# Network-Coded Relaying in Multiuser Multicast D2D Network

Authors: Mingjun Dai; Shengli Zhang; Hui Wang; Bin Chen; Xiaohui Lin; Hongwei Liu; Li Zhang
Journal: International Journal of Antennas and Propagation
 (2014)

---

## Abstract

D2D communication trades short-range communication for achieving high communication rate and short communication latency. Relay aided D2D communication can further tackle the problem of intermediate obstacles blocking the communication. In this work, multidevice multicast communication via a layer of parallel relay nodes is considered. Two relaying strategies, respectively, called the conventional relaying (CR) and network-coded relaying (NCR), are proposed. The throughput of these two schemes is analytically derived and evaluated through numerical study. Theoretically, NCR shows advantage over CR in twofold: one is higher throughput and the other is requiring less relay nodes and, hence, consuming less aggregate power. Numerical studies verify the analysis and show that the throughput performance gap between the two schemes increases significantly, actually exponentially with the number of devices.


---

## Body

## 1. Introduction

The main idea of D2D communication is to use short-range communication to trade for high rate communication, short delivery latency, and low aggregate consumption power [1–3]. Sometimes, due to the mobility of devices (devices move out of the communication coverage) or due to the fact that intermediate obstacles are blocking the communication, the communication link may be down intermittently [4]. These facts render relay-aided D2D communication necessary.For relay-aided communication networks [5, 6], there are mainly two categories of strategies adopted by intermediate relay, including physical-layer relaying and network-coded relaying. For simple topology, for example, the point-to-point D2D communication via intermediate relays [7], capacity is given by the physical-layer technique in certain scenarios [8–10]. However, for complex topology, physical-layer technique tends to be unwieldy, while network coding (NC) [11, 12] shows significant advantages. Typical topology examples include interference network [13], multicast multihop network [14], intracell uplink relay network [15], and multiple unicasts networks [16] including three-node Alice-and-Bob relay network, two unicasts X-topology, cross topology, and wheel topology [17–25].The full-duplex relay assumption in the above investigations is not practical [26]. In practice, if “cheap” relay is adopted, namely, half-duplex relay [27], which cannot transmit and receive simultaneously, the advantage of NC shown in the above investigations may be lost due to the orthogonal transmission nature of the multiple transmitters.In this paper, we investigate a more general and practical model by considering the scenario where multiple devices exchange information via a layer of intermediate relays, namely, relay-aided D2D multicasting communication. For this model, we propose two relaying protocols based on the physical-layer and NC technique, respectively. Details of each technique are illustrated below.Physical-Layer Technique.Firstly, for the special case where there is a single relay, the communication stage when all the devices transmit simultaneously can be viewed as multiple access channel (MAC) [28]. After this MAC stage, the relay broadcasts superimposed signals originating from the devices, and the corresponding channel can be viewed as broadcast channel (BC). In the BC stage, power allocation among all the information streams originating from different devices may degrade the system performance. In addition, due to heavy burden on the single relay, multiple cheap relays cooperately share the burden become a preference. One natural solution is that a lot of parallel relays are deployed between the devices, with each relay serving a single device, respectively, a device, a relay that serves it, and the other destination devices form a multicast single relay network. We call this strategy conventional multicast relaying (CMCR). The most desirable feature of this scheme is the simple operation at each relay. However, since each relay only decodes one device’s information while treating the other devices’ signals as interference, the performance of CMCR is interference-limited [29].NC Technique. After the MAC stage, each relay firstly applies NC operation on some information flows originating from different devices and then multicasts the resultant information flow to all the devices. Each device can employ the idea of side information-aided decoding [30]. More specifically, each device performs decoding utilizing the message originating from itself. We call this strategy network-coded multicast relaying (NCMCR). The most desirable feature of NCMCR is that the interference can be reduced to some extent, since more devices’ signals are decoded at each relay and hence less devices’ signals are treated as interference. Besides, this does not involve power allocation among the information streams at each relay node, which also embodies NC’s advantage over the physical relaying technique.


## 2. System Model

Refer to Figure1. Consider M devices, denoted as Ui for i={1,2,…,M}≜M, exchanging information over a layer of N parallel relays, denoted as Ri for i={1,2,…,N}≜N. Define device terminal set U≜{U1,U2,…,UM} and relay node set R≜{R1,R2,…,RN}. Device Ui multicasts information mi to the other devices U/{Ui}, i∈M. The devices are assumed to be so far away that the wireless link between them can be neglected. We assume that the communication between any pair of devices experiences two hops, that is, through certain relay nodes. Each node in the network is assumed to have one single antenna and operates in half-duplex mode. Node i∈U∪R is subject to power constraint Pi. For simplicity, we consider additive white Gaussian noise (AWGN) at the receiver.Figure 1
The general model.The communication between any pair of devices is performed in two time slots. During the first time slot, the devices simultaneously distribute their data while the relays listen. During the second time slot, the relays forward the messages to the devices.LetXi[m], Yi[m], and wi[m] be, respectively, the transmitted symbol from node i, the received symbol, and the thermal noise at node i, at time m, respectively, i∈U∪R. The first hop’s input-output relationship is represented by the following formula:
(1)YRi[m]=∑j∈U‍XUj[m]+wi[m],i∈N,j∈M,
which is subject to
(2)Var⁡{XUj[m]}≤PUj.
For simplicity, we assume that Pi=P and wi[m]~CN(0,σ2) for all i∈U∪R. Define signal-to-noise ratio (SNR) Γ≜P/σ2. The second hop’s input-output relationship depends on the selected transmission scheme and is hence detailed in the following sections. Define Rmi and Rmi,mj as the data rate of information flow originating from source Ui and the data rate of information flows originating from sources Ui and Uj, respectively, i,j∈U, i≠j.Remark 1.
Throughout this paper, we assume that matched-filter receiver is used; hence, the achievable rate in each hop can be denoted by its capacity, and we letC(x)≜log⁡2(1+x).


## 3. Proposed Transmission Protocols

For easy understanding, we first consider the special case where there are three devices in Section3.1. We then generalize the above results to an arbitrary number of devices together with mathematical analysis, that is, to M>3, in Section 3.2.


### 3.1. Proposed Schemes for Three Devices

We describe the conventional method, the proposed method, and the refined proposed method, respectively.


#### 3.1.1. Conventional Multicast Relaying (CMCR)

Refer to Figure2. Relay Ri adopts decode-and-forward (DF) strategy to relay device Ui’s signal to the other devices, i∈{1,2,3}≜U3. In the following, we derive the achievable rate of each device. Note that there are totally two stages of transmission and the achievable rate is the minimum of these two stages.Figure 2
Conventional relaying.In the first stage, since relayRi only decodes the message originating from device Ui, the signals originating from other devices’ are treated as interference, i∈U3. In this case, the achievable rate region is
(3)Rmi≤C(PUiσ2+∑j∈U3,j≠iPUj),
which is simplified to
(4)Rmi≤C(11/Γ+2).In the second stage, deviceUi intends to decode the messages from relays Rj’s for j∈U3/{i}. This stage can be viewed as a two-user MAC by treating the signal from relay Ri as interference. In this case, the achievable rate region is
(5)Rmj≤C(PRjσ2+PRi),j∈U3/{i},Rmj+Rmk≤C(PRj+PRkσ2+PRi),j,k∈U3/{i},j≠k.
From (5), we obtain that the achievable rate region of the message rate originating from device i in the second stage is
(6)Rmi≤12C(21/Γ+1).Combining (4) and (6), we obtain that the achievable rate region of each device by CMCR is
(7)RmiCMCR≤min⁡(C(11/Γ+2),12C(21/Γ+1))(8)=C(11/Γ+2).


#### 3.1.2. Network-Coded Multicast Relaying (NCMCR)

Refer to Figure3. Device Ui selects relays Rj for all j∈U3∖{i} as the intermediate relays to forward its message to the other devices. Relay Rj performs the following procedures. Firstly, it decodes the messages from devices Ui for i∈U3∖{j}. Afterwards, it performs exclusive OR (XOR) operation on the decoded information bits. Finally, it encodes the resultant information bits into new codeword and sends the resultant codeword out. We derive the achieved rate region in the following.Figure 3
Network-coded relaying.In the first stage, relayRi needs to decode the signals from devices Uj’s for j∈U3∖{i}. This stage can be viewed as a two-user MAC. The achievable rate region is
(9)Rmj≤C(PUjσ2+PUi),j∈U3/{i},Rmj+Rmk≤C(PUj+PUkσ2+PUi),j,k∈U3/{i},j≠k.
From (9), we obtain the achievable rate region of device Ui in the first stage as
(10)Rmi≤12C(21/Γ+1).In the second stage, since deviceUi intends to decode the messages from relays Rj’s for j∈U3∖{i}, the channel can also be viewed as a two-user MAC. The achievable rate region at decoder Ui is
(11)Rmi,mj≤C(PRjσ2+PRi),j∈U3∖{i},Rmi,mj+Rmi,mk≤C(PRj+PRkσ2+PRi),j,k∈U3∖{i},j≠k.
Note that
(12)Rmi,mj=min⁡(Rmi,Rmj),i,j∈U3,i≠j.
From (11) and (12), we obtain that the achievable rate region of device Ui in the second stage is
(13)Rmi≤12C(21/Γ+1).Combining (10) and (13), we obtain that the achievable rate region of device Ui by NCMCR is
(14)RmiNCMCR≤12C(21/Γ+1).


#### 3.1.3. Refined NCMCR

Combining (8) and (14), we can obtain
(15)RmiNCMCR>RmiCMCR,
which shows the advantage of NCMCR over CMCR. However, for more than 3 devices, the number of relays needed by intuitively applying NCMCR should be CM2, while CMCR requires only M relays.In the following, we propose a method based on NCMCR to reduce the number of relays required to be even less than the number of devices. Without loss of generality, we first illustrate the case forM=3. We remove relay R1 and keep the operations performed on the other relay nodes the same as those described in Section 3.1.2 (refer to Figure 4). Obviously, the decoding at device U1 can be made the same as before. Now, let us consider the decoding operation at devices U2 and U3. Without loss of generality, we consider node U2 only. It intends to decode m1 and m3. Since m2 is available to U2 and hence can be viewed as side information [31], node U2 first performs XOR operation on m2 with m1⊕m2 which is forwarded by relay R2, obtaining m1. Afterwards, U2 performs XOR operation again on m1 and m1⊕m3 which is forwarded by relay R3, obtaining m3. We name this protocol as refined network-coded multicast relaying (RNCMCR). In the following, we derive the achievable rate region of RNCMCR.Figure 4
Two relays.In the first stage, consider relayR2. It needs to decode the message from devices U1 and U2, respectively. The channel can be viewed as a two-user MAC. The achievable rate region is, therefore,
(16)Rmj≤C(PUjσ2+PU3),j∈{1,2},Rm1+Rm2≤C(PU1+PU2σ2+PU3).
Similarly, for relay R3 in the first stage, we have
(17)Rmj≤C(PUjσ2+PU2),j∈{1,3},Rm1+Rm3≤C(PU1+PU3σ2+PU2).
We then obtain that the achievable rate region of device Ui in the first stage is
(18)Rmi≤12C(21/Γ+1).In the second stage, for deviceU2, we have
(19)Rm1,m2≤C(PR2σ2)=C(Γ),Rm1,m3≤C(PR3σ2)=C(Γ),Rm1,m2+Rm1,m3≤C(PR2+PR3σ2)=C(2Γ).
We then obtain that the achievable rate region of device Ui in the second stage is
(20)Rmi≤12C(2Γ).Combining (18) with (20), we obtain that the achievable rate region of device Ui by RNCMCR is
(21)RmiRNCMCR≤12C(21/Γ+1),
which is the same as (14). It naturally indicates a conjecture that, for M devices, M-1 relays are enough for RNCMCR without performance penalty with respect to that composed of CM2 relays. In the next subsection, we analytically verify this conjecture and compare RNCMCR with CMCR.


### 3.2. Analysis for General Number of Devices

We first derive the achievable rate region obtained by CMCR and RNCMCR, respectively. We then compare these two schemes.


#### 3.2.1. CMCR

Following the above derivation steps, we can obtain that the achievable rate region by CMCR forM devices and M relays in each stage is as follows.Stage  1. Consider
(22)R≤C(1t+M-1),
where t≜1/Γ.Stage  2. Consider
(23)R≤1M-1C(M-1t+1).By jointly considering these two stages, we obtain(24)RCMCR≤min⁡(C(1t+M-1),1M-1C(M-1t+1))(25)=C(1t+M-1),
where the proof of (25) is given in The Appendix.


#### 3.2.2. RNCMCR

Following the above derivation steps, we can obtain that the achievable rate region by RNCMCR forM devices and M-1 relays in each stage as follows.Stage  1. Consider
(26)R≤1M-1C(M-1t+1).Stage  2. Consider
(27)R≤1M-1C(M-1t).By jointly considering these two stages, we obtain(28)RRNCMCR≤min⁡(1M-1C(M-1t+1),1M-1C(M-1t))(29)=1M-1C(M-1t+1).


#### 3.2.3. Comparison

Comparing (25) with (29), we obtain
(30)RRNCMCR>RCMCR.
It reveals the fact that RNCMCR outperforms CMCR twofold. On one hand, it uses less relays and hence needs less aggregate power budget. On the other hand, according to (30), RNCMCR outperforms CMCR in terms of achievable rate.


## 3.1. Proposed Schemes for Three Devices

We describe the conventional method, the proposed method, and the refined proposed method, respectively.


### 3.1.1. Conventional Multicast Relaying (CMCR)

Refer to Figure2. Relay Ri adopts decode-and-forward (DF) strategy to relay device Ui’s signal to the other devices, i∈{1,2,3}≜U3. In the following, we derive the achievable rate of each device. Note that there are totally two stages of transmission and the achievable rate is the minimum of these two stages.Figure 2
Conventional relaying.In the first stage, since relayRi only decodes the message originating from device Ui, the signals originating from other devices’ are treated as interference, i∈U3. In this case, the achievable rate region is
(3)Rmi≤C(PUiσ2+∑j∈U3,j≠iPUj),
which is simplified to
(4)Rmi≤C(11/Γ+2).In the second stage, deviceUi intends to decode the messages from relays Rj’s for j∈U3/{i}. This stage can be viewed as a two-user MAC by treating the signal from relay Ri as interference. In this case, the achievable rate region is
(5)Rmj≤C(PRjσ2+PRi),j∈U3/{i},Rmj+Rmk≤C(PRj+PRkσ2+PRi),j,k∈U3/{i},j≠k.
From (5), we obtain that the achievable rate region of the message rate originating from device i in the second stage is
(6)Rmi≤12C(21/Γ+1).Combining (4) and (6), we obtain that the achievable rate region of each device by CMCR is
(7)RmiCMCR≤min⁡(C(11/Γ+2),12C(21/Γ+1))(8)=C(11/Γ+2).


### 3.1.2. Network-Coded Multicast Relaying (NCMCR)

Refer to Figure3. Device Ui selects relays Rj for all j∈U3∖{i} as the intermediate relays to forward its message to the other devices. Relay Rj performs the following procedures. Firstly, it decodes the messages from devices Ui for i∈U3∖{j}. Afterwards, it performs exclusive OR (XOR) operation on the decoded information bits. Finally, it encodes the resultant information bits into new codeword and sends the resultant codeword out. We derive the achieved rate region in the following.Figure 3
Network-coded relaying.In the first stage, relayRi needs to decode the signals from devices Uj’s for j∈U3∖{i}. This stage can be viewed as a two-user MAC. The achievable rate region is
(9)Rmj≤C(PUjσ2+PUi),j∈U3/{i},Rmj+Rmk≤C(PUj+PUkσ2+PUi),j,k∈U3/{i},j≠k.
From (9), we obtain the achievable rate region of device Ui in the first stage as
(10)Rmi≤12C(21/Γ+1).In the second stage, since deviceUi intends to decode the messages from relays Rj’s for j∈U3∖{i}, the channel can also be viewed as a two-user MAC. The achievable rate region at decoder Ui is
(11)Rmi,mj≤C(PRjσ2+PRi),j∈U3∖{i},Rmi,mj+Rmi,mk≤C(PRj+PRkσ2+PRi),j,k∈U3∖{i},j≠k.
Note that
(12)Rmi,mj=min⁡(Rmi,Rmj),i,j∈U3,i≠j.
From (11) and (12), we obtain that the achievable rate region of device Ui in the second stage is
(13)Rmi≤12C(21/Γ+1).Combining (10) and (13), we obtain that the achievable rate region of device Ui by NCMCR is
(14)RmiNCMCR≤12C(21/Γ+1).


### 3.1.3. Refined NCMCR

Combining (8) and (14), we can obtain
(15)RmiNCMCR>RmiCMCR,
which shows the advantage of NCMCR over CMCR. However, for more than 3 devices, the number of relays needed by intuitively applying NCMCR should be CM2, while CMCR requires only M relays.In the following, we propose a method based on NCMCR to reduce the number of relays required to be even less than the number of devices. Without loss of generality, we first illustrate the case forM=3. We remove relay R1 and keep the operations performed on the other relay nodes the same as those described in Section 3.1.2 (refer to Figure 4). Obviously, the decoding at device U1 can be made the same as before. Now, let us consider the decoding operation at devices U2 and U3. Without loss of generality, we consider node U2 only. It intends to decode m1 and m3. Since m2 is available to U2 and hence can be viewed as side information [31], node U2 first performs XOR operation on m2 with m1⊕m2 which is forwarded by relay R2, obtaining m1. Afterwards, U2 performs XOR operation again on m1 and m1⊕m3 which is forwarded by relay R3, obtaining m3. We name this protocol as refined network-coded multicast relaying (RNCMCR). In the following, we derive the achievable rate region of RNCMCR.Figure 4
Two relays.In the first stage, consider relayR2. It needs to decode the message from devices U1 and U2, respectively. The channel can be viewed as a two-user MAC. The achievable rate region is, therefore,
(16)Rmj≤C(PUjσ2+PU3),j∈{1,2},Rm1+Rm2≤C(PU1+PU2σ2+PU3).
Similarly, for relay R3 in the first stage, we have
(17)Rmj≤C(PUjσ2+PU2),j∈{1,3},Rm1+Rm3≤C(PU1+PU3σ2+PU2).
We then obtain that the achievable rate region of device Ui in the first stage is
(18)Rmi≤12C(21/Γ+1).In the second stage, for deviceU2, we have
(19)Rm1,m2≤C(PR2σ2)=C(Γ),Rm1,m3≤C(PR3σ2)=C(Γ),Rm1,m2+Rm1,m3≤C(PR2+PR3σ2)=C(2Γ).
We then obtain that the achievable rate region of device Ui in the second stage is
(20)Rmi≤12C(2Γ).Combining (18) with (20), we obtain that the achievable rate region of device Ui by RNCMCR is
(21)RmiRNCMCR≤12C(21/Γ+1),
which is the same as (14). It naturally indicates a conjecture that, for M devices, M-1 relays are enough for RNCMCR without performance penalty with respect to that composed of CM2 relays. In the next subsection, we analytically verify this conjecture and compare RNCMCR with CMCR.


## 3.1.1. Conventional Multicast Relaying (CMCR)

Refer to Figure2. Relay Ri adopts decode-and-forward (DF) strategy to relay device Ui’s signal to the other devices, i∈{1,2,3}≜U3. In the following, we derive the achievable rate of each device. Note that there are totally two stages of transmission and the achievable rate is the minimum of these two stages.Figure 2
Conventional relaying.In the first stage, since relayRi only decodes the message originating from device Ui, the signals originating from other devices’ are treated as interference, i∈U3. In this case, the achievable rate region is
(3)Rmi≤C(PUiσ2+∑j∈U3,j≠iPUj),
which is simplified to
(4)Rmi≤C(11/Γ+2).In the second stage, deviceUi intends to decode the messages from relays Rj’s for j∈U3/{i}. This stage can be viewed as a two-user MAC by treating the signal from relay Ri as interference. In this case, the achievable rate region is
(5)Rmj≤C(PRjσ2+PRi),j∈U3/{i},Rmj+Rmk≤C(PRj+PRkσ2+PRi),j,k∈U3/{i},j≠k.
From (5), we obtain that the achievable rate region of the message rate originating from device i in the second stage is
(6)Rmi≤12C(21/Γ+1).Combining (4) and (6), we obtain that the achievable rate region of each device by CMCR is
(7)RmiCMCR≤min⁡(C(11/Γ+2),12C(21/Γ+1))(8)=C(11/Γ+2).


## 3.1.2. Network-Coded Multicast Relaying (NCMCR)

Refer to Figure3. Device Ui selects relays Rj for all j∈U3∖{i} as the intermediate relays to forward its message to the other devices. Relay Rj performs the following procedures. Firstly, it decodes the messages from devices Ui for i∈U3∖{j}. Afterwards, it performs exclusive OR (XOR) operation on the decoded information bits. Finally, it encodes the resultant information bits into new codeword and sends the resultant codeword out. We derive the achieved rate region in the following.Figure 3
Network-coded relaying.In the first stage, relayRi needs to decode the signals from devices Uj’s for j∈U3∖{i}. This stage can be viewed as a two-user MAC. The achievable rate region is
(9)Rmj≤C(PUjσ2+PUi),j∈U3/{i},Rmj+Rmk≤C(PUj+PUkσ2+PUi),j,k∈U3/{i},j≠k.
From (9), we obtain the achievable rate region of device Ui in the first stage as
(10)Rmi≤12C(21/Γ+1).In the second stage, since deviceUi intends to decode the messages from relays Rj’s for j∈U3∖{i}, the channel can also be viewed as a two-user MAC. The achievable rate region at decoder Ui is
(11)Rmi,mj≤C(PRjσ2+PRi),j∈U3∖{i},Rmi,mj+Rmi,mk≤C(PRj+PRkσ2+PRi),j,k∈U3∖{i},j≠k.
Note that
(12)Rmi,mj=min⁡(Rmi,Rmj),i,j∈U3,i≠j.
From (11) and (12), we obtain that the achievable rate region of device Ui in the second stage is
(13)Rmi≤12C(21/Γ+1).Combining (10) and (13), we obtain that the achievable rate region of device Ui by NCMCR is
(14)RmiNCMCR≤12C(21/Γ+1).


## 3.1.3. Refined NCMCR

Combining (8) and (14), we can obtain
(15)RmiNCMCR>RmiCMCR,
which shows the advantage of NCMCR over CMCR. However, for more than 3 devices, the number of relays needed by intuitively applying NCMCR should be CM2, while CMCR requires only M relays.In the following, we propose a method based on NCMCR to reduce the number of relays required to be even less than the number of devices. Without loss of generality, we first illustrate the case forM=3. We remove relay R1 and keep the operations performed on the other relay nodes the same as those described in Section 3.1.2 (refer to Figure 4). Obviously, the decoding at device U1 can be made the same as before. Now, let us consider the decoding operation at devices U2 and U3. Without loss of generality, we consider node U2 only. It intends to decode m1 and m3. Since m2 is available to U2 and hence can be viewed as side information [31], node U2 first performs XOR operation on m2 with m1⊕m2 which is forwarded by relay R2, obtaining m1. Afterwards, U2 performs XOR operation again on m1 and m1⊕m3 which is forwarded by relay R3, obtaining m3. We name this protocol as refined network-coded multicast relaying (RNCMCR). In the following, we derive the achievable rate region of RNCMCR.Figure 4
Two relays.In the first stage, consider relayR2. It needs to decode the message from devices U1 and U2, respectively. The channel can be viewed as a two-user MAC. The achievable rate region is, therefore,
(16)Rmj≤C(PUjσ2+PU3),j∈{1,2},Rm1+Rm2≤C(PU1+PU2σ2+PU3).
Similarly, for relay R3 in the first stage, we have
(17)Rmj≤C(PUjσ2+PU2),j∈{1,3},Rm1+Rm3≤C(PU1+PU3σ2+PU2).
We then obtain that the achievable rate region of device Ui in the first stage is
(18)Rmi≤12C(21/Γ+1).In the second stage, for deviceU2, we have
(19)Rm1,m2≤C(PR2σ2)=C(Γ),Rm1,m3≤C(PR3σ2)=C(Γ),Rm1,m2+Rm1,m3≤C(PR2+PR3σ2)=C(2Γ).
We then obtain that the achievable rate region of device Ui in the second stage is
(20)Rmi≤12C(2Γ).Combining (18) with (20), we obtain that the achievable rate region of device Ui by RNCMCR is
(21)RmiRNCMCR≤12C(21/Γ+1),
which is the same as (14). It naturally indicates a conjecture that, for M devices, M-1 relays are enough for RNCMCR without performance penalty with respect to that composed of CM2 relays. In the next subsection, we analytically verify this conjecture and compare RNCMCR with CMCR.


## 3.2. Analysis for General Number of Devices

We first derive the achievable rate region obtained by CMCR and RNCMCR, respectively. We then compare these two schemes.


### 3.2.1. CMCR

Following the above derivation steps, we can obtain that the achievable rate region by CMCR forM devices and M relays in each stage is as follows.Stage  1. Consider
(22)R≤C(1t+M-1),
where t≜1/Γ.Stage  2. Consider
(23)R≤1M-1C(M-1t+1).By jointly considering these two stages, we obtain(24)RCMCR≤min⁡(C(1t+M-1),1M-1C(M-1t+1))(25)=C(1t+M-1),
where the proof of (25) is given in The Appendix.


### 3.2.2. RNCMCR

Following the above derivation steps, we can obtain that the achievable rate region by RNCMCR forM devices and M-1 relays in each stage as follows.Stage  1. Consider
(26)R≤1M-1C(M-1t+1).Stage  2. Consider
(27)R≤1M-1C(M-1t).By jointly considering these two stages, we obtain(28)RRNCMCR≤min⁡(1M-1C(M-1t+1),1M-1C(M-1t))(29)=1M-1C(M-1t+1).


### 3.2.3. Comparison

Comparing (25) with (29), we obtain
(30)RRNCMCR>RCMCR.
It reveals the fact that RNCMCR outperforms CMCR twofold. On one hand, it uses less relays and hence needs less aggregate power budget. On the other hand, according to (30), RNCMCR outperforms CMCR in terms of achievable rate.


## 3.2.1. CMCR

Following the above derivation steps, we can obtain that the achievable rate region by CMCR forM devices and M relays in each stage is as follows.Stage  1. Consider
(22)R≤C(1t+M-1),
where t≜1/Γ.Stage  2. Consider
(23)R≤1M-1C(M-1t+1).By jointly considering these two stages, we obtain(24)RCMCR≤min⁡(C(1t+M-1),1M-1C(M-1t+1))(25)=C(1t+M-1),
where the proof of (25) is given in The Appendix.


## 3.2.2. RNCMCR

Following the above derivation steps, we can obtain that the achievable rate region by RNCMCR forM devices and M-1 relays in each stage as follows.Stage  1. Consider
(26)R≤1M-1C(M-1t+1).Stage  2. Consider
(27)R≤1M-1C(M-1t).By jointly considering these two stages, we obtain(28)RRNCMCR≤min⁡(1M-1C(M-1t+1),1M-1C(M-1t))(29)=1M-1C(M-1t+1).


## 3.2.3. Comparison

Comparing (25) with (29), we obtain
(30)RRNCMCR>RCMCR.
It reveals the fact that RNCMCR outperforms CMCR twofold. On one hand, it uses less relays and hence needs less aggregate power budget. On the other hand, according to (30), RNCMCR outperforms CMCR in terms of achievable rate.


## 4.  Numerical Study

In this section, we compare the achievable throughput of different schemes, where throughput is defined as the sum rate of all devices in the network. We setσ=1.ForM=3, the achievable throughput of NCMCR and CMCR is plotted in Figure 5. We can see that the performance improvement in the high SNR region is around 33%.Figure 5
Throughput comparison at different SNRs,M=3.In Figure6, we consider the case where M=100. We can see that the performance improvement is much more significant than the M=3 case.Figure 6
Throughput comparison at different SNRs,M=100.To show the advantage under different number of devices, we plot Figure7 with the horizontal axis representing the number of devices. We can see that the performance advantage increases linearly with respect to the logarithm of M. This indicates that our proposed scheme scales well with the network size (in terms of the number of devices).Figure 7
Throughput comparison with respect toM, Γ=50 dB.


## 5. Conclusion

The throughput of a D2D communication aided by a multidevice multicast two-hop Gaussian parallel relay network is analyzed. Both conventional relaying (CR) strategy and network-coded relaying (NCR) strategy together with refined version are proposed. Their achievable rates are evaluated theoretically and numerically. Comparison results show that NCR outperforms CR; the advantage increases linearly with respect to the logarithm of the number of devices, which indicates that the proposed scheme scales well with the network size.



---

*Source: 589794-2014-08-12.xml*