**Optimization of Packet Length for Two Way Relaying with Energy Harvesting**

Ghassan Alnwaimi *,Hatem Boujemaa **, Kamran Arshad ***

(*) King Abdulaziz University, Kingdom of Saudi Arabia

(**) University of Carthage, Sup’Com, COSIM Laboratory, Tunisia

(***) College of Engineering, Ajman University

**Abstract**

In this article, we suggest optimizing packet length for two way relaying with energy harvesting. In the rst transmission phase, two source nodes N_{1} and N_{2} are transmitting data to each others through a selected relay R. In the second phase, the selected relay will amplify the sum of the signals received signals from N_{1} and N_{2}. The selected relay amplies the received signals using the harvested energy from Radio Frequency (RF) signals transmitted by nodes N_{1} and N_{2}. Finally, N_{1} will remove, from the relay’s signal, its own signal to be able to decode the symbol of N_{2}. Similarly, N_{2} will remove, from the relay’s signal, its own signal to be able to decode the symbol of N_{1}. We derive the outage probability, packet error probability and throughput at N_{1} and N_{2}. We also optimize packet length to maximize the throughput at N_{1}or N_{2}.

**Index Terms :**

Cooperative systems, Optimal packet length, Rayleigh fading channels.

**1. Introduction**

In Two-Way Relaying (TWR), two nodes N_{1} and N_{2} simultaneously transmit data to each other using a selected relay [1-5]. The communication process contains two phases. In the first one, N_{1} and N_{2} transmit data to some relays. Each relay will receive the sum of signals transmitted by N_{1} and N_{2}. In the second phase, a selected relay amplifies the received signal. Then, N_{1} will remove, from the relay’s signal, its own signal to be able to decode the symbol of N_{2}. Similarly, N_{2} will remove, from the relay’s signal, its own signal to be able to decode the symbol of N_{1}.

Two way relaying for Multiple Input Multiple Output (MIMO) systems has been considered in [1-5]. Receive and transmit diversity improves the performance of TWR. At the receiver, the best antenna can be selected (Selection Combining SC). The corresponding Signal to Noise Ratio (SNR) is the maximum of SNRs over all antennas. It is also possible to combine the signals of all antennas using Maximum Ratio Combining (MRC). The SNR will be the sum of all SNRs [1-5]. TWR with Energy Harvesting (EH) consists to use the Radio Frequency (RF) signal to charge the battery of nodes [6-10]. Relays with EH capabilities has been studied in [6-10]. In order to enhance the throughput especially at low SNRs, channel coding is required in TWR [11-13]. Secure two way relaying has been suggested in [14-20]. Security aspects of TWR should be studied to avoid data recovery by a malicious node. The main contribution of the paper is to optimize packet length so that the throughput at node N1 or N2 is maximized. In all previous studies, a Fixed Packet Length (FPL) is used [1-20]. This is the first paper to suggest an Optimal Packet Length (OPL) for TWR with Energy Harvesting.

The system model is presented in section 2. Section 3 gives the Cumulative Distribution Function (CDF) of SNR. Section 4 derives the PEP while section 5 gives the expression of OPL. Some numerical results are given in section 6. Conclusions are presented in section 7.

**2. System model**

The system model is shown in Fig. 1. There are two nodes N_{1} and N_{2} communicating information to each other through a relay R. Node N_{1} transmits data to node N_{2} and at the same time node N_{2} is also communicating data to N_{1} through relay R. N_{1} and N_{2} transmit over the same channel.

**Figure 1**. Two way relaying with Energy harvesting.

The frame with duration T is decomposed in three parts :

– The first slot with duration T is dedicated to energy harvesting. Relay R harvests energy from RF signal transmitted by nodes N_{1} and N_{2}.

The harvested energy is written as

where0< < 1 is harvesting duration percentage, Pi (resp. Ei) is the transmit power (resp. symbol energy) of node Ni and hN1R (respectively hN2R) is channel coefficient between nodes N_{1} (respectively N_{2}) and R: p = T=Ts is the number of symbols per frame T. We have EX = TsPX

– During the second time slot with duration (1 − )T=2, N_{1} and N_{2} transmit data to node R over the same channel. This is the multiple access phase. The received signal at R is written as

**where Ei is the transmitted energy per symbol of node i with 1≤ i ≤ 2, xi(j) is the j-th transmitted symbol by node Ni and nR(j) is an Addivite White Gaussian Noise (AWGN)**

with variance N0. A Rayleigh block fading channel is assumed where the channel remains constant over all the time frame with duration T.

– During the third time slot with duration (1 − )T=2, R transmits amplifies the received signal to nodes N_{1} and N_{2}. This is the broadcast phase.

Relay R uses the harvested energy E to amplify the received signal yR(j) to N_{1} and N_{2}: The transmit symbol energy of R is equal to the harvested energy E devided by the number of transmitted symbols during (1−)T=2 seconds i.e. (1−)T=(2Ts) = (1−)p=2 with p = Ts=T :

Using (43), the amplification factor G used by relay R is written as

**2.1** **SNR at node N _{1}**

The received signal at N_{1} is written as

where n_{1}(j) is an AWGN with variance N_{0}. Using (43), we deduce

Node N_{1} removes the self interference,**since it knows the value of symbol x1(j): After removing self interference, we obtain**

The SNR at N_{1} is written as

Using the expression of amplification factor G (45), we deduce

We assume that channels are reciprocal i.e. h_{N1R} = h_{RN1} . By neglecting the term in N^{2}_{ 0} and using (44), the SNR at node N_{1} lower bounded by

This upper bound is tight at high average SNR as the term N2 0 can be neglected. We can write

**2.2 SNR at node N _{2}**

The received signal at N_{2} is written as

where n_{2}(j) is an AWGN with variance N_{0}. Using (43), we

Using the expression of amplification factor G (45), we deduce

**2.3 Two way relaying in the presence of multiple relays**

Fig. 2 shows the principle of TWR in the presence of K relays. The selected relay offers the largest SNR at node N_{1} or N_{2}. When the selected relay maximizes the SNR at node N_{1}, the CDF of SNR is the products of CDF of SNRs of different relays

where Γ^{k}_{1} is the SNR at node N_{1} when relay R_{k} is the active relay. Γ^{k}_{1} is given in (9).

**Figure 2**. Two way relaying with Energy harvesting in the presence of K relays

**3. ****CDF of SNR**

**3.1 CDF of SNR at node N _{1}**

The SNR at node N_{1} is lower bounded by

** 3.2 CDF of SNR at node N _{2}**

The SNR at node N_{2} is lower bounded by

We use (34), to deduce

**4. PEP**

In this section, we derive the expression of the average Packet Error Probability (PEP).The PEP can be tightly upper bounded by [21]

where F_{Γ }(γ) is the Probability Density Function (PDF) of SNR Γ and w_{0} is a waterfall threshold.** **

Equation (13) shows that the PEP for a given instantaneous SNR, γ≤ w_{0}, can be approximated to 1. However, the PEP for a given instantaneous SNR, γ> w_{0} can be approximated to 0 [21].

Hence,

**4.1 PEP for uncoded transmission**

For uncoded M-QAM modulation, we have

Using (45) and (46), the SEP is approximated by

**4.2 PEP with Channel Coding**

** 4.3 Waterfall Threshold**

**5. Optimal Packet Length for TWR**

The average number of attempts of HARQ protocols is equal to

**6. Theoretical and simulation results**

Simulation results were obtained using MATLAB as a simulation environment.

Simulation results were performed by measuring the Packet Error Rate (PER) to deduce the throughput. The packet error rate is the number of erroneous packets/number of transmitted packets. We made simulation until 1000 packets are erroneously received.

Fig. 3 and 4 show the throughput at node *N*1 for α = 1/3, a QPSK modulation for average SNR 10 and 20 dB. The distance between all nodes is equal to 1. We notice that we can maximize the throughput by choosing the packet length. Also, the throughput increases as the number of relays increase due to cooperative diversity. In fact, we always select the relay with the largest SNR. Finally, by comparing Fig. 3 and 4, we observe that packet length should be increased as the average SNR increases. There is good accordance between theoretical and simulation results.

**Figure 3**.Throughput at node N_{1} versus packet length at SNR=10 dB : 64 QAM modualtion.

**Figure 4**.Throughput at node N_{1} versus packet length at SNR=20 dB: 64 QAM modualtion.

Fig. 5 shows that OPL offers higher throughput than Fixed Packet Length (FPL)

as studied in [1-20]. These results correspond to throughput of *N*1 for α = 1*=*3. They were obtained using MATLAB for a 64 QAM modulation. In fact, the proposed optimal packet length allows maximizing the throughput. If the SNR is low, the packet length is decreased. However, at high SNR, we can increase packet length.

**Figure 5**.Throughput at node *N*1 for OPL and FPL :64 QAM modualtion.

Fig. 6 shows the OPL for QPSK, 16 QAM and 64 QAM modulation. We observe that packet length should be increases when we use a small modulation such as QPSK. When 64 QAM modulation is used, packet length should be reduced since the PEP is high. Also packet length should be increased (respectively decreased) at high (respectively low) SNR.

**Figure 6**.Optimal packet length versus SNR.

**7. Conclusion**

In this paper, we suggested enhancing the throughput of Two Way Relaying (TWR) with energy harvesting. We derive the best packet length that yields the largest throughput at node N_{1 }or N_{2}. Our study is valid for energy harvesting systems where the relay harvest energy from RF signals transmitted by source nodes N_{1} and N_{2}. We have shown that the proposed TWR with best packet length offers better throughput than previous studies. Also, the throughput can be enhanced by increasing the number of relays. The proposed optimal packet length can be used in Wireless Sensor Networks (WSN) with two way relaying.

**Appendix A** : We have

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