# Effects of Co-Channel Interference on Sum-Rate Based Relay Selection Method for A Dual-Hop Multiple Full-Duplex Two-Way Wireless Relaying Networks.

Abstract: This paper investigates the co-channel interference
effects on sum-rate based relay selection strategy. The investigation
considers a dual-hop multiple full-duplex bi-directional wireless relay
in the system model. According to analytical, asymptotic and the
Monte-Carlo simulation results, sum-rate based relay section strategy
outperforms the max-min based strategy. Results also show that the
co-channel interference degrades the achievable diversity order from N
to 0 and also causes system coding gain losses in high signal-to-noise
ratio regimes. On the other hand, the co-channel interference also
severely affects the system achievable rate performance and degrades the
performance curves at high SNR regimes.

Keywords: Full-Duplex Relay, Co-Channel Interference, Sum-rate.

1. Introduction

Cooperative communication has got a lot attention in the cellular communications in recent years [1-9]. This is because, communications with the aid of wireless repeaters provide additional advantages on information exchange process. Forefront of these advantages can be improvement on the signal fading effects. Rapid increases of number of mobile users also increase the number of wireless repeaters in cellular coverage area. N available active wireless repeaters increases the system overhead severely. This process can be minimized with the help of a prudent relay selection process. Prudent relay selection strategy also provides an information exchange process in low signal-to-noise ratio (SNR) regimes. The relay terminal's operating modes, which are half-duplex (HD) or full-duplex (FD), also affect this process. In the case that the relay terminal operates in HD mode, the information exchange process can be completed in two phases, which are multiple access and broadcast phase. However, in the case that the relay terminal operates in FD mode, the information exchange process can be completed in a single phase. FD wireless relays, besides doubling the capacity also suffers from the loop-interference (LI), caused by transmitting and receiving the information exchange process at the same time period. The LI can be minimize with an efficient antenna design and/or advanced signal processing techniques. Co-channel interference (CCI) is another factor, which affects the system performance. In the case that the CCI is high, this severely affects the system performance and does not provide a reliable communications.

Literature contains several types of studies in this area and forefront of these studies can be summarized as follows: [10] considers that source terminal communicates with the destination terminal with the aid of a selected FD relay among N available amplify-and-forward (AF) one-way relay (OWR) terminal. [10] Investigates a various FD relay selection strategies. [10] also considers that relay terminals have two antennas in such a system model. [11] also considers a similar system model structure as [10]. Differently from [10], [11] considers decode-and-forward (DF) based relay terminal and also considers Nakagami-m fading environment in such a system model. [11] also considers that there is a direct-link between source and destination terminals. In addition, [11] proposes two relay selection strategy. As distinct from [10, 11], [12] considers a different system model structure, where M source terminal communicates with the destination terminal via N available AF based FD relay terminal. [12] also considers a joint source and relay selection strategy in such a system model. In addition, [12] also assumes that there is a direct-link between source and destination terminals. [12]'s joint source and relay selection strategy is based on instantaneous SNR value between source and destination terminals. In addition, relay selection strategy is based on classical max-min (MM) strategy. [13] and its extended version [14], consider a similar system model structure as [10, 11]. Differently from [10, 11], [13, 14] consider that relay terminals posses N receive and transmit antennas. In addition, [13, 14] also investigates the effects of the joint relay and antenna selection technique on the system performance of such a system model. [13, 14] considers Rayleigh fading environment for the performance analysis of such a system model. [15] considers a similar system model structure as [10]. Differently from [10], [15] assumes that the FD relays operate in two-way. In addition, [15] investigates the effective SNR based best relay selection strategy on the system performance of such a system model. [16, 17] considers that source and destination terminals conduct information exchange with the help of FD based relay terminals in the system model. In addition, [16, 17] also considers that CCI affects the system model structure and investigates the effects of CCI on such a system model. [17] considers a Rayleigh fading environment, while [16] considers Nakagami-m fading environment.

At a glance to the aforementioned literature, in order to minimize the system overhead and provides an information exchange at low SNR regimes, most of the studies considers different types of FD relay selection strategies. Our earlier study [18], considers sum-rate based opportunistic relay selection strategy. [19-21] also employs sum-rate based strategy for the user-pair selection process. [22] and [23] considers the same system structure and the selection strategy as [18] but considers more realistic scenarios that the channel state information is imperfect because of the channel estimation error and feedback delay, respectively. [24] extends [18] by taking into consideration the CCI effects on such a system model. This paper extends [24] by providing the achievable rate analytical derivation and the diversity order analysis of such a system model.

The remainder of this paper is organized as follows. Section II describes the system model and channel statistics. Section III presents the performance analysis results including both exact and high SNR situations for the SR-ORS and MM methods. Section IV provides numerical results and the paper concludes in section V.

Notations: This paper uses [f.sub.h] (.) and [F.sub.h] (.) to denote the probability density function (PDF) and cumulative distribution function (CDF) of a random variable (RV) h, respectively. The operator E[.] stands for expectation, while Pr(.) denotes probability.

2. System Model and Channel Statistics

Figure 1 depicts a dual-hop FD TWR system structure. Here, [S.sub.1] and [S.sub.2] conduct information exchange with the help of N available FD wireless relaying. Since the relay terminal operates in FD mode, the information exchange process can be completed in a single phase.

This paper also considers that each terminal has an omnidirectional antenna. [h.sub.j], [g.sub.j], [f.sub.j], [k.sub.j] and [m.sub.j], [[for all].sub.j] = 1,..., N, represents the channel impulse response between [S.sub.1] [right arrow] R, [S.sub.2] [right arrow] R, [a.sub.j] [right arrow] [S.sub.1], [b.sub.j] [right arrow] R and [c.sub.j] [right arrow] [S.sub.2], respectively. [h.sub.j] is a complex Gaussian RVs. with zero mean and variance [mathematical expression not reproducible]. Likewise, [mathematical expression not reproducible], [mathematical expression not reproducible], [mathematical expression not reproducible] and [mathematical expression not reproducible]. d, [e.sub.j] and p, [mathematical expression not reproducible], [mathematical expression not reproducible] and [mathematical expression not reproducible], represent the loop-interferences at [S.sub.1], R and [S.sub.2], respectively. Amplitudes of all channels are distributed according to Rayleigh distribution.

The received signal at relay terminal can be written as

[mathematical expression not reproducible] (1)

Here, [P.sub.s], [P.sub.r] and [P.sub.j] are cooresponding transmit powers of mobile, relay terminals and [j.sup.th] interference terminal respectively. x, y and [b.sub.j] are corresponding transmit information, with a unit energy E[[| x |.sup.2]] = 1, E[[| y |.sup.2]] = 1 and E[[| b j |.sup.2]] = 1, of [S.sub.1], [S.sub.2] and [j.sup.th] interference terminal, respectively. [mathematical expression not reproducible] is the additive white Gaussian noise (AWGN) with zero mean and variance [[sigma].sup.2] (i.e. [mathematical expression not reproducible]). Since the relay terminal operates in AF mode, the G amplification factor can be calculated as

[mathematical expression not reproducible] (2)

After the amplification process, the received signals at [S.sub.1] and [S.sub.2] can be calculated as (3) and (4), respectively.

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

By using (3) and (4), the received signal-to-interference noise ratios (SINRs) at [S.sub.1] and [S.sub.2] can be calculated as

[mathematical expression not reproducible] (5)

[mathematical expression not reproducible] (6)

Substituting the [G.sub.j], (2), into (5) and (6) and also doing some mathematical manipulations, [mathematical expression not reproducible] and [mathematical expression not reproducible] can be represent as in (7) and (8), respectively.

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

Here,

[mathematical expression not reproducible]

(7) and (8) can be re-written as in (9) and (10), respectively.

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

3. Performance Analysis

This subsection investigates the system performance by using the outage probability and achievable rate performance metrics.

3.1. The Outage Probability

The outage probability defines as the probability that the achievable capacity cannot support the pre-defined target rate, R in bps/Hz. In other words, the outage probability is the CDF of received SNR/SINR evaluated at target threshold rate, [[gamma].sub.th] [26]. In this regard, (9) and (10), with the help of [mathematical expression not reproducible], can be upper-bounded as in (11) and (12), respectively.

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

Here, [mathematical expression not reproducible]

With the help of (11) and (12), the end-to-end (e2e) SINR can be calculated as

[mathematical expression not reproducible] (13)

Since the best relay selection strategy is based on the SR expression, (14), by using the logarithm properties and with the help of [18,20,27], (14) can be approximated to min(X, Y), (15), expression.

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

The CDF expression of [SR.sup.FD] is given by the following proposition.

Proposition 1: [mathematical expression not reproducible] can be calculated as

[mathematical expression not reproducible] (16)

Proof: See Appendix A.

Here, [[OMEGA].sub.h], [[OMEGA].sub.g], [mathematical expression not reproducible] [mathematical expression not reproducible], [mathematical expression not reproducible], [mathematical expression not reproducible], [mathematical expression not reproducible],[[OMEGA].sub.d] and [[OMEGA].sup.p] are mean of [|h|.sup.2], [|g|.sup.2], [[|k.sub.j]|.sup.2], [[|f.sub.j]|.sup.2] [[|m.sub.j].sup.2], [[|e.sub.j]|.sup.2], [[|c.sub.j]|.sup.2], [|d|.sup.2] and [|p|.sup.2], respectively. If the MM based relay selection strategy is used, the target threshold rate can be calculated as: [mathematical expression not reproducible].

3.2. Asymptotic Analysis

By using the Taylor series expansions exp term can be written as: exp(x) = 1 + x for x [right arrow] 0 [28]. By doing variable changes in (16), the asymptotic CDF expression can be calculated as

[mathematical expression not reproducible] (17)

3.3. Achievable Rate Analysis

This subsection now focuses on achievable rate analysis of such a system model. By using the Jensens' inequality the upper-bounded achievable rate expression can be formulated with the help of [26, Eq. (25)] as

[mathematical expression not reproducible] (18)

By using [26, Eq. (26)], [mathematical expression not reproducible] can be formulated as

[mathematical expression not reproducible] (19)

In order to continue analysis, (19) requires the [mathematical expression not reproducible] expression. [mathematical expression not reproducible] can be calculated as in following proposition.

Proposition 2: [mathematical expression not reproducible] can be calculated as

[mathematical expression not reproducible] (20)

Proof: See Appendix B.

Substituting (20) into (19), following expression can be obtained.

[mathematical expression not reproducible] (21)

By using partial fraction decomposition techniques, (21) can be written as

[mathematical expression not reproducible] (22)

(22) can be written as

[mathematical expression not reproducible] (23)

Where, [mathematical expression not reproducible]

By using [29, Eq. (10,11)] and solving the integral expression with the help of [30], (23) can be obtained as

[mathematical expression not reproducible] (24)

Please note that [alpha] term is set to 1 in (24). Following the same procedures, [mathematical expression not reproducible] can be calculated. This derivation is omitted because of the space limitation. Substituting (24) and derived [mathematical expression not reproducible] into (18), the final upper-bounded achievable rate expression can be calculated for such a system model.

3.4. The Diversity Order Analysis

The asymptotic CDF expression, (17), can be rewritten as

[mathematical expression not reproducible] (25)

Interpreting (25), the diversity order, which is the least power of [mathematical expression not reproducible], is the zero. This result has an agreement with figure 2. Since the performance curves saturate at high SNR regimes. In other words, the performance curves' slope at high SNRs is zero.

4. Main Results

This section validates the theoretical analysis by means of the Monte-Carlo simulation results. Figure 2 provides two different information: The first one, which is colored with black, is related to sum-rate based relay selection. The other one, which is colored with blue, is related to MM based relay selection strategy. The LI variances, [mathematical expression not reproducible], [mathematical expression not reproducible] and, [mathematical expression not reproducible] are equal to each other and set to [10.sup.3]. The target rate, R, is set to 1.00 bps/Hz. M is set to 1 in such a system model performance analysis. The transmit powers of CCI are set to : [P.sub.j]= [P.sub.s] /100.

Interpreting the figure 2 based on these assumptions, in low SNR regimes, the SR based relay selection outperforms the MM based relay selection strategy. In high SNR regimes, both selection strategy achieves the diversity order. The SR based relay selection strategy provides better performance values in terms of coding gain in comparison to MM based strategy. On the other hand, the CCI degrades the achievable diversity order from N to 0 and causes system coding gain losses in both selection strategy. A large number of FD relay provides better performance values than a system model that contains a small number of relay terminals.

Figure 3 provides achievable rate performance analysis comparisons of such a system model. Figure 3 provides two types of information: The first one, which is colored with red is related to system model contains two and four relays and under effect of a single CCI. The second one, which is colored with black is related to the syetem model that contains two and four relays and under effect of two CCI.

A large number of relay terminals provides better performance values in comparison to a small number of relay terminals in low and high SNR regimes. CCI severely affects the system achievable rate performance and degrades the performance curves in high SNR regimes.

5. Conclusions

This paper has investigated the CCI effects on SR based opportunistic relay selection strategy. The investigation has also considered a dual-hop multiple full-duplex two-way wireless relaying networks. According to Monte-Carlo simulation results, the SR based relay selection strategy outperforms the MM based relay selection strategy in terms of the end-to-end outage probability in such a system model. The CCI severely affects the system performance and degrades the achievable diversity order from N to 0 and also causes system coding gain losses. On the other hand, CCI also severely affects the system achievable rate performance and degrades the performance curves in high SNR regimes.

Appendix A

Proof of Proposition 1

Starting with (14) and (15) and following the same procedure of Appendix III of [25] and also assuming that the variables are independent to each other following expression can be obtained.

[mathematical expression not reproducible] (26)

The integral expressions in (26) can be solved with the help of [28, (Eq. [3.310.sup.11], [3.351.sup.3])]. In addition, with the help of order statistics [31], (16) can be obtained.

Appendix B

Proof of Proposition 2

Starting with (11) and following the same procedures as Appendix III of [20] and also assuming that the variables are independent to each other following expression can be obtained

[mathematical expression not reproducible] (27)

First and forth lines integrals and also second and third lines integrals in (27) can be solved with help of [28, (Eq. ([3.310.sup.11])] and [28, (Eq. ([3.351.sup.3])], respectively. The final expression can be obtained as in (20).

6. References

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Volkan Ozduran (1), Ehsan Soleimani-Nasab (2), B. Siddik Yarman (1)

(1) Istanbul University, Department of Electrical and Electronics Engineering, Avcilar, Istanbul, Turkey

(2) Faculty of Electrical and Computer Engineering, Graduate University of Advanced Technology, Kerman, Iran volkan@istanbul.edu.tr, ehsan.soleimani@kgut.ac.ir, yarman@istanbul.edu.tr

Received on: 29.06.2017

Accepted on: 21.07.2017

Volkan Ozduran graduated from department of Electronics at Soke Technical High School, Aydin, Turkey in 1997. He received his A.Sc. degree in Industrial Electronics, B.Sc., M.Sc. and Ph.D. degrees are in Electrical and Electronics Engineering from Istanbul University, Istanbul, Turkey in 2002, 2005, 2008 and 2015 respectively. During his Ph.D. studies he visited Stanford University, Stanford, CA, USA department of Electrical Engineering, Space, Telecommunications, and Radioscience Laboratory, Dynamic Spectrum Management (DSM) Research group as a visiting student researcher between April 2012 and September 2012 under the supervision of Prof. Dr. John M. CIOFFI, the best known " father of DSL". Prof. CIOFFI was his formal second advisor in his PhD studies. During his Ph.D. studies he also had some short visits to California Institue of Technology (CALTECH), Pasadena, CA, USA and Princeton University, Princeton, NJ, USA, department of Electrical Engineering in April 2013 and November 2014, respectively. His current research interests area more on cooperative communications, interference mitigation, massive MIMO and signal processing for wireless communications. He is an active Reviewer in various IEEE conferences and Transactions Journals

Ehsan Soleimani-Nasab was born in Kerman, Iran, in 1984. He received the B.Sc. degree in electrical engineering from Iran University of Science and Technology, Tehran, Iran, in 2006 and the M.Sc. and Ph.D. degrees in communication systems from K. N. Toosi University of Technology, Tehran, in 2009 and 2013, respectively.

From April to October 2012, he was a Visiting Researcher with the Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden. From June to August 2014, he was with the Department of Electrical and Electronics Engineering, Ozyegin University, Istanbul, Turkey, working as a Research Associate. Since September 2014, he has been with the Graduate University of Advanced Technology, Kerman, Iran, where he is currently an Assistant Professor. He is the author or coauthor of around 30 journal and conference publications. His research interests include optical wireless communications, resource allocation in wireless communications, and cooperative and cognitive radio networks. Dr. Soleimani-Nasab has served on the technical program committees for various IEEE conferences. He is an active Reviewer for various IEEE TRANSACTIONS and other journals.

Binboga Siddik Yarman received his BSc Degree from Technical University of Istanbul in 1974; MSc. Degree from Stevens Institute of Technology (1977), Hoboken, N.J., USA, and Ph.D. Degree from Cornell University (1982), Ithaca, NY, USA. He was a Member of Technical Staff at General David Sarnoff Microwave Technology and Research Center, in Princeton, NJ, USA. He served as professor and administrator at various Universities: Anadolu University, Middle East Technical University, Istanbul University, Isik University of Turkey; Ruhr University of Germany, Tokyo Institute of Technology of Japan, Wuhan Technology University of China. He is one of the founders of Savronik Group of Companies and recently serves as the Chairman of the Board of Directors. He was the founding president of Isik University (1996-2004). Currently serves as the Chairman of Board of Trustees. He published numerous papers in the field of Microwave Engineering, Circuit and Systems, Signal Processing, Mathematical Modeling and Decision Making. He has published 4 books: Design of Ultra - Wideband Antenna Matching Networks by Springer (2008), Design of Ultra - Wideband Power Transfer Networks by Wiley (2010), Intelligence Based Decision Making by Nobel Press of Turkey (2014) and He is one of the co-author of the book titled "Broadband Microwave and RF Power Amplifiers" by CRC (November 2015).

He holds 4 US and 9 Turkish Patents.

He was the recipient of Young Turkish Scientist Award (1986), Technology Award (1987) of Scientific and Research Council of Turkey.

He is a member of New York Academy of Science (1994), selected as the Man of the year in Science and Technology of Cambridge Biography (1999).He is a Fellow of IEEE; Alexander Von Humboldt Fellow and Salzburg Fellow of USIS.

Keywords: Full-Duplex Relay, Co-Channel Interference, Sum-rate.

1. Introduction

Cooperative communication has got a lot attention in the cellular communications in recent years [1-9]. This is because, communications with the aid of wireless repeaters provide additional advantages on information exchange process. Forefront of these advantages can be improvement on the signal fading effects. Rapid increases of number of mobile users also increase the number of wireless repeaters in cellular coverage area. N available active wireless repeaters increases the system overhead severely. This process can be minimized with the help of a prudent relay selection process. Prudent relay selection strategy also provides an information exchange process in low signal-to-noise ratio (SNR) regimes. The relay terminal's operating modes, which are half-duplex (HD) or full-duplex (FD), also affect this process. In the case that the relay terminal operates in HD mode, the information exchange process can be completed in two phases, which are multiple access and broadcast phase. However, in the case that the relay terminal operates in FD mode, the information exchange process can be completed in a single phase. FD wireless relays, besides doubling the capacity also suffers from the loop-interference (LI), caused by transmitting and receiving the information exchange process at the same time period. The LI can be minimize with an efficient antenna design and/or advanced signal processing techniques. Co-channel interference (CCI) is another factor, which affects the system performance. In the case that the CCI is high, this severely affects the system performance and does not provide a reliable communications.

Literature contains several types of studies in this area and forefront of these studies can be summarized as follows: [10] considers that source terminal communicates with the destination terminal with the aid of a selected FD relay among N available amplify-and-forward (AF) one-way relay (OWR) terminal. [10] Investigates a various FD relay selection strategies. [10] also considers that relay terminals have two antennas in such a system model. [11] also considers a similar system model structure as [10]. Differently from [10], [11] considers decode-and-forward (DF) based relay terminal and also considers Nakagami-m fading environment in such a system model. [11] also considers that there is a direct-link between source and destination terminals. In addition, [11] proposes two relay selection strategy. As distinct from [10, 11], [12] considers a different system model structure, where M source terminal communicates with the destination terminal via N available AF based FD relay terminal. [12] also considers a joint source and relay selection strategy in such a system model. In addition, [12] also assumes that there is a direct-link between source and destination terminals. [12]'s joint source and relay selection strategy is based on instantaneous SNR value between source and destination terminals. In addition, relay selection strategy is based on classical max-min (MM) strategy. [13] and its extended version [14], consider a similar system model structure as [10, 11]. Differently from [10, 11], [13, 14] consider that relay terminals posses N receive and transmit antennas. In addition, [13, 14] also investigates the effects of the joint relay and antenna selection technique on the system performance of such a system model. [13, 14] considers Rayleigh fading environment for the performance analysis of such a system model. [15] considers a similar system model structure as [10]. Differently from [10], [15] assumes that the FD relays operate in two-way. In addition, [15] investigates the effective SNR based best relay selection strategy on the system performance of such a system model. [16, 17] considers that source and destination terminals conduct information exchange with the help of FD based relay terminals in the system model. In addition, [16, 17] also considers that CCI affects the system model structure and investigates the effects of CCI on such a system model. [17] considers a Rayleigh fading environment, while [16] considers Nakagami-m fading environment.

At a glance to the aforementioned literature, in order to minimize the system overhead and provides an information exchange at low SNR regimes, most of the studies considers different types of FD relay selection strategies. Our earlier study [18], considers sum-rate based opportunistic relay selection strategy. [19-21] also employs sum-rate based strategy for the user-pair selection process. [22] and [23] considers the same system structure and the selection strategy as [18] but considers more realistic scenarios that the channel state information is imperfect because of the channel estimation error and feedback delay, respectively. [24] extends [18] by taking into consideration the CCI effects on such a system model. This paper extends [24] by providing the achievable rate analytical derivation and the diversity order analysis of such a system model.

The remainder of this paper is organized as follows. Section II describes the system model and channel statistics. Section III presents the performance analysis results including both exact and high SNR situations for the SR-ORS and MM methods. Section IV provides numerical results and the paper concludes in section V.

Notations: This paper uses [f.sub.h] (.) and [F.sub.h] (.) to denote the probability density function (PDF) and cumulative distribution function (CDF) of a random variable (RV) h, respectively. The operator E[.] stands for expectation, while Pr(.) denotes probability.

2. System Model and Channel Statistics

Figure 1 depicts a dual-hop FD TWR system structure. Here, [S.sub.1] and [S.sub.2] conduct information exchange with the help of N available FD wireless relaying. Since the relay terminal operates in FD mode, the information exchange process can be completed in a single phase.

This paper also considers that each terminal has an omnidirectional antenna. [h.sub.j], [g.sub.j], [f.sub.j], [k.sub.j] and [m.sub.j], [[for all].sub.j] = 1,..., N, represents the channel impulse response between [S.sub.1] [right arrow] R, [S.sub.2] [right arrow] R, [a.sub.j] [right arrow] [S.sub.1], [b.sub.j] [right arrow] R and [c.sub.j] [right arrow] [S.sub.2], respectively. [h.sub.j] is a complex Gaussian RVs. with zero mean and variance [mathematical expression not reproducible]. Likewise, [mathematical expression not reproducible], [mathematical expression not reproducible], [mathematical expression not reproducible] and [mathematical expression not reproducible]. d, [e.sub.j] and p, [mathematical expression not reproducible], [mathematical expression not reproducible] and [mathematical expression not reproducible], represent the loop-interferences at [S.sub.1], R and [S.sub.2], respectively. Amplitudes of all channels are distributed according to Rayleigh distribution.

The received signal at relay terminal can be written as

[mathematical expression not reproducible] (1)

Here, [P.sub.s], [P.sub.r] and [P.sub.j] are cooresponding transmit powers of mobile, relay terminals and [j.sup.th] interference terminal respectively. x, y and [b.sub.j] are corresponding transmit information, with a unit energy E[[| x |.sup.2]] = 1, E[[| y |.sup.2]] = 1 and E[[| b j |.sup.2]] = 1, of [S.sub.1], [S.sub.2] and [j.sup.th] interference terminal, respectively. [mathematical expression not reproducible] is the additive white Gaussian noise (AWGN) with zero mean and variance [[sigma].sup.2] (i.e. [mathematical expression not reproducible]). Since the relay terminal operates in AF mode, the G amplification factor can be calculated as

[mathematical expression not reproducible] (2)

After the amplification process, the received signals at [S.sub.1] and [S.sub.2] can be calculated as (3) and (4), respectively.

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

By using (3) and (4), the received signal-to-interference noise ratios (SINRs) at [S.sub.1] and [S.sub.2] can be calculated as

[mathematical expression not reproducible] (5)

[mathematical expression not reproducible] (6)

Substituting the [G.sub.j], (2), into (5) and (6) and also doing some mathematical manipulations, [mathematical expression not reproducible] and [mathematical expression not reproducible] can be represent as in (7) and (8), respectively.

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

Here,

[mathematical expression not reproducible]

(7) and (8) can be re-written as in (9) and (10), respectively.

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

3. Performance Analysis

This subsection investigates the system performance by using the outage probability and achievable rate performance metrics.

3.1. The Outage Probability

The outage probability defines as the probability that the achievable capacity cannot support the pre-defined target rate, R in bps/Hz. In other words, the outage probability is the CDF of received SNR/SINR evaluated at target threshold rate, [[gamma].sub.th] [26]. In this regard, (9) and (10), with the help of [mathematical expression not reproducible], can be upper-bounded as in (11) and (12), respectively.

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

Here, [mathematical expression not reproducible]

With the help of (11) and (12), the end-to-end (e2e) SINR can be calculated as

[mathematical expression not reproducible] (13)

Since the best relay selection strategy is based on the SR expression, (14), by using the logarithm properties and with the help of [18,20,27], (14) can be approximated to min(X, Y), (15), expression.

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

The CDF expression of [SR.sup.FD] is given by the following proposition.

Proposition 1: [mathematical expression not reproducible] can be calculated as

[mathematical expression not reproducible] (16)

Proof: See Appendix A.

Here, [[OMEGA].sub.h], [[OMEGA].sub.g], [mathematical expression not reproducible] [mathematical expression not reproducible], [mathematical expression not reproducible], [mathematical expression not reproducible], [mathematical expression not reproducible],[[OMEGA].sub.d] and [[OMEGA].sup.p] are mean of [|h|.sup.2], [|g|.sup.2], [[|k.sub.j]|.sup.2], [[|f.sub.j]|.sup.2] [[|m.sub.j].sup.2], [[|e.sub.j]|.sup.2], [[|c.sub.j]|.sup.2], [|d|.sup.2] and [|p|.sup.2], respectively. If the MM based relay selection strategy is used, the target threshold rate can be calculated as: [mathematical expression not reproducible].

3.2. Asymptotic Analysis

By using the Taylor series expansions exp term can be written as: exp(x) = 1 + x for x [right arrow] 0 [28]. By doing variable changes in (16), the asymptotic CDF expression can be calculated as

[mathematical expression not reproducible] (17)

3.3. Achievable Rate Analysis

This subsection now focuses on achievable rate analysis of such a system model. By using the Jensens' inequality the upper-bounded achievable rate expression can be formulated with the help of [26, Eq. (25)] as

[mathematical expression not reproducible] (18)

By using [26, Eq. (26)], [mathematical expression not reproducible] can be formulated as

[mathematical expression not reproducible] (19)

In order to continue analysis, (19) requires the [mathematical expression not reproducible] expression. [mathematical expression not reproducible] can be calculated as in following proposition.

Proposition 2: [mathematical expression not reproducible] can be calculated as

[mathematical expression not reproducible] (20)

Proof: See Appendix B.

Substituting (20) into (19), following expression can be obtained.

[mathematical expression not reproducible] (21)

By using partial fraction decomposition techniques, (21) can be written as

[mathematical expression not reproducible] (22)

(22) can be written as

[mathematical expression not reproducible] (23)

Where, [mathematical expression not reproducible]

By using [29, Eq. (10,11)] and solving the integral expression with the help of [30], (23) can be obtained as

[mathematical expression not reproducible] (24)

Please note that [alpha] term is set to 1 in (24). Following the same procedures, [mathematical expression not reproducible] can be calculated. This derivation is omitted because of the space limitation. Substituting (24) and derived [mathematical expression not reproducible] into (18), the final upper-bounded achievable rate expression can be calculated for such a system model.

3.4. The Diversity Order Analysis

The asymptotic CDF expression, (17), can be rewritten as

[mathematical expression not reproducible] (25)

Interpreting (25), the diversity order, which is the least power of [mathematical expression not reproducible], is the zero. This result has an agreement with figure 2. Since the performance curves saturate at high SNR regimes. In other words, the performance curves' slope at high SNRs is zero.

4. Main Results

This section validates the theoretical analysis by means of the Monte-Carlo simulation results. Figure 2 provides two different information: The first one, which is colored with black, is related to sum-rate based relay selection. The other one, which is colored with blue, is related to MM based relay selection strategy. The LI variances, [mathematical expression not reproducible], [mathematical expression not reproducible] and, [mathematical expression not reproducible] are equal to each other and set to [10.sup.3]. The target rate, R, is set to 1.00 bps/Hz. M is set to 1 in such a system model performance analysis. The transmit powers of CCI are set to : [P.sub.j]= [P.sub.s] /100.

Interpreting the figure 2 based on these assumptions, in low SNR regimes, the SR based relay selection outperforms the MM based relay selection strategy. In high SNR regimes, both selection strategy achieves the diversity order. The SR based relay selection strategy provides better performance values in terms of coding gain in comparison to MM based strategy. On the other hand, the CCI degrades the achievable diversity order from N to 0 and causes system coding gain losses in both selection strategy. A large number of FD relay provides better performance values than a system model that contains a small number of relay terminals.

Figure 3 provides achievable rate performance analysis comparisons of such a system model. Figure 3 provides two types of information: The first one, which is colored with red is related to system model contains two and four relays and under effect of a single CCI. The second one, which is colored with black is related to the syetem model that contains two and four relays and under effect of two CCI.

A large number of relay terminals provides better performance values in comparison to a small number of relay terminals in low and high SNR regimes. CCI severely affects the system achievable rate performance and degrades the performance curves in high SNR regimes.

5. Conclusions

This paper has investigated the CCI effects on SR based opportunistic relay selection strategy. The investigation has also considered a dual-hop multiple full-duplex two-way wireless relaying networks. According to Monte-Carlo simulation results, the SR based relay selection strategy outperforms the MM based relay selection strategy in terms of the end-to-end outage probability in such a system model. The CCI severely affects the system performance and degrades the achievable diversity order from N to 0 and also causes system coding gain losses. On the other hand, CCI also severely affects the system achievable rate performance and degrades the performance curves in high SNR regimes.

Appendix A

Proof of Proposition 1

Starting with (14) and (15) and following the same procedure of Appendix III of [25] and also assuming that the variables are independent to each other following expression can be obtained.

[mathematical expression not reproducible] (26)

The integral expressions in (26) can be solved with the help of [28, (Eq. [3.310.sup.11], [3.351.sup.3])]. In addition, with the help of order statistics [31], (16) can be obtained.

Appendix B

Proof of Proposition 2

Starting with (11) and following the same procedures as Appendix III of [20] and also assuming that the variables are independent to each other following expression can be obtained

[mathematical expression not reproducible] (27)

First and forth lines integrals and also second and third lines integrals in (27) can be solved with help of [28, (Eq. ([3.310.sup.11])] and [28, (Eq. ([3.351.sup.3])], respectively. The final expression can be obtained as in (20).

6. References

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[10] I. Krikidis, H. A. Suraweera, P. J. Smith and C. Yuen, "Full-Duplex Relay Selection for Amplify-and-Forward Cooperative Networks," in IEEE Trans. on Wireless Commun., vol. 11, no. 12, pp. 4381-4393, Dec. 2012. doi:10.1109/TWC.2012.101912.111944

[11] Y. Wang, Y. Xu, N. Li, W. Xie, K. Xu and X. Xia, "Relay selection of full-duplex decode-and-forward relaying over Nakagami-m fading channels," in IET Commun., vol. 10, no. 2, pp. 170-179, 1 21 2016. doi: 10.1049/iet-com.2015.0524

[12] Y. Tang, H. Gao, X. Su and T. Lv, "Joint source-relay selection in two-way full-duplex relay network," 2016 IEEE Intern. Conf. on Commun. Workshops (ICC), Kuala Lumpur, 2016, pp. 577-582. doi: 10.1109/ICCW.2016.7503849

[13] M. Toka, O. Kucur and B. S. Tezekici, "Performance of joint relay and antenna selection in the Full-Duplex Amplify-and-Forward relay networks," 2015 23nd Signal Processing and Commun. Applications Conf. (SIU), Malatya, 2015, pp. 661-664. doi: 10.1109/SIU.2015.7129911

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[15] H. Cui, M. Ma, L. Song, and B. Jiao, "Relay selection for two-way full duplex relay networks with amplify-and-forward protocol," IEEE Trans. on Wireless Commun., vol. 13, no. 7, pp. 3768-3777, July 2014.

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Volkan Ozduran (1), Ehsan Soleimani-Nasab (2), B. Siddik Yarman (1)

(1) Istanbul University, Department of Electrical and Electronics Engineering, Avcilar, Istanbul, Turkey

(2) Faculty of Electrical and Computer Engineering, Graduate University of Advanced Technology, Kerman, Iran volkan@istanbul.edu.tr, ehsan.soleimani@kgut.ac.ir, yarman@istanbul.edu.tr

Received on: 29.06.2017

Accepted on: 21.07.2017

Volkan Ozduran graduated from department of Electronics at Soke Technical High School, Aydin, Turkey in 1997. He received his A.Sc. degree in Industrial Electronics, B.Sc., M.Sc. and Ph.D. degrees are in Electrical and Electronics Engineering from Istanbul University, Istanbul, Turkey in 2002, 2005, 2008 and 2015 respectively. During his Ph.D. studies he visited Stanford University, Stanford, CA, USA department of Electrical Engineering, Space, Telecommunications, and Radioscience Laboratory, Dynamic Spectrum Management (DSM) Research group as a visiting student researcher between April 2012 and September 2012 under the supervision of Prof. Dr. John M. CIOFFI, the best known " father of DSL". Prof. CIOFFI was his formal second advisor in his PhD studies. During his Ph.D. studies he also had some short visits to California Institue of Technology (CALTECH), Pasadena, CA, USA and Princeton University, Princeton, NJ, USA, department of Electrical Engineering in April 2013 and November 2014, respectively. His current research interests area more on cooperative communications, interference mitigation, massive MIMO and signal processing for wireless communications. He is an active Reviewer in various IEEE conferences and Transactions Journals

Ehsan Soleimani-Nasab was born in Kerman, Iran, in 1984. He received the B.Sc. degree in electrical engineering from Iran University of Science and Technology, Tehran, Iran, in 2006 and the M.Sc. and Ph.D. degrees in communication systems from K. N. Toosi University of Technology, Tehran, in 2009 and 2013, respectively.

From April to October 2012, he was a Visiting Researcher with the Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden. From June to August 2014, he was with the Department of Electrical and Electronics Engineering, Ozyegin University, Istanbul, Turkey, working as a Research Associate. Since September 2014, he has been with the Graduate University of Advanced Technology, Kerman, Iran, where he is currently an Assistant Professor. He is the author or coauthor of around 30 journal and conference publications. His research interests include optical wireless communications, resource allocation in wireless communications, and cooperative and cognitive radio networks. Dr. Soleimani-Nasab has served on the technical program committees for various IEEE conferences. He is an active Reviewer for various IEEE TRANSACTIONS and other journals.

Binboga Siddik Yarman received his BSc Degree from Technical University of Istanbul in 1974; MSc. Degree from Stevens Institute of Technology (1977), Hoboken, N.J., USA, and Ph.D. Degree from Cornell University (1982), Ithaca, NY, USA. He was a Member of Technical Staff at General David Sarnoff Microwave Technology and Research Center, in Princeton, NJ, USA. He served as professor and administrator at various Universities: Anadolu University, Middle East Technical University, Istanbul University, Isik University of Turkey; Ruhr University of Germany, Tokyo Institute of Technology of Japan, Wuhan Technology University of China. He is one of the founders of Savronik Group of Companies and recently serves as the Chairman of the Board of Directors. He was the founding president of Isik University (1996-2004). Currently serves as the Chairman of Board of Trustees. He published numerous papers in the field of Microwave Engineering, Circuit and Systems, Signal Processing, Mathematical Modeling and Decision Making. He has published 4 books: Design of Ultra - Wideband Antenna Matching Networks by Springer (2008), Design of Ultra - Wideband Power Transfer Networks by Wiley (2010), Intelligence Based Decision Making by Nobel Press of Turkey (2014) and He is one of the co-author of the book titled "Broadband Microwave and RF Power Amplifiers" by CRC (November 2015).

He holds 4 US and 9 Turkish Patents.

He was the recipient of Young Turkish Scientist Award (1986), Technology Award (1987) of Scientific and Research Council of Turkey.

He is a member of New York Academy of Science (1994), selected as the Man of the year in Science and Technology of Cambridge Biography (1999).He is a Fellow of IEEE; Alexander Von Humboldt Fellow and Salzburg Fellow of USIS.

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Author: | Ozduran, Volkan; Soleimani-Nasab, Ehsan; Yarman, B. Siddik |
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Publication: | Istanbul University - Journal of Electrical & Electronics Engineering |

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Date: | Jul 1, 2017 |

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