Low-Complexity Estimation for Spatially Modulated Physical-Layer Network Coding Systems

This paper proposes a low-complexity signal estimator at the relay node for a spatially modulated physical-layer network coding system. In the considered system, the two terminal nodes use spatial modulation to transmit their signals to the relay node during themultiple access phase. Based on the channel quantizationmethod, we propose a low-complexity estimatorwhich can detect both antenna indices andM-QAM symbols using successive interference cancellation (SIC). Moreover, we design signal constellations for a combined signal component at the relay for arbitraryM-QAMmodulation.Theobtained constellations allow further reduction of the computational complexity of the estimator. Performance evaluations show that the proposed estimator can achieve nearoptimal error performance while requiring significantly less computational complexity compared with the maximum-likelihood detector, particularly with high-order modulation.


Introduction
Recently, two-way relay systems have received much attention [1][2][3][4] as they not only can extend system coverage but also increase transmission efficiency. Among various two-way relay schemes, physical-layer network coding (PNC) [1][2][3][4][5] is known as an effective scheme as it allows the two terminal nodes to transmit at the same time and same frequency during multiple access phase, thus reducing the number of exchange phases to two. As a consequence, throughput and spectral efficiency of the PNC system are higher than those of the traditional network coding (NC) [6]. However, the PNC system often requires higher complexity to obtain network coded symbols under effect of cochannel interference (CCI). This requirement leads to increase in transmission delay and energy consumption [6], which needs to be minimized for real-time applications, especially for Internet of Things (IoT) since many IoT devices are often powered by a limited-energy source such as battery. In order to perform PNC, maximum likelihood (ML) estimation was used to separate individual symbols from the two terminal nodes before combining them at the relay. Since the optimal ML estimation requires excessive complexity, especially for PNC systems with highorder QAM modulation, suboptimal estimators with less computational operations are often a better replacement. Aiming at enhancing system performance, PNC was also proposed to combine with multiple-input multiple-output (MIMO) transmission techniques such as spatial multiplexing [7] or space-time coding [8]. However, using MIMO transmission amounts to employment of multiple radio frequency (RF) chains, causing problems in not only strict antenna synchronization but also power consumption. Spatial modulation (SM) is another MIMO technique, which can avoid these problems by activating only one antenna at a time. While enjoying this advantage SM can also increase the spectral efficiency by using the activated antenna indices to convey information bits [9,10]. Obviously, combination of PNC and SM would provide more merits and this motivated several previous works.

Related Works.
In order to improve the spectral efficiency of two-way relay systems, SM was proposed to combine with PNC in [11][12][13][14]. The work in [11] considered the combination 2 Wireless Communications and Mobile Computing of space-time coding spatial modulation using coordinate interleaved orthogonal designs. This scheme achieves better symbol error performance compared with the traditional network coding for both the cases in which SM is used at the relay node during the broadcast phase and at all the nodes during both the multiple access and the broadcast phase. Despite this advantage, the proposed scheme still requires high complexity due to using ML estimation for joint decoding. Moreover, this scheme cannot be applied to the two-way relay systems with more than 2 antennas at all nodes. In [10] the author proposed two spatial modulation schemes for the two-way relay channel, where either only simple SM or combined space-time coding and SM are applied at the relay node. The paper successfully derived the system capacity for both the systems in terms of achievable rate region and sum rate. The work in [12] proposed a combined SM and PNC system where simple bit-wise XOR operation was used for network coding at the relay node. Thanks to this XOR operation the proposed system can be easily extended to the case of any arbitrary number of antennas at all nodes. However, similar to that in [10], this system uses the optimal ML decoding and thus exhibits highest computational complexity. In a similar work, paper [13] proposed combining SM with PNC but adding convolutional code for error correction. The proposed scheme uses optimal ML estimation for symbol estimation and then either separate decoding or direct decoding can be employed to attain the transmitted packets from the two terminal nodes. In order to improve the error performance of the two-way relay system using network coding and spatial modulation, the work in [14] proposed using precoding with signal constellation rotation at the terminal nodes and simple XOR network coding at the relay. The proposed system with the optimized rotation angle achieves better error performance over the three-phase network coding system. However, this system needs the knowledge of channel state information (CSI) at the terminal nodes.

Contributions of the Paper.
In this paper, based on the channel quantization-based SIC estimation in [4] we propose a low-complexity estimation scheme which achieves nearoptimal error performance for the combined PNC and SM scheme proposed in [12]. Compared with the previous works, our main contributions can be summarized as follows: (i) First, a new signal constellation for ( ) = (1) + (2) , where is the channel quantization value, is proposed for arbitrary -QAM modulated symbols (1) and (2) . Our constellation relaxes the limitation of QPSK modulation in [4].
(ii) In order to estimate a signal point in the constellation of ( ) , we propose a low-complexity scheme by estimating only the positive real and imaginary parts of ( ) and using a simple sign function. The complexity of the proposed estimation scheme depends less on the modulation order but mainly the number of antennas .
(iii) Based on the improved SIC scheme in [4], we to estimating the transmit antenna index and the -QAM modulated symbols successively. This proposed scheme differs from those used in [10,12,15] in that the previous first two schemes used ML to jointly estimate both active antenna index and modulated symbols and the last scheme used QR decomposition to estimate antenna index together with an ML estimator to estimate the modulated symbols.
The rest of paper is organized as follows. Section 2 presents an overview on spatial modulation with NC using the ML estimation method. The low-complexity estimation method is discussed in details in Section 3. Section 4 analyzes the computational complexity of the proposed scheme. Performance evaluations using simulated results are shown in Section 5. Finally, Section 6 concludes the paper.
Throughout this paper, we use the following mathematical notations. Bold lower-case letter presents a vector, bold upper-case letter is used for a matrix, and italic normal letter is for a variable. C × denotes a matrix of rows and columns. Notations (⋅) , (⋅) * , | ⋅ |, ‖ ⋅ ‖ are for transpose, conjugate transpose, absolute value, and Frobenious norm, respectively.

Spatial Modulation with Network Coding Using ML Estimation
A typical two-way relay system using spatial modulation is illustrated in Figure 1 [12]. In this model, two terminal nodes T 1 and T 2 transmit data to each other simultaneously via a relay node R. All nodes are equipped with , ( = 2 , ≥ 1) antennas for spatial modulation upon transmission and for signal combination upon reception. In order to implement two-way transmission, network coding by XOR mapping is used at the relay node R. Channels between each pair of transmit and receive antennas are assumed flat and slow Rayleigh fading, which are modeled by complex Gaussian distributed random variables ∼ N (0, 1). The signal reception at each node is affected by additive white Gaussian noise which is modeled by a complex random variable ∼ N (0, 2 ). The two-way transmission involves two phases, namely, the multiple access (MA) when T 1 and T 2 transmit their data to R and the broadcast (BC) when R forwards a network coded symbol to both T 1 and T 2 . antenna is activated and the symbol (1 + ) is transmitted over it.
The received signal at antennas of R y (R) = [ (R) 1 , .., (R) ] in the MA phase is given by where 1/ √ is a normalized power factor to ensure is the channel vector between the th ( = 1, . . . , ) active antenna of T and the antennas of R. Assuming that the channel state information (CSI) is perfectly known at the receiver, an ML detector is used to jointly detect the transmitted symbols, including the -QAM modulated symbols ( (1) , (2) ) and the active antenna indices ( , V) at T as follows [10,12]: where Ω denotes the -QAM constellation. The estimated symbols are then demapped to bit sequences for network coding using XOR operation as follows: where M −1 1 (⋅) and M −1 2 (⋅) are the demapping function of M 1 (⋅) and M 2 (⋅), respectively; ⊕ denotes the bitwise XOR operation.

BC Operation.
In the BC phase, the relay node R first maps the estimated bit sequences b (R) id and b (R) sb to the network coded symbols that consist of the transmit antenna index of the relay node (R) = M 1 (b (R) id ) and the -QAM symbol . These symbols are then broadcast to the two terminal nodes using the SM technique. The received signal at the terminal node T is expressed as follows: where is the channel vector between the th ( = 1, . . . , ) active antenna of R and the antennas of T ; z ( ) = [ ( ) 1 , ⋅ ⋅ ⋅ , ( ) ] denotes the thermal noise vector at the receive antennas. At the terminal node T , an ML estimator is utilized to estimate the network coded symbols that consist of the transmit antenna index and the -QAM symbol as follows [10,12]: Based on the estimated network coded symbols and using its transmitted bits in the MA phase b ( ) id , b ( ) sb , each terminal node T can estimate the bit sequence from its counterpartner. For instance, the operation of the terminal node T 1 is given as follows:

Proposed Low-Complexity Estimation at Relay
Recasting (1) in the matrix form, we have where ] , 4

Wireless Communications and Mobile Computing
The channel matrix H ( ,V) in (7) can be decomposed using a QR factorization as follows: where Q ( ,V) is a unitary matrix with Q ( ,V) Q ( ,V) = I × and ] with ( , ) ( ,V) , ( = 1, 2) being a real number and (1,2) ( ,V) a complex number. Multiplying both sides of (7) by Q ( ,V) gives us where n = [ 1 , . . . , ] ≜ Q ( ,V) n. Therefore, the ML estimation applied to (10) can be expressed as follows [16]: It is worth noting that in order to perform ML estimation in (2) or (11), the required computational complexity is 2 × 2 .
The larger the size of the signal constellation is, the more estimation complexity it requires. As a consequence, this results in increased transmission delay and large consumed energy for processing.

Proposed Low-Complexity Estimation Using SM-QSIC.
In this section, we propose a low-complexity estimation method by combining the channel quantization and the SIC technique, abbreviated as SM-QSIC. The proposed method consists of two stages, namely, channel quantization and SICbased estimation, as follows.
Denoting (sum) ≜ (1) + (2) , (12) can be rewritten as follows: Using the SIC estimation (2) is first estimated from (2) . Then (sum) can be detected by removing the interference component (2) in (1) . In fact, to estimate (sum) , its constellation must be stored in advance at the receiver. In order to limit memory size for storing the constellation of (sum) , we derive the following lemma.

SIC-Based
Estimation. This estimation method involves two steps: estimation of the active antenna index and estimation of the -QAM modulated symbol.

(i) Estimation of the Active Antenna Index
Step 1. Soft estimation of (2) in the second equation of (14) using (2) .
Step 3. Remove (sum) and estimate (2) using the maximum ratio combining (MRC) as follows: Step 4. Calculate the estimation error and detect the active antenna index of the terminals. The estimation error of (̃( sum) ( ,V) ,̃( 2) ( ,V) ) can be calculated as follows: where R (sum) Finally, a pair of active antenna indices is detected as follows:

(ii) Estimation of the -QAM Modulated Symbol
The total estimation of (1) and (2) from the received signal in (12) is performed as follows.
Step 1. Estimate the modulated symbol (1) in the -QAM constellation. Cancel (2) in (1) (̂,V) in (12) to estimate the modulated symbol (1) . Different from the conventional SIC method which uses the estimatẽ( 2) (̂,V) obtained from ( ) . (23) Step 2. Estimate the modulated symbol (2) . In (19), we only estimate a part of the signal (2) . Therefore, the total estimation of the modulated symbol (2) in the received signal can be estimated as follows: (2) = ( Similar to [12], the pair of active antennas (̂,V) and the pair of -QAM modulated symbols (̂( 1) ,̂( 2) ) will be mapped to network coded symbols. Then the relay uses spatial modulation to broadcast these symbols to the two terminal nodes as in Section 2.

Constellation Design for ( (1) + (2) ) and Decision Func-tion̂(⋅).
To decide the signal (sum) in (18), we first study the constellation of the signal (sum) and then create a decision rule for the function̂(⋅). Because of limited space, we only focus our presentation on the case with ≤ 64. The remaining case with > 64 can be extended in a straightforward way.

Proposed Decision Function̂(⋅).
Since ∈ {0, ±1, ± , ±1 ± }, the decision function̂(⋅) is performed as follows: (a) Case ∈ {0, ±1, ± }: Because the -QAM constellation is square, we can perform separate estimation for the real part and the imaginary part of the signal ( (1) + (2) ) to reduce complexity when estimating this signal. On the other hand, because the constellation is symmetric, we only need to estimate signal points in the first quarter of the constellation using the sign function. The decision function̂(⋅) in (18) is given bỹ where sign(⋅) represents the sign function and , denote the real and imagine part of the complex variable , respectively. The real and the imaginary part of̂( sum) are determined as follows:̂( sum) = arg min where A is the set of values for different and as given in Table 1. Notice that, in the case = 4, = 0 we can select (sum) =̂( sum) = 1.
Let us consider the simple case with = 16 and = 0 and assume that we need to estimate ( (1 ) , (1 ) ). The estimation values correspond to the point C 1 (−1.5, −1.5) as illustrated in Figure 5. For the ML estimation, there are a total of 16 calculations to decide the point C 1 (−1.5, −1.5) to the constellation point (−1, −1). If we estimate the real and the imagine part, separately, then it reduces to 8 calculations. Meanwhile, our proposed method only needs 4 calculations to obtain the magnitude and 1 calculation to decide the sign. The more increases, the more the complexity can be reduced compared with that of the conventional estimation methods.
(b) Case ∈ {±1 ± }: It can be seen from Figures 2(c), 3(c), and 4(c) that the constellation is truncated at all four quadrants. At the first quadrant, the constellation is bounded by a line + −4 = 0 when = 4 and + −12 = 0, when = 16 and + − 28 = 0, and when = 64, where , , respectively, represents the arbitrary real and imaginary values that satisfy the zero condition of the equation. Therefore, in the case ∈ {±1 ± } we can perform estimation in two steps as follows.
Step 2 (estimation). (i) If is on the left of the boundary line + − = 0, the estimation function is same as the case ∈ {0, ±1, ± }. However, the set of A in (26) is taken from Table 2.
(ii) If is on the right of the boundary line + − = 0; at this point, we only need to estimate the points on the boundary line as follows: where B is the set of points on the boundary line as given in the Table 3. Finally, substitutê( sum) into (25) to get̃( sum) .

Complexity Analysis
In order to show the advantage of the proposed method in terms of computational complexity, we estimate the floating   point operations (flop). Similar to [19], all real algebraic operation is considered as 1flop, a complex multiplication 6flops, a complex division 11flops, and a complex addition or subtraction 2flops.
(a) Case ∈ {±1 ± }, because the complexity of function̂(⋅) when ∈ {±1 ± } varies depending on the value (| (1 ) |, | (1 ) |) lying on the left or on the right of the boundary. Therefore, in this case the complexity is given as an average value based on the probability that the received signal lying on the right or left side of the boundary. Assuming that the probability that ( (1 ) , (1 ) ) lying close to an arbitrary constellation point of the square constellation (not blocked by the boundary) is the same. Letting Φ = | (1 ) | + | (1 ) | − , based on the geometric area we can determine the probability that the received signal lying on the right side of the boundary (Φ > 0) and the probability of the remaining points P r (Φ ≤ 0) = 1 − P r (Φ > 0) as shown in Figure 6. These probabilities are presented in Table 4.
A , is an average value of the set A over at a certain value of , and B is the sum of the values of the set B. The values of A , and B are summarized in Table 5.
As a result, the total complexity required for signal estimate at R for SM-QSIC is given by The total complexity of SM-ML in [10,12] calculated in (2) is given by The total complexity for the EGA and EQRP method in [15] are estimated as follows: ) flops, It is clear that the complexity of SM-ML [10,12] and both EGA and EQRP in [15] depend mainly on the modulation order of the -QAM constellation. Therefore, when the modulation order is high, these methods do not achieve high efficiency compared to the proposed method.
Wireless Communications and Mobile Computing 9

Simulation Results and Performance Evaluation
This section presents simulation results of BER and complexities of different estimation methods. The simulation results of the proposed SM-QSIC method are compared with that of the ML estimation method in [10,12] (denoted by SM-ML), the ML one in the conventional SIMO system (denoted by SIMO-ML), and the EGA, EQRD ones [15]. Let × × denote the system configuration, where is the number of antennas of the terminal node and is the number of antennas at the relay node. For fair comparison, performance is evaluated at the relay for the proposed method and the SM-ML in [12]. Assume that the relay knows one of the two signals from the terminal nodes and that power of each node is equal to , ( 1 = 2 = = ). Figures 7, 8, and 9 show the BER performances obtained at the spectral efficiency of 4 bps/Hz, 5 bps/Hz, and 6 bps/Hz. It can be seen that performance of the proposed method is close to that of the ML one [12] for the same spatial modulation configuration. Furthermore, the proposed method obtains higher SNR gain than the ML one in the SIMO system for the same transmission rate and the number of receive antennas at the relay. Particularly, at BER = 10 −3 , the SM-QSIC offers about 4 dB SNR gain at the spectral efficiency of 4 bps/Hz, about 7.5 dB SNR gain at 5 bps/Hz, and about 5 dB SNR gain at 6 bps/Hz over the ML in the SIMO system. This SNR gain increases with SNR or .
Next, we compare the end-to-end BER of the SM-QSIC and the ML [10]. For fair comparison, we assume that the transmit power of all terminals is the same and equal to half of the transmit power of the relay ( = ). Figures 10 and 11 compare the BER performances at the spectral efficiency of 4 bps/Hz and 6 bps/Hz. It can be seen that the proposed method outperforms the SIMO system without the spatial modulation. Particularly, at BER = 10 −3 and the spectral efficiency of 4 bps/Hz, the proposed method achieves about 7 dB SNR gain compared with the SIMO one. More SNR gain can be achieved at higher SNR or higher spectral efficiency. Compared with the ML method [10], the proposed one loses only about 0.7 dB SNR gain at low spectral efficiency ( Figure 10) but achieves the same performance at high spectral efficiency (Figure 11).
The processing efficiencies in terms of flops/symbol of the related schemes are compared in Table 7. It can be clearly seen that the proposed SM-QSIC scheme has the complexity that depends only slightly on the modulation order but more on the number of the transmit antennas . Moreover, the proposed scheme has much lower complexity compared with that of the ML estimation in [10,12], especially for high modulation order. Although it has higher complexity than the EGA and the EQRP in [15] for = 4, the proposed scheme becomes much more effective for > 4. Therefore, the proposed scheme is more suitable for high rate transmission systems.

Conclusions
This paper proposed a low-complexity estimation method using channel quantization and SIC for spatially modulated PNC systems. In our scheme, SIC was implemented first to estimate the activated antenna indices and then the modulated -QAM symbol. We also designed signal constellations for the combined signal ( (1) + (2) ) and derived a decision function̂(⋅) which facilitated a reduced-complexity estimator at the relay for arbitrary -QAM modulation. Using simulation results and complexity analysis we showed that the proposed scheme achieves near-optimal performance of the ML estimation while requiring less computational complexity. The proposed scheme is thus a prospective candidate for those applications which require low computational complexity such as IoT systems.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.