2013 IEEE 24th International Symposium on Personal, Indoor and Mobile Radio Communications: Fundamentals and PHY Track
Symbol Error Rate of Two-way Decode-and-Forward Relaying with Co-channel Interference Youyun Xu, Xiaochen Xia, Kui Xu, and Yande Chen PLA University of Science and Technology, Nanjing, China Email:
[email protected]
Abstract—In this paper, we analyze the performance of twoway relaying (TWR) protocol in Rayleigh fading channels, where the terminals and relay are interfered by a finite number of co-channel interferers. The relay is assumed to operate in the decode-and-forward mode. The average symbol error rate (SER) performance for binary phase shift keying (BPSK) is analyzed. To make the analysis mathematically tractable, two approximations are adopted to deal with the problem of correlations between the received SINRs of the relay and terminal and a tight approximate expression of the average SER is derived in closedform. Moreover, it can be shown that the result can be simply applied in straightforward network coding protocol with cochannel interference. Based on the analytic results, we study the impacts of system parameters, such as interference power, number of interferers and relay placement, on the average SER performance. Finally, the correctness of our analytic results is validated through Monte Carlo simulations. Index Terms—Two-way relaying, decode-and-forward, cochannel interference, symbol error rate.
I. I NTRODUCTION In recent years, two-way relaying (TWR) or bi-directional relaying has emerged as a powerful technique to improve the spectral efficiency of wireless network [1]. A number of relaying protocols have been proposed, such as amplify-andforward (AF), decode-and-forward (DF) and compress-andforward (CF). Several previous works have investigated the two-way relaying protocol in Rayleigh fading channels, where the relay and terminals are only perturbed by the additive white Gaussian noise (AWGN) [2]-[7]. In [2], the authors have studied the physical network coding and presented an algorithm to select the modulation mapping for several typical modulations at the relay in order to optimize the end-to-end throughput. Lower bound of outage probability for AF based time division broadcasting (TDBC) protocol in Rayleigh fading channels was analyzed in [5], [6] and the diversity-multiplexing tradeoff (DMT) was also obtained. In [7], the authors considered relay selection scheme for analog network coding (ANC) protocol and TDBC protocol, and analyzed the outage performance with optimal relay selection. However, signals of terminals (or relay) are often corrupted by co-channel interference (CCI) from other sources that share the same frequency resources in wireless networks [8]. Moreover, CCI often dominates AWGN in wireless networks with dense frequency reuse. Therefore, it
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is necessary to take the effect of CCI into serious consideration in the analysis and design of the practical TWR protocol. In our previous work [9], we have considered outage performance of AF based TDBC protocol corrupted by CCI and presented the closed-form expression of outage probability. The outage performance and average bit error rate of ANC protocol with a set of equal-power interferers were also studied in [10]. In this paper, we focus on the DF based TWR protocol in which all the nodes are interfered by a finite numbers of co-channel interferers. The tight approximate expression of average SER for BPSK modulation is derived in closed-form. The rest of this paper is organized as follow. The system model is described in the next section. In section III, the closedform expression average SER is considered and analytically derived. In section IV, the effects of interference power, number of interferers and relay placement on the average SER performance are studied. At last, some conclusions will be drawn in section V. II. S YSTEM MODEL We consider the TWR network which consists of two terminals and a relay node, as shown in Fig. 1, where terminals 𝑇1 and 𝑇2 exchange information with the help of a relay 𝑅. Each node is equipped with a single antenna and operates in the half-duplex mode, that is, a node cannot transmit and receive simultaneously. It is assumed that both terminals and the relay are interfered by a finite number of co-channel interferers. Here we let 𝐿𝑅 , 𝐿1 and 𝐿2 denote the total numbers of interferers that affect node 𝑅, 𝑇1 and 𝑇2 , respectively. Let ℎ, 𝑓 and 𝑔 denote the channel coefficients between 𝑇1 and 𝑅, 𝑇2 and 𝑅, and 𝑇1 and 𝑇2 with variances Ω𝐻 , Ω𝐹 and Ω𝐺 , respectively. Moreover, let 𝑐𝑁,𝑘 ∈ 𝒞𝒩 (0, 1) denote the channel coefficient between node 𝑁 ∈ {𝑇1 , 𝑇2 , 𝑅} and the 𝑘th interferer that affects node 𝑁 . All the channels are assumed to be reciprocal and independent Rayleigh fading and the channel coefficients do not change within one round of the information exchange. 1 This protocol is first considered in [11]. Moreover, the protocol can be simply transformed into the traditional straightforward network coding (SNC) [3] by neglecting the direct link between terminals. Therefore, the results in this paper can be used for SNC with little modification.
138
I
network coding. Therefore, the received SINR at 𝑇1 is actually the received SINR of link 𝑅 → 𝑇1 which can be written as
I
𝐸𝑅 𝐻 , (2) Γ1 + 1 ∑𝐿 1 2 2 where 𝐻 = ∣ℎ∣ and Γ1 = is the 𝑘=1 𝑃𝑇1 ,𝑘 ∣𝑐𝑇1 ,𝑘 ∣ instantaneous total interference power at 𝑇1 . Similarly, the received SINR at 𝑇2 can be expressed as 𝛾1𝐶1 =
R
I
I
T1
I Fig. 1: The TWR network with a finite number of co-channel interferers, where 𝐼 denotes the co-channel interferer. The TWR protocol1 can be achieved within three time slots, i.e., 𝑇1 transmits during the first time slot, while 𝑇2 and 𝑅 listen. In the second time slot, 𝑇2 transmits while 𝑇1 and 𝑅 listen. The received signals at the relay dur[1] ing the first two time slots can be expressed as 𝑦𝑅 = √ √ ∑ 𝐿𝑅 √ [1] [1] [2] 𝐸1 ℎ𝑆1 + 𝑘=1 𝑃𝑅,𝑘 𝑐𝑅,𝑘 𝐼𝑅,𝑘 +𝑛𝑅 and 𝑦𝑅 = 𝐸2 𝑓 𝑆2 + ∑𝐿 𝑅 √ [2] [2] 𝑃𝑅,𝑘 𝑐𝑅,𝑘 𝐼𝑅,𝑘 + 𝑛𝑅 . Wherein, 𝑃𝑁,𝑘 indicates the 𝑘=1 transmitted power of the 𝑘th interferer that affects node 𝑁 ∈ {𝑇1 , 𝑇2 , 𝑅}. 𝑆𝑖 and 𝐸𝑖 , 𝑖 = 1, 2, 𝑅, denote the unit-power transmitted symbols and transmitted powers of [𝑚] [𝑚] nodes 𝑇1 , 𝑇2 and 𝑅, respectively. Moreover, 𝑦𝑁 , 𝐼𝑁,𝑘 and [𝑚] 𝑛𝑁 ∈ 𝒞𝒩 (0, 1) represent the received signal, the unit-power interference signal of the 𝑘th interferer and the AWGN at node 𝑁 ∈ {𝑇1 , 𝑇2 , 𝑅} during the 𝑚th time slot, respectively, where 𝑚 ∈ {1, 2, 3}. Meanwhile, the received signals at during the first two time slots can be written 𝑇1 and 𝑇2 √ ∑ 𝐿1 √ [2] [2] [2] as 𝑦𝑇1 = 𝐸2 𝑔𝑆2 + 𝑘=1 𝑃𝑇1 ,𝑘 𝑐𝑇1 ,𝑘 𝐼𝑇1 ,𝑘 + 𝑛𝑇1 and √ √ ∑ [1] [1] [1] 𝐿2 𝑦𝑇2 = 𝐸1 𝑔𝑆1 + 𝑘=1 𝑃𝑇2 ,𝑘 𝑐𝑇2 ,𝑘 𝐼𝑇2 ,𝑘 +𝑛𝑇2 , respectively. In time-slot 3, the relay broadcasts the combined information based on the received signals during the first two time slots. In this paper, it is assumed that the relay operates in the DF mode. Let 𝐶𝐷 denote a subset of {𝑇1 , 𝑇2 } that consists of the terminals whose symbols were decoded successfully at 𝑅, then the relay will process the information extracted from the received signals of 𝑇1 and 𝑇2 according to 𝐶𝐷 . Case 1 (𝐶𝐷 = {𝑇1 , 𝑇2 }): In this case, both 𝑆1 and 𝑆2 were decoded successfully. The relay can combine the decoded symbols using network coding (i.e., bit-wise XOR), then the received signal at the terminal 𝑇1 can be expressed as [3]
𝑦𝑇1 =
√
𝐸𝑅 ℎ (𝑆1 ⊕ 𝑆2 ) +
𝐿1 ∑ √ 𝑘=1
[3]
𝐸𝑅 𝐹 , (3) Γ2 + 1 ∑ 𝐿2 2 2 where 𝐹 = ∣𝑓 ∣ and Γ2 = is the 𝑘=1 𝑃𝑇2 ,𝑘 ∣𝑐𝑇2 ,𝑘 ∣ instantaneous total interference power at 𝑇2 . Case 2 (𝐶𝐷 = {𝑇2 }): In this case, only the symbol of 𝑇2 is broadcasted by√the relay. The received signal at 𝑇1 can be ∑ 𝐿1 √ [3] [3] [3] written as 𝑦𝑇1 = 𝐸𝑅 ℎ𝑆2 + 𝑘=1 𝑃𝑇1 ,𝑘 𝑐𝑇1 ,𝑘 𝐼𝑇1 ,𝑘 + 𝑛𝑇1 . Upon performing the MRC, the received SINR at 𝑇1 can be expressed as 𝐸𝑅 𝐻 + 𝐸 2 𝐺 𝛾1𝐶2 = , (4) Γ1 + 1 𝛾2𝐶1 =
T2
[3]
𝑃𝑇1 ,𝑘 𝑐𝑇1 ,𝑘 𝐼𝑇1 ,𝑘 + 𝑛𝑇1 . (1)
Note that the maximal ratio combining (MRC) cannot be employed at 𝑇1 in this case since the transmitted symbols of the relay to terminal link and the direct link, i.e., the terminal to terminal link, may be different due to the application of
2
where 𝐺 = ∣𝑔∣ , and the received SINR at 𝑇2 is given by 𝛾2𝐶2 =
𝐸1 𝐺 . Γ2 + 1
(5)
Case 3 (𝐶𝐷 = {𝑇1 }): Similar to Case 2, the relay broadcasts the symbol of 𝑇1 in this case, then received SINRs at 𝑇1 and 𝑇2 can be given by 𝛾1𝐶3 =
𝐸2 𝐺 𝐸𝑅 𝐹 + 𝐸1 𝐺 , 𝛾 𝐶3 = . Γ1 + 1 2 Γ2 + 1
(6)
Case 4 (𝐶𝐷 = ∅): The relay keeps silent in this case and the received SINRs at 𝑇1 and 𝑇2 are actually the received SINRs of links 𝑇2 → 𝑇1 and 𝑇1 → 𝑇2 , respectively, which can be written as 𝐸2 𝐺 𝐸1 𝐺 , 𝛾 𝐶4 = . (7) 𝛾1𝐶4 = Γ1 + 1 2 Γ2 + 1 III. P ERFORMANCE A NALYSIS In this section, the average SER for the DF based TWR protocol in the presence of CCI are analyzed for BPSK modulation. For brevity of analysis and without loss of generality, we only consider the SER performance of terminal 𝑇1 . Attempting to derive the closed-form expression of SER, where the error detection at relay is performed in a frameby-frame fashion, is difficult. Therefore as in [13], [14], we assume that the error detection process can be performed symbol-by-symbol at 𝑅. According to the assumption in the above and the total probability theorem, the conditioned average SER can be written as ( ) 𝑃𝑇𝐸1 𝛾1𝐶1 , 𝛾1𝐶2 , 𝛾1𝐶3 , 𝛾𝑅,𝑇1 , 𝛾𝑅,𝑇2 ( ) = 𝒫 𝛾1𝐶1 (1 − 𝒫 (𝛾𝑅,𝑇1 )) (1 − 𝒫 (𝛾𝑅,𝑇2 )) ) ( (8) + 𝒫 𝛾1𝐶2 𝒫 (𝛾𝑅,𝑇1 ) (1 − 𝒫 (𝛾𝑅,𝑇2 )) ( 𝐶3 ) + 𝒫 𝛾1 𝒫 (𝛾𝑅,𝑇2 ) .
139
Wherein, 𝛾𝑅,𝑇1 and 𝛾𝑅,𝑇2 are the received SINRs at the relay during the first and second time slots, respectively, which are given by 𝛾𝑅,𝑇1 =
𝐸1 𝐻 𝐸2 𝐹 , 𝛾𝑅,𝑇2 = , Γ𝑅 + 1 Γ𝑅 + 1
(9)
where Γ𝑅 is the ∑ instantaneous total interference power at the 𝐿𝑅 2 𝑃𝑅,𝑘 ∣𝑐𝑅,𝑘 ∣ . 𝒫 (Δ) indicates the conrelay and Γ𝑅 = 𝑘=1 ditioned SER for BPSK modulation with the received SINR Δ ∈ {𝛾1𝐶1 , 𝛾1𝐶2(, 𝛾1𝐶3 ,)𝛾𝑅,𝑇1 , 𝛾𝑅,𝑇2 }, which can be computed √ 2Δ [15]. Wherein, 𝒬(⋅) is the Gaussian-𝒬 by 𝒫 (Δ) = 𝒬 ( 2) ∫∞ function which is defined as 𝒬 (𝑥) = √12𝜋 𝑥 exp − 𝑡2 𝑑𝑡. Then (8) can be rewritten as ( ) 𝑃𝑇𝐸1 𝛾1𝐶1 , 𝛾1𝐶2 , 𝛾1𝐶3 , 𝛾𝑅,𝑇1 , 𝛾𝑅,𝑇2 (√ ) ( (√ )) ( ( √ )) 1 − 𝒬 2 𝛾𝑅,𝑇2 =𝒬 2𝛾1𝐶1 1 − 𝒬 2𝛾𝑅,𝑇1
𝑃𝐸,1 (𝛾1𝐶1 ,𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 ) (√ ) )( (√ )) (√ +𝒬 2𝛾1𝐶2 𝒬 2𝛾𝑅,𝑇1 1 − 𝒬 2𝛾𝑅,𝑇2
𝑃𝐸,2 (𝛾1𝐶2 ,𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 ) (√ ) ) (√ +𝒬 2𝛾1𝐶3 𝒬 2𝛾𝑅,𝑇2 .
𝑃𝐸,3 (𝛾1𝐶3 ,𝛾𝑅,𝑇2 ) (10) The average SER can be obtained by averaging the conditioned SER with respect to the joint PDF of 𝛾1𝐶𝑖 and 𝛾𝑅,𝑇𝑗 for 𝑖 = 1, 2, 3 and 𝑗 = 1, 2, that is, ∫ ∞∫ ∞∫ ∞ 𝑃𝑇𝐸1 = 𝑃𝐸,1 (𝑋, 𝑌, 𝑍) 0
∫ +
∞
∫
0
∫ +
0
0 ∞
0
∞
∫
×𝑓𝛾 𝐶1 ,𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 (𝑋, 𝑌, 𝑍) 𝑑𝑋𝑑𝑌 𝑑𝑍 ∫ ∞1 𝑃𝐸,2 (𝑋, 𝑌, 𝑍) 0
∞ 0
0
×𝑓𝛾1𝐶2 ,𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 (𝑋, 𝑌, 𝑍) 𝑑𝑋𝑑𝑌 𝑑𝑍 𝑃𝐸,3 (𝑋, 𝑍) 𝑓𝛾1𝐶3 ,𝛾𝑅,𝑇2 (𝑋, 𝑍) 𝑑𝑋𝑑𝑍,
(11) where 𝑓𝑋1 ,𝑋2 ,⋅⋅⋅ ,𝑋𝑚 (𝑥1 , 𝑥2 , ⋅ ⋅ ⋅ , 𝑥𝑚 ) is the joint PDF of RVs 𝑋1 , 𝑋2 , ⋅ ⋅ ⋅ , 𝑋𝑚 . However, an exact closed-form solution of (11) is still difficult to be derived due to the correlations between 𝛾1𝐶𝑖 , 𝛾𝑅,𝑇1 and 𝛾𝑅,𝑇2 . To make the integrals mathematically tractable, two approximations will be adopted in this paper. In the first approximation, we assume that the 𝛾1𝐶𝑖 is independent with 𝛾𝑅,𝑇1 , then we have2
top of the next page, where 𝛽 = 𝐸2 Ω𝐺 /(𝐸2 Ω𝐺 − 𝐸𝑅 Ω𝐻 ), Ψ (𝑥, 𝑦, 𝑧) is the confluent hypergeometric function of the ∏𝐿 1 1 second kind [16], 𝜙𝑇1 ,𝑗 = 𝑖=1,𝑖∕=𝑗 𝑃𝑇1 ,𝑗 −𝑃𝑇1 ,𝑖 , 𝜙𝑅,𝑘 = ∏𝐿𝑅 1 𝑖=1,𝑖∕=𝑘 𝑃𝑅,𝑘 −𝑃𝑅,𝑖 and 𝐽 can be expressed as 1 𝐽= 4𝜋 ∫∞ 0
−1 𝜑1 2
( [ exp − 1 +
0 − 12
1 𝑃𝑅,𝑘
+
𝜑2
𝜑1 𝐸 1 Ω𝐻
+
𝜑2 𝐸 2 Ω𝐹
] ) 1 𝜑1 𝐸1 Ω𝐻
( [ exp − 1 +
] ) 1 𝜑2 𝑑𝜑1 𝑑𝜑2 . 𝐸2 Ω𝐹
(15) In order to solve the integral 𝐽, we present the second approximation, i.e., ) ( 2 1 𝜑1 𝜑2 1 𝜑1 𝜑2 + + ≈ + + 𝑃𝑅,𝑘 𝐸1 Ω𝐻 𝐸2 Ω𝐹 2 𝑃𝑅,𝑘 𝐸1 Ω𝐻 𝐸2 Ω𝐹 ( ) 12 ( ) 12 1 1 𝜑1 𝜑2 ≥2 + + , 𝑃𝑅,𝑘 𝐸1 Ω𝐻 𝑃𝑅,𝑘 𝐸2 Ω𝐹 (16) where the first step is because we consider the case that the average interference power is smaller than useful power (otherwise the terminal will try to decode the interference). The second step is due to (𝐴 + 𝐵)2 ≥ 4𝐴𝐵. Substituting (16) into (15) and using the integral reported in [16, 9.211.4], we can obtain √ ( ) 1 𝐸1 Ω𝐻 + 1 1 𝐸1 𝐸2 Ω𝐻 Ω𝐹 Ψ , 1, 𝐽= 2 16 2 𝑃𝑅,𝑘 (17) ( ) 1 𝐸2 Ω𝐹 + 1 ×Ψ , 1, . 2 𝑃𝑅,𝑘 Since 𝛾1𝐶3 is independent with 𝛾𝑅,𝑇2 , the third term of the RHS of (11) (denoted as 𝑃𝐸,3 ) can be rewritten as ∫ ∞ (√ ) 𝑃𝐸,3 = 𝒬 2𝑋 𝑓𝛾1𝐶3 (𝑋) 𝑑𝑋 ∫ 0∞ (√ ) × 𝒬 2𝑍 𝑓𝛾𝑅,𝑇2 (𝑍) 𝑑𝑍 0 √ ∫ ∞ (18) ) ( 𝐶3 1 exp (−𝜑0 ) 𝑑𝜑 = Pr 𝛾 < 𝜑 √ 0 0 1 4𝜋 0 𝜑0 √ ∫ ∞ 1 exp (−𝜑2 ) × Pr (𝛾𝑅,𝑇2 < 𝜑2 )𝑑𝜑2 √ 4𝜋 0 𝜑2 Using the similar approach as in the above, it is easy to show that 𝑃𝐸,3 can be solved into ∑∑ 𝑃𝐸,3 = 𝜙1𝑗 𝜙𝑅𝑘 (
𝑓𝛾1𝐶𝑖 ,𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 (𝑋, 𝑌, 𝑍) = 𝑓𝛾1𝐶𝑖 (𝑋) 𝑓𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 (𝑌, 𝑍) . (12) According to this approximation and the results presented in the Appendix, the first two terms of the RHS of (11) (denoted as 𝑃𝐸,1 and 𝑃𝐸,2 ) can be rewritten as in (13) and (14) at the 2 Note that similar approximation is also employed in [5] to determine the outage probability of a multiuser two-way relaying systems.
∫∞
× ( ×
𝑘
( )) 1 1 𝐸2 Ω𝐺 + 1 𝐸2 Ω𝐺 𝑃𝑇1 ,𝑗 Ψ , , (19) 4 2 2 𝑃𝑇1 ,𝑗 √ ( )) 1 1 𝐸2 Ω𝐹 + 1 𝐸2 Ω𝐹 𝑃𝑇1 ,𝑗 − Ψ , , . 4 2 2 𝑃𝑅,𝑘
𝑃𝑇1 ,𝑗 − 2 𝑃𝑅,𝑘 2
𝑗
√
Finally, the closed-form expression of average SER for the DF based TWR protocol at 𝑇1 in the presence of CCI
140
{
)} 1 1 𝐸𝑅 Ω𝐻 + 1 , , 𝑃𝐸,1 ≈ 𝜙1𝑗 𝜙𝑅𝑘 2 2 𝑃𝑇1 ,𝑗 𝑗 𝑘 { } √ ( ( ) √ ) 𝑃𝑅𝑘 𝐸1 Ω𝐻 𝑃𝑅,𝑘 𝐸2 Ω𝐹 𝑃𝑅,𝑘 1 1 𝐸1 Ω𝐻 + 1 1 1 𝐸2 Ω𝐹 + 1 × + Ψ , , Ψ , , + +𝐽 4 16 2 2 𝑃𝑅,𝑘 16 2 2 𝑃𝑅,𝑘 ∑∑
𝑃𝐸,2
𝑃𝑇1 ,𝑗 − 2
√
𝐸𝑅 Ω𝐻 𝑃𝑇1 ,𝑗 Ψ 4
(
(13)
{
√ √ ( ( ) )} 1 1 𝐸𝑅 Ω𝐻 + 1 1 1 𝐸2 Ω𝐺 + 1 𝐸𝑅 Ω𝐻 𝑃𝑇1 ,𝑗 𝐸2 Ω𝐺 𝑃𝑇1 ,𝑗 𝑃𝑇1 ,𝑗 + (𝛽 − 1) Ψ , , Ψ , , ≈ 𝜙1𝑗 𝜙𝑅𝑘 −𝛽 2 4 2 2 𝑃𝑇1 ,𝑗 4 2 2 𝑃𝑇1 ,𝑗 𝑗 𝑘 { } √ ( ( ) √ ) 1 1 𝐸1 Ω𝐻 + 1 1 1 𝐸2 Ω𝐹 + 1 𝐸1 Ω𝐻 𝑃𝑅,𝑘 𝐸2 Ω𝐹 𝑃𝑅,𝑘 𝑃𝑅,𝑘 × − Ψ , , Ψ , , + − 2𝐽 4 16 2 2 𝑃𝑅,𝑘 16 2 2 𝑃𝑅,𝑘 (14) ∑∑
Fig. 2: Symbol error rate at 𝑇1 against the transmitted power 𝐸, 𝐷1 = 0.5, 𝜑 = 7dB and 𝑁 ∈ {𝑇1 , 𝑇2 , 𝑅}. for BPSK modulation can be obtained by substituting (13), (14), (17) and (19) into (11), appropriately. Moreover, the approximate average SER at 𝑇1 with BPSK modulation for the straightforward network coding protocol with CCI can be 𝐶 = 𝑃𝐸,1 . directly expressed as 𝑃𝑇𝐸−𝑆𝑁 1 IV. S IMULATION R ESULTS In this section, we present the simulation results to verify our theoretical analyses on the average SER. Without loss of generality, we assume that the terminals and relay are placed in a straight line and the relay is set between 𝑇1 and 𝑇2 . The normalized distance between 𝑇1 and 𝑇2 is set to one. Meanwhile, denote 𝐷1 as the normalized distance between 𝑇1 and 𝑅. As a result, the variances of ℎ, 𝑓 and 𝑔 can be computed −𝑣 by Ω𝐻 = 𝐷1−𝑣 , Ω𝐹 = (1 − 𝐷1 ) and Ω𝐺 = 1, respectively, where 𝑣 indicates the path loss exponent which is set to 4 [17] in this paper. Moreover, equal power allocation between terminals and relay, i.e., 𝐸1 = 𝐸2 = 𝐸𝑅 = 𝐸, is assumed for the sake of simplicity. Finally, the interference powers of the interferers that affect 𝑁 ∈ {𝑇1 , 𝑇2 , 𝑅} are assumed to
Fig. 3: Symbol error rate at 𝑇1 versus 𝐸/𝐸𝐼 , where 𝐸𝐼 = 5dB, 𝐷1 = 0.5 and 𝜑 = 7dB be evenly distributed on the interval [0.1𝐸𝐼,𝑁 , 𝐸𝐼,𝑁 ], then we 0.9𝐸𝐼,𝑇𝑖 (𝑘−1) and 𝑃𝑅,𝑘 = 0.1𝐸𝐼,𝑅 + have 𝑃𝑇𝑖 ,𝑘 = 0.1𝐸𝐼,𝑇𝑖 + 𝐿𝑖 −1 0.9𝐸𝐼,𝑅 (𝑘 − 1). 𝐿𝑅 −1 Fig. 2 depicts the average SER performance at terminal 𝑇1 as a function of 𝐸 for different numbers of interferers and different interference powers. The SER performance of the interference-free scenario is also presented as a benchmark. As shown in the figure, the SER expression derived in section III based on the approximations in (12) and (16) yields results in good agreement with the exact SER derived by Monte Carlo simulations in the whole observation interval. Moreover, it can be seen that, in both figures, the performance curves are steep in the low SNR region (𝐸 < 10dB) due to the dominant role of the noise power. One the other hand, performance floors can be observed in the high SNR region. Fig. 3 studies the average SER performance at 𝑇1 against 𝐸/𝐸𝐼 , where 𝐸𝐼,𝑇1 = 𝐸𝐼,𝑇2 = 𝐸𝐼,𝑅 = 𝐸𝐼 . From Fig. 3, it can be seen that the average SER increases as the numbers of interferers as well as 𝐸𝐼 /𝐸 increase as expected. Fig. 4 studies the effect of relay placement on the SER
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Similarly, the PDF of Γ𝑅 can be given by 𝑓Γ𝑅 (𝑡) =
𝐿𝑅 ∑
⎡ ⎣
𝑘=1
𝐿𝑅 ∏ 𝑖=1,𝑖∕=𝑘
⎤ ( ) 1 ⎦ exp − 1 𝑡 . 𝑃𝑅,𝑘 − 𝑃𝑅,𝑖 𝑃𝑅,𝑘
(21) (20) should be Note that for the special case of( 𝐿1 = 1, ) replaced by 𝑓Γ1 (𝑡) = 𝑃𝑇1 ,1 exp − 𝑃𝑇1 ,1 𝑡 . Moreover, (21) 1 1 ( ) 1 1 should be replaced by 𝑓Γ𝑅 (𝑡) = 𝑃𝑅,1 exp − 𝑃𝑅,1 𝑡 when 𝐿𝑅 = 1. By substituting (12) and the expression of 𝒬(⋅) into the first term of (11), 𝑃𝐸,1 can be rewritten as 𝑃𝐸,1 Fig. 4: SER performance against relay placement, where 𝐸 = 20dB, 𝐿1 = 𝐿2 = 𝐿𝑅 = 5, and 𝜑 = 7dB.
1 ≈√ 2𝜋
∫∞ ∫∞ exp(− 0
∫∞ ∫∞
⎛
∫∞
⎞ 𝑡21
2𝑍
Replacing
𝑡2𝑖 2
(22)
with 𝜑𝑖 , for 𝑖 = 0, 1, 2, using the equality
𝑓𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 (𝑌, 𝑍) ∫ ∞ = 𝑓Γ𝑅 (𝑟)𝑓 𝛾𝑅,𝑇 ∣Γ𝑅 (𝑌 )𝑓 𝛾𝑅,𝑇 ∣Γ𝑅 (𝑍) 𝑑𝑟, 1 2 0 (23) and interchanging the integration order, we can obtain √
V. C ONCLUSIONS The effect of CCI on the DF based TWR protocol is considered in this paper for Rayleigh fading channels. The approximate closed-form expression of the average SER is obtained when BPSK is utilized and is shown to provide a perfect match with the Monte Carlo simulations. The expression is valid for arbitrary positive numbers of interferers with different interference powers. Moreover, the results can be simply applied in straightforward network coding protocol with CCI. A PPENDIX Since Γ1 is the sum of a finite number of exponential random variables (RVs), the probability density function (PDF) of Γ1 can be written according to [12] as (Herein, we will only consider the scenario that the transmitted power of interferer differs from one another, i.e., 𝑃𝑁,𝑖 ∕= 𝑃𝑁,𝑘 , ∀𝑖 ∕= 𝑘.) ⎡ ⎤ ( ) 𝐿1 𝐿1 ∑ ∏ 1 1 ⎣ ⎦ 𝑓Γ1 (𝑡) = exp − 𝑡 . 𝑃𝑇1 ,𝑘 − 𝑃𝑇1 ,𝑖 𝑃𝑇1 ,𝑘 𝑘=1
2𝑋
1 ⎟ ⎜ 𝑓𝛾𝑅,𝑇1 ,𝛾𝑅,𝑇2 (𝑌, 𝑍) ⎝1 − √ exp(− )𝑑𝑡1 ⎠ 2 2𝜋 √ 0 0 2𝑌 ⎞ ⎛ ∞ ∫ 𝑡2 1 ⎟ ⎜ exp(− 2 )𝑑𝑡2 ⎠ 𝑑𝑌 𝑑𝑍. ⎝1 − √ 2 2𝜋 √ ×
performance, where the average total interference powers at 𝑇1 and 𝑇2 are unbalanced. More specifically, we ∑ set the average 𝐿1 total interference powers at 𝑇1 and 𝑅 to 𝑘=1 𝑃𝑇1 ,𝑘 = ∑𝐿 𝑅 the average total interference 𝑘=1 𝑃𝑅,𝑘 = 10dB while let ∑ 𝐿2 power at 𝑇2 , i.e., 𝑃𝑇2 = 𝑘=1 𝑃𝑇2 ,𝑘 increase from 10, 15 to 20dB. Then we examine the terminal with poor SER 𝐸 . As shown in the figure, the relay placeperformance 𝑃𝑚𝑎𝑥 ment that achieves the best average SER performance (i.e. the intersection of the two terminals’ SER curves) moves to the terminal with higher interference power (i.e., 𝑇2 ). The reason is that the relay needs to balance the received SINRs at 𝑇1 𝐸 and 𝑇2 to minimize 𝑃𝑚𝑎𝑥 .
√
𝑡20 )𝑓 𝐶1 (𝑋) 𝑑𝑡0 𝑑𝑋 2 𝛾1
𝑖=1,𝑖∕=𝑘
(20)
∫ ∞ ∫ ∞ ) ( 1 exp (−𝜑0 ) 𝑃𝐸,1 ≈ Pr 𝛾1𝐶1 < 𝜑0 𝑑𝜑0 𝑓Γ𝑅 (𝑟) √ 4𝜋 0 𝜑0 0 ) ( √ ∫ 1 ∞exp (−𝑏𝜑1 ) Pr ( 𝛾𝑅,𝑇1 < 𝜑1 ∣ Γ𝑅 = 𝑟)𝑑𝜑1 × 1− √ 4𝜋 0 𝜑1 ( √ ∫ ) 1 ∞exp (−𝑏𝜑2 ) × 1− Pr ( 𝛾𝑅,𝑇2 < 𝜑2 ∣ Γ𝑅 = 𝑟)𝑑𝜑2 𝑑𝑟 √ 4𝜋 0 𝜑2 (24) Since 𝐻 and 𝐹 are exponential random variables with means Ω𝐻 and Ω𝐹 , respectively, we have ( ) (Γ𝑅 + 1) 𝜑1 Pr ( 𝛾𝑅,𝑇1 < 𝜑1 ∣ Γ𝑅 = 𝑟) = 1 − exp − 𝐸1 Ω𝐻 ( ) (Γ𝑅 + 1) 𝜑2 . Pr ( 𝛾𝑅,𝑇2 < 𝜑2 ∣ Γ𝑅 = 𝑟) = 1 − exp − 𝐸2 Ω𝐹 (25) Inserting (25) and the PDF of Γ𝑅 into (24) and using the result in [16, 9.211.4], it can be shown that (24) can be expressed as in (13). Similarly, using the approximation (12) on the second term of the RHS of (11) (denoted as 𝑃𝐸,2 ) and interchanging the
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integration order, we can yield 𝑃𝐸,2 ≈ ∫∞ 0 ⎛
1 4𝜋
∫∞ 0
) ( exp (−𝜑0 ) Pr 𝛾1𝐶2 < 𝜑0 𝑑𝜑0 √ 𝜑0
[15] J. G. Proakis, Digital Communications, 5th ed. McGraw-Hill, 2007. [16] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition. Academic Press, 2007. [17] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice Hall, 2002.
∫∞ 𝑓Γ𝑅 (𝑟) 0
exp (−𝜑1 ) Pr ( 𝛾𝑅,𝑇1 < 𝜑1 ∣ Γ𝑅 = 𝑟)𝑑𝜑1 × √ 𝜑1
⎝1 −
√
1 4𝜋
∫∞ 0
⎞ exp (−𝜑2 ) Pr ( 𝛾𝑅,𝑇2 < 𝜑2 ∣ Γ𝑅 = 𝑟)𝑑𝜑2 ⎠ 𝑑𝑟. √ 𝜑2
(26) Inserting (25) and the PDF of Γ𝑅 into (26) and using the result in [16, 9.211.4], one can arrive at the approximate expression of 𝑃𝐸,2 in (14). ACKNOWLEDGMENT This work is supported by the Jiangsu Province Natural Science Foundation (BK2011002), Major Special Project of China (2010ZX03003-003-01), National Natural Science Foundation of China (No. 60972050) and Jiangsu Province Natural Science Foundation for Young Scholar (BK2012055). R EFERENCES [1] S. J. Kim, N. Devroye, P. Mitran, and V. Tarokh, “Achievable Rate Regions and Performance Comparison of Half Duplex Protocols,” IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6405-6418, Oct. 2011. [2] T. K. Akino, P. Popoviki and V. Tarokh, “Optimized constellations for two-way wireless relaying with physical network coding,” in IEEE Journal on Sel. Areas in Commun., pp. 358-365, 2006. [3] S. Zhang, S. C. Liew, and P. P. Lam, “Hot topic: physical layer network coding,” in Proc. 12th MobiCom, pp. 358-365, 2006. [4] M. Zaeri-Amirani, S. Shahbazpanahi, T. Mirfakhraie, K. Ozdemir, “Performance tradeoffs in amplify-and-forward bidirectional network beamforming,” IEEE Trans. Signal Processing, vol. 60, no. 8, pp. 4196-4209, Aug. 2012 [5] H. Ding, J. Ge and D. B. Costa, “Two birds with one stone: exploiting direct links for multiuser two-way relaying system,” IEEE Trans. Wireless. Commun., vol. 11, pp. 54-59, Jan. 2012. [6] Z. Yi, M. Ju and I. M. Kim, “Outage probability and optimum combining for time division broadcast protocol,” IEEE Trans. Wireless. Commun., vol. 10, pp. 1362-1367, May. 2011. [7] M. Ju, and I. M. Kim, “Relay selection with ANC and TDBC protocols in bidirectional relay networks,” IEEE Trans. Commun., vol. 58, no. 12, pp 3500-3511, Dec. 2010. [8] P.A. Hoeher, S. Badri-Hoeher, W. Xu, C. Krakowski, “Single-antenna co-channel interference cancellation for TDMA cellular radio systems,” IEEE Wireless Commun., vol. 12, no. 2, pp. 30-37 , April 2005. [9] X. Xia, Y. Xu, K. Xu, D. Zhang, and N. Li, “Outage Performance of AF-based Time Division Broadcasting Protocol in the Presence of Co-channel Interference,” accepted by IEEE WCNC 2013, available at http://arxiv.org/abs/1212.6686. [10] S. S. Ikki, and S. Aissa, “Performance analysis of two-way amplifyand-forward relaying in the presence of co-channel interferences,” IEEE Trans. Commun., vol. 60, no. 4, pp. 933-939, April 2012. [11] Q. Li, S. H. Ting, A. Pandharipande, and Y. Han, “Cognitive spectrum sharing with two-way relaying systems,” IEEE Trans. Vech., vol. 60, no. 3, pp. 764-777, Feb. 2010. [12] H. V. Khuong and H. Kong, “General expression for pdf of a sum of independent exponential random variables,” IEEE Commun. Lett., vol. 10, no. 3, pp. 159-161, Mar 2006. [13] J. Hu and N. C. Beaulieu “Closed-form expressions for the outage and error probabilities of decode-and-forward relaying in dissimilar rayleigh fading channels ,” in Proc. IEEE ICC, pp. 5553-5557, Jun. 2007. [14] I. Lee and D. Kim “BER analysis for decode-and-forward relaying in dissimilar rayleigh fading channels,” IEEE Commun. Lett., vol. 11, no. 1, pp. 52-54, Jan. 2007.
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