Beam-domain hybrid time-switching and power-splitting SWIPT in full-duplex massive MIMO system

We consider a FD massive MIMO SWIPT system that the BS simultaneously serves a set of downlink users (cellular users) K _D ={1,2,⋯,K _d } and a set of fixed uplink sensors K _U ={1,2,⋯,K _u } which are uniformly distributed in the coverage area of the BS, as shown in Fig. 1. The number of downlink users is K _d , and the number of sensors is K _u . The BS is equipped with a separate massive antenna array. The total number of antennas at the BS is 2N of which N antennas are dedicated to the downlink and N antennas are used for the uplink. We assume that all users and sensors are work in HD mode and have a single antenna. Each sensor employs a rectanna for EH and uses the harvested energy to power its subsequent data transmission.

We consider frame-based transmissions over Rayleigh fading channels. The length of one frame is fixed to T seconds, which is assumed to be less than the coherence interval of the channel. We use \(\mathbf {h}_{k_{u}}\in \mathbb {C}^{N \times 1}\) and \(\mathbf {h}_{k_{d}}\in \mathbb {C}^{N \times 1}\) to denote the uplink UE-BS channel vector from the user k _u to the receive antenna array of BS and the downlink UE-BS channel vector from the transmit antenna array of BS to the user k _d . Let \(\mathbf {h}^{e}_{k_{u}}\in \mathbb {C}^{N \times 1}\) and \(\mathbf {h}^{e}_{k_{d}}\in \mathbb {C}^{N \times 1}\) denote the uplink sensor-BS channel vector from the sensor node k _d to the receive antenna array of the BS and the downlink sensor-BS channel vector from the transmit antenna array of the BS to the sensor node k _d . We use \(\mathbf {H}_{SI}\in \mathbb {C}^{N \times N}\) to denote the SI channel matrix from the transmit antenna array to the receive antenna array of the BS.

Note that the channel model between the user and the BS is the same as the channel model between the sensor and the BS; in the following sections, we only introduce the uplink and downlink channel model between the user and the BS.

In order to reduce the number of dimensions of the channel to be estimated as well as the amount of feedback, we resort to the BEM [56]. Using the BEM, the BD channel representation can be obtained by projecting the channel vector (matrix) on common bases. In this way, the channel can be compressed in the BD under certain selected basis spaces and the channel dimension required to be estimated can be reduced greatly. Moreover, by exploiting the BD structure of SI channel, it is possible to eliminate the SI without using the instantaneous SI channel knowledge and hence realize efficient SWIPT transmission.

where \(B_{k_{\xi }}\) is the selected active beam set (ABS) of uplink/downlink which contains the indexes of beams with non-negligible BD channel gains. All the elements selected in the set \(B_{k_{\xi }}\) is related to the DOA/DOD, and the detailed method for determining all the elements in the set of \(B_{k_{\xi }}\) will be described in the following section. \(\mathbf {F}^{\{B_{k_{\xi }}\}}\) is the corresponding active beam spaces, whose columns are consisted of the beams in \(B_{k_{\xi }}\). The reduced-dimension BD channel vector \(\widetilde {\mathbf {h}}_{{k_{\xi }}}^{\left \{ {{B_{{k_{\xi }}}}} \right \}} \in {\mathbb {C}^{\left | {{B_{{k_{\xi }}}}} \right | \times 1}}\) is called the BDEC. Note that (7) holds with equality as N→∞.

where ϖ<1 is the threshold.

In the same way, we can decompose the SI channel in the BD form, that is \({\tilde {\mathbf {H}}_{SI}} = {\mathbf {F}^{H}}{\mathbf {H}_{SI}}\mathbf {F}\). Since the average BD SI channel gain has non-negligible value when the beam angle of receive side f _m lies in \( \cup _{i = 1}^{{M_{SI}}}\left ({\sin \theta _{R,i}^{\min } - \varepsilon,\sin \theta _{R,i}^{\max } + \varepsilon } \right)\), and the beam angle of transmit side f _n lies in \( \cup _{i = 1}^{{M_{SI}}}\left ({\sin \theta _{T,i}^{\min } - \varepsilon,\sin \theta _{T,i}^{\max } + \varepsilon } \right)\), where ε approaches to zero in the large N regime. Hence, the average BD SI channel gain \(\mathbb {E}\left [ {{{\left | {{{\left [ {{\tilde {\mathbf {H}}_{SI}}} \right ]}_{m,n}}} \right |}^{2}}} \right ] = \mathbb {E}\left [ {{{\left | {\mathbf {f}_{m}^{H}{\mathbf {H}_{SI}}{\mathbf {f}_{n}}} \right |}^{2}}} \right ]\).x

In this section, we propose a BD scheme to realize SWIPT in the cellular system with FD massive MIMO BS. In order to accomplish the channel estimation, the transmit and receive antenna arrays of the BS are assumed to be shared antennas, i.e., the transmit and receive RF chains share the same antenna [10]. In the frame-based BD massive MIMO FD SWIPT system, each frame is divided into two phases, phase I: estimation and EH phase and phase II: BD FD transmission phase, as shown in Fig. 1.

In the phase I of time period αT (0≤α≤1), users transmit pilots for channel estimation at the BS, while the BS simultaneously transmits energy signals to the sensor nodes for EH and channel estimation. Due to the separate antenna configuration at the BS; therefore, the reciprocity of uplink and downlink channels does not hold. The BS uses the transmit antenna array to receive the pilot signals of users for uplink UE-BS channel estimation and uses the receive antenna array to transmit the energy signals. We assume that the sensor opens the receive circuit during the interval αT, and part of the received energy signal is used for channel estimation and part of the received energy signal is used for EH; hence, the sensors can obtain the uplink sensor-BS channel. Specifically, the BS uses the transmit antenna array to receive the pilot signals of UEs, that is \(y_{t}^{U}\) as shown in Fig. 2. Thus, the uplink UE-BS channel can be estimated during phase I. Simultaneously, the BS uses receive antenna array to transmit the energy signals to all the sensors. The signal received by sensor node can be expressed as \(y_{r}^{S}\). Part of the received energy signal, \(\sqrt {\beta }y_{r}^{S}\), is used for EH and part of the received signal, \(\sqrt {1-\beta }y_{r}^{S}\), is used for downlink sensor-BS channel estimation, where β denotes the PS ratio. Note that the detailed training signals (pilot signals and energy signals) design will illustrated in the next subsection.

In the phase II of time period (1−α)T, the BS obtains the downlink UE-BS CSI based on channel reciprocity and then forms the beams for information transmission to the users. Moreover, the sensors feedback the uplink sensor-BS CSI to the BS and transmit their data during the interval (1−α)T using the harvested energy. Specifically, the sensors transmit signal \(\sqrt {\beta \eta }y_{t}^{S}\) to the BS, where η<1 denotes the energy conversion efficiency. Note that the resource consumption for CSI feedback of sensors are omitted in this paper.

The key idea of the proposed BD FD massive MIMO SWIPT scheme lies in partitioning users and sensors according to their ABSs to realize efficient energy and information transmission. In particular, we divide the users and sensors into groups under the following two criterions:

• Criterion 1: The sensors or users with the same ABS are collected in the same sensor/user group. The ABSs of different sensor/user groups are non-overlapping with certain guard interval \(\mho \). Let \(B_{g_{u}}\) and \(B_{g_{u}'}\) be the ABSs of two arbitrary user groups g _u and gu′, and \(B_{g_{u}}\) and \(B_{g_{u}'}\) should satisfy \(\mathcal {D}(B_{g_{u}},B_{g_{u}'})\geq \mho \), as shown in Fig. 3. Let \(B_{g_{d}}\) and \(B_{g_{d}'}\) be the ABSs of two arbitrary sensor groups g _d and gd′, and \(B_{g_{d}}\) and \(B_{g_{d}'}\) should satisfy \(\mathcal {D}(B_{g_{d}},B_{g_{d}'})\geq \mho \).

• Criterion 2: The complete ABS of all the user groups or sensor groups is non-overlapping with the receive ABS or transmit ABS of the SI channel under certain guard interval \(\mho \). Let G _u and G _d be the complete set of sensor groups and user groups, respectively. The ABSs \(B_{g_{u}}\) and \(B_{g_{d}}\) satisfy \(\mathcal {D}(\bigcup _{g_{u}\in G_{u}}B_{g_{u}}, B_{SI,R})> \mho \) or \(\mathcal {D}(\bigcup _{g_{d}\in G_{d}}B_{g_{d}}, B_{SI,T})> \mho \).

In the practical implementation, the users and sensors with different ABSs must be partitioned so that the above two conditions are satisfied as close as possible, as shown in Fig. 3. For all the users or sensors, the ABS of the SI channel at the receive side B _SI,R and transmit side B _SI,T should be took into consideration when carry out grouping. In the following, we proposed a BD user and sensor grouping scheme, which consists of the following two steps.

Step 1: Remove the ABS of the SI channel at the receive side B _SI,R and transmit side B _SI,T from the set of all the beams Γ with a certain guard interval \(\mho \).

Step2: Remove all the beams that satisfy the adjacent \(\mho \) beams including itself which are not occupied by any user or sensor. We obtain serval beam sets, each of which contains several continuous beams occupied by different users or sensors. For each beam set with several continuous beams, the corresponding users or sensors form a user and sensor group.

In this way, the self-interference at the BS can be mitigated effectively based on the statistical CSI of SI channel, that is, the ABS of SI channel, rather than the instantaneous CSI of SI channel. Note that users and sensors should be grouped independently. Meanwhile, to mitigate the interference between the uplink and downlink, the user group or sensor group with overlap beams cannot be allocated with the same frequency band for transmission.

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 αT can be determined by αT=τ _d ≥τ _u . It is worth noting that when αT≥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.

6.3 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. \({\mathcal {P}} = \left \{ {{p_{{g_{d,1}}}}, \cdots,{p_{{g_{d,{K_{d}}}}}},{p_{g_{1}^{e}}}, \cdots,{p_{g_{K_{g}^{E}}^{e}}}} \right \}\).

6.3.1 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 p1, p2, p3, and p4 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 c1 and c2 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.

6.3.2 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)

6.3.3 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.

In this section, the performance of BD hybrid TS and PS SWIPT protocol in FD massive MIMO system is evaluated using the 3GPP LTE simulation model for macro-cell environment [62]. The center frequency is 2.4 GHz and system bandwidth is set to 20 MHz. Thermal noise density is set to − 174 dBm/Hz. The channel coherent time is 200 symbol times. The path loss between BS and users/sensors are modeled as 2.7 + 42.8 log10(R) (dB), and the path loss between users/sensors are modeled as 55.78 + 40 log10(R) (dB), where R denotes the distance. The number of scattering clusters is set to M _u = M _d = 1. It is assumed that the passive SIC scheme for infrastructure nodes proposed in [63] has been employed at the BS. In such scheme, the suppression is from two parts, namely (1) the path loss introduced by the 20 m separation between transmit and receive antenna arrays and (2) an additional cancelation of 45 dB provided by techniques, such as radio frequency absorber material and cross-polarization. No other active SIC scheme is used in this paper.

In this paper, we assume that the BS can obtain the perfect DOA and DOD information when users access the system. In the following simulations, we consider the fixed sensors, hence it is reasonable to assume that the BS can obtain the perfect DOA and DOD information of sensors. We first consider a scenario where the sensors and users gather perfectly in three groups and the DOA/DOD regions of users in each group are identical. We assume that each group contains 5 uplink/downlink users. The DOA/DOD regions of three user groups are [− 45°,− 35°], [12°,22°] and [42°,52°], respectively. Since we assume M _u = 1, the DOA region of uplink group is [a,b], that is \(\left [ {\theta _{{g_{u,k},1}}^{\min },\theta _{{g_{u,k},1}}^{\max }} \right ] = \left [ {a,b} \right ]\). Similarly, the DOA/DOD regions of three sensor groups are [− 42°,− 32°], [16°,26°] and [39°,49°], respectively. The DOA and DOD regions of SI channel between the transmit antenna array and receive antenna array are set to [− 12°,− 22°], [56°,66°] and [− 56°,− 46°], [27°,37°], respectively. The resulting ABSs for all the groups satisfy the user and sensor grouping criteria.

Figure 4 illustrates the SE of the proposed BD hybrid TS and PS SWIPT protocol for FD massive MIMO system with perfect user and sensor grouping. The transmit powers of BS is 20.2 dBm. The distance between users/sensors and BS is set to 500 m. This setup ensures the average downlink receive SNR is 3 dB. The energy conversion efficiency η is set to 0.8, and the power ratio used for energy harvesting β is set to 0.8. For comparison, existing SWIPT protocols designed for massive MIMO (MM) system are also demonstrated, such as TDD SWIPT with MF precoder in [49], TDD SWIPT with ZF precoder in [50], and TDD SWIPT with MRT precoder in [51]. Meanwhile, traditional FDD MM with JSDM [7], FD MM with linear transceiver [64], FD MM with spatial SI suppression [17], and traditional BD FD MM transmission scheme [52] are also extended to the SWIPT scenario and demonstrated for comparison. Note that the FD MM with spatial SI suppression in [17], the BS requires instantaneous CSI of SI channel in order to perform SI cancelation in spatial domain. In the simulations, the uplink transmit power in [7, 17, 52, 64] are also harvested from the BS downlink transmit signal. With estimated effective BD CSI, the SE of the proposed BD hybrid TS and PS SWIPT protocol outperforms the existing transmission schemes. The reason is that, the proposed BD hybrid TS and PS SWIPT transmission scheme can reduce the required length of pilot sequence significantly, hence significant SE gain can be achieved by the proposed scheme over the TDD and FDD MM systems and traditional FD MM. The SE of the proposed BD hybrid TS and PS SWIPT protocol with perfect CSI and the capacity are also depicted in Fig. 4.

In Fig. 5, we test the robustness of the proposed BD hybrid TS and PS SWIPT protocol for FD MM system under the presence of DOA/DOD offset. We can see from Fig. 5 that the proposed BD hybrid TS and PS SWIPT protocol obtains the best performance. Moreover, the SE performance decreases with the increasing of DOA/DOD offset. This is because the proposed BD SWIPT protocol requires accurate DOA/DOD information when users and sensors are partitioned into groups. For the traditional SWIPT protocols in [49–51], linear precoders are adopted so that the user/sensor DOA/DOD information is not required.

In Fig. 6, the SE performance of the proposed BD hybrid TS and PS SWIPT protocol for FD MM system with different PS ratio β is demonstrated. In Fig. 7, the SE performance of the proposed BD hybrid TS and PS SWIPT protocol for FD MM system with different energy conversion efficiency η. We can see from Figs. 6 and 7 that the proposed BD hybrid TS and PS SWIPT protocol obtains the best performance.

Figure 8 depicts the SE of the proposed BD hybrid TS and PS SWIPT protocol for FD MM system with imperfect user and sensor grouping. In Fig. 8, there are 50 users and 50 sensors within the coverage of BS which is equipped with N = 128 transmit/receive antennas. The signal of each user/sensor is within a 10° DOA/DOD region which is randomly distributed in the sectors. The distance between uplink/downlink UE and BS is randomly distributed in the interval [200, 1000] meters. After user and sensor grouping, three user groups and three sensor groups are formed, and the users and sensors in different groups are allocated orthogonal time-frequency resources. Since users and sensors are grouped into three groups, respectively, the SE is defined as the average SEs for three groups. The number of scattering cluster for SI channel is set to M _SI = 2. We can see from Fig. 8 that the proposed BD hybrid TS and PS SWIPT protocol for FD MM system obtains the best performance. For the FD MM with linear transceiver [64], we assume that there is no downlink reciprocity, hence the SE performance is greatly reduced as a result of the heavy cost of pilot resources.

Figure 9 depicts the SE of the proposed BD hybrid TS and PS SWIPT protocol for FD MM system under the condition that the number of scattering clusters MSI is increase from 2 to 10. The SI signal from each scattering cluster is within a 10° DOA/DOD region which is randomly distributed [− 90°,90°]. Interestingly, with the increase of the number of scattering clusters M_SI, the SE of the proposed BD hybrid TS and PS SWIPT protocol for FD MM transmission scheme increases first and then decreases when M_SI>6. Meanwhile, the proposed BD hybrid TS and PS SWIPT protocol for FD MM transmission scheme obtains the best SE performance. The SE of FD MM scheme in [52] approaches to that of the TDD/FDD MM as M _SI increases. This is because the numbers of uplink groups and the active beams for each uplink group decrease as M_SI increases.