On Constellations for Physical Layer Network Coded TwoWay Relaying
Abstract
Modulation schemes for twoway bidirectional relay network employing two phases: Multiple access (MA) phase and Broadcast (BC) phase and using physical layer network coding are currently studied intensively. Recently, adaptive modulation schemes using Latin Squares to obtain network coding maps with the denoise and forward protocol have been reported with good endtoend performance. These schemes work based on avoiding the detrimental effects of distance shortening in the effective receive constellation at the end of the MA phase at the relay. The channel fade states that create such distance shortening called singular fade states, are effectively removed using appropriate Latin squares. This scheme as well as all other known schemes studied so far use conventional regular PSK or QAM signal sets for the end users which lead to the relay using different sized constellations for the BC phase depending upon the fade state. In this work, we propose a 4point signal set that would always require a 4ary constellation for the BC phase for all the channel fade conditions. We also propose an 8point constellation that gives better SER performance (gain of 1 dB) than 8PSK while still using 8ary constellation for BC phase like the case with 8PSK. This is in spite of the fact that the proposed 8point signal set has more number of singular fade states than for 8PSK.
I Background and Preliminaries
Physical layer network coding for wireless networks is a promising new area of research and a lot of work has been done in this area in recent times. Wireless twoway relay network shown in Fig 1, in which bidirectional data transfer takes place between the end nodes A and B with the help of the relay R is considered in this work. All the wireless links are assumed to be halfduplex. The Denoise and Forward (DNF) protocol for this case which consists of two phases: the multiple access (MA) phase, during which both A and B simultaneously transmit to R and the broadcast (BC) phase during which R transmits the network coded information to A and B is considered. Network coding map, also called the denoising map, is used at R during the BC phase to broadcast a symbol to A and B in such a way that A (B) can decode the message of B (A), given that A (B) knows its own message. This scheme, however, results in Multiple Access Interference (MAI) at the relay node and strategies to mitigate MAI for two way relay channel were studied in [1]. Design of coding schemes and network coding maps for uncoded transmission was studied in [2], and that for coded transmission was reported in [3] and [4].
It was observed in [2] that for uncoded transmission, the network coding map used at the relay needs to be changed adaptively according to the channel fade coefficients, in order to optimize performance. Such Adaptive Network Coding (ANC) maps can be either obtained through computer search, as done in [2] (called CNC algorithm), or analytically using Latin Squares, as studied in [5], [6] and [7]. The Latin Square approach focuses on the removal of the harmful effects of a finite number of deep channel fade conditions, termed singular fade states, that can occur during the MA phase. It was also shown in [5] that the Latin Square scheme gives better endtoend performance than the CNC approach in [2].
Performance analysis of ANC schemes in [2] and [5] was done in [8], where the connection between the removal of the harmful effect of singular states and the average endtoend Symbol Error Rate (SER) performance was explained. Upper bounds were also given in [8] for SER in terms of the average error probability of pointtopoint (SISO) fading channel and a term to account for the choice of network coding maps. From this, it is clear that by just using signal sets that give good performance in a SISO fading channel scenario, good endtoend SER performance for wireless twoway relay network cannot be guaranteed. Also, it was observed in [7] that using QAM instead of PSK at the end nodes gives a considerable improvement in performance. This suggests that the study of signal sets (possibly certain unconventional ones) for the MA phase assumes significance to ensure better endtoend SER performance. However, to the best of our knowledge, there has been no prior work in this direction.
Ia Signal Model
Throughout, a quasistatic block fading scenario is assumed with perfect channel state information available at the receivers. All the fade coefficients are assumed to be Rician distributed with a Rician factor . The fade coefficients associated with the AR and BR links are denoted by and respectively and and denote the fading coefficients associated with the RA and RB links respectively. The ratio denoted as , where and , is referred as the fade state and for simplicity, also denoted by . All the additive noises are assumed to be , which denotes the circularly symmetric complex Gaussian random variable with mean zero and variance .
Let denote a signal set of size used at A and B. Assume that A (B) wants to transmit to B (A). With denoting the additive noise at R, the received signal at during the MA phase is given by . At the end of the MA phase, R evaluates , which denotes the Maximum Likelihood (ML) estimate of based on the received complex number , i.e.,
(1) 
Depending on the value of , R chooses a map , where is the signal set (of size between and ) used by R during phase and broadcasts . The received signals at A and B during the BC phase are respectively given by, and , where is the additive noise at A (B).
In order to ensure that A (B) is able to decode B’s (A’s) message, the map should satisfy the exclusive law [2], i.e.,
Node A (B) can decode the message from B (A) by observing through ML decoding, since A (B) knows and satisfies the exclusive law.
IB Removal of Singular Fade States by ANC
Let denote the effective constellation (normalized by ) seen at R, i.e., . The minimum distance between the points in is given by
(2) 
Let . The elements of are called the singular fade states [5]. The set depends on the signal set used. The singular fade states are of the form
(3) 
and is obtained by equating (2) to zero. With the difference constellation of defined as , the singular fade states can be written as , for some .
The elements in which are mapped on to the same complex number in by the map are said to form a cluster. The set of all such clusters is denoted by , to indicate that it is a function of . Since the map clusters the points in , we interchangeably use the terms clustering and network coding map to denote . The number of clusters for a given channel fade state , is the size of the set which is the same as . The minimum cluster distance of the clustering is defined in [5] as
A clustering is said to remove a singular fade state , if . For a singular fade state , let denote a clustering which removes the singular fade state , and denote the corresponding signal set used.
The relay clusterings satisfying the exclusive law form Latin Squares [5]. The entry of the Latin square (corresponding to the singular fade state ) is the index of the signal point from that is transmitted by R during the BC phase where the indices of and are and respectively. Let denote the value of the largest entry in the Latin square of .
For channel fade states other than the singular fade states, among all the network maps which remove the singular fade states, the one which optimizes the performance is chosen. Since the network maps which remove the singular fade states are known to all the three nodes and are finite in number, the clustering used for a particular realization of the fade state can be indicated by R to A and B using overhead bits.
The main contributions of this paper are as follows:

The desirable features of the ary signal set (for a general ) used in the MA phase for wireless bidirectional twoway relay network to ensure good performance are identified to be the following

large value for the minimum Euclidean distance of .

minimum number of clusterings for each singular fade state .


It is shown that signal sets having fewer number of singular fade states do not invariably give better endtoend SER performance; examples of signal sets having larger number of singular fade states and yet giving better endtoend SER performance are given.

In [2] and [5] it was noted that when QPSK is used for MA phase, the BC phase required a 5ary constellation for certain channel conditions. We identify an alternate 4ary constellation for the MA phase which requires only 4ary signal set for all channel conditions. This constellation is also shown to give better endtoend SER performance than 4PSK with the requirement that only 4ary constellation is used by the relay.

An 8ary signal set is proposed as an alternative for the existing 8PSK signal set, which results in better endtoend SER performance.
The remaining content is organized as follows: In Section II we briefly summarize the factors determining the performance of physical layer network coding schemes for twoway relay network and list out the desirable features for the choice of the constellation used for MA phase. In Section III, an alternate 4point signal set which gives clear advantages over 4PSK is presented. An alternate 8point signal set that gives better SER performance than 8PSK is given in Section IV. Simulation results are given in Section VI.
Ii Performance determining factors
In this section, we summarize the error analysis results for wireless twoway relaying scenario described in [8]. Following this, the desirable features in the signal sets used for MA phase are described.
Iia Average endtoend SER
Let denote the ML estimate of at node A(B) at the end of the BC phase. For a given channel realisation , the error event can be upper bounded by the sum of two conditional error events viz., one occurring due to the relay R mapping to a wrong cluster (termed cluster error) and hence transmitting a wrong symbol during BC phase and the second occurring in spite of the fact that R sent the correct symbol. The first term is denoted by and the second by . Taking expectation over , the endtoend average SER is upper bounded by the sum of the average Cluster Error Probability (CEP), and the average probability that an error occurs during BC phase given that no cluster error occurred at R, denoted as .
With denoting the average SER of the pointtopoint fading channel using the signal set , it was shown in [8] that a tight upper bound for at high SNR values is equal to . The signal sets used for BC phase determine . Let denote the fraction of time the signal set is used in the BC phase. Then .
IiB Desirable features for selecting the signal set
With an average energy constraint , the factor that determines at high SNR values for a signal set is the minimum Euclidean distance of the constellation. However, choosing signal set for MA phase that gives smaller values for need not necessarily ensure better endtoend SER performance for the twoway relay network. This is because, such a signal set may require more number of clusterings (hence requiring larger sized ) for some singular fade state . This in turn increases the cluster error probability in addition to degrading the performance in the BC phase through the factor . Note that the minimum value for is . The number of singular fade states, given by , as such does not play any role in determining the endtoend SER performance as long as the clusterings remove all the singular fade states. However, a large value for would require the use of more number of overhead bits to indicate the choice of the network coding map to the end nodes A and B. We summarize the desirable features for the choice of the signal set used by the end nodes during the MA phase as follows

Larger value for is preferred so as to get improved performance in MA phase. This ensures that the average Cluster Error Probability is reduced at the relay.

The number of clusterings for each must be minimal i.e., itself. With this, the average probability that an error occurs during the BC phase, assuming the relay transmitted the correct network coded information, is minimized.
With these features satisfied, we come up with two constellations, one for and another for in the next two sections.
Iii An Alternate 4point Constellation
The only 4ary constellation that has been considered in the context of physical layer network coded twoway relaying in [2], [5] and [7] is the conventional 4PSK (same as 4QAM with an energy constraint). In this section we present an alternate constellation with 4 points that gives advantages over 4PSK as described below.
Iiia Issues with 4PSK signal set
It was noted in [2] and [5] that when 4PSK is used for MA phase, in order to overcome MAI at relay R, for several clusterings (8 out of the total 12 clusterings), an unconventional 5point signal set has to be used for the BC phase. After channel quantization, both [2] and [5] identify regions in the complex plane corresponding to values of where the BC phase requires a 5point constellation. In particular, this scenario arose in [5] because certain singular fade states for 4PSK needed Latin Squares with . For the proposed 4point constellation, the Latin Squares always have and hence the same 4point constellation can be used at the relay. In particular, the use of the unconventional 5point signal set can be totally avoided.
The alternate 4point constellation, denoted by is shown in Fig. 2. This constellation appears in [9] in the context of bandlimited pointtopoint AWGN channel. In Fig. 2, the quantity is chosen so as to meet an average energy constraint .
The geometry of the signal set is such that the sets of points and (in Fig. 2) form vertices of an equilateral triangle. The constellation has 18 singular fade states viz., which is more than that for 4PSK (12 singular fade states). The singular fade states of lie on three concentric circles centered at the origin and of radii as shown in Fig. 3. Table I summarizes the key aspects of the 4PSK and which are relevant in our context.
Feature  4PSK  

(for )  
Number of  
singular fade states  12  18 
Max. size of constellation  
required for BC phase  5  4 
Next we provide the Latin Squares for in Fig. 4 to show that no singular fade state of has . This ensures that the relay R can always use a 4point signal set during the BC phase. The Latin squares of the singular fade states lying on the circle with radius can be obtained by taking the transpose of the corresponding singular fade states (inverses) lying on the circle with radius , and from the Latin Square corresponding to a singular fade state , those corresponding to and (conjugate of ) can be obtained using row/column permutations. This is because the signal set is such that if , then . Also the same Latin square (XOR network coding map) can remove the singular fade states and . So from the 3 Latin Squares shown in Fig. 4, the Latin Squares for all the remaining 12 singular fade states can be derived using row and/or column permutation(s).
For the purpose of comparison with 4PSK, notice that for 4PSK, the clusterings which require 4ary signal set for BC phase can remove only the four singular fade states . The remaining 8 singular fade states require the relay to invariably use 5ary signal set for BC phase. Simulation results in Section VI show that when we impose an additional constraint that the relay can use only 4ary constellation for the BC phase in the physical layer network coded twoway relay scenario (thus only removing 4 singular fade states for 4PSK), the SER performance of is better than that of 4PSK.



Iv The proposed 8point signal set
In this section we consider the case when =8 point signal set is used by the end users. The conventional 8PSK was studied in [5] and was shown to require only 8 point constellation for BC phase for all channel realisations. We compare this with the 8cross QAM constellation (shown in Fig. 5) and propose an 8point constellation which is shown to give the best performance in terms of the average endtoend SER performance for =8. The proposed 8point constellation is shown in Fig. 6 and is denoted by . The advantage of 8PSK that it always requires the relay R to use 8point signal set for the BC phase carries forward to also. While the number of singular fade states for 8PSK is 104, that for is 108. The key features of all the three above mentioned constellations are summarized in Table II. From Table II we can see that while all the three signal sets require only 8 point constellations during the BC phase, has the largest value for and hence gives the best endtoend SER performance. Simulation results in Section VI validate this.
Feature  8PSK  8cross QAM  

(for )  0.7653  0.8944  0.9194 
Number of  
singular fade states  104  116  108 
Max. size of constellation  
required for BC phase  8  8  8 
From Fig. 6, points form QPSK signal points and the sets of points , , and form the vertices of an equilateral triangle. This constellation finds mention in [10]. The 108 singular fade states for are distributed in the complex fade state space as shown in Fig. 7.
Due to the symmetry of with respect to rotation of degrees, the technique used in [7] can be applied to obtain the Latin squares corresponding to all the 108 singular fade states from those corresponding to the singular fades lying on or outside the unit circle and in the angular interval which comes out to be 10 in number. These are given in Fig. 8 and Fig. 9.










V Channel Quantization
In the physical layer network coding schemes employing Latin Squares to obtain network maps, channel quantization is done to partition the complex plane representing the fade state and to specify which Latin Square is to be used for a generic value of the fade state realization . In this section we derive the channel quantization for and .
The values of for which any network coding map satisfying the exclusive law gives the same minimum cluster distance value is referred to as the clustering independent region. The region in the complex plane other than the clustering independent region is called the clustering dependent region. It was shown in [6] and [7] that the clustering independent region, denoted by consists of those values of which satisfy the following condition,
(4) 
The region in where is denoted by and the region where is denoted by . Since can be obtained as the complex inversion of , finding out one, the other can be easily evaluated [6].
Associated with each singular fade state , a region is identified in the clustering dependent region, in which the clustering is used. The region is obtained essentially by plotting the curves , which is referred to as the pairwise transition boundary between the regions and .
Let and be defined as follows:
Then the pair wise transition curve , is any one of the following:

if , a circle with center and radius , where

if , a straight line of the form , where
Channel Quantization for : Recalling that the singular fade states of lie on three concentric circles centered at the origin and with radii , the region is the exterior of the six unit circles centered at the singular fade states lying on the outermost circle (of radius ). This is obtained by evaluating the expression . The shaded region in Fig. 10(a) shows . The region is obtained by the complex inversion of the region and is given by the region lying exterior to the circles of radius and centered at locations . is shown as the shaded region in Fig. 10(b).
As an approximation, in order to reduce the computational complexity at R, the relay can choose to treat all those values of , that fall outside the circle of radius (C1 in Fig. 11) or inside the circle of radius (C2 in Fig. 11), to constitute the Clustering independent region and hence use any network coding map of Fig. 4. In the clustering dependent region, by plotting the pairwise transition curves , , we get the complete channel quantization for as shown in Fig. 12.
Channel Quantization for : First notice that the maximum and the minimum values for for are and respectively. The region is the exterior of the unit circles centered at the singular fade states with . Fig. 13(a) shows the region and the region obtained as the complex inversion of is shown in Fig. 13(b). Finally, the complete channel quantization for is shown in Fig. 15.
The relay can choose to treat all those values of , that fall outside the circle of radius (C1 in Fig. 14) or inside the circle of radius (C2 in Fig. 14), to constitute the clustering independent region and hence use any network coding map of Fig. 8. The relay needs to adaptively switch between the network coding maps only if the fade state value falls in the region between the circles C1 and C2 in Fig. 14. This reduces the computational complexity incurred at R.
Vi Simulation Results
In this section we present the simulation results showing the average endtoend SER for the proposed constellations and and compare them with those for 4PSK and 8PSK under different fading scenarios. First consider the case when the end nodes A and B use 4point constellations. The average SER curves as a function of SNR in dB for and 4PSK when the fade coefficients and are Rayleigh distributed (Rician fading with K=0) and when the fade coefficients are Rician distributed with Rician factor K=5 are given in Fig. 16. Fig 16 also shows the performance of 4PSK signal set when the relay R uses only 4point constellation for BC phase. In this scenario gives roughly 1 dB advantage over 4PSK. The same performance as that given by can be achieved by 4PSK only if the relay is allowed to use 5point constellation. It may be noted here that there is no degradation of performance for due to the fact that it has more number of singular fade states than 4PSK.
When the end nodes A and B use 8point constellations, the average SER vs SNR curves for , 8cross QAM and 8PSK are given in Fig. 17 for the case when the fade coefficients and are Rayleigh distributed (Rician fading with K=0) and also when K=5. Table II shows that 8PSK has the least number of singular fade states among the three. However, from Fig. 17, it is seen that 8PSK has the worst SER performance. So, the number of singular fade states has no effect on the SER performance. It is seen that gives 0.8 dB better average endtoend SER performance than 8PSK for Rayleigh fading scenario and also outperforms 8cross QAM. For Rician fading scenario with Rician factor K=5, gives 1.7 dB better average endtoend SER performance than 8PSK. Hence is the best choice among =8 signal sets.
Vii Discussion
In this paper we discuss the desirable features in the signal set used during the MA phase for bidirectional wireless relaying using physical layer network coding. 4PSK requires the use of 5ary constellation at R for several fade states. We propose a 4point constellation that always requires a 4point constellation during the BC phase and whose SER performance can be achieved by 4PSK only if the relay is allowed to use a 5ary constellation. We show that the number of singular fade states does not play any role in determining the endtoend SER performance in bidirectional wireless relaying. We propose an 8point signal set that always performs better than 8PSK. Not only does it require 8point constellation for the BC phase like 8PSK for all channel realisations, but also has just four singular fade states more than that of 8PSK. Further studies can focus on the construction of signal sets giving best endtoend SER performance for arbitrary values of .
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