Research Article  Open Access
LowComplexity Estimation for Spatially Modulated PhysicalLayer Network Coding Systems
Abstract
This paper proposes a lowcomplexity signal estimator at the relay node for a spatially modulated physicallayer network coding system. In the considered system, the two terminal nodes use spatial modulation to transmit their signals to the relay node during the multiple access phase. Based on the channel quantization method, we propose a lowcomplexity estimator which can detect both antenna indices and QAM symbols using successive interference cancellation (SIC). Moreover, we design signal constellations for a combined signal component at the relay for arbitrary QAM modulation. The obtained 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 maximumlikelihood detector, particularly with highorder modulation.
1. Introduction
Recently, twoway relay systems have received much attention [1â€“4] as they not only can extend system coverage but also increase transmission efficiency. Among various twoway relay schemes, physicallayer network coding (PNC) [1â€“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 realtime applications, especially for Internet of Things (IoT) since many IoT devices are often powered by a limitedenergy 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 multipleinput multipleoutput (MIMO) transmission techniques such as spatial multiplexing [7] or spacetime 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.
1.1. Related Works
In order to improve the spectral efficiency of twoway relay systems, SM was proposed to combine with PNC in [11â€“14]. The work in [11] considered the combination of spacetime 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 twoway relay systems with more than 2 antennas at all nodes. In [10] the author proposed two spatial modulation schemes for the twoway relay channel, where either only simple SM or combined spacetime 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 bitwise 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 twoway 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 threephase network coding system. However, this system needs the knowledge of channel state information (CSI) at the terminal nodes.
1.2. Contributions of the Paper
In this paper, based on the channel quantizationbased SIC estimation in [4] we propose a lowcomplexity 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 , where is the channel quantization value, is proposed for arbitrary QAM modulated symbols and . 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 lowcomplexity 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 lowcomplexity 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 lowercase letter presents a vector, bold uppercase letter is used for a matrix, and italic normal letter is for a variable. denotes a matrix of rows and columns. Notations are for transpose, conjugate transpose, absolute value, and Frobenious norm, respectively.
2. Spatial Modulation with Network Coding Using ML Estimation
A typical twoway relay system using spatial modulation is illustrated in Figure 1 [12]. In this model, two terminal nodes and transmit data to each other simultaneously via a relay node . All nodes are equipped with antennas for spatial modulation upon transmission and for signal combination upon reception. In order to implement twoway transmission, network coding by XOR mapping is used at the relay node . 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 . The signal reception at each node is affected by additive white Gaussian noise which is modeled by a complex random variable .
(a) Multiple access phase
(b) Broadcast phase
The twoway transmission involves two phases, namely, the multiple access (MA) when and transmit their data to and the broadcast (BC) when forwards a network coded symbol to both and .
2.1. MA Operation
In the MA phase, the two terminal nodes , send their length bit sequences to . The first bits, denoted by , are used to activate one out of transmit antennas, while the remaining bits, denoted by , for the QAM modulation. The resulting data rate is bits/time slot. Let and , where and is the mapping function which maps a bit sequence to an active antenna index; , where maps a bit sequence to an QAM symbol. For example, the terminal node with antennas needs to transmits 4 information bits . Since and , the second antenna is activated and the symbol is transmitted over it.
The received signal at antennas of R in the MA phase is given bywhere is a normalized power factor to ensure ; denotes the thermal noise vector at receive branches of ; , , where is the channel vector between the th active antenna of and the antennas of . 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 and the active antenna indices at as follows [10, 12]:where denotes the QAM constellation. The estimated symbols are then demapped to bit sequences for network coding using XOR operation as follows:where and are the demapping function of and , respectively; denotes the bitwise XOR operation.
2.2. BC Operation
In the BC phase, the relay node first maps the estimated bit sequences and to the network coded symbols that consist of the transmit antenna index of the relay node 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 is expressed as follows:where and is the channel vector between the th active antenna of and the antennas of ; denotes the thermal noise vector at the receive antennas. At the terminal node , 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 , each terminal node can estimate the bit sequence from its counterpartner. For instance, the operation of the terminal node is given as follows:
3. Proposed LowComplexity Estimation at Relay
Recasting (1) in the matrix form, we havewhere
The channel matrix in (7) can be decomposed using a QR factorization as follows:where is a unitary matrix with and and is an upper triangle matrix, where with , being a real number and a complex number. Multiplying both sides of (7) by gives uswhere .
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 . 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.
3.1. Proposed LowComplexity Estimation Using SMQSIC
In this section, we propose a lowcomplexity estimation method by combining the channel quantization and the SIC technique, abbreviated as SMQSIC. The proposed method consists of two stages, namely, channel quantization and SICbased estimation, as follows.
3.1.1. Channel Quantization
Because has only the first nonzero two rows, the first two elements ofâ€‰â€‰ in (10) can be rewritten as follows:
To simplify the presentation the subscripts will be omitted in the following parts. in (12) can be decomposed as follows:
Let denote the quantization level such that , where denotes the rounding operation. is regarded as the deviation between and . It is also often referred to as quantization error or residual interference.
Denoting , (12) can be rewritten as follows:
Using the SIC estimation is first estimated from . Then can be detected by removing the interference component in . In fact, to estimate , its constellation must be stored in advance at the receiver. In order to limit memory size for storing the constellation of , we derive the following lemma.
Lemma 1. If , belongs to the set .
Proof. Because is a real and is a complex number, where , from the above assumptions, we have or and . Moreover, . This means that , or .
On the other hand, from (14), it can be seen that estimation reliability of depends on . Therefore, we propose an effective algorithm (Algorithm 1) to satisfy the condition and reduce the magnitude of .

3.1.2. SICBased 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 in the second equation of (14) using .
(i) For the QAM constellation, the soft estimation is done as follows [17]:(ii) For other cases, i.e., , the soft estimation is given by [18]Note that the estimation in (15) and (16) is performed separately for the real and imaginary parts.
Step 2. Cancel the residual interference in (14) to estimate the signal . Using the conventional SIC, the estimate of in (1), denoted by , is used to remove the noise components in (12) before estimating . If the estimate is correct, the noise component is removed and it does not affect the estimation of . In contrast, if the estimate is erroneous, the removal of results in an additional interference component which affects the estimation of significantly. The larger , the more significant the impact of this interference. In order to lessen this impact, instead of complete removing , we use the SIC method in [4] to remove only a part of it, namely, as follows.
Let us defineThe decision function of is given bywhere is the decision function for the signal constellation . Details on this decision function will be presented in the next part.
Step 3. Remove and estimate 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 can be calculated as follows:whereFinally, a pair of active antenna indices is detected as follows:
(ii) Estimation of the QAM Modulated Symbol
The total estimation of and from the received signal in (12) is performed as follows.
Step 1. Estimate the modulated symbol in the QAM constellation. Cancel in in (12) to estimate the modulated symbol . Different from the conventional SIC method which uses the estimate obtained from in (12), our scheme uses the estimate from the MRC in (19) as follows:
Step 2. Estimate the modulated symbol . In (19), we only estimate a part of the signal . Therefore, the total estimation of the modulated symbol in the received signal can be estimated as follows:Similar to [12], the pair of active antennas and the pair of QAM modulated symbols 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.
3.2. Constellation Design for and Decision Function
To decide the signal in (18), we first study the constellation of the signal and then create a decision rule for the function . Because of limited space, we only focus our presentation on the case with . The remaining case with can be extended in a straightforward way.
3.2.1. Constellation Design for
It can be seen from the above section that belongs to one of the values: . Meanwhile, the signals and belong to the QAM constellation. Therefore, the constellation of the signal can be described for various values of and as shown in Figures 2, 3, and 4.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
3.2.2. Proposed Decision Function
Since , the decision function is performed as follows:
(a) Case : Because the QAM constellation is square, we can perform separate estimation for the real part and the imaginary part of the signal 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 function in (18) is given bywhere represents the sign function and denote the real and imagine part of the complex variable , respectively. The real and the imaginary part of are determined as follows:where is the set of values for different and as given in Table 1. Notice that, in the case , we can select .

Let us consider the simple case with and and assume that we need to estimate . The estimation values correspond to the point as illustrated in Figure 5. For the ML estimation, there are a total of 16 calculations to decide the point to the constellation point . 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 : 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 when and , when and , and when , where , respectively, represents the arbitrary real and imaginary values that satisfy the zero condition of the equation. Therefore, in the case we can perform estimation in two steps as follows.
Step 1 (decide estimation domain). The signal points are determined to the right or to the left of the boundary line as givenwhere if and and if and , respectively.
Step 2 (estimation). (i) If is on the left of the boundary line , the estimation function is same as the case . However, the set of in (26) is taken from Table 2.
(ii) If is on the right of the boundary line ; at this point, we only need to estimate the points on the boundary line as follows:where is the set of points on the boundary line as given in the Table 3. Finally, substitute into (25) to get .


4. 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 , a complex multiplication , a complex division , and a complex addition or subtraction .
4.1. Complexity of Decision Functions and
First we analyze the complexity of the decision function in (18). From the above section, it can be seen that, for a certain modulation order , the signal belongs only to one of three constellations corresponding to , , or . Therefore, the probability of falls into one of the three constellations that can be given bywhere .
(a) Case , because the complexity of function when varies depending on the value lying on the left or on the right of the boundary. Theref