other cells' rate.

\underset{{P}_{t,i}:{P}_{t,i}\ge 0}{max}\frac{Wlog\left\{1+\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}+\sum _{j=1,j\ne i}^{M}{P}_{t,j}|{h}_{j,i}{f}_{j}{|}^{2}}\right\}}{{P}_{\mathsf{\text{total}},i}+\sum _{j=1,j\ne i}^{M}{P}_{\mathsf{\text{total}},j}}.

(12)

In order to optimize (12), inter-cell interference {\sum}_{j=1,j\ne i}^{M}{P}_{\mathsf{\text{t}},j}|{h}_{j,i}{f}_{j}{|}^{2} and other BSs' power consumption {\sum}_{j=1,j\ne i}^{M}{P}_{\mathsf{\text{total}},j} are required except for the own cell's CSIT. Fortunately, the noise and inter-cell interference level of the previous slot can be measured at the user side, so only other BSs' power consumption {\sum}_{j=1,j\ne i}^{M}{P}_{\mathsf{\text{total}},j} affects the optimization (12). Motivated by Björnson et al. [20], we provide two simple strategies to meet this challenge, which both lead to maximizing LEE at each BS. In the first strategy, each BS should assume that the BS itself is the only BS in the cluster, thus it should be set as {\sum}_{j=1,j\ne i}^{M}{P}_{\mathsf{\text{total}},j}=0 in the denominator. Although the assumption is simple and sub-optimal, it is robust because the effect of other BSs' power parts are all excluded whether their impact is positive or negative. In the second strategy, the system should be assumed to be symmetrical at each BS, which means the user in each BS experiences the similar channel condition. Thus, the optimized power at each BS should be the same in the symmetrical scenario and it is set that *P*_{total,j}= *P*_{total,i}*,*∀*j ≠ i*. Interestingly, for the above both strategies, the optimization object at BS *i* is equivalent to the LEE after some simple calculation, which can be denoted as follows:

\underset{{P}_{t,i}}{max}\mathsf{\text{LE}}{\mathsf{\text{E}}}_{i}=\frac{Wlog\left\{1+\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}+\sum _{j=1,j\ne i}^{M}{P}_{t,j}|{h}_{j,i}{f}_{j}{|}^{2}}\right\}}{{P}_{\mathsf{\text{total}},i}}.

(13)

When each BS optimizes LEE according to above equation, the interference level of other cells would change, and hence, the other BSs' LEE would be affected. Thus, when each BS optimizes its own LEE, Pareto-efficient Nash equilibrium, which is defined as the point where no BS can unilaterally improve its LEE without decreasing any other BS's LEE, is expected to be achieved. Fortunately, we find that the optimization (13) is similar with the uplink multi-cell systems [17]. Therefore, the practical non-cooperative power control strategy based on the game theory in [17] can be directly applied here to achieve the Pareto-efficient Nash equilibrium. During the power control procedure, no cooperation is needed and each BS only need to get the interference level and then maximize its own LEE.

We should notice that here although other BSs' power consumption part is left out to help the distributed optimization (13) at each BS, the NEE in (11) should be employed as the performance metric to express the systems' EE. In the simulation, we optimize the power according to (13) and then calculate the NEE based on (11). The same principle is applied in the other schemes in the rest of the article.

#### 3.2.2 Inter-cell interference cancellation

ICIC is a scheme in which each BS proactively cancel its own interference to other cells. Only local CSIT is required and no data sharing is needed. Zero-forcing precoding is considered to cancel the inter-cell interference and *J* ≥ *M* should be assumed to guarantee the matrices' degree of freedom. Denote {\widehat{H}}_{i}={\left[{h}_{i,1}^{\mathsf{\text{T}}},...,{h}_{i,i-1}^{\mathsf{\text{T}}},{h}_{i,i+1}^{\mathsf{\text{T}}},...{h}_{i,M}^{\mathsf{\text{T}}}\right]}^{\mathsf{\text{T}}}. The precoding vector **f**_{
i
}in ICIC is the normalized version of the following vector

{w}_{i}=\left(I-\frac{{\widehat{H}}_{i}^{H}{\widehat{H}}_{i}}{\left|\right|{\widehat{H}}_{i}|{|}^{2}}\right){h}_{i,i}^{\mathsf{\text{H}}},

(14)

and it can be denoted as {f}_{i}=\frac{{w}_{i}}{\left|\right|{w}_{i}\left|\right|}. As perfect CSIT is assumed at the transmitter, the inter-cell interference can be perfectly canceled, and then the SINR can be denoted as :

\mathsf{\text{SIN}}{\mathsf{\text{R}}}_{i}=\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}}.

(15)

In this case, problem \mathbb{P}1 can be rewritten as:

\mathbb{P}1:\underset{{\left\{{P}_{t,i}\right\}}_{i=1}^{M}:{P}_{t,i}\ge 0}{max}\frac{\sum _{i=1}^{M}Wlog\left\{1+\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}}\right\}}{\sum _{i=1}^{M}{P}_{\mathsf{\text{total}},i}}.

(16)

Different from IA-GT, changing transmit power *P*_{
t,i
}would not change other cells' interference level here, and hence, would not affect SINR_{
j
}*, j* ≠ *i*. Therefore, for each BS, the optimal transmit power derivation should be based on the following criteria.

\underset{\left\{{P}_{t,i}\right\}:{P}_{t,i}\ge 0}{max}\frac{Wlog\left\{1+\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}}\right\}}{{P}_{\mathsf{\text{total}},i}+\sum _{j=1j\ne i}^{M}{P}_{\mathsf{\text{total}},j}}.

(17)

In order to perform the above optimization, the other cells' power consumption information is required, which is similar with the optimization in IA-GT (12). In order to realize it in a distributed manner, we apply the same strategies as in section 3.2.1, i.e., setting {\sum}_{j=1,j\ne i}^{M}{P}_{\mathsf{\text{total}},j}=0 or assuming a symmetrical scenario with *P*_{total ,j}= *P*_{total,i}*,*∀*j ≠ i*. For both strategies, the optimization object is changed as LEE_{
i
}again which can be denoted as follows.

\underset{\left\{{P}_{t,i}\right\}:{P}_{t,i}\ge 0}{max}\mathsf{\text{LE}}{\mathsf{\text{E}}}_{i}=\frac{Wlog\left\{1+\frac{{P}_{t,i}|{h}_{i,i}{f}_{i}{|}^{2}}{{N}_{0}}\right\}}{{P}_{\mathsf{\text{total}},i}}.

(18)

The LEE optimization of a MIMO channels can be directly applied here. For more details, the readers can be referred in our previous study [13].

It is worthwhile here that the interference cannot be fully canceled if the CSIT is not perfect. In that case, the SINR formula of ICIC should not be (15) but be (10) and non-cooperative power control strategy based on the game theory in [17] is applicable in order to optimize NEE, which is similar with section 3.2.1. Another critical issue in the imperfect CSIT case is that the capacity cannot be perfectly known before the transmission, the so-called capacity estimation mechanism is important for the capacity predication and for the EE optimization. About the capacity estimation, [13] discussed it in the single cell MIMO systems in detail and it can be simply extended here.

#### 3.2.3 Multi-cell joint processing

Full CSIT and data sharing are assumed in MC-JP. As full cooperation is available in MC-JP, the multi-cell system can be viewed as a multi-user MIMO system which consists of a single "super-BS" deployed with JM transmit antennas and *M* single antenna receivers. CU gathers the whole data and CSIT information and then controls each BS's precoding and power allocation. Globally zero-forcing beamforming is applied.

Denote the channel matrix from all BSs to the *M* users as **H** ∈ ℂ^{M × MJ}and then the precoding matrix is denoted as :

F={H}^{\mathsf{\text{H}}}{\left(H{H}^{\mathsf{\text{H}}}\right)}^{-1}.

(19)

And then the SINR of user *i* is

\mathsf{\text{SIN}}{\mathsf{\text{R}}}_{i}=\frac{{P}_{t,i}{\lambda}_{i}}{{N}_{0}},

(20)

in which {\lambda}_{i}=\frac{1}{{\left(H{H}^{\mathsf{\text{H}}}\right)}_{i,i}^{-1}} and here *P*_{
t,i
}is the total power for user *i*. The NEE optimization problem with MC-JP can be rewritten as:

\mathbb{P}1:\underset{{\left\{{P}_{t,i}\right\}}_{j=1}^{M}:{P}_{t,i}\ge 0}{max}\frac{\sum _{i=1}^{M}Wlog\left(1+\frac{{P}_{t,i}{\lambda}_{i}}{{N}_{0}}\right)}{\sum _{i=1}^{M}{P}_{\mathsf{\text{total}},i}}.

(21)

As full CSIT and data sharing are available at the CU, NEE with different transmit power can be calculated. This feature in MC-JP indicates that the power control can be applied cooperatively. Fortunately, the maximizing NEE problem (21) is equivalent to the LEE maximizing in the frequency selective channels [4]. And then the binary search assisted ascent (BSAA) algorithm in [4] should be applied directly here. Compared with ICIC and IA-GT, MC-JP benefits from two aspects. For one thing, cooperative precoding can fully exploit the interference to further increase the SINR. For another, cooperative power control can better balance the capacity and power consumption. And hence MC-JP leads to higher NEE.