Multiple DF relaying modes selection criterion for sum-capacity maximization
In designing the DF relaying scheme, a key issue is to design the decode-and-forward mode at the second phase for the RS. During the second phase, the RS can decode and forward the data streams from both the BS and MS, only from BS, only from MS, or none of them, which are referred to as bidirectional DF (BDF) mode, downlink DF (DDF) mode, uplink DF (UDF) mode, and no DF (NF) mode, respectively.
Let us denote the SNRs of the channels from the BS to RS, from the BS to MS, and from the MR to RS by
$${} \begin{aligned} {\text{SNR}_{{\text{BR}}}} &= \frac{{{P_{\mathrm{B}}} \cdot {{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{2}}},\\ {\text{SNR}_{{\text{BM}}}} &= \frac{{{P_{\mathrm{B}}} \cdot {{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{2}}},\\ {\text{SNR}_{{\text{MR}}}} &= \frac{{{P_{\mathrm{M}}} \cdot {{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{2}}}, \end{aligned} $$
(9)
respectively.
For the case that SNRBM is much smaller than SNRMR and SNRBR, the RS can dramatically improve the performances of both the uplink and downlink; therefore, the BDF is the best among the four modes. Contrarily, for the case that SNRMR and SNRBR are much smaller than SNRBM, the RS is unnecessary, thus direct transmission between the BS and MS, i.e., the NF mode, is the best. The DDF mode has its advantage if SNRMR is much smaller than SNRBM and SNRBR. In this case, the uplink data is only transmitted directly from MS to BS at the first phase, without the assistance of RS. Consequently, the uplink data rate is not limited by the channel fading from MS to RS, which would be a limiting factor to the capacity when SNRMR is very small. Similarly, the UDF mode offers the maximum capacity if SNRBR is much smaller than SNRBM and SNRMR. In this case, as SNRBR is very small, to decode the downlink data at RS requires a very low coding rate, thus limiting the uplink capacity. Therefore, the best strategy is to decode and forward only the uplink data at RS, i.e., the UDF mode.
Selecting the best relaying mode from the above four modes, the system can obtain a selection diversity gain. In this paper, we propose a multiple relaying modes selection criterion to maximize the sum-capacity. The proposed criterion can be expressed as
$$\begin{array}{@{}rcl@{}} {\Re^{*}} = \mathop {\arg }\limits_{\Re \in \Phi} \max {C_{\Re} }, \end{array} $$
(10)
where Φ={BDF,DDF,UDF,NF} denotes the relaying mode set and C
ℜ denotes the sum-capacity of mode ℜ. In the following subsections, we will present the achievable rate region, the sum-capacity, and the transmission and reception modes to achieve the sum-capacity for the four relaying modes.
Bidirectional DF mode
In the BDF mode, the RS decodes both the uplink and downlink data from the received signal by using a successive decoding method, in which the RS first decodes one data stream while treating the other one as a noise, and then subtracts the decoded signal from the received signal, thus then decodes the rest data stream without any interference. After that, the RS encodes the data stream using network coding, i.e., performs an XOR operation on the decoded information bits from BS and MS. Finally, the RS broadcasts them to BS and MS. Meanwhile, the BS transmits another independent downlink signal, which is shown in Fig. 2.
At the first phase, the relay channel can be modeled as a multiple access channel (MAC) [40], therefore, the capacity region can be given by
$$ {\left\{ \begin{array}{l} {R_{{\text{UL}}}^{{\text{BDF}}}[1] < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right]},\\ {R_{{\text{DL}}}^{{\text{BDF}}}[1] < \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right]},\\ {R_{{\text{DL}}}^{{\text{BDF}}}[1] + R_{{\text{UL}}}^{{\text{BDF}}}[1] < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right]}. \end{array} \right.} $$
(11)
The above upper bound can be achieved by using superposition coding with time sharing or rate splitting at transmitter and the successive decoding at receiver [41]. For uplink, the BS decodes the uplink data by jointly using the received signals at the first and second phases (Fig. 3). Therefore, the achievable uplink rate is upper bounded by [42]
$$\begin{array}{@{}rcl@{}} R_{{\text{UL}}}^{{\text{BDF}}}[2] < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right] + \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right]. \end{array} $$
(12)
For downlink, as MS receives two signals from the BS and the RS, respectively. At the second phase, the capacity region can be expressed as
$$ {\left\{ \begin{array}{l} {{R_{{\mathrm{BS-MS}}}^{{\text{BDF}}}[{\mathrm{2}}]} < \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \right]}\\ {{R_{{\mathrm{RS-MS}}}^{{\text{BDF}}}[{\mathrm{2}}]} < \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \right]}\\ {{R_{{\mathrm{BS-MS}}}^{{\text{BDF}}}[{\mathrm{2}}]} + {R_{{\mathrm{RS-MS}}}^{{\text{BDF}}}[{\mathrm{2}}]} < \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}} + {P_{\mathrm{R}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \right]}, \end{array} \right.} $$
(13)
where RRS−MSBDF[2] denotes the rate from the RS to MS and RBS−MSBDF[2] denotes the rate from the BS to MS at the second phase. Due to that, the forward information bits from the RS to MS at the second phase should be less than or equal to the transmitted bits from the BS at the first phase, and RBS−MSBDF[2]+RRS−MSBDF[2]=RDLBDF[2], the downlink data rate at the second phase is upper bounded by
$${} \left\{\begin{array}{llll} &{R_{{\text{DL}}}^{{\text{BDF}}}[{\mathrm{2}}] < R_{{\text{DL}}}^{\text{BDF}}[{\mathrm{1}}]{\mathrm{+ }} \gamma \Big[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \Big]}, \\ &\text{if}~~{R_{{\text{DL}}}^{\text{BDF}}[{\mathrm{1}}] < \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}} \right]};\\ &{R_{{\text{DL}}}^{{\text{BDF}}}[{\mathrm{2}}] < \gamma \Big[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}} + {P_{\mathrm{R}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \Big]}, & \text{otherwise}. \end{array}\right. $$
(14)
Taking consideration of all the upper bounds in (11), (12), and (14), rate regions in terms of R
DL, R
UL, and R
UL+R
DL can be obtained by using the Fourier-Motzkin elimination (FME) method (see Appendix D of [43]). The achievable rate region is finally given by
$${} {\left\{ \begin{array}{l} \!{R_{{\text{UL}}}^{{\text{BDF}}} < \min \left\{ { \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right],\!\gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right] + \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right]} \right\}},\\ \!\!{R_{{\text{DL}}}^{{\text{BDF}}} \!<\! \gamma\! \left[\!{\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \right]\,+\,\min \left\{ { \gamma\! \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right],\!\gamma \left[\! {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}} \right]} \right\}},\\ \!{R_{{\text{DL}}}^{{\text{BDF}}} + R_{{\text{UL}}}^{{\text{BDF}}} < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right]}. \end{array} \right.} $$
(15)
It is observed from (15) that the discussion of the achievable rate region depends on two special boundaries, which are represented by
$$ \begin{aligned} {\beta_{1}} &= \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}} \right], \\ \beta_{2} &= \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} \right] + \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} \right]. \end{aligned} $$
(16)
According to different channel conditions, the rate region is bounded in four cases as shown in Fig. 4, and the sum-capacity can be achieved at the boundary points, C
i
(i=1,2,3,4). Finally, the sum-capacity of the BDF mode can be written by
$$\begin{array}{@{}rcl@{}}{} {C^{{\text{BDF}}}} \,=\, \min \!\left\{\! {C_{{\text{UL}}}^{{\text{BDF}}}\! +\! C_{{\text{DL}}}^{{\text{BDF}}},\!\gamma\! \left[ \!{\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}}\!\right]}\! \right\}, \end{array} $$
(17)
where
$${} {\begin{aligned} C_{{\text{UL}}}^{{\text{BDF}}} &= \min \left\{ {\gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right],\gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} \right] + \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} \right]} \right\},\\ C_{{\text{DL}}}^{{\text{BDF}}} &= \min \left\{ \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right]{\mathrm{+ }}\gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \right],\right.\\ &\qquad \quad \left. \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{\mathrm{2}}}}} \right] \right\}. \end{aligned}} $$
(18)
For cases a, b, and c in Fig. 4, the sum-capacity is achieved only at one point, i.e., there is only one optimal uplink and downlink rate pair; however, for case d, the sum-capacity is achieved for any rate pair on the segment of
$$\begin{array}{@{}rcl@{}} R_{{\text{DL}}}^{{\text{BDF}}} + R_{{\text{UL}}}^{{\text{BDF}}} < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{2}}}} \right]. \end{array} $$
(19)
Downlink DF mode
In the DDF Mode, the RS decodes and forwards only the downlink signal. At the first phase, as shown in Fig. 3, the link from MS to RS is an interference link. This can be modeled as the so-called Z-channel [44], in which the BS received signal is interference-free, whereas the RS receives a combination of the desired downlink signal and the interfering signal. The received signals at the first phase in (1) and (2) can be rewritten in a standard form of Gaussian Z-channel [45] as
$$\begin{array}{*{20}l} y_{\mathrm{B}}^{\mathrm{*}} = \sqrt {\frac{{{P_{\mathrm{M}}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} {h_{{\text{BM}}}}{x_{\mathrm{M}}}[1] + n_{\mathrm{B}}^{\mathrm{*}}, \end{array} $$
(20)
$$\begin{array}{*{20}l} y_{\mathrm{R}}^{\mathrm{*}} = \sqrt {\frac{{{P_{\mathrm{B}}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} {h_{{\text{BR}}}}{x_{\mathrm{B}}}[1] + \sqrt {\frac{{{P_{\mathrm{M}}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} {h_{{\text{BM}}}}{\eta_{\inf }}{x_{\mathrm{M}}}[1] + n_{\mathrm{R}}^{\mathrm{*}}, \end{array} $$
(21)
where \(n_{\mathrm {B}}^{\mathrm {*}}\) and \(n_{\mathrm {R}}^{\mathrm {*}}\) are additive Gaussian noises with zero mean and unit variance and η
inf denotes the interference coefficient defined as \({\eta _{\inf }} = \frac {{{h_{{\text {MR}}}}\cdot {\sigma _{\mathrm {B}}}}}{{{h_{{\text {BM}}}}}\cdot {\sigma _{\mathrm {R}}}}\).
The rate region of the Gaussian Z-channel is upper bounded by [46]. We have
$${} {\left\{ \begin{array}{l} R_{{\text{UL}}}^{{\text{DDF}}}[1] < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right];\\ R_{{\text{DL}}}^{{\text{DDF}}}[1] < \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{2}}}} \right];\\ R_{{\text{UL}}}^{{\text{DDF}}}[1] + R_{{\text{DL}}}^{{\text{DDF}}}[1] < \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}} + {P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{2}}}} \right] \;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\frac{1}{2}{\log_{2}}\left({\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}} + \sigma_{\mathrm{B}}^{2}}}{{\min \left\{ {{\eta_{\inf }}^{2},1} \right\} \cdot {P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}} + \sigma_{\mathrm{B}}^{2}}}} \right). \end{array} \right.} $$
(22)
At the second phase, the SNR of the received signal at the MS is maximized by applying beamforming technique, thus the downlink rate is upper bounded by
$$ R_{{\text{DL}}}^{{\text{DDF}}}[2] < \gamma \left[ {\frac{{{{\left({\sqrt {{P_{\mathrm{B}}}} {h_{{\text{BM}}}} + \sqrt {{P_{\mathrm{R}}}} {h_{{\text{MR}}}}} \right)}^{2}}}}{{\sigma_{\mathrm{M}}^{2}}}} \right]. $$
(23)
According to (22) and (23), the achievable rate region is
$${} {\left\{ \begin{array}{l} R_{{\text{UL}}}^{{\text{DDF}}} < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right];\\ R_{{\text{DL}}}^{{\text{DDF}}} < \min \left\{ {\gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{2}}}} \right],\gamma \left[ {\frac{{{{\left({\sqrt {{P_{\mathrm{B}}}} {h_{{\text{BM}}}} + \sqrt {{P_{\mathrm{R}}}} {h_{{\text{MR}}}}} \right)}^{2}}}}{{\sigma_{\mathrm{M}}^{2}}}} \right]} \right\};\\ [8pt] R_{{\text{UL}}}^{{\text{DDF}}} + R_{{\text{DL}}}^{{\text{DDF}}} = R_{{\text{UL}}}^{{\text{DDF}}}[1] + R_{{\text{DL}}}^{{\text{DDF}}}[1]. \end{array} \right.} $$
(24)
The sum-capacity of the first phase can be expressed in three cases according to the value of η
inf. Figure 5 shows the relationship between the upper bound of Z-channel in (22) and the upper bound of (23) for the three cases of Z-channel. By taking the minimum upper bound for each case, the sum-capacity C
DDF can be expressed as
$${} {C^{{\text{DDF}}}} = \left\{\begin{array}{ll} {\gamma \left({\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} \right) + \min \left\{ {\gamma \left({\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}} + {P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}} \right)\;,R_{{\text{DL}}}^{{\text{DDF}}}[2]} \right\}}, &\text{if} \;{{\eta_{\inf }} \le 1};\\ {\gamma \left({\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right) + \min \left\{ {\gamma \left({\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}} + {P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}} \right),R_{{\text{DL}}}^{{\text{DDF}}}[2]} \right\}}, &\text{if} \; {1 \le {\eta_{\inf }} \le \sqrt {1 + \frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} };\\ {\gamma \left({\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{\mathrm{2}}}}} \right) + \min \left\{ {\gamma \left({\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} \right),R_{{\text{DL}}}^{{\text{DDF}}}[2]} \right\}}, &\text{if} \; {{\eta_{\inf }} \ge \sqrt {1 + \frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{\mathrm{2}}}}} }; \end{array}\right. $$
(25)
where RDLDDF[2] is given by (23).
Uplink DF mode
In the UDF relaying mode, at the second phase, the RS only decodes the uplink signal, and then re-encodes and forwards it. Meanwhile, the BS transmits another downlink signal.
At the first phase, the RS detects the uplink signal by treating the downlink signal as an additive Gaussian noise; therefore, the uplink data rate should be upper bounded by
$$ R_{{\text{UL}}}^{{\text{UDF}}}[1] < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}} \right]. $$
(26)
Note that the assumption of Gaussian distribution used here is only for capacity analysis, but not a mandatory requirement for practical application of the proposed scheme. At the second phase, the RS forwards the decoded uplink signal, and the MS receives the downlink signal from the BS. As the MS knows the forwarded uplink signal from RS, the MS subtracts it from the received signal, thus the MS’s received signal is interference-free. Therefore, the channel in Fig. 3
b can be transformed to a parallel fading channel.
The BS recovers the uplink message by using the signals received at the first and second phases. The downlink signal is only transmitted and received at the second phase without the assistance of RS. Thus, the uplink and downlink rates are expressed as
$$ {\left\{ \begin{array}{l} R_{{\text{UL}}}^{{\text{UDF}}}[2] < \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right] + \gamma \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{R}}^{2}}}} \right]\\ R_{{\text{DL}}}^{{\text{UDF}}}[2] < \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{2}}}} \right] \end{array} \right.}. $$
(27)
The achievable rate region of UDF mode is
$$ {\left\{ \begin{array}{l} R_{{\text{DL}}}^{{\text{UDF}}} = R_{{\text{DL}}}^{{\text{UDF}}}[2]\\ R_{{\text{UL}}}^{{\text{UDF}}} = \min \left\{ {R_{{\text{UL}}}^{{\text{UDF}}}[1],R_{{\text{UL}}}^{{\text{UDF}}}[2]} \right\} \end{array} \right.}. $$
(28)
The sum-capacity is achieved when both the uplink and downlink data rates are maximized and can be expressed as
$$ \begin{aligned} {C^{{\text{UDF}}}} &= \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{2}}}} \right] + \min \left\{ \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{MR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2} + {P_{\mathrm{B}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}} \right],\right.\\ &\left. \qquad \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right] + \left[ {\frac{{{P_{\mathrm{R}}}{{\left| {{h_{{\text{BR}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{2}}}} \right] \right\}. \end{aligned} $$
(29)
The rate region for UDF mode is shown in Fig. 6, in which point A or D denotes the achievable point with the maximum sum-capacity.
No DF mode
In the NF mode, the BS and MS exchange their information directly without the assistance of RS. The uplink signal is transmitted at the first phase, and the downlink signal is transmitted at the second phase. The sum-capacity is
$$ {C^{{\text{NF}}}} = \gamma \left[ {\frac{{{P_{\mathrm{M}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{B}}^{2}}}} \right] + \gamma \left[ {\frac{{{P_{\mathrm{B}}}{{\left| {{h_{{\text{BM}}}}} \right|}^{2}}}}{{\sigma_{\mathrm{M}}^{2}}}} \right]. $$
(30)