We consider an MIMO cooperative network consisting of I cells. As illustrated in Fig. 1, the BS of cell i∈{1,…,I} serves a group of D DLUs in the downlink (DL) channel and a group of U ULUs in the uplink (UL) channel. Each BS operates in the FD mode and is equipped with \({N} \triangleq N_{1}+N_{2}\) antennas, where N
1 antennas are used to transmit and the remaining N
2 antennas to receive signals. In cell i, DLU (i,j
D
) and ULU (i,j
U
) operate in the HD mode and each is equipped with N
r
antennas. Similar to other works on precoding and interference suppression (see, e.g., [6, 12, 13, 15–17, 29] and references therein), it is assumed in this paper that there are high-performance channel estimation mechanisms in place and a central processing unit is available to collect and disseminate the relevant CSI.
In the DL, a complex-valued vector \(\mathbf {s}_{i,j_{\mathsf {D}}}\in \mathbb {C}^{d_{1}}\) is the symbols intended for DLU (i,j
D
), where \(\mathbb {E}\left [\mathbf {s}_{i,j_{\mathsf {D}}}(\mathbf {s}_{i,j_{\mathsf {D}}})^{H}\right ]=\mathbf {I}_{d_{1}}\), d
1 is the number of concurrent data streams, and d
1≤ min{N
1,N
r
}. Denote by \(\mathsf {V}_{i,j_{\mathsf {D}}}\in \mathbb {C}^{{N}_{1}\times d_{1}}\) the complex-valued precoding matrix for DLU (i,j
D
). Similarly, in the UL, \(\mathbf {s}_{i,j_{\mathsf {U}}}\in \mathbb {C}^{d_{2}}\) is the symbols sent by ULU (i,j
U
), where \(\mathbb {E}\left [\mathbf {s}_{i,j_{\mathsf {U}}}(\mathbf {s}_{i,j_{\mathsf {U}}})^{H}\right ]=\mathbf {I}_{d_{2}}\), d
2 is the number of concurrent data streams, and d
2≤ min{N
2,N
r
}. The precoding matrix of ULU (i,j
U
) is denoted as \(\mathsf {V}_{i,j_{\mathsf {U}}}\in \mathbb {C}^{N_{r}\times d_{2}}\). Define
$$ \begin{array}{lll} {\mathcal{I}} &\triangleq& \{1,2,\dots,I\}; \quad {\mathcal{D}} \triangleq \{1_{\mathsf{D}},2_{\mathsf{D}},\dots,D_{\mathsf{D}}\};\\ {\mathcal{U}} &\triangleq& \{1_{\mathsf{U}},2_{\mathsf{U}},\dots,U_{\mathsf{U}}\};\quad {\mathcal{S}}_{1} \triangleq {\mathcal{I}}\times {\mathcal{D}}; \quad {\mathcal{S}}_{2} \triangleq {\mathcal{I}}\times {\mathcal{U}};\\ \mathsf{V} &\triangleq& [\!\mathsf{V}_{i,j}]_{(i,j)\in {\mathcal{S}}_{1}\cup{\mathcal{S}}_{2}}. \end{array} $$
(1)
In the DL channel, the received signal at DLU (i,j
D
) is expressed as:
$$\begin{array}{@{}rcl@{}} y_{i,j_{\mathsf{ D}}} &\triangleq& \underbrace{\mathbf{H}_{i,i,j_{\mathsf{D}}}\mathsf{V}_{i,j_{\mathsf{D}}} \mathbf{s}_{i,j_{\mathsf{D}}}}_{{\mathsf{desired\,\, signal}}} +\underbrace{\sum_{(m,\ell_{\mathsf{D}})\in{\mathcal S}_{1}\setminus (i,j_{\mathsf{D}})} \mathbf{H}_{m,i,j_{\mathsf{D}}}\mathsf{V}_{m,\ell_{\mathsf{D}}} \mathbf{s}_{m,\ell_{\mathsf{D}}}}_{{\mathsf{DL\,\, interference}}}\\ &&+\underbrace{\sum_{\ell_{\mathsf{U}}\in{\mathcal U}}\mathbf{H}_{i,j_{\mathsf{D}},\ell_{\mathsf{U}}}\mathsf{V}_{i,\ell_{\mathsf{U}}}\mathbf{s}_{i,\ell_{\mathsf {U}}}}_{{\mathsf{UL intracell\,\, interference}}} + \mathbf{n}_{i,j_{\mathsf{D}}}, \end{array} $$
(2)
where \(\mathbf {H}_{m,i,j_{\mathsf {D}}}\in \mathbb {C}^{N_{r}\times {N}_{1}}\) and \(\mathbf {H}_{i,j_{\mathsf {D}},\ell _{\mathsf {U}}}\in \mathbb {C}^{N_{r}\times N_{r}}\) are the channel matrices from BS m to DLU (i,j
D
) and from ULU (i,ℓ
U
) to DLU (i,j
D
), respectively. Also, \(\mathbf {n}_{i,j_{\mathsf {D}}}\) is the additive white Gaussian noise (AWGN) sample, modeled as circularly symmetric complex Gaussian random variable with variance \(\sigma _{\mathsf {D}}^{2}\). Suppose that each BS i employs dirty-paper coding (DPC)-based transmission strategy (see, e.g., [31]) in broadcasting signals to its users. Then the corresponding DL throughput for user (i,j
D
) is [32, eq. (4)]
$$ f_{i,j_{\mathsf{D}}}(\mathsf{V}) \triangleq \ln\left|\mathbf{I}_{N_{r}}+{\mathcal L}_{i,j_{\mathsf{D}}}(\mathsf{V}_{i,j_{\mathsf{D}}}){\mathcal L}_{i,j_{\mathsf{D}}}^{H}(\mathsf{V}_{i,j_{\mathsf {D}}}) \Psi_{i,j_{\mathsf{D}}}^{-1}(\mathsf{V})\right|, $$
(3)
where \({\mathcal {L}}_{i,j_{\mathsf {D}}}(\mathsf {V}_{i,j_{\mathsf {D}}}) \triangleq \mathbf {H}_{i,i,j_{\mathsf {D}}}\mathsf {V}_{i,j_{\mathsf {D}}}\) and
$$\begin{array}{@{}rcl@{}} {\mathcal{L}}_{i,j_{\mathsf{D}}}(\mathsf{V}_{i,j_{\mathsf{D}}}){\mathcal L}^{H}_{i,j_{\mathsf{D}}}(\mathsf{V}_{i,j_{\mathsf{D}}}) &=& \mathbf{H}_{i,i,j_{\mathsf{D}}}\mathsf{V}_{i,j_{\mathsf{D}}}\mathsf{V}_{i,j_{\mathsf{D}}}^{H}\mathbf{H}_{i,i,j_{\mathsf{D}}}^{H}, \end{array} $$
(4)
$$\begin{array}{@{}rcl@{}} \Psi_{i,j_{\mathsf{D}}}(\mathsf{V}) &\triangleq &\sum_{(m,\ell_{\mathsf{D}})\in{\mathcal S}_{1}\setminus\{(i,\ell_{\mathsf{D}}), \ell=j, \ldots,D\}} \mathbf{H}_{m,i,j_{\mathsf{D}}}\mathsf{V}_{m,\ell_{\mathsf{D}}}\mathsf{V}_{m,\ell_{\mathsf{D}}}^{H}\mathbf{H}_{m,i,j_{\mathsf {D}}}^{H} \\ &&+\sum_{\ell_{\mathsf{U}}\in{\mathcal{U}}}\mathbf{H}_{i,j_{\mathsf{D}},\ell_{\mathsf{U}}}\mathsf{V}_{i,\ell_{\mathsf {U}}}\mathsf{V}_{i,\ell_{\mathsf{U}}}^{H}\mathbf{H}_{i,j_{\mathsf{D}},\ell_{\mathsf{U}}}^{H}+ \sigma_{\mathsf{D}}^{2}\mathbf{I}_{N_{r}}. \end{array} $$
(5)
Note that DPC-based broadcasting is a capacity achieving transmission, which enables user (i,j
D
) view the term \(\sum _{k_{{\mathsf {D}}}<j_{{\mathsf {D}}}}\mathbf {H}_{i,i,j_{{\mathsf {D}}}}\mathsf {V}_{i,k_{{\mathsf { D}}}}\mathbf {s}_{i,k_{{\mathsf {D}}}}\) as known non-causally and thus reduces it from the interference in (3) [33, Lemma 1]. This term is still present under conventional broadcast, for which the interference mapping \(\Psi _{i,j_{\mathsf {D}}}(\mathsf {V})\) in (3) becomes
$$\begin{array}{@{}rcl@{}} \Psi_{i,j_{\mathsf{D}}}(\mathsf{V}) &\triangleq &\sum_{(m,\ell_{\mathsf{D}})\in{\mathcal S}_{1}\setminus\{(i,j_{\mathsf{D}})\}} \mathbf{H}_{m,i,j_{\mathsf{D}}}\mathsf{V}_{m,\ell_{\mathsf{D}}}\mathsf{V}_{m,\ell_{\mathsf{D}}}^{H}\mathbf{H}_{m,i,j_{\mathsf{D}}}^{H} \\ &&+\sum_{\ell_{\mathsf{U}}\in{\mathcal U}}\mathbf{H}_{i,j_{\mathsf{D}},\ell_{\mathsf{U}}}\mathsf{V}_{i,\ell_{\mathsf{U}}}\mathsf{V}_{i,\ell_{\mathsf{U}}}^{H}\mathbf{H}_{i,j_{\mathsf{D}},\ell_{\mathsf{U}}}^{H}+ \sigma_{\mathsf{D}}^{2}\mathbf{I}_{N_{r}}. \end{array} $$
(6)
It is pointed out that our below development is still applicable to the case of conventional broadcast.
In the UL channel, the received signal at BS i can be expressed as
$${\kern-16.5pt} \begin{aligned} y_{i} \triangleq& \underbrace{\sum_{\ell_{\mathsf{U}}\in{\mathcal U}} \mathbf{H}_{i,\ell_{\mathsf{U}},i}\mathsf{V}_{i,\ell_{\mathsf{U}}} \mathbf{s}_{i,\ell_{\mathsf{U}}}}_{{\mathsf{desired\,\, signal}}} +\underbrace{\sum_{m\in{\mathcal I}\setminus \{i\}}\sum_{\ell_{\mathsf{U}}\in{\mathcal U}}\mathbf{H}_{m,\ell_{\mathsf{U}},i}\mathsf{V}_{m,\ell_{\mathsf{U}}} \mathbf{s}_{m,\ell_{\mathsf{U}}}}_{{\mathsf{UL\,\, interference}}}\\ &\!\!+\underbrace{\mathbf{H}_{i}^{{\mathcal S}{\mathcal I}}\sum_{\ell_{\mathsf{D}}\in{\mathcal D}}\mathsf{V}_{i,\ell_{\mathsf{D}}} \tilde{\mathbf{s}}_{i,\ell_{\mathsf{D}}}}_{{\mathsf{residual\,\, SI}}} +\!\underbrace{\sum_{m\in{\mathcal I}\setminus\{i\}}\!\mathbf{H}_{m,i}^{{\mathcal B}}\sum_{j_{\mathsf{D}}\in{\mathcal D}}\!\mathsf{V}_{m,j_{\mathsf{D}}}\mathbf{s}_{m,\mathbf{s}_{\mathsf{D}}}}_{{\mathsf {DL intercell\,\, interference}}}+ \mathbf{n}_{i}, \end{aligned} $$
(7)
where \(\mathbf {H}_{m,\ell _{\mathsf {U}},i}\in \mathbb {C}^{{N}_{2}\times N_{r}}\) and \(\mathbf {H}^{{\mathcal B}}_{m,i}\in \mathbb {C}^{{N}_{2}\times {N}_{1}}\) are the channel matrices from ULU (m,ℓ
U
) to BS i and from BS m to BS i, respectively. The channel matrix \(\mathbf {H}_{i}^{{\mathcal S}{\mathcal I}}\in \mathbb {C}^{{N}_{2}\times {N}_{1}}\) represents the residual self-loop channel from the transmit antennas to the receive antennas at BS i after all real-time interference cancelations in both analog and digital domains [22, 34] are accounted for (more detailed discussion on modelling the SI channel can be found in [22, 34]). The additive Gaussian noise vector \(\tilde {\mathbf {s}}_{i,\ell _{\mathsf {D}}}\) with \(\mathbb {E}\left [\tilde {\mathbf {s}}_{i,j_{\mathsf {D}}}(\tilde {\mathbf {s}}_{i,j_{\mathsf {D}}})^{H}\right ]=\sigma _{\text {SI}}^{2}\mathbf {I}_{d_{1}}\) models the effects of the analog circuit’s non-ideality and the limited dynamic range of the analog-to-digital converter (ADC) [19, 23, 34, 35]. The SI level \(\sigma _{\text {SI}}^{2}\) is the ratio of the average SI powers before and after the SI cancelation process. Lastly, n
i
is the AWGN sample, modeled as circularly-symmetric complex Gaussian random variable with variance \(\sigma _{\mathsf {U}}^{2}\).
By treating the entries of the self-loop channel \(\mathbf {H}_{i}^{{\mathcal {S}}{\mathcal {I}}}\) in (7) as independent circularly symmetric complex Gaussian random variables with zero mean and unit variance, the power of the residual SI in (7) is
$${} {{\begin{aligned} \sigma_{\text{SI}}^{2}\mathbb{E}\left\{\mathbf{H}_{i}^{{\mathcal{S}}{\mathcal I}}\left(\sum_{\ell_{D}\in{\mathsf{D}}}\mathsf{V}_{i,\ell_{D}}\mathsf{V}_{i,\ell_{D}}^{H}\right)(\mathbf{H}_{i}^{{\mathcal S}{\mathcal I}})^{H}\right\} = \sigma_{\text{SI}}^{2}\left(\sum_{\ell_{D}\in{\mathsf{D}}}||\mathsf{V}_{i,\ell_{D}}||^{2}\right) \mathbf{I}_{N_{r}}. \end{aligned}}} $$
(8)
It is important to point out that the above power expression only depends on the BS transmit power, and it cannot be changed by precoder matrices \(\mathsf {V}_{i,\ell _{\sf D}}\).
Given that the minimum mean square error–successive interference cancelation (MMSE-SIC) detector is the most popular detection method in uplink communications, this type of receiver is also adopted in this paper. Under the MMSE-SIC receiver, the achievable uplink throughput at BS i is given as [36]
$$\begin{array}{*{20}l} f_{i}(\mathsf{V}) &\triangleq \ln\left|\mathbf{I}_{{N}_{2}} + {\mathcal L}_{i}(\mathsf{V}_{{\mathsf{U}} i}){\mathcal L}_{i}^{H}(\mathsf{V}_{{\mathsf{U}} i})\Psi_{i}^{-1}(\mathsf{V})\right|, \end{array} $$
(9)
where
$$ \begin{aligned} \mathsf{V}_{{\mathsf{U}} i} &\triangleq (\mathsf{V}_{i,\ell_{\mathsf{U}}})_{\ell_{\mathsf{U}}\in{\mathcal U}}, {\mathcal L}_{i}(\mathsf{V}_{{\mathsf{U}} i})\\ &\triangleq \left[ \mathbf{H}_{i,1_{\mathsf{U}},i}\mathsf{V}_{i,1_{\mathsf{U}}}, \mathbf{H}_{i,2_{\mathsf{U}},i}\mathsf{V}_{i,2_{\mathsf{U}}}, \dots, \mathbf{H}_{i,U_{\mathsf{U}},i}\mathsf{V}_{i,U_{\mathsf{U}}} \right], \end{aligned} $$
(10)
and
$$ \begin{aligned} {\mathcal{L}}_{i}(\mathsf{V}_{{\mathsf{U}} i}){\mathcal L}_{i}^{H}(\mathsf{V}_{{\mathsf{U}} i}) &= \sum_{\ell=1}^{U}\mathbf{H}_{i,\ell_{\mathsf{U}},i}\mathsf{V}_{i,\ell_{\mathsf{U}}}\mathsf{V}_{i,\ell_{\mathsf{U}}}^{H}\mathbf{H}_{i,\ell_{\mathsf{U}},i}^{H}, \end{aligned} $$
(11)
$${\kern-16.5pt} \begin{aligned} \Psi_{i}(\mathsf{V}) \triangleq& \sum_{m\in{ I}\setminus \{i\}}\sum_{\ell_{\mathsf{U}}\in{\mathcal U}}\mathbf{H}_{m,\ell_{\mathsf{U}},i}\mathsf{V}_{m,\ell_{\mathsf{U}}} \mathsf{V}_{m,\ell_{\mathsf{U}}}^{H}\mathbf{H}_{m,\ell_{\mathsf{U}},i}^{H}\\&+\sigma_{\text{SI}}^{2}(\sum_{\ell_{D}\in{\mathsf{D}}}||\mathsf{V}_{i,\ell_{D}}||^{2}) \mathbf{I}_{N_{r}}\\ &+\sum_{m\in{\mathcal I}\setminus\{i\}}\mathbf{H}_{m,i}^{{\mathcal B}}\!\left(\sum_{j_{\mathsf{D}}\in{\mathcal D}}\mathsf{V}_{m,j_{\mathsf{D}}}\mathsf{V}_{m,j_{\mathsf{D}}}^{H}\!\!\right)\!(\mathbf{H}_{m,i}^{{\mathcal B}})^{H} +\sigma_{\mathsf{U}}^{2}\mathbf{I}_{{N}_{2}}. \end{aligned} $$
(12)
Following [37], the consumed power \(P^{\text {tot}}_{i}\) of cell i can be modeled as
$$ P^{\text{tot}}_{i}(\mathsf{V}) = \zeta P^{t}_{i}(\mathsf{V}) + P^{\text{BS}} + UP^{\text{UE}}, $$
(13)
where \(P^{t}_{i}(\mathsf {V})\triangleq \underset {{j_{\mathsf {D}}\in {\mathcal D}}}{\sum } ||\mathsf {V}_{i,j_{\mathsf {D}}}||^{2}+\underset {j_{\mathsf {U}}\in {\mathcal U}}{\sum } ||\mathsf {V}_{i,j_{\mathsf {U}}}||^{2}\) is the total transmit power of BS and UEs in cell i and ζ is the reciprocal of drain efficiency of power amplifier. Alo P
BS=N
1
P
b
and P
UE=N
r
P
u
are the circuit powers of BS and UE, respectively, where P
b
and P
u
represent the per-antenna circuit power of BS and UEs, respectively. Consequently, the energy efficiency of cell i is defined by
$$ \frac{\underset{j_{\sf D}\in{\mathcal D}}{\sum}f_{i,j_{\sf D}}(\mathsf{V})+f_{i}(\mathsf{V})}{P^{\text{tot}}_{i}(\mathsf{V})}. $$
(14)
In this paper, we consider the following precoder design to optimize the network’s energy efficiency:
$$\begin{array}{*{20}l} \max_{\mathsf{V}}\ \min_{i\in {\mathcal I}}\frac{\underset{j_{\mathsf{D}}\in{\mathcal D}}{\sum}f_{i,j_{\mathsf{D}}}(\mathsf{V})+f_{i}(\mathsf{V})} {P^{\text{tot}}_{i}(\mathsf{V})} & \mathrm{s.t.} \end{array} $$
(15a)
$$\begin{array}{*{20}l} \sum_{j_{\mathsf{D}}\in{\mathcal D}}|| \mathsf{V}_{i,j_{\mathsf{D}}}||^{2} \le P^{\max}_{\text{BS}}, \ i\in{\mathcal I},& \end{array} $$
(15b)
$$\begin{array}{*{20}l} ||\mathsf{V}_{i,j_{\mathsf{U}}}||^{2} \le P^{\max}_{\text{UE}}, \ (i,j_{\mathsf{U}})\in{\mathcal S}_{2},& \end{array} $$
(15c)
$$\begin{array}{*{20}l} f_{i,j_{\mathsf{D}}}(\mathsf{V}) \ge r_{i,j_{\sf D}}^{\min}, \ (i,j_{\mathsf{D}})\in{\mathcal S}_{1} & \end{array} $$
(15d)
$$\begin{array}{*{20}l} f_{i}(\mathsf{V}) \ge r_{i}^{{\mathsf{U}},\min}, \ i\in{\mathcal I},& \end{array} $$
(15e)
where (15b)-(15c) limit the transmit powers for each BS and ULU, while (15d)-(15e) are the QoS constraints for both downlink and uplink transmissions.
On the other hand, the problem of optimizing the energy efficiency in DL transmission only is formulated as follows:
$$\begin{array}{*{20}l} {}\max_{\mathsf{V}^{\text{DL}}=[\mathsf{V}_{i,j_{{\mathsf{D}}}}]_{(i,j_{{\mathsf{D}}})\in{\mathcal S}_{1}}}\min_{i\in {\mathcal I}}\frac{\underset{j_{\mathsf{D}} \in{\mathcal D}}{\sum}f_{i,j_{\mathsf{D}}}^{\text{DL}}(\mathsf{V}^{\text{DL}})} {\zeta\underset{j_{\mathsf{ D}}\in{\mathcal D}}{\sum}||\mathsf{V}_{i,j_{\mathsf{ D}}}||^{2} + P^{\text{BS}}}\ \mathrm{s.t.}\,\,\,(15b),& \end{array} $$
(16a)
$$\begin{array}{*{20}l} f_{i,j_{\mathsf{ D}}}^{\text{DL}}(\mathsf{V}^{\text{DL}})\ge r_{i,j_{\mathsf{ D}}}^{\min}, \ (i,j_{\mathsf{ D}})\in{\mathcal S}_{1}& \end{array} $$
(16b)
with
$$\begin{array}{@{}rcl@{}} f_{i,j_{\mathsf{D}}}^{\text{DL}}(\mathsf{V}^{\text{DL}})&\triangleq&\ln\left|\mathbf{I}_{N_{r}}+\mathbf{H}_{i,i,j_{\mathsf {D}}}\mathsf{V}_{i,j_{\mathsf{D}}}\mathsf{V}_{i,j_{\mathsf{D}}}^{H} \mathbf{H}_{i,i,j_{\mathsf{D}}}^{H}\right.\\ &&\times (\sum_{(m,\ell_{\mathsf{ D}})\in{\mathcal S}_{1}\setminus\{(i,\ell_{\mathsf {D}}), \ell=j, \ldots,D\}} \mathbf{H}_{m,i,j_{\mathsf{ D}}}\mathsf{V}_{m,\ell_{\mathsf{ D}}}\\ &&\times \left.\left.\mathsf{V}_{m,\ell_{\mathsf{ D}}}^{H}\mathbf{H}_{m,i,j_{\mathsf{ D}}}^{H}+\sigma_{\mathsf{ D}}^{2}\mathbf{I}_{N_{r}} \right)^{-1}\right|. \end{array} $$
(17)
Likewise, the problem of optimizing the energy efficiency in the UL transmission only is
$$\begin{array}{*{20}l} \max_{\mathsf{V}^{\text{UL}}=[\mathsf{V}_{i,\ell_{{\mathsf{ U}}}}]_{(i,\ell_{{\mathsf{ U}}})\in{\mathcal S}_{2}}}\min_{i\in {\mathcal I}}\frac{f_{i}^{\text{UL}}(\mathsf{V}^{\text{UL}})} {\zeta\underset{\ell_{\sf U}\in{\mathcal U}}{\sum} ||\mathsf{V}_{i,\ell_{\mathsf{U}}}||^{2} + UP^{\text{UE}}}\ \ \mathrm{s.t.}\,\,\,\, {(15c)},& \end{array} $$
(18a)
$$\begin{array}{*{20}l} f_{i}^{\text{UL}}(\mathsf{V}^{\text{UL}})\geq r_{i}^{{\mathsf{ U}},\min}, \ i\in{\mathcal I},& \end{array} $$
(18b)
with
$${} {{\begin{aligned} f_{i}^{\text{UL}}(\mathsf{V}^{\text{UL}})&\triangleq \ln\left|{\vphantom{\left(\underset{m\in{\mathcal I}\setminus \{i\}}{\sum}\sum_{\ell_{\mathsf{ U}}\in{\mathcal U}}\mathbf{H}_{m,\ell_{\mathsf{ U}},i}\mathsf{V}_{m,\ell_{\mathsf{ U}}} \mathsf{V}_{m,\ell_{\mathsf{U}}}^{H}\mathbf{H}_{m,\ell_{\mathsf{ U}},i}^{H}+\sigma_{\mathsf{ U}}^{2}\mathbf{I}_{{N}_{2}}\right)}}\mathbf{I}_{{N}_{2}} +\sum_{\ell=1}^{U}\mathbf{H}_{i,\ell_{\mathsf{ U}},i}\mathsf{V}_{i,\ell_{\mathsf{U}}}\mathsf{V}_{i,\ell_{\mathsf{ U}}}^{H}\mathbf{H}_{i,\ell_{\mathsf{U}},i}^{H}\right.\\ &\left.\quad\!\times\left(\underset{m\in{\mathcal I}\setminus \{i\}}{\sum}\sum_{\ell_{\mathsf{ U}}\in{\mathcal U}}\!\mathbf{H}_{m,\ell_{\mathsf{ U}},i}\mathsf{V}_{m,\ell_{\mathsf{ U}}} \mathsf{V}_{m,\ell_{\mathsf{U}}}^{H}\mathbf{H}_{m,\ell_{\mathsf{ U}},i}^{H}+\sigma_{\mathsf{ U}}^{2}\mathbf{I}_{{N}_{2}}\!\!\right)^{-1}\right|. \end{aligned}}} $$
(19)
As discussed before, for the downlink EE optimization problem (16), references [29] and [30] apply zero-forcing precoders so that all the interference terms in (5) are completely canceled, making \(f_{i,j_{\sf D}}^{\text {DL}}(\mathsf {V}^{\text {DL}})=\ln \left |\mathbf {I}_{N_{r}}+\mathbf {H}_{i,i,j_{\mathsf {D}}}\mathsf {V}_{i,j_{\mathsf { D}}}\mathsf {V}_{i,j_{\mathsf { D}}}^{H} \mathbf {H}_{i,i,j_{\mathsf { D}}}^{H}/\sigma _{{\mathsf { D}}}^{2}\right |\). Then by making the variable change \(\mathsf {X}_{i,j_{\mathsf { D}}}=\mathsf {V}_{i,j_{\mathsf { D}}}\mathsf {V}_{i,j_{\mathsf { D}}}^{H}\), the EE optimization for zero-forcing precoders becomes:
$$\begin{array}{*{20}l} &\!\!\!\!\!\!\!\!\!\!\!\!\max_{\mathsf{X}^{\text{DL}}=[\mathsf{X}_{i,j_{{\mathsf{D}}}}]_{(i,j_{{\mathsf{ D}}})\in{\mathcal S}_{1}}}\min_{i\in {\mathcal I}}\\ &\times\frac{\underset{j_{\sf D}\in{\mathcal D}}{\sum} \ln\left|\mathbf{I}_{N_{r}}+\mathbf{H}_{i,i,j_{\mathsf{ D}}}\mathsf{X}_{i,j_{\mathsf{ D}}}^{H} \mathbf{H}_{i,i,j_{\mathsf{ D}}}^{H}/\sigma_{{\mathsf{ D}}}^{2}\right|} {\zeta\underset{j_{\sf D}\in{\mathcal D}}{\sum}\text{Trace}(\mathsf{X}_{i,j_{\mathsf {D}}}) + P^{\text{BS}}}\quad \mathrm{s.t.}\quad \end{array} $$
(20a)
$$\begin{array}{*{20}l} &\sum_{j_{\mathsf {D}}\in{\mathcal D}}\text{Trace}(\mathsf{X}_{i,j_{\mathsf{ D}}}) \le P^{\max}_{\text{BS}}, \ i\in{\mathcal I}, \end{array} $$
(20b)
$$\begin{array}{*{20}l} &\ln\left|\mathbf{I}_{N_{r}}+\mathbf{H}_{i,i,j_{\mathsf{ D}}}\mathsf{X}_{i,j_{\mathsf{ D}}}^{H} \mathbf{H}_{i,i,j_{\mathsf{D}}}^{H}/\sigma_{{\mathsf{ D}}}^{2}\right|\ge r_{i,j_{\mathsf{ D}}}^{\min}, \ (i,j_{\mathsf{D}})\in{\mathcal S}_{1}, \end{array} $$
(20c)
$$\begin{array}{*{20}l} &\mathsf{X}^{\text{DL}}\in Z_{zf}, \end{array} $$
(20d)
where the last linear constraint (20d) is to explicitly specify a zero-forcing precoder. Since the numerator in the objective (20a) is concave in \(\mathsf {X}_{i,j_{\sf D}}\), the problem expressed in (20) is maximin optimization of concave-convex function ratios. To solve such problem, references [29] and [30] use the Dinkelbach-type algorithm [25]. Specifically, the optimal value of (20) is found as the maximum of γ for which the optimal value of the following convex program is nonnegative:
$${} {{\begin{aligned} \max_{\mathsf{X}^{\text{DL}}=[\mathsf{X}_{i,j_{{\mathsf{ D}}}}]_{(i,j_{{\mathsf {D}}})\in{\mathcal S}_{1}}}\min_{i\in {\mathcal I}} \left[\sum_{j_{\sf D}\in{\mathcal D}} \ln\left|\mathbf{I}_{N_{r}}+\mathbf{H}_{i,i,j_{\mathsf{D}}}\mathsf{X}_{i,j_{\mathsf{ D}}}^{H} \mathbf{H}_{i,i,j_{\mathsf{ D}}}^{H}/\sigma_{{\mathsf{ D}}}^{2}\right|\right.\\\left. -\gamma\left(\zeta\sum_{j_{\mathsf{ D}}\in{\mathcal D}}\text{Trace}(\mathsf{X}_{i,j_{\mathsf {D}}}) + P^{\text{BS}}\right)\right] \quad \mathrm{s.t.}\ \ \ (20b), (20c), (20d). \end{aligned}}} $$
(21)
It should be noted that, although being convex for fixed γ, the program (21) is still computationally difficult. This is because the concave objective function and convex constraints (20c) in (21) involve log-det functions. In fact, no polynomial-time algorithms are known to find the solution. Another issue is that the zero-forcing constraint (20d) in (21) would rule out the effectiveness of the optimization, unless the total number (N·I) of the BSs’ antennas is much larger than the total number (I·D·N
r
) of DLUs’ antennas.
The optimal value of (7) is still the maximum of γ>0 such that the optimal value of the following program is nonnegative
$${\kern-15.9pt} {{\begin{aligned} \max_{\mathsf{V}}\ \min_{i\in {\mathcal I}}\ \left[\underset{j_{\mathsf{D}}\in{\mathcal D}}{\sum}f_{i,j_{\mathsf{ D}}}(\mathsf{V})+f_{i}(\mathsf{V})-\gamma P^{\text{tot}}_{i}(\mathsf{V})\right]\quad\mathrm{s.t.}\quad \protect{(15b)-(15e)}. \end{aligned}}} $$
(22)
However, problem (22) is a very difficult nonconvex optimization even for a fixed γ>0 because its objective function is obviously nonconcave while its constraints (15e) are highly nonconvex. In fact, one can see that (22) for a fixed γ is not easier than the original nonconvex optimization problem (15). In the next section we will develop a path-following procedure for computing the solution of (15) that avoids the setting (22).