We carry out the achievable rate analysis and optimization in this section. In this paper, the part of the time used for channel estimation at the BS and energy harvesting *α**T* can be determined by *α**T*=*τ*_{
d
}≥*τ*_{
u
}. It is worth noting that when *α**T*≥*τ*_{
u
}, more time is allocated to the sensors for EH which as a result will increase the harvested energy and hence increase the transmit power of sensors. But for the users, less time is allocated for transmitting and hence reduce the SE.

### Downlink achievable rate

#### Downlink achievable rate with perfect CSI

With perfect CSI, the downlink achievable rate of user *g*_{
d,k
} can be expressed as

$$ {\ddot R_{{g_{d,k}}}}\left(\alpha \right) = \left({1 - \alpha} \right){\log_{2}}\left({1 + {{\ddot \gamma }_{{g_{d,k}}}}} \right) $$

(34)

where \({{\ddot \gamma }_{{g_{d,k}}}}\) denotes the SINR of user *g*_{
d,k
} and can be written as

$$ {{\ddot \gamma }_{{g_{d,k}}}} = \frac{{{p_{{g_{d,k}}}}\mathbb{E}\left[ {{{\left[ {\Upsilon_{{g_{d}}}^{- 1}} \right]}_{k,k}}} \right]}}{{\sum\limits_{g'_{d} \in {{{G_{d}}} \left/\right. {\left\{ {{g_{d}}} \right\}}}} {{{\left\| {{{\left({\tilde{\mathbf{h}}_{{g_{d,k}}}^{\left\{ {{B_{g'_{d}}}} \right\}}} \right)}^{H}}{\mathbf{W}_{g'_{d}}}{\Lambda_{g'_{d}}}} \right\|}^{2}}} + 1}} $$

(35)

#### Downlink achievable rate with imperfect CSI

Since perfect channel estimation does not exist in real systems, hence we let \({\Delta \tilde {\mathbf {h}}_{{g_{d,k}},LM}^{\left \{ {{B_{{g_{d}}}}} \right \}}}=\mathbf {h}_{{g_{d,k}},LM}^{\left \{ {{B_{{g_{d}}}}} \right \}}-\tilde {\mathbf {h}}_{{g_{d,k}},LM}^{\left \{ {{B_{{g_{d}}}}} \right \}}\) denote the channel estimation error of LMMSE estimator. According to (25) and using the bounding technique in [58], the average achievable rates at user *g*_{
d,k
} can be expressed as (36) where \({p_{{g_{d,k}}}} = \mathbb {E}\left [ {{{\left | {{s_{{g_{d,k}}}}} \right |}^{2}}} \right ]\) denotes the transmit powers. The powers of channel estimation error (CEE), IUI within the group and IGI, are explained in the equation.

$$ \begin{aligned} \begin{array}{l} {R_{{g_{d,k}}}}(\alpha) = \left({1 - \alpha} \right){\log_{2}}\left({1 + {p_{g_{d,k}^{}}}\mathbb{E}\left[ {{{\left[ {\Upsilon_{{g_{d}}}^{- 1}} \right]}_{k,k}}} \right]\left({\underbrace {{p_{{g_{d,k}}}}{{\left| {{{\left({\Delta \tilde{\mathbf{h}}_{{g_{d,k}},LM}^{\left\{ {{B_{{g_{d}}}}} \right\}}} \right)}^{H}}{\mathbf{w}_{{g_{d,k}}}}} \right|}^{2}}}_{{{\text{CEE}_{g_{d,k}}}}}} \right.} \right.\\ \ \ \qquad\quad + \underbrace {\sum\limits_{k' = 1,k' \ne k}^{{K_{{g_{d}}}}} {{p_{{g_{d,k'}}}}{{\left| {{{\left({\Delta \tilde{\mathbf{h}}_{{g_{d,k}},LM}^{\left\{ {{B_{{g_{d}}}}} \right\}}} \right)}^{H}}{\mathbf{w}_{{g_{d,k'}}}}} \right|}^{2}}} }_{\text{IUI}_{g_{d,k}}} \\ \ \ \qquad\quad +\left. \left. \underbrace {\sum\limits_{g'_{d} \in {{{G_{d}}} \left/\right. {\left\{ {{g_{d}}} \right\}}}}^{} {{{\left\| {{{\left({\tilde{\mathbf{h}}_{{g_{d,k}}}^{\left\{ {{B_{{g_{'d}}}}} \right\}}} \right)}^{H}}{\mathbf{W}_{g'_{d}}}{\Lambda_{g'_{d}}}} \right\|}^{2}}} }_{\text{IGI}_{g_{d,k}}} + 1 \right)^{- 1} \right) \end{array} \end{aligned} $$

(36)

### Uplink achievable rate

#### Uplink achievable rate with perfect CSI

With perfect CSI, the uplink achievable rate of sensor \(g^{e}_{u,k}\) can be written as

$$ {{\ddot R}_{g_{u,k}^{e}}}\left(\alpha \right) = \left({1 - \alpha} \right){\log_{2}}\left({1 + {{\ddot \gamma }_{g_{u,k}^{e}}}} \right) $$

(37)

where \({{{\ddot \gamma }_{g_{u,k}^{e}}}}\) denotes the SINR of the sensor \(g^{e}_{u,k}\). Since perfect CSI is available, \({{\mathbf {\Xi }_{g_{u,k}^{e}}}}\) in (32) can be rewritten as

$$ \begin{array}{l} {}{{\mathbf{\Xi }}_{g_{u,k}^{e}}} ={{\ddot{\mathbf{\Xi}}}_{g_{u,k}^{e}}} + {\sigma^{2}}{\mathbf{I}_{b_{u}^{e}}}\\ {\kern1pt}=\sum\limits_{{g'}_{u}^{e} \in {{{G_{u}}} \left/\right. {\left\{ {g_{u}^{e}} \right\}}}} {\tilde{\mathbf{H}}_{{g'}_{u}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}\Lambda_{{g'}_{u}^{e}}^{{1 \left/\right. 2}}{{\left({\tilde{\mathbf{H}}_{{g'}_{u}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}\Lambda_{{g'}_{u}^{e}}^{{1 \left/\right. 2}}} \right)}^{H}}} \\ \quad + \sum\limits_{g'_{d} \in {G_{d}}} {\tilde{\mathbf{H}}_{SI}^{\left\{ {{B_{g_{u}^{e}}},{B_{g'_{d}}}} \right\}}} {\mathbf{W}_{{g'}{_{d}}}}\Lambda_{{g'}{_{d}}}^{{1 \left/\right. 2}}{\left({\tilde{\mathbf{H}}_{SI}^{\left\{ {{B_{g_{u}^{e}}},{B_{{g'}{_{d}}}}} \right\}}{\mathbf{W}_{{g'}{_{d}}}}\Lambda_{{g'}{_{d}}}^{{1 \left/\right. 2}}} \right)^{H}}\\ \quad + {\sigma^{2}}{\mathbf{I}_{b_{u}^{e}}} \end{array} $$

(38)

and (32) is in generalized Rayleigh quotient form. The maximum value of (32) is obtained when

$$ {{\ddot{\mathbf{W}}}_{g_{u,k}^{e}}} = \frac{{{{\left({{{\ddot{\mathbf{\Xi} }}_{g_{u,k}^{e}}}} + {\sigma^{2}}{\mathbf{I}_{b_{u}^{e}}} \right)}^{- 1}}\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}}}{{\left\| {{{\left({{{\ddot{\mathbf{\Xi}}}_{g_{u,k}^{e}}}}+ {\sigma^{2}}{\mathbf{I}_{b_{u}^{e}}} \right)}^{- 1}}\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}} \right\|}} $$

(39)

The maximum SINR of sensor \(g^{e}_{u,k}\) can be expressed as

$$ \begin{array}{l} {{\ddot \gamma }_{g_{u,k}^{e}}} = {p_{g_{u,k}^{e}}}{\left({\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}} \right)^{H}}{\left({{{\ddot{\mathbf{\Xi}}}_{g_{u,k}^{e}}}} + {\sigma^{2}}{\mathbf{I}_{b_{u}^{e}}}\right)^{- 1}}\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}\\ \qquad = \frac{\alpha }{{1 - \alpha }}\hat p_{g_{d,k}^{e}}^{H}{\left({\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}} \right)^{H}}{\left({{{\ddot{\mathbf{\Xi}}}_{g_{u,k}^{e}}}}+ {\sigma^{2}}{\mathbf{I}_{b_{u}^{e}}} \right)^{- 1}}\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}} \end{array} $$

(40)

With the help of Sherman-Morrison formula [59], the achievable rate of sensor \(g^{e}_{u,k}\) can be rewritten as

$$ \begin{aligned} {{\ddot R}_{g_{u,k}^{e}}}\left(\alpha \right) = \left({1 - \alpha} \right){\log_{2}}\left({1 + \frac{{\alpha \hat p_{g_{d,k}^{e}}^{H}}}{{\left({1 - \alpha} \right){\sigma^{2}}}}\left({{{\left\| {\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}} \right\|}^{2}} - {\vartheta_{g_{u,k}^{e}}}} \right)} \right) \end{aligned} $$

(41)

where \({{\vartheta _{g_{u,k}^{e}}}}\) can be expressed as

$$ {\vartheta_{g_{u,k}^{e}}} = \frac{{{{\left({\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}} \right)}^{H}}{{\ddot{\mathbf{\Xi}}}_{g_{u,k}^{e}}}\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}}}{{{\sigma^{2}} + tr\left({{{\ddot{\mathbf{\Xi}}}_{g_{u,k}^{e}}}} \right)}} $$

(42)

In this way, the achievable rate of sensor \(g^{e}_{u,k}\) can be further rewritten as

$$ \ddot{R}_{g_{u,k}^{e}}\left(\alpha \right) = \left({1 - \alpha} \right){\log_{2}}\left({1 + \frac{{\alpha \hat p_{g_{d,k}^{e}}^{H}}}{{\left({1 - \alpha} \right){\sigma^{2}}}}{{\left\| {\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}} \right\|}^{2}}} \right) $$

(43)

when \({{\ddot {\mathbf {\Xi }}_{g_{u,k}^{e}}}}\) is completely eliminated, and hence, we have \({{\vartheta _{g_{u,k}^{e}}}}=0\).

#### Uplink achievable rate with imperfect CSI

According to (29) and using the bounding technique in [58], the average achievable rates at the downlink user *g*_{
u,k
} can be expressed as (44) at the top of the next page. The transmit power of sensor \( g^{e}_{d,k}\) is set to \(p_{g_{u,k}^{e}}={p^{\mathrm {H}}_{{g^{e}_{d,k}}}}\). The powers of CEE, IUI, IGI, and SI are explained in the equation.

$$ \begin{aligned} \begin{array}{l} {\kern-4.5pt}{R_{{g^{e}_{u,k}}}}(\alpha) = \left({1 - \alpha} \right){\log_{2}}\left(1 + {p_{g_{u,k}^{e}}}\|\tilde{\mathbf{h}}_{g_{u,k}^{e}}^{\left\{ {{B_{g_{u}^{e}}}} \right\}}\|^{2}\left(\underbrace {{p_{g_{u,k}^{e}}}{{\left| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\Delta \tilde{\mathbf{h}}_{{g^{e}_{u,k}}}^{\left\{ {{B_{{g^{e}_{u}}}}} \right\}}} \right|}^{2}}}_{{\text{CEE}_{g^{e}_{u,k}}}} \right.\right.\\ \ \ \qquad + \underbrace {\sum\limits_{k' = 1,k' \ne k}^{{K_{{g^{e}_{u}}}}} {{p_{{g^{e}_{u,k'}}}}{{\left| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\Delta \tilde{\mathbf{h}}_{{g^{e}_{u,k'}},LM}^{\left\{ {{B_{{g^{e}_{u}}}}} \right\}}} \right|}^{2}}} }_{\text{IUI}_{g^{e}_{u,k}}} +\underbrace {\sum\limits_{{g^{\prime}}_{u} \in {{{G_{u}}} \left/\right. {\left\{ {{g^{e}_{u}}} \right\}}}}^{} {{{\left\| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\tilde{\mathbf{H}}_{g^{{\prime}{e}}_{u}}^{\left\{ {{B_{{g^{e}_{u}}}}} \right\}} \Lambda_{g^{{e}}_{u}}^{{1 \left/\right. 2}}} \right\|}^{2}}}}_{\text{IGI}_{g^{e}_{u,k}}} \\ \ \ \qquad \left. {{{\left. {+ \underbrace {\sum\limits_{g'_{d} \in {G_{d}}}^{} {{{\left\| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\tilde{\mathbf{H}}_{SI}^{\left\{ {{B_{{g^{e}_{u}}}},{B_{g'_{d}}}} \right\}}{\mathbf{W}_{g'_{d}}}\Lambda_{g'_{d}}^{{1 \left/\right. 2}}} \right\|}^{2}}} }_{\text{SI}_{g^{e}_{u,k}}}} + \|\mathbf{w}_{g_{u,k}^{e}}\|^{2}\right)}^{- 1}}} \right) \end{array} \end{aligned} $$

(44)

### Optimization of the achievable sum rate

In particular system, the assumption of having perfect instantaneous CSI is idealistic due to the fact that the CSI at the BS, users, and sensors are obtained by estimation or feedback. Hence, the CSI is subject to estimation, feedback, delay, and quantization errors. In this subsection, we consider the problem of optimizing the system achievable sum rate with imperfect CSI.

The achievable sum rate of the system with imperfect CSI can be written as

$$ {{R}_{{\text{SUM}}}}(\alpha)=\sum\limits_{{g_{d,k}} \in {\mathbf{K}_{D}}}^{} {{{R}_{{g_{d,k}}}}}(\alpha) {\mathrm{+ }}\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{} {{{{R}}_{g_{u,k}^{e}}}}(\alpha) $$

(45)

where **K**_{
D
} denotes the set of all the users and **K**_{
U
} denotes the set of all the sensors. The downlink rate \({{R_{{g_{d,k}}}}}\) of user *g*_{
d,k
} and the uplink rate \({{R}}_{g_{u,k}^{e}}\) of sensor \(g_{u,k}^{e}\) are given in (36) and (44), respectively. The system achievable sum rate maximization problem can be written as

$$ \begin{array}{l} \mathop {\max }\limits_{\mathcal{P},\alpha} \sum\limits_{{g_{d,k}} \in {\mathbf{K}_{D}}}^{} {{{R}_{{g_{d,k}}}}}(\alpha)+\sum\limits_{{g^{e}_{u,k}} \in {\mathbf{K}_{U}}}^{}{{{{{{R}}}_{g_{u,k}^{e}}}}}(\alpha)\\ {\mathrm{s}}{\mathrm{.t}}{.}\left\{ {\begin{array}{*{20}{l}} {p1:1 >\alpha \ge {\tau_{u}}/T}\\ {p2:\sum\limits_{d = 1}^{{K_{g}^{E}}} {{p_{g_{d}^{e}}} \le P_{d}^{E}} }\\ {p3:\sum\limits_{k = 1}^{K_{d}} {{p_{{g_{d,k}}}}{{\left\| {{\mathbf{w}_{{g_{d,k}}}}} \right\|}^{2}}} \le {P_{d}}{}}\\ {p4:{p^{e}_{{g_{d,k}}}}, {p_{{g_{d,k'}}}} \ge 0, k \in {{\mathbf{K}_{U}}}, k' \in {{\mathbf{K}_{D}}}} \end{array}} \right. \end{array} $$

(46)

where \({K_{g}^{E}}\) denotes the number of sensor groups. \(P^{\mathrm {E}}_{d}\) and *P*_{
d
} denote the maximum transmit power constraint at the BS in *phase I* and *phase II*, respectively. \({\mathcal {P}} = \left \{ {{p_{{g_{d,1}}}}, \cdots,{p_{{g_{d,{K_{d}}}}}},{p_{g_{1}^{e}}}, \cdots,{p_{g_{K_{g}^{E}}^{e}}}} \right \}\).

#### Optimize \(\mathcal {P}\) for fixed *α*

We assume that the length of the pilot signals and energy signals are fixed, that is, *τ*_{
u
} and *τ*_{
d
} are fixed. The downlink rate \({{R_{{g_{d,k}}}}}\) can be written as \({{R_{{g_{d,k}}}}} = (1-\alpha){\log _{2}}\left ({1 +\gamma _{g_{d,k}}} \right)\), and the SINR at user *g*_{
d,k
} can be written as \(\gamma _{g_{d,k}}=\frac {{{p_{g_{d,k}}}\mathbb {E}\left [ {{{\left [ {\Upsilon _{{g_{d}}}^{- 1}} \right ]}_{k,k}}} \right ]}}{{{\text {CE}}{{\mathrm {E}}_{{g_{d,k}}}} + {\text {IU}}{{\mathrm {I}}_{{g_{d,k}}}} + {\text {IG}}{{\mathrm {I}}_{{g_{d,k}}}} + 1}}\). The uplink rate \({{R_{g^{e}_{u,k}}}}\) can be written as \({R_{g^{e}_{u,k}}} = (1-\alpha){\log _{2}}\left ({1 + \gamma _{g^{e}_{u,k}}} \right) \), and the SINR of user \(g^{e}_{u,k}\) signal can be written as \(\gamma _{g^{e}_{u,k}}=p_{g^{e}_{u,k}}{b_{g^{e}_{u,k}}}\left ({{c_{g^{e}_{u,k}} + d_{g^{e}_{u,k}}}}\right)^{-1}\), wherein \(p_{g_{u,k}^{e}}=\frac {\alpha }{(1-\alpha)}p_{g_{d}^{e}} q_{g_{d,k}^{e}}\) and \(b_{g^{e}_{u,k}},c_{g^{e}_{u,k}},d_{g^{e}_{u,k}}\) are positive variables and are unrelated to *α*. Specifically, \(b_{g^{e}_{u,k}} = {\left \| {\tilde {\mathbf {h}}_{g^{e}_{u,k}}^{\left \{ {{B_{g_{u}}}} \right \}}} \right \|^{2}}\), \(c_{g^{e}_{u,k}} = {\text {CE}}{{\mathrm {E}}_{g^{e}_{u,k}}} + {\text {IU}}{{\mathrm {I}}_{g^{e}_{u,k}}} + {\text {IG}}{{\mathrm {I}}_{g^{e}_{u,k}}}\), and \(d_{g^{e}_{u,k}} = {\mathrm {S}}{{\mathrm {I}}_{g^{e}_{u,k}}} + \|\mathbf {w}_{{g^{e}_{u,k}}}^{H}\|^{2}\). We omit the user group index, and the sum rate optimization problem can be rewritten as

$$ \begin{aligned} \begin{array}{l} \mathop {\max }\limits_{\mathcal{P}} \prod\limits_{k = 1}^{{K_{u}}} {\left({1 + {\gamma^{e}_{u,k}}} \right)} \prod\limits_{k = 1}^{{K_{d}}} {\left({1 + {\gamma_{d,k}}} \right)} \\ = \mathop {\min }\limits_{\mathcal{P}} {\left[ {\prod\limits_{k = 1}^{{K_{u}}} {\left({1 + {\gamma^{e}_{u,k}}} \right)} \prod\limits_{k = 1}^{{K_{d}}} {\left({1 + {\gamma_{d,k}}} \right)}} \right]^{- 1}}\\ {\mathrm{s}}{\mathrm{.t}}.\left\{ {\begin{array}{*{20}{l}} {c1:{\gamma_{{g^{e}_{u,k}}}} \le \frac{\alpha}{(1-\alpha)}q_{g_{d,k}^{e}} p_{g_{d}^{e}}{b_{{g^{e}_{u,k}}}}{{\left({{c_{{g^{e}_{u,k}}}} + {d_{{g^{e}_{u,k}}}}} \right)}^{- 1}},}k = 1, \cdots,{K_{u}}\\ {c2:{\gamma_{{g_{d,k}}}} \le \frac{{{p_{{g_{d,k}}}}\mathbb{E}\left[ {{{\left[ {\Upsilon_{{g_{d}}}^{- 1}} \right]}_{k,k}}} \right]}}{{{\text{CE}}{{\mathrm{E}}_{{g_{d,k}}}} + {\text{IU}}{{\mathrm{I}}_{{g_{d,k}}}} +{\text{IG}}{{\mathrm{I}}_{{g_{d,k}}}}+ 1}},}k = 1, \cdots,{K_{u}}\\ {p1,p2,p3,p4} \end{array}} \right. \end{array} \end{aligned} $$

(47)

where *p*1, *p*2, *p*3, and *p*4 are power constraints given by (46). The form of SE optimization problem in (47) is also close to a geometric programming (GP) except that the target function is not in the posynomial form [60]. In this paper, we apply the technique in [61] to approximate the target and solve it with the help of convex optimization tools. Specifically, 1+*γ*_{
a,k
} can be approximated by \({\lambda _{a,k}}\gamma _{a,k}^{{\mu _{a,k}}}\) close to a point \({{\tilde \gamma }_{a,k}}\), where \({\mu _{a,k}} = {{\tilde \gamma }_{a,k}}{\left ({1 + {{\tilde \gamma }_{a,k}}} \right)^{- 1}}\), \({\lambda _{a,k}} = \tilde \gamma _{a,k}^{- {\mu _{a,k}}}\left ({1 + {{\tilde \gamma }_{a,k}}} \right)\), and *a*∈{*u,d*}. As a result, (47) can be rewritten as

$$ \begin{aligned} \begin{array}{l} \mathop {\min }\limits_{P} \prod\limits_{k = 1}^{{K_{u}}} {{{\left({{\lambda^{e}_{u,k}}} \right)}^{- 1}}(\gamma^{e}_{u,k})^{- {\mu_{u,k}}}} \prod\limits_{k = 1}^{{K_{d}}} {{{\left({{\lambda_{d,k}}} \right)}^{- 1}}\gamma_{d,k}^{- {\mu_{d,k}}}} \\ {\mathrm{s}}{\mathrm{.t}}.\left\{ {\begin{array}{*{20}{l}} {c1:{\gamma_{{g^{e}_{u,k}}}}\left({{c_{{g^{e}_{u,k}}}} + {d_{{g^{e}_{u,k}}}}} \right){{\left({\frac{\alpha}{(1-\alpha)}q_{g_{d,k}^{e}} p_{g_{d}^{e}}{b_{{g^{e}_{u,k}}}}} \right)}^{- 1}} \le 1,}k = 1, \cdots,{K_{u}}\\ {c2:\frac{{\gamma_{{g_{d,k}}}}\left({{\text{CE}}{{\mathrm{E}}_{{g_{d,k}}}} + {\text{IU}}{{\mathrm{I}}_{{g_{d,k}}}} + {\text{IG}}{{\mathrm{I}}_{{g_{d,k}}}} + 1} \right)}{{{ {{p_{{g_{d,k}}}}\left[ {{{\left[ {\Upsilon_{{g_{d}}}^{- 1}} \right]}_{k,k}}} \right]} }}} \le 1,}k = 1, \cdots,{K_{d}}{\mathrm{ }}\\ {p1,p2,p3,p4} \end{array}} \right. \end{array} \end{aligned} $$

(48)

Constraints *c*1 and *c*2 can be rewritten as (49) and (50) at the top of this page.

$$ \begin{array}{l} c1:\frac{1}{{{{\left\| {\tilde{\mathbf{h}}_{{g^{e}_{u,k}}}^{\left\{ {{B_{{g^{e}_{u}}}}} \right\}}} \right\|}^{2}}}}\left({\gamma_{{g^{e}_{u,k}}}}{{\left| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\Delta \tilde{\mathbf{h}}_{{g^{e}_{u,k}},LM}^{\left\{ {{B_{{g^{e}_{u}}}}} \right\}}} \right|}^{2}} \right.\\ + {\gamma_{{g^{e}_{u,k}}}}p_{{g^{e}_{u,k}}}^{- 1}\sum\limits_{k' = 1,k' \ne k}^{{K_{{g^{e}_{u}}}}} {{p_{{g^{e}_{u,k'}}}}{{\left| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\Delta \tilde{\mathbf{h}}_{{g^{e}_{u,k'}},LM}^{\left\{ {{B_{{g^{e}_{u}}}}} \right\}}} \right|}^{2}}}\\ + {\gamma_{{g^{e}_{u,k}}}}p_{{g^{e}_{u,k}}}^{- 1}\sum\limits_{g{^{{\prime}e}_{u}} \in {{{G_{u}}} \left/\right. {\left\{ {{g^{e}_{u}}} \right\}}}}^{} {{{\sum\limits_{k' = 1}^{{K_{g{^{{\prime}e}_{u}}}}} {p_{{g^{{\prime}e}_{u,k'}}}\left| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\tilde{\mathbf{h}}_{g{^{{\prime}e}_{u,k'}}}^{\left\{ {{B_{{g^{e}_{u}}}}} \right\}}} \right|} }^{2}}}\\ + {\gamma_{{g^{e}_{u,k}}}}p_{{g^{e}_{u,k}}}^{- 1}\sum\limits_{g'_{d} \in {G_{d}}}^{} {{{\sum\limits_{k' = 1}^{{K_{g'_{d}}}} {p_{g'_{d,k'}}^{}\left| {\mathbf{w}_{{g^{e}_{u,k}}}^{H}\tilde{\mathbf{H}}_{SI}^{\left\{ {{B_{{g^{e}_{u}}}},{B_{g'_{d}}}} \right\}}\mathbf{w}_{g'_{d,k'}}^{}} \right|} }^{2}}} \\ +\left. {\gamma_{{g^{e}_{u,k}}}}p_{{g^{e}_{u,k}}}^{- 1}\|\mathbf{w}_{{g^{e}_{u,k}}}^{H}\|^{2} \right) \le 1\\ k=1,\cdots,K_{u} \end{array} $$

(49)

$$ \begin{aligned} \begin{array}{l} c2:\frac{1}{{\mathbb{E}\left[ {{{\left[ {\Upsilon_{{g_{d}}}^{- 1}} \right]}_{k,k}}} \right]}}\left({\gamma_{{g_{d,k}}}}{{\left| {{{\left({\Delta \tilde{\mathbf{h}}_{{g_{d,k}},LM}^{\left\{ {{B_{{g_{d}}}}} \right\}}} \right)}^{H}}\mathbf{w}_{{g_{d,k}}}^{}} \right|}^{2}}\right.\\ + {\gamma_{{g_{d,k}}}}p_{{g_{d,k}}}^{- 1}\sum\limits_{k' = 1,k' \ne k}^{{K_{{g_{d}}}}} {{p_{{g_{d,k'}}}}{{\left| {{{\left({\Delta \tilde{\mathbf{h}}_{{g_{d,k}},LM}^{\left\{ {{B_{{g_{d}}}}} \right\}}} \right)}^{H}}\mathbf{w}_{{g_{d,k'}}}^{}} \right|}^{2}}} \\ + {\gamma_{{g_{d,k}}}}p_{{g_{d,k}}}^{- 1}\left. {\sum\limits_{g'_{d} \in {{{G_{d}}} \left/\right. {\left\{ {{g_{d}}} \right\}}}}^{} {{{\sum\limits_{k' = 1}^{{K_{g'_{d}}}} {p_{g'_{d,k'}}^{}\left| {{{\left({\tilde{\mathbf{h}}_{{g_{d,k}}}^{\left\{ {{B_{g'_{d}}}} \right\}}} \right)}^{H}}\mathbf{w}_{g'_{d,k'}}^{H}} \right|} }^{2}}} + {\gamma_{{g_{d,k}}}}p_{{g_{d,k}}}^{- 1}} \right) \le 1\\ k=1,\cdots,K_{d} \end{array} \end{aligned} $$

(50)

The SE optimization problem in (48) becomes a standard GP and can be solved by the **Algorithm 1**.

#### Optimize *α* for fixed \(\mathcal {P}\)

Since the downlink rate \({{R_{{g_{d,k}}}}}(\alpha)\) can be rewritten as \({R_{{g_{d,k}}}}(\alpha)= (1-\alpha) \zeta _{g_{d,k}}\), where \(\zeta _{g_{d,k}} = {\log _{2}}\left ({1 + \frac {{{p_{d,k}}\mathbb {E}\left [ {{{\left [ {\Upsilon _{{g_{d}}}^{- 1}} \right ]}_{k,k}}} \right ]}}{{{\text {CE}}{{\mathrm {E}}_{{g_{d,k}}}} + {\text {IU}}{{\mathrm {I}}_{{g_{d,k}}}} + 1}}} \right)\) is a variable unrelated to *α*. The ratio used for information transmission can be adjusted by changing the during of *phase II*, that is (1−*α*)*T*. The uplink rate \({{R_{g_{u,k}^{e}}}}(\alpha)\) can be rewritten as

$$ {R_{g_{u,k}^{e}}}(\alpha) = \left({1 - \alpha} \right){\log_{2}}\left({1 + \gamma_{g_{u,k}^{e}}(\alpha)} \right) $$

(51)

where \(\gamma _{g_{u,k}^{e}}(\alpha)={\frac {\alpha \tilde {{b}}_{g_{u,k}^{e}}}{{\left ({1 - \alpha } \right)}}}\left ({{\frac {\alpha \tilde {{c}}_{g_{u,k}^{e}}}{{\left ({1 - \alpha } \right)}} + d_{g_{u,k}^{e}}}}\right)^{-1}\) and \(\tilde {b}_{g_{u,k}^{e}},\tilde {{c}}_{g_{u,k}^{e}},d_{g_{u,k}^{e}}\) are positive variables and are unrelated to *α*. Specifically, \(\tilde {b}_{g_{u,k}^{e}} = q_{g_{d,k}^{e}} p_{g_{d}^{e}}{\left \| {\tilde {\mathbf {h}}_{g_{u,k}^{e}}^{\left \{ {{B_{g_{u}^{e}}}} \right \}}} \right \|^{2}}\), \(\tilde {{c}}_{g_{u,k}^{e}} = {\left ({{\text {CEE}}_{g_{u,k}^{e}}} + {\text {IU}}{{\mathrm {I}}_{g_{u,k}^{e}}} + {\text {IG}}{{\mathrm {I}}_{g_{u,k}^{e}}} \right)} (1-\alpha)/\alpha \), and \(d_{g_{u,k}^{e}} = {\mathrm {S}}{{\mathrm {I}}_{g_{u,k}^{e}}} + 1\) are variables unrelated to *α*. The achievable sum rate maximization problem of *α* can be rewritten as

$$ \begin{array}{l} \mathop {\max }\limits_{\alpha} {{R}_{{\text{SUM}}}}(\alpha)=(1-\alpha) \sum\limits_{{g_{d,k}} \in {\mathbf{K}_{D}}}^{}\zeta_{g_{d,k}}\\ \qquad\qquad\qquad\quad+\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{}\left({1 - \alpha} \right){\log_{2}}\left({1 + \gamma_{g_{u,k}^{e}}}(\alpha) \right) \\ {\mathrm{s}}{\mathrm{.t}}.1 >\alpha \ge {\tau_{u}}/T \end{array} $$

(52)

Let \(\zeta =\sum \limits _{{g_{d,k}} \in {\mathbf {K}_{D}}}^{}\zeta _{g_{d,k}}\) and (52) can be rewritten as

$$ \begin{array}{l} \mathop {\max }\limits_{\alpha} {{R}_{{\text{SUM}}}}(\alpha)=(1-\alpha) \zeta\\ \qquad\qquad\qquad\quad+\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{}\left({1 - \alpha} \right){\log_{2}}\left({1 + \gamma_{g_{u,k}^{e}}}(\alpha) \right) \\ {\mathrm{s}}{\mathrm{.t}}.1 >\alpha \ge {\tau_{u}}/T \end{array} $$

(53)

The partial derivative of function *R*_{SUM}(*α*) can be expressed as

$$ \begin{aligned} \frac{\partial {{R}_{{\text{SUM}}}}(\alpha)}{\partial \alpha} &=-\zeta -\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{}{\log_{2}}\left({1 + \gamma_{g_{u,k}^{e}}}(\alpha) \right)\\ &\quad {\mathrm{+ }}\left({{\mathrm{1}} - \alpha} \right)\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{} {\frac{{{{\tilde b}_{g_{u,k}^{e}}}{d_{g_{u,k}^{e}}}}}{{\ln \left(2 \right)\left({1 + {\gamma_{g_{u,k}^{e}}}(\alpha)} \right){{\left({\alpha {{\tilde c}_{g_{u,k}^{e}}} + \left({1 - \alpha} \right){d_{g_{u,k}^{e}}}} \right)}^{2}}}}} \end{aligned} $$

(54)

The two order derivative of function *R*_{SUM}(*α*) can be expressed as (55) at the top of this page.

$$ \begin{aligned} \begin{array}{l} \frac{{{\partial^{2}}{{R}_{{\text{SUM}}}}(\alpha)}}{{\partial {\alpha^{2}}}} = \- \sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{} {\frac{{2{{\tilde b}_{g_{u,k}^{e}}}{d_{g_{u,k}^{e}}}}}{{\ln \left(2 \right)\left({1 + {\gamma_{g_{u,k}^{e}}}(\alpha)} \right){{\left({\alpha {{\tilde c}_{g_{u,k}^{e}}} + \left({1 - \alpha} \right){d_{g_{u,k}^{e}}}} \right)}^{2}}}}} \\ - \left({{\mathrm{1}} - \alpha } \right)\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{} {\frac{{{{\tilde b}_{g_{u,k}^{e}}}{d_{g_{u,k}^{e}}}\left({{{\tilde b}_{g_{u,k}^{e}}}{d_{g_{u,k}^{e}}} + 2\left({1 + {\gamma_{g_{u,k}^{e}}}(\alpha)} \right)\left({\alpha {{\tilde c}_{g_{u,k}^{e}}} + \left({1 - \alpha} \right){d_{g_{u,k}^{e}}}} \right)\left({{{\tilde c}_{g_{u,k}^{e}}} - {d_{g_{u,k}^{e}}}} \right)} \right)}}{{\ln \left(2 \right){{\left({1 + {\gamma_{g_{u,k}^{e}}}(\alpha)} \right)}^{2}}{{\left({\alpha {{\tilde c}_{g_{u,k}^{e}}} + \left({1 - \alpha} \right){d_{g_{u,k}^{e}}}} \right)}^{4}}}}} \end{array} \end{aligned} $$

(55)

Consistent with the assumption in [52], in this paper, we assume that the power of CEE, IUI, and IGI is much larger than SI and AWGN. Hence, we have \(\frac {\partial ^{2}{{R}_{{\text {SUM}}}}(\alpha)}{\partial ^{2} \alpha }<0\) for 0<*α*<1 and we can obtain the optimum *α* of *R*_{SUM}(*α*) when \(\frac {\partial {{R}_{{\text {SUM}}}}(\alpha)}{\partial \alpha }=0\), that is

$$ {{} \begin{aligned} \begin{array}{l} \underbrace {{{\left({{\mathrm{1}} - \alpha} \right)}^{2}}\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{} {\frac{{{{\tilde b}_{g_{u,k}^{e}}}{d_{g_{u,k}^{e}}}}}{{\ln \left(2 \right)\left({1 + {\gamma_{g_{u,k}^{e}}}(\alpha)} \right){{\left({\alpha {{\tilde c}_{g_{u,k}^{e}}} + \left({1 - \alpha} \right){d_{g_{u,k}^{e}}}} \right)}^{2}}}}} }_{A\left(\alpha \right)}\\ = \left({{\mathrm{1}} - \alpha} \right)\zeta + \left({{\mathrm{1}} - \alpha} \right)\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{} {{{\log }_{2}}} \left({1 + {\gamma_{g_{u,k}^{e}}}(\alpha)} \right)\\ ={{R}_{{\text{SUM}}}}(\alpha)|_{\mathcal{P}} \end{array} \end{aligned}} $$

(56)

Equation (56) is a transcendental equation, and we also can solve it numerically or resort the Newton iterative method to solve the problem, that is, \(\hat {\alpha }_{k+1}=\hat {\alpha }_{k}-\frac {\partial {{R}_{{\text {SUM}}}}(\alpha)}{\partial \alpha }/\frac {\partial ^{2}{{R}_{{\text {SUM}}}}(\alpha)}{\partial ^{2} \alpha }|_{\alpha =\hat {\alpha }_{k}}\). Let \(\hat {\alpha }^{\text {Opt}}\) denote the solution of (56) and if 1>*α*^{Opt}>*τ*_{
u
}/*T*, then the solution of the problem in (53) is *α*^{Opt}. Moreover, it is interesting to see that the maximum sum rate of the system is \({{R}^{\text {OPT}}_{{\text {SUM}}}}=A\left (\alpha ^{\text {Opt}}\right)\). Otherwise, the solution of the problem in (53) is *τ*_{
u
}/*T*, and the maximum sum rate of the system is

$$ {{R}_{{\text{SUM}}}^{\text{Opt}}}=\sum\limits_{{g_{d,k}} \in {\mathbf{K}_{D}}}^{} {{{R}_{{g_{d,k}}}}}(\tau_{u}/T) {\mathrm{+ }}\sum\limits_{g_{u,k}^{e} \in {\mathbf{K}_{U}}}^{} {{{R}_{g_{u,k}^{e}}}}(\tau_{u}/T) $$

(57)

#### Joint optimization of *α* and \(\mathcal {P}\)

The joint optimal *α* and \(\mathcal {P}\) can be obtained by finding the optimum \(\mathcal {P}\) for each *α* and then selecting the found \(\mathcal {P}\) and *α* that maximize the problem in (46). Hence, a one-dimensional search over *α* is needed. We can conclude that the required one-dimensional search can be limited to a small region of *α* by exploiting the structure of the problem (46) and the properties obtained in (54)–(56). Hence, the computational complexity for solving the joint optimization problem can be greatly reduced. The proposed solution to jointly optimize *α* and \(\mathcal {P}\) is summarized in Algorithm 2.