Performance Analysis for Analog Network Coding with Imperfect CSI in FDD Two Way Channels

1 Introduction

Cooperative relaying has been shown to be a practical technique to enhance the communication range of wireless networks. In some practical scenarios where data flows in both directions, a relay improves the performance of both transmission directions simultaneously. This pragmatic approach has been modeled as a two-way relay channel, and has recently attracted significant interests. Of particular interest is an analog network coding (ANC) protocol for this channel, which is a well-known amplify-and-forward (AF)-based two-way relaying protocol^[1,2] However, there works assume that receivers have the perfect knowledge of channel state information (CSI). With the impact of system noise and imperfect channel estimation algorithms, receivers hardly have the perfect knowledge of CSI. Therefore, performance analysis that take into account such CSI uncertainties play an important role in the design of practical systems.

Very recently, some related works on ANC have taken into consideration the impact of imperfect CSI^[3–5]. These works consider a time division duplexing (TDD) system, where any given node exploits reciprocity to estimate the forward channel gain from the reverse link. This condition can’t be met in the FDD system since all channel gains are independent of each other. Thereby results of TDD systems in [3-5] aren’t generated to FDD systems either due to more random variables. To the best of our knowledge, a performance analysis in term of outage probability in the FDD two way channel is still an open issue.

In this letter, we investigate the outage performance for ANC with imperfect CSI in the two way channel with FDD. Approximating expressions of outage probabilities for non-orthogonal and orthogonal ANC are derived in the high SNR regime. It is validated that Monte Carlo simulation results are in agreement with the derived outage probabilities.

2 System Model

In this letter, we consider an FDD two-way system with two sources S₁, S₂and a relay R. Sources S₁and S₂desire to exchange their information with the help of relay R. We assume that all terminals are equipped with a single antenna and operate in a half-duplex mode. Suppose transmissions in this system suffer from frequency nonselective fading and additive noises. We denote channel gains from S₁to R, S₂to R, R to S₁, and R to S₂by hs1r,hs2r,hrs1andhrs2, respectively. They are modeled as zero-mean, circularly symmetric complex Gaussian random variables, denoted by hs1r∼CN(0,Ωh,s1r),hs2r∼CN(0,Ωh,s2r),hrs1∼CN(0,Ωh,rs1)andhrs2∼CN(0,Ωh,rs2). These channel gains are assumed to be independent of each other, i.e. any given node cannot exploit reciprocity to estimate the forward channel gain from the reverse link in the FDD system. Considering channel estimation errors, we denote the estimations of channel gains by h^s1r,h^s2r,h^rs1andh^rs2,respectively. We model the channel estimation by hab=h^ab+eab,wherea,b∈{s1,s2,r},h^ab∼CN(0,Ωh^,ab),eab∼CN(0,Ωe,ab) is estimation error for channel gain h_ab. We also assume that all channel gains estimation errors are independent of each other. Without loss of generality, the transmit power of each terminal is set to be P, each additive noise power is Ω₀. We also assume source node S_i transmits with a fixed rate v_i bits/Sec/Hz.

2.1 Non-orthogonal ANC

Due to the half-duplex constraint and the complete separation between the two source nodes, the communication process of non-orthogonal ANC is organized into two transmission phases in Figure 1, where two different transmission phases work in different frequency bands.

Figure 1

The two-way system where sources S₁ and S₂ desire to exchange their information via a relay R

In the first phase, both source nodes transmit simultaneously. The received signal at the relay R is expressed as Yr=∑i=12hsirPXi+nr,where X_i is the unit energy transmit symbol from S_i and n_r ~ 𝓒 𝓝(0, Ω₀) is an additive noise at relay R. In the other phase, the relay amplifies its received symbol Y_r and then broadcasts toward both source nodes. The received signal at the source node S_i is given by Yi=hrsiαnYr+ni,where α_n is a power-scaled gain at relay R and n_i ~ 𝓒𝓝(0, Ω₀)is an additive noise at source S_i. We consider the short-term power constraint at the relay node R, and thus the power-scaled gain, α_n, is written as

αn=P|h^s1r|2P+|h^s2r|2P+Ωe,s1rP+Ωe,s2rP+Ω0(1)

The signal Y_i can be rewritten as

Yi=h^sjrh^rsiαnPXj+h^sirh^rsjαnPXi(Self−interferenceterm)+(h^sjrersiαn+h^rsiesjrαn+esjrersiαn)PXj+(h^sirersiαn+h^rsiesirαn+esirersiαn)PXi+h^rs1nrαn+ersinrαn+ni(2)

where j is the other element in set 𝓐 = {1, 2}, with a given element i ∈ 𝓐 (e.g., if i = 1, then j = 2). Note that the definitions of i and j are used throughout this letter. Here we assume that the estimation of channel gains h^s1r,h^s2r,h^rsi,are obtained at source S_i by using some channel estimation techniques. We assume both sources simultaneously transmit pilot symbols, and the relay amplifies and forwards the symbols to both sources. Source S_i simultaneously estimates h^s1r,h^s2randh^rsiby using the received pilot symbols (Alternatively, the relay estimates h^s1randh^s2r by using pilot symbols, then forwards them to both sources^[2]. Source S_i also obtains hrsiby using pilot symbols from the relay). The power scaled gain α_n is also derived according to (1). Therefore the self-interference term h^sirh^rsjαnPXi in (2) can be known and canceled at source S_i. Therefore, the mutual information per transmission from source S_j to S_i is given by

Isjsin=12log1+|h^sjr|2|h^rsi|2P2b1i(|h^sir|2+|h^sjr|2)+b2i|h^rsi|2+b3i(3)

where parameters b_1i, b_2i and b_3i are denoted by

b1i=Ωe,rsiP2+Ω0Pb2i=Ωe,sjrP2+Ωe,sirP2+Ω0Pb3i=Ωe,sjrΩe,rsiP2+Ωe,sirΩe,rsiP2+Ωe,rsiΩ0P  +Ωe,sirΩ0P+Ωe,sjrΩ0P+Ω02(4)

3 Outage Probability Analysis

As clearly shown in (3) and (8), the forms of mutual information per transmission are not easily tractable due to the existence of imperfect channel gains and thus we approximately analyze their outage probabilities.

We define the system suffers from an outage if there exists one of both source nodes which decodes its received signals with error, and denote outage probabilities achieved by non-orthogonal ANC and orthogonal ANC by PoutnandPouto, respectively. The outage probability is expressed as

where x ∈ {o, n} and v_j denotes the transmission rate of source node S_j. In the high SNR regime, system outage probability is approximated by its upper bound^[6], i.e.,

4 Simulation Results

Consider a two-way network topology where the distance between two sources is 100m, and the relay node is located on the line segment between two source nodes. We assume that the distance from source S₁ to the relay node is Lm, and thus the distance from source S₂ to the relay is (100 - L)m. The signal fading follows the quasi-static Rayleigh distribution and the average channel gain equals d^–β[8], where d denotes the distance from transmitter to receiver and β is the path loss index. Let the path loss index β be 3 and the additive noise power Ω₀ be –70 dbm. We also assume that ratio Ωh^,abΩh,ab=0.995 for transmitter a and receiver b where a ∈ {S₁, S₂, R} and b ∈ {S₁, S₂, R} in this channel.

Figure 2 shows the system outage performance (OP) versus distance L for non-orthogonal and orthogonal ANC, where P = 20dbm and v₁ = v₂ = 0.5 bits/Sec/Hz. It is shown that the derived OP for the two protocols match the simulated results, which validates the accuracy of our derived expressions. We also observe that orthogonal ANC outperforms non-orthogonal ANC due to higher diversity gain by using the link between source S_i to S_j , and FDD system achieves lower OP than TDD for both the protocols. Figure 3 shows OP versus transmission power P in different transmission-rate scenarios, where L = 50m. Our derived curves are in agreement with the Monte Carlo simulation results. It is feasible to achieve the lower OP with smaller transmission rates. We also observe that OP reaches a fixed level, called error floor, due to the imperfection in the channel estimation.

Figure 2

OP versus distance L for non-orthogonal and orthogonal ANC, where P = 20dbm and v₁ = v₂ = 0.5 bits/Sec/Hz

Figure 3

OP versus transmission power P in different transmission-rate scenarios, where L = 50m

5 Conclusion

We have investigated the impact of imperfect CSI on the performance of non-orthogonal and orthogonal ANC in the FDD two way channel. The closed-formed expressions of outage probabilities achieved by non-orthogonal and orthogonal ANC have been derived in the high SNR regime. By using Monte Carlo simulation, it validates that the simulation results are in agreement with the derived results, and FDD system outperforms TDD in terms of OP.