Transmit Antenna Selection for Physical-Layer Network Coding Based on Euclidean Distance

# Transmit Antenna Selection for Physical-Layer Network Coding Based on Euclidean Distance

Vaibhav Kumar1, Barry Cardiff2, and Mark F. Flanagan3
School of Electrical and Electronic Engineering,
University College Dublin, Belfield, Dublin 4, Ireland
Email: 1vaibhav.kumar@ucdconnect.ie, 2barry.cardiff@ucd.ie, 3mark.flanagan@ieee.org
###### Abstract

Physical-layer network coding (PNC) is now well-known as a potential candidate for delay-sensitive and spectrally efficient communication applications, especially in two-way relay channels (TWRCs). In this paper, we present the error performance analysis of a multiple-input single-output (MISO) fixed network coding (FNC) system with two different transmit antenna selection (TAS) schemes. For the first scheme, where the antenna selection is performed based on the strongest channel, we derive a tight closed-form upper bound on the average symbol error rate (SER) with -ary modulation and show that the system achieves a diversity order of 1 for . Next, we propose a Euclidean distance (ED) based antenna selection scheme which outperforms the first scheme in terms of error performance and is shown to achieve a diversity order lower bounded by the minimum of the number of antennas at the two users.

## I Introduction

Wireless PNC has received a lot of attention among researchers in recent years due to its inherent desirable properties of delay reduction, throughput enhancement and better spectral efficiency. The advantage of PNC can easily be seen in a TWRC, where bidirectional information exchange takes place in the half-duplex mode between two users and with the help of a relay . In a TWRC, PNC requires only two time slots to exchange the information between the users compared to three time slots required by traditional network coding [1]. In the first time slot, also termed the multiple access (MA) phase, both users and simultaneously transmit their data to the relay . Based on its received signal, the relay forms the maximum-likelihood (ML) estimate of the pair of transmitted user constellation symbols. This estimate of the pair of user symbols is then mapped to a network-coded constellation symbol using the denoise-and-forward (DNF) protocol [2] and the relay broadcasts this to both users in the next time slot, called the broadcast (BC) phase. User constellation symbol pairs which are mapped to the same complex number in the network-coded constellation are said to form a cluster. Using its own message transmitted in the previous MA phase, can decode the message transmitted from and vice versa.

In the case of a fixed network coding (FNC) system, the network code applied at the relay is always fixed and does not depend on channel conditions. One of the bottlenecks in the FNC system limiting the error performance is the existence of singular fade states [3] that result in shortening the distance between clusters and making the relay vulnerable to erroneous mapping. In order to mitigate the distance shortening phenomenon, adaptive network coding (ANC) has been proposed in [4], where the network coding applied at the relay varies with the channel conditions. The network coding scheme at the relay in ANC systems depends on the ratio of the channel coefficients between the - and - links. For QPSK modulation in the MA phase, an unconventional 5-ary modulation scheme has been suggested in [4] for the BC phase and a computer search algorithm called closest-neighbor clustering (CNC) has been proposed to obtain adaptive codes. An analytical treatment for the ANC scheme has been presented in [3] considering a Rician fading model. Although the ANC scheme alleviates the problem of distance shortening between the clusters in an efficient way, the related system complexity increases significantly.

In [5], the error performance of a multiple-antenna based PNC system has been investigated in detail considering a Rayleigh fading model and BPSK modulation. A transmit antenna selection (TAS) scheme based on the strongest channel (in terms of signal-to-noise ratio) between the user and the relay has been applied in [5] where the relay, which is assumed to have perfect channel state information (CSI), decides on the indices of the antenna to be used at and , and shares this information with the users via a error-free, low bandwidth feedback channel. It has been shown in [5] that for the case when both users and are equipped with multiple antennas and the relay has only one antenna (MISO), the diversity order is equal to the minimum of the number of antennas at the two users. We show analytically that this result is true only for the case of binary modulation, while for higher-order modulation the antenna selection scheme based on the strongest channel fails to exploit the advantage of multiple antennas at the user end to leverage diversity gain.

The contribution made in the present paper is twofold. First, we analyze the error performance of the system presented in [5] with -ary modulation for the MISO case and present a closed-form expression for a tight upper bound on the average SER at the end of the MA phase. Second, we propose a new TAS scheme that maximizes the minimum ED between different clusters at the relay for the fixed network coded PNC system. To the best of our knowledge, the analysis of a PNC system with a ED based antenna selection scheme is not yet available in the open literature. We also prove analytically that the diversity order of such a PNC system is lower bounded by the minimum of the number of antennas at the user end. Since our system uses a fixed network coding scheme, the implementation complexity is lower than that involved in ANC systems, and we do not require the use of any nonstandard (e.g. 5-ary) modulation scheme in the BC phase, reducing the complexity associated with the constellation design.

## Ii System Model

The system model is shown in Fig. 1, where the users and are equipped with () and () antennas respectively, while the relay has only a single antenna. During every transmission, only one of the antennas from each user is used for transmission, and the choice of antennas is based on feedback received from the relay. The channel between user and the relay is modeled as slow Rayleigh fading with perfect CSI available at only. We assume that the channel remains constant during a frame transmission and changes independently from one frame to another. Hence the channel coefficient between the antenna of user and the relay is distributed according to . Suppose that both users employ the same unit-energy -ary constellation and denotes the difference constellation set of , defined as . Letting denote the message symbol at user , and denote the corresponding transmitted constellation symbol, the signal received at the relay during the MA phase is

 y=√EshAxA+√EshBxB+n (1)

where denotes the noise at the relay and is assumed to be distributed according to , denotes the energy of the transmitted signal and is the channel coefficient of the link between the selected antenna of user and the relay (based on the antenna selection scheme). The ML estimate of the transmitted symbol pair is given by

 (^xA,^xB)=argmin(xA,xB)∈S2∣∣y−√EshAxA−√EshBxB∣∣. (2)

The relay then maps this estimate to the network coded symbol using the PNC map . Using their own message , transmitted in the previous MA phase, each user can decode the other user’s message, provided that the map satisfies the exclusive law [4]. Table I shows the network coding operation for QPSK modulation, where and denotes bitwise addition (XOR) in .

The performance analysis of the two TAS schemes is presented in detail in the next section.

## Iii Transmit Antenna Selection

This section presents the analysis of two different TAS schemes for the PNC system. In the first scheme (TAS1), the index of the selected antenna at each user is the one having the highest signal-to-noise ratio (SNR). The channel coefficient between the selected antenna of user and the relay is given by

 hm=argmax1≤i≤Nm|hm,i|2 (3)

where and is the channel coefficient between the antenna of user and relay .

In contrast to this, in TAS2 the transmit antenna of each user is selected such that the minimum ED between the clusters at the relay is maximized. A similar scheme has been discussed in [6] for spatial modulation (SM). Let be the set which enumerates all of the possible combinations of selecting one antenna from each user. Among these combinations, the set of transmit antennas that maximizes the minimum ED between the clusters is obtained as [6, eqn. (2)]

 IED=argmaxI∈I⎧⎪ ⎪⎨⎪ ⎪⎩minx,x′∈S2M(x)≠M(x′)∥∥HI(x−x′)∥∥2⎫⎪ ⎪⎬⎪ ⎪⎭ (4)

where , , and is the optimal channel vector.

To understand the performance superiority of TAS2 over TAS1, we first consider a simple example of one transmission slot where the users transmit their messages using QPSK modulation. Suppose that and , , and . In this case, since the - link is stronger (i.e., has a higher SNR) than the - link, and similarly the - link is stronger than the - link, TAS1 will choose the antenna combination . With this combination the minimum distance between the clusters at the relay becomes very small, which can lead to an incorrect ML estimate at the relay. Fig. 2 shows a plot, for TAS1, of the noise-free received signal at the relay, i.e., (here we assume ), together with the corresponding network coded symbols, where each -tuple in the figure represents . In contrast to this, the proposed antenna selection scheme (TAS2) chooses as the optimal combination and the resulting network coded symbols are shown in Fig. 3. It is clear that TAS2 overcomes the distance shortening phenomenon.

In the following subsections, we present the analysis of the error performance for the two TAS schemes.

### Iii-a TAS1: Antenna selection based on the maximum SNR

For the error performance analysis of TAS1, we use the union-bound approach given in [3] rather that the approach given in [5] which applies only for binary modulations. For FNC, the average SER at the end of the MA phase is given by [3, eqn. (6)]

 Pe =1M2⎡⎢⎣∑(xA,xB)∈S2∑xA≠x′A∈SE[P{(^xA,^xB)=(x′A,xB)|hA,hB}] +∑(xA,xB)∈S2∑xB≠x′B∈SE[P{(^xA,^xB)=(xA,x′B)|hA,hB}] +∑(xA,xB)∈S2∑xA≠x′A∈SxB≠x′B∈SE[P{^xA,^xB)=(x′A,x′B),M(xA,xB)≠M(x′A,x′B)|hA,hB}] (5)

where is the modulation order and is the expectation operator. An upper bound on the average SER can be given as

 Pe≤1M2 ∑(xA,xB)∈S2∑(xA,xB)≠(x′A,x′B)∈S2M(xA,xB)≠M(x′A,x′B)E[Q(√Es2N0∣∣(hAxA+hBxB)−(hAx′A+hBx′B)∣∣)] (6)

where is the Gaussian Q-function. The Chernoff bound on the Q-function used in [3] results in a loose upper bound for the present case and hence we use the Chiani approximation [7, eqn. (14)] instead, yielding

 Pe≲ +14E{exp(−Es3N0|hAΔxA+hBΔxB|2)}] = (7)

Now we analyze the three different terms on the right-hand side of (5) separately as follows:

### Case I: When xA≠x′A and xB=x′B

In this case and . Hence,

 E(Υ1)=∫∞0exp(−Es4N0|hAΔxA|2)f(|hA|)d|hA| (8)

where denotes the probability density function. Using the fact that each is Rayleigh distributed, the distribution of the magnitude of can easily be derived as [5, eqn. (37)]

 f(|hm|)=Nm∑k=1(Nmk)(−1)k−12k|hm|exp(−k|hm|2). (9)

Substituting into (8) we can write

 E(Υ1)= NA∑k=1(NAk)(−1)(k−1)k∫∞02|hA|exp{−(k+Es|ΔxA|24N0)|hA|2} d|hA| = NA∑k=1(NAk)(−1)(k−1)(1+Es|ΔxA|24kN0)−1. (10)

Similarly,

 E(Υ2)=NA∑k=1(NAk)(−1)(k−1)(1+Es|ΔxA|23kN0)−1. (11)

Substituting the values of and from (10) and (11) respectively into (7), the average SER arising from the case when and can be written as

 Pe{(^xA,^xB)=(x′A,xB)} ≲1M2∑(xA,xB)∈S2∑xA≠x′A∈SNA∑k=1(NAk)(−1)(k−1)[(12ΨA,k)−1+(4ΞA,k)−1] = 1M2∑(xA,xB)∈S2∑xA≠x′A∈S(ζ1+ζ2) (12)

where

 ΨA,k=1+Es|ΔxA|24kN0,ΞA,k=1+Es|ΔxA|23kN0, (13) ζ1=NA∑k=1(NAk)(−1)(k−1)12ΨA,k,ζ2=NA∑k=1(NAk)(−1)(k−1)4ΞA,k. (14)

Using the binomial expansion, can be written as

 ζ1=112NA∑k=1(NAk)(−1)(k−1)1+Es|ΔxA|24kN0 = 112NA∑k=1(NAk)(−1)(k−1)Es|ΔxA|24kN0⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1+1Es|ΔxA|24kN0⎞⎟ ⎟ ⎟ ⎟ ⎟⎠−1 = 112NA∑k=1(NAk)(−1)(k−1)Es|ΔxA|24kN0∞∑n=1⎛⎜ ⎜ ⎜ ⎜ ⎜⎝−1Es|ΔxA|24kN0⎞⎟ ⎟ ⎟ ⎟ ⎟⎠n−1 = ∞∑n=1(EsN0)−nNA∑k=1(NAk)(|ΔxA|24k)−n(−1)k+n−212C−n = ∞∑n=NA(EsN0)−nC−n (∵C−n=0 ∀ 1≤n

where is the Landau symbol. Similarly, for , it can be shown that

 (16)

where

 C′−n=NA∑k=1(NAk)(|ΔxA|23k)−n(−1)k+n−24. (17)

From (12), (15) and (16), it is clear that the average symbol error probability arising from the case when and decays as for higher values of .

### Case II: When xA=x′A and xB≠x′B

In this case and analogous to the previous case, the average symbol error probability can be given by

 Pe{(^xA,^xB)=(xA,x′B)}≲1M2∑(xA,xB)∈S2∑xB≠x′B∈S NB∑k=1(NBk)(−1)(k−1)[(12ΨB,k)−1+(4ΞB,k)−1] (18)

where

 ΨB,k=1+Es|ΔxB|24kN0,ΞB,k=1+Es|ΔxB|23kN0. (19)

Analogous to the previous case, the average error probability due to the case when and decays as for higher values of .

### Case III: When xA≠x′A, xB≠x′B and M(xA,xB)≠M(x′A,x′B)

This case is possible only for , because for the case of binary modulation (e.g., BPSK), if and , both and will lie in the same cluster for fixed network coding, i.e., and hence a confusion among these pairs will not cause a symbol error event in the MA phase. For , in this case can be written using (7) as

 E(Υ1)=E{exp(−Es4N0|hAΔxA+hBΔxB|2)} =∫∞0∫∞0∫π−πexp(−Es|ΔxAhA|24N0)exp(−Es4N0|ΔxBhB|2) ×exp(−Escosθ|ΔxAΔxBhAhB|2N0)f(θ)f(|hA|)f(|hB|) dθ d|hA| d|hB| (20)

where is a random variable uniformly distributed over . Using the fact that is an even function of , integration w.r.t in (20) can be solved as [8, p. 376]

 E(Υ1) =∫∞0exp(−Es|ΔxAhA|24N0)∫∞0exp(−Es|ΔxBhB|24N0) ×[I0(Es|ΔxAΔxB|2N0|hAhB|)]f(|hA|)f(|hB|)d|hA|d|hB|

where is the modified Bessel function of the first kind. Substituting for in the above equation yields

 E(Υ1) =∫∞0exp(−Es|ΔxAhA|24N0)NB∑l=1(NBl)(−1)l−1[l∫∞02|hB|exp{−(l+Es|ΔxB|24N0) ×|hB|2}]I0(Es|ΔxAΔxB|2N0|hA||hB|)d|hB|]f(|hA|)d|hA|. (21)

Solving the inner integral using [9, p. 306] yields

 E(Υ1)=∫∞0exp(−Es|ΔxAhA|24N0)NB∑l=1(NBk)(−1)k−1(ΨB,l)−1exp⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(Es|ΔxA||ΔxB|2N0|hA|)24lΨB,l⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ ×f(|hA|)d|hA| =NA∑k=1NB∑l=1(NAk)(NBl)(−1)(k+l−2)(ΨB,l)−1k =NA∑k=1NB∑l=1(NAk)(NBl)(−1)(k+l−2)(ΨB,l)−111+η/k =NA∑k=1NB∑l=1(NAk)(NBl)(−1)(k+l−2)(ΨA,kΨB,l−14klΘA,B) (22)

where, . Solving in a similar fashion for , we obtain

 E(Υ2)= NA∑k=1NB∑l=1(NAk)(NB)l(−1)(k+l−2)(ΞA,kΞB,l−14klΦA,B) (23)

where, . Hence, the average SER arising from the case when , and becomes

 Pe{(^xA,^xB)=(x′A,x′B),M(xA,xB)≠M(x′A,x′B)} ≲1M2∑(xA,xB)∈S2∑xA≠x′A∈S,xB≠x′B∈SM(xA,xB)≠M(x′A,x′B)NA∑k=1NB∑l=1(NAk)(NBl)(−1)(k+l−2) ×⎧⎨⎩112(ΨA,kΨB,l−ΘA,B4kl)−1+14(ΞA,kΞB,l−ΦA,B4kl)−1⎫⎬⎭ =1M2∑(xA,xB)∈S2∑xA≠x′A∈S,xB≠x′B∈SM(xA,xB)≠M(x′A,x′B)(ξ1+ξ2) (24)

where

 ξ1=NA∑k=1NB∑l=1(NAk)(NBl)(−1)(k+l−2)12(ΨA,kΨB,l−ΘA,B4kl), (25)
 (26)

Substituting the values of , and into (25), becomes

 ξ1= NA∑k=1NB∑l=1(NAk)(NB)l×(−1)(k+l−2)12(1+Es|ΔxA|24kN0+Es|ΔxB|24lN0) =112NA∑k=1NB∑l=1(NAk)(NBl)(−1)(k+l−2)g(k,l)(Es/N0)(1+1g(k,l)(Es/N0))−1 (27)

where

 g(k,l)=|ΔxA|24k+|ΔxB|24l. (28)

Using the binomial expansion, can be rewritten as

 ξ1=112NA∑k=1NB∑l=1(NAk)(NBl)(−1)(k+l−2)g(k,l)(Es/N0)∞∑m=1(−1)m−1(1g(k,l)(Es/N0))m−1
 = ∞∑m=1(EsN0)−m112NA∑k=1NB∑l=1(NAk)(NBl)(−1)k+l+m−3(g(k,l))−mB−m = ∞∑m=1B−m(EsN0)−m

Similarly, for it can be shown that

 ξ2

where

 B′−m=14NA∑k=1NB∑l=1(NAk)(NBl)(−1)k+l+m−3(g′(k,l))−m, (31) g′(k,l)=|ΔxA|23k+|ΔxB|23l. (32)

From (24), (29) and (30), it is clear that the average symbol error probability due to the case when , and