Analysis and performance of network decoding strategies for cooperative network coding

Physical Communication 6 (2013) 48–61

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Analysis and performance of network decoding strategies for cooperative network coding✩ Jawad Manssour a , Tafzeel ur Rehman Ahsin b , Slimane Ben Slimane b , Afif Osseiran a,∗ a b

Ericsson AB, Stockholm, Sweden Royal Institute of Technology (KTH), Stockholm, Sweden

article

info

Article history: Received 2 February 2011 Received in revised form 2 March 2012 Accepted 3 May 2012 Available online 11 May 2012 Keywords: Network coding Network decoding Multiple-access relay channel Detection algorithms Relaying User pairing

abstract In this work, we present two new low-complexity network decoding strategies for cooperative network coding in a multiple-access relay channel scenario. For these two strategies, Selection and Soft Combining and Majority Vote Network Decoding, along with the optimal joint network decoding, we derive expressions for bit error probability performance as a function of the signal-to-noise ratio (SNR) of the different Rayleigh fading links, and show the tightness of the derived bounds through simulation results. The two proposed schemes provide a similar bit error probability (BEP) performance compared to the optimal scheme, while having significantly lower complexity. Further, we study the effect of user pairing on the error performance by considering different SNRs on the user and relay links towards the destination. It is also shown that the error performance of the different schemes follows the same trend for a given user pairing strategy. © 2012 Elsevier B.V. All rights reserved.

1. Introduction One of the main limitations of wireless networks is the unreliability of the wireless channel. This limitation is usually overcome by introducing diversity into the system. One way to introduce diversity in the received signal is the use of relaying [1–3]. In a relaying system, the information sent to an intended destination is conveyed through various routes and combined at the destination, where each route can consist of one or more hops utilizing the relay nodes (RNs). In particular, a multiple-access relay channel (MARC) [4] refers to a specific topology in which a common RN assists several transmitters in relaying their data to a common destination. For such a scenario, a total of four transmissions are needed in order to relay the information of the two users to the destination: in the first transmission, the first user transmits its signal; in the second transmission, the relay forwards the first user’s signal to the destination; in the third transmission, the second user transmits its signal; and in the fourth transmission, the relay forwards the second user’s signal to the destination. Consequently, each of the transmitted signals can achieve a diversity order up to two, as it will be received through two (independently fading) routes at the destination. Another intriguing research area that has gained a lot of interest is network coding (NC). This area was introduced in [5] as a method that allows manipulation of the data at intermediate nodes inside the network in order to improve throughput, delay, and/or robustness. Although initially targeting lossless wired networks, NC gained a lot of interest within the wireless communications research community [6–10]. In the last few years, implementations of the MARC based on NC, also known as cooperative NC, have been investigated in the literature. The most intriguing characteristic of such implementations is

✩ A part of the current paper is based on the conference paper [20]. Herein substantial additional technical details as well new results have been added.

∗

Corresponding author. Tel.: +46 107132670. E-mail addresses: [email protected] (J. Manssour), [email protected] (T.u.R. Ahsin), [email protected] (S. Ben Slimane), [email protected] (A. Osseiran). 1874-4907/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.phycom.2012.05.006

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that, by virtue of applying NC at the RN on the data of the two users, the total number of transmissions needed by the relaying node to forward the signals of the two users to the destination can be decreased from two to one, thus requiring a total of three transmissions as opposed to four in the conventional MARC to complete the data transmission:

• in the first transmission, the first user transmits its signal, • in the second transmission, the second user transmits its signal, • and in the third transmission, the relaying node transmits the network coded signal which contains the information of both users. Among other gains, such as the reduction in the transmit power at the RN, this decrease in the number of transmissions can lead to an asymptotic capacity gain of 4/3. A considerable amount of research work has targeted the NC-based MARC; some of these works are [11–15], just to mention a few. A common denominator among work in that field is to show that, although using one less transmission at the RN, the diversity performance of the NC-based MARC is not lower than that of the conventional MARC, thus achieving the promised capacity gains without jeopardizing the diversity of such a system. For instance, [13] investigates the diversity gain of the NC-based MARC by studying the outage probability. This is done by looking at the probability of correct reception of the different signals at the destination. Another investigation of outage probability was presented in [15] where, unlike in other works, the authors consider the case of non-ideal user–relay links. The relay would then transmit a network-coded combination only when it correctly receives the signals of both users; otherwise, it would only send the correctly received signal. Similar works related to code design for systems in which several nodes relay each other’s information in a network coding fashion can be found in [16,17]. In short, most of the previous works focus either on the study of outage probability or on the design of the network coding operation at the transmitter. One of the topics that has not received its fair share of attention in the literature is how the network decoding of the users’ signals in the NC-based MARC is performed at the destination. Such an area is interesting, since the network decoding strategy plays a crucial role in both the performance and the complexity. That is because the destination would have three signals with non-identical information content (i.e. the two signals of the users and the network-coded signal transmitted by the relay) that are used to estimate the two signals transmitted by the users. A common (implicit) assumption is that joint network decoding of the three signals is used at the destination as mentioned in [14], which is known to have very high complexity. In this work, we present two low-complexity network decoding strategies for the NC-based MARC, namely Selection and Soft Combining and Majority Vote Network Decoding. For these two strategies, and for the well-known joint network decoding, we derive expressions for bit error probability performance as a function of the signal-to-noise ratio (SNR) of the different Rayleigh fading links, as well as their computation complexity. These expressions are compared to simulation results to confirm their tightness. We also examine how different user pairing strategies affect the performance of the different network decoding schemes. The rest of this paper is organized as follows. The studied system model is presented in Section 2. Section 3 presents the different network decoding schemes and further contains the BEP derivation. Section 4 includes the simulation results and discussion. Finally, conclusions are given in Section 5. 2. System model The system model studied in this work is that of the NC-based MARC topology. The system consists of two users (U1 and U2 ) that want to transmit data to a common destination (D) with the help of a common RN. Such a system, in addition to the data transmitted in different transmission times, is shown in Fig. 1. Ideal links between the two users and the RN are assumed, meaning that transmissions from the users to the common destination are always correctly received by the RN. We further assume an uncoded system in which both users are using the same modulation scheme. Linear modulation is considered with a modulation mapping denoted by M (·). The transmission structure of the system studied consists of transmission frames, where each frame is further divided into three equal transmission slots T1 , T2 , and T3 . During each slot, one of the transmitting nodes (i.e. the two users and the relay) transmits one symbol. We define the transmitted symbol of link i during a certain symbol interval as mi , where mi is the symbol in the bit domain and si = M (i) is the resultant modulated symbol in the complex domain. Hence, during the first slot T1 , the first user U1 transmits its modulated symbol s1 = M (m1 ). In the second slot T2 , the second user U2 transmits its modulated symbol s2 = M (m2 ). The relay detects the two user symbols, encodes them using network coding, and then ˜ i the estimated symbol at the RN, the transmitted modulated symbol transmits the encoded symbol during T3 . Denoting by m during the third time slot is s3 = M (m1 ⊕ m2 ). Network coding allows the RN to forward information about the two user symbols in one time slot instead of two time slots. This provides a saving of one transmission as compared to the conventional MARC system. In the following, we will denote xˆ and x˜ the estimated value and decided estimated value of a transmitted symbol x, respectively. Assuming a flat fading Rayleigh channel on each transmitted symbol, the received signals at the destination for a given symbol interval from the three links (direct and relay) are given by r i = hi s i + z i ;

i = 1, 2, 3,

(1)

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J. Manssour et al. / Physical Communication 6 (2013) 48–61

Fig. 1. Topology and transmission phases of the NC-based MARC.

Fig. 2. Generic system structure: user nodes (U1 and U2 ), relay node, and destination.

where ri is the received signal sample, si is the normalized transmitted complex modulated symbol with E {|si |2 } = 1, and zi is the complex Gaussian noise at the receiver with double-sided power spectral density N0 . The coefficient hi represents the channel gain between node i and the destination including the transmitted signal amplitude, i.e. hi has an average power of Ei , where Ei is the average energy per transmitted symbol of node i. Using the received samples {r1 , r2 , r3 }, the destination tries to estimate the transmitted user symbols {m1 , m2 } assuming it has full knowledge of the channel state. The generic system structure of the user nodes, RN, and destination are shown in Fig. 2, where the different data entities explained above are illustrated. At the RN, the network coding operation is performed as a bit-wise XOR operation on the symbols of U1 and U2 . The dashed lines refer to a wireless transmission of the data; the effects of channel fading and noise were excluded from the figure for clarity. The three network decoding schemes studied in this work utilize the same structure for the user and RNs, but differ in the network decoder structure at the destination. 2.0.1. Definitions Let us first define the following terms which will be useful when explaining both of the different network decoding schemes.

• Pre-detection SNR is the SNR of the signals r1 , r2 ,and r3 at the destination prior to any network-decoding or combining operations.

• Post-detection SNR is the SNR of the estimated symbols mˆ1 and mˆ2 at the destination after performing network-decoding • • • • • •

and combining operations. Strong user is the user with the higher pre-detection SNR (i.e. between the two users U1 and U2 ). Weak user is the user with the lower pre-detection SNR (i.e. between the two users U1 and U2 ). Strong link is the link with the highest pre-detection SNR (i.e. among the three nodes U1 , U2 , and RN). Weak link is the link with the weakest pre-detection SNR (i.e. among the three nodes U1 , U2 , and RN). Middle link is the link that is neither the strong link nor the weak link (i.e. among the three nodes U1 , U2 and RN). Direct transmission is the transmission from a user (i.e. U1 or U2 ) that is received by the destination D. In other words, direct transmission refers to the signals r1 and r2 received at D after transmission from U1 or U2 .

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• Network decoding (ND) refers to the process of using (an estimate of) the relay signal and (an estimate of) the signal of one of the users to obtain (an estimate of) the signal of the second user. Network decoding is performed as a bitwise XOR on the estimated symbols mi which represent the symbols in the bit domain. 3. Network decoding strategies and receiver structures This section analyzes the different network decoding schemes studied for estimating the information of the cooperating users at the destination. The destination receives three different samples carrying two information symbols originating from the two users. The question is how to use the redundancy provided by the RN without considerably increasing the receiver complexity. 3.1. Joint network decoding Joint network decoding (JD) is the optimal and commonly utilized approach for performing network decoding in the NC-based MARC. The destination has estimates of three signals with different information content (r1 , r2 , and r3 ), and the detector will simply try all possible output alternatives for the different signals before choosing the alternative with the smallest error metric to estimate the transmitted symbols m1 and m2 . However, JD is also the most complex network decoding scheme, due to its exhaustive search nature. In the case of JD, the receiver uses the three received samples jointly in order to decide on the symbols transmitted by the two users. Specifically, for every three time slots, the receiver computes the metric

ˆ 1, m ˆ 2) = C (m

3    rk − hk sˆk 2

(2)

k =1

for all possible symbols, chooses the set of symbols that have the minimum metric, and declares them as the two transmitted symbols by the two users. Assuming uncorrelated links, and for a given fading channel observation h = {h1 , h2 , h3 }, the conditional pairwise error probability can be written as

   3 2     Γi si − sˆi    i=1  , Pc (s → sˆ|h) = Q    2  

(3)

where sˆi is the estimate of the symbol si and

Γi =

|hi |2 N0

,

i = 1, 2, 3.

(4)

For a normalized Rayleigh fading channel, the probability density function (PDF) of the instantaneous received signal-tonoise ratio of link i, Γi , is given by fi (γ ) =

dFi (γ )

=

dγ

1

γi

e−γ /γi ,

γ ≥ 0.

(5)

where Fi (γ ) is the cumulative density function (CDF) and Ei

γi = E {Γi } =

(6)

N0

is the average received signal-to-noise ratio (SNR) of link i. Averaging (3) over the fading PDFs, the pairwise error probability can be derived, and, after some analytical manipulations, it is given by 1− P2 (s → sˆ) = 2



γ2 δ22 γ1 δ12



γ1 γ1 +1/δ12



−1

γ3 δ32 γ1 δ12

1−

+ −1

2



γ1 δ12 γ2 δ22



−1

γ2 γ2 +1/δ22



γ3 δ32 γ2 δ22

1−

+ −1

 2

γ1 δ12 γ3 δ32



γ3 γ3 +1/δ32

 −1

γ2 δ22 γ3 δ32

,

(7)

−1

where

δ = 2 i

  si − sˆi 2 4

.

(8)

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J. Manssour et al. / Physical Communication 6 (2013) 48–61

Fig. 3. Selection and Soft Combining receiver structure assuming that U1 is the strong user.

Using the expression in (7), an upper bound on the average bit error probability for U1 can be written as (1)

Pb

≤

1



M 2 log2 (M )

ad (s1 , sˆ1 )P2 (s → sˆ),

(9)

sˆ1 ̸=s1 s2 ,ˆs2

where M is the modulation level and ad (s1 , sˆ1 ) is the number of bits in error when choosing sˆ1 instead of s1 . Similarly, the average bit error probability of U2 can be written as in (9) by simply replacing 1 by 2 and vice versa. Example To further investigate the performance of JD let us, for simplicity, consider the case of coherent Binary phase-shift keying (BPSK) modulation.1 Based on the expressions in (7) and (9), an upper bound on the bit error probability for user k can be written as (k)

Pb

≤

      γk γ3 γk 1 − 1+γ − γ 3 1− 1+γ3 k 2 (γk − γ3 )

+

      γ1 γ2 γ1 1 − 1+γ − γ 2 1− 1+γ2 1 2 (γ1 − γ2 )

.

(10)

It is observed from (9) and (10) that the performance of each user is affected by the three non-ideal links (i.e. the links from the two users towards the destination, and from the relay towards the destination). At high SNR, the upper bound can be rewritten as (k)

Pb

≤

1 2 (1 + γk ) (1 + γ3 )

+

1 2 (1 + γ1 ) (1 + γ2 )

,

(11)

showing that a diversity gain of order 2 can be achieved. However, it is also noticeable that, if one of the links fails (i.e. γk → 0, for k = 1, 2, or 3) the order of this diversity reduces to 1 for both users. By examining (2), one can see that the complexity of JD grows exponentially with the modulation order of the transmitted information. Such a high complexity could be a deterrent factor for any practical implementation of the NC-based MARC in real-world applications. For the simplicity of illustration, and unless explicitly stated otherwise, we assume in the remainder of this text that U1 is the strong user and U2 is the weak user. 3.2. Selection and Soft Combining Selection and Soft Combining (SSC) is the first of the two low-complexity network decoding schemes we propose, and its corresponding receiver structure is shown in Fig. 3, where for simplicity it is assumed that U1 is the strong user. The core idea of this scheme is to detect one of the users based on its direct transmission, while the other user’s data is obtained based on a combination of that user’s direct transmission with the network-decoded relay transmission. The first step in the Selection and Soft Combining scheme consists of determining the strong user. This strong user is then detected solely based on its direct transmission. Assuming without loss of generality that U1 is the strong user, the detector estimates the symbol transmitted by U1 by finding the symbol that minimizes the following metric: C1 (mˆ1 ) = |r1 − h1 sˆ1 |2 ,

1 Similar expressions can be derived for higher-level modulations.

(12)

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where s1 represents the set of all possible symbols that could have been transmitted. In the second step, the receiver exploits the knowledge of the strong symbol sˆ1 to network decode the relay’s signal, thus obtaining a copy of U2 ’s signal and consequently combining it with the signal directly received from U2 , as shown in (13): C2 (mˆ2 ) = |r2 − h2 sˆ2 |2 + |r3 − h3 sˆ3 |2 ,

(13)

˜ 1 ⊕m ˆ 2 ), and M(m ˜ 1 ) is the decided estimated symbol of the strong user during the first stage. Unlike the case where sˆ3 = M (m of JD, the complexity does not increase exponentially, as the different symbols are detected separately, while still providing diversity for both users: the strong user inherently benefits from selection diversity by being the strong user, whereas the weak user benefits from transmit diversity by combining its direct transmission with the network-decoded RN transmission. We proceed now with the derivation of the BEP of the SSC scheme. Denoting by Pstrong the error probability of the strong user and by Pweak the error probability of the weak user, the pairwise error probability of user i can be written as follows: (i)

P2

s → sˆ = αi Pstrong + (1 − αi )Pweak ,





(14)

where α1 = Pr (Γ1 > Γ2 ) and α2 = 1 − α1 = Pr (Γ2 ≥ Γ1 ). The average bit error probability of user i can be obtained from its pairwise error probability as 1

(i)

Pb ≤



M log2 (M )

si

(i)

ad (si , sˆi )P2

si → sˆi .





(15)

sˆi ̸=si

The error probability of the strong user is based on selection diversity. Denoting by ss the symbol of the strong user and by sw the symbol of the weak user during a certain symbol interval, the error probability of the strong user can be written as follows:



 2  Γmax ss − sˆs  , Pstrong {E |h1 , h2 } = Q 

(16)

2

where Γmax = max {Γ1 , Γ2 }. For Rayleigh fading channels with uncorrelated channel coefficients, the error event probability of the strong user is obtained by averaging the expression in (16) over the PDFs of the two fading amplitudes, and it can be written as Pstrong = where δs2 =

1



 1−

2

|ss −ˆss |2 4

γ1 δs2 − 1 + γ1 δs2

and γi =



γ2 δs2 + 1 + γ2 δs2



γ1 γ2 δs4 γ1 δs2 + γ2 δs2 + γ1 γ2 δs4

 ,

(17)

Ei . N0

Denoting by s˜s the estimated symbol of the strong user during the first stage, the conditional error event probability of the weak user can be written as

   2 2  2         Γmin sw − sˆw + Γ3 s3 − sˆ3 − s3 − s˜3  , Pweak {E |h1 , h2 } = Q      2   2      2 Γmin sw − sˆw + Γ3 s˜3 − sˆ3 

(18)

˜ s ⊕ mw ), and sˆ3 = M(m ˜s⊕m ˆ w ). We notice from the expression where Γmin = min {Γ1 , Γ2 } , s3 = M (ms ⊕ mw ), s˜3 = M (m in (18) that the error in the first stage will increase the error probability of the weak user. In fact it is noticeable that error propagation can make the argument of the Q -function negative, thus resulting in an error probability of 50% for the weak user. Again, assuming uncorrelated Rayleigh fading coefficients and averaging the expression of (18) over the PDFs of the fading amplitudes, the error event probability of the weak user is written as        ≤ 1,  γ 3 ν3 γ e νw Pweak = γe νw 1 − 1+γe νw − γ3 ν3 1 − 1+γ3 ν3   , 2 (γe νw − γ3 ν3 ) |sw −ˆsw |2 |s −ˆs |2 γ2 where νw = , ν3 = 3 4 3 , γ3 = NE30 , and γe = γγ11+γ 4 2

Error propagation (19) No error propagation,

Using (17) and (19) in (14), the pairwise error probability of user i for SSC can be rewritten as follows: (i)

P2



si → sˆi ≤ αi Pstrong + (1 − αi ) ×



 

1 − Pstrong





       γ 3 ν3 γ e νw γe νw 1 − 1+γ + γ ν 1 − 3 3 1+γ3 ν3  e νw  ×  + Pstrong . 2 (+(γe νw − γ3 ν3 ))

(20)

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J. Manssour et al. / Physical Communication 6 (2013) 48–61

Fig. 4. Majority Vote Network Decoding receiver structure assuming that U1 is the strong user.

Example Considering the case of coherent BPSK modulation, an upper bound on the average bit error probability for user i is obtained by substituting (20) in (15), and is given by

       γ3 γe γe 1 − 1+γ − γ 3 1− 1+γ3  e    (1 − ψ)   + ψ , 2 (γe − γ3 ) 

γi γi ψ + 1− γ1 + γ2 γ1 + γ2 

(i)

Pb ≤





(21)

where 1

ψ=

2



 1−

γ1 − 1 + γ1



γ2 + 1 + γ2



γ1 γ2 γ1 + γ2 + γ1 γ2



.

(22)

3.3. Majority Vote Network Decoding Majority Vote Network Decoding (MVD) is the second low-complexity network decoding scheme we propose, and its corresponding receiver structure is shown in Fig. 4, where for simplicity it is assumed that U1 is the strong user. Similar to selection diversity, where only the best link or antenna is typically used, the core idea of this scheme is to use only the two strongest links at the receiver out of the three links; hence the nomenclature Majority Vote. Let us examine in more detail how the network decoding operation in this scheme works, and explain the proposed receiver structure presented above. Similar to the SSC scheme, the first stage in MVD is to determine the strong link. The strong link is detected solely based on its received sample. The detector estimates the transmitted symbol based on the minimization of the following metric:

ˆ s ) = |rs − hs sˆs |2 , C1 (m

(23)

where s = arg maxi {Γi }. It is clear that the obtained symbol sˆs from this first stage could be an estimate of s1 , s2 , or s3 , depending on the status of the different links. At the second stage, the receiver uses the middle link to get an estimate of the second symbol. Here, the receiver chooses the modulated symbol that minimizes the following metric:

2

ˆ m ) = rm − hm sˆm  , C2 (m 

(24)

where m = arg maxi̸=s {Γi } and sˆm is an estimate of one of the other two modulated symbols. Using the two obtained symbols {ˆss , sˆm } from the two stages, the user symbols {s1 , s2 } can be obtained either directly if link 3 (relay link) is the weakest link or via network decoding if link 3 is the strong or the middle link. It is clear that the information carried by the weakest link is always discarded, resulting in a lower complexity as compared to JD and SSC. Let us proceed now with the derivation of the error probability of the user links when this network decoding scheme is employed. Based on the above methodology, each link has one of three possible SNR states strong, middle, or weak, where these states are based on the links’ instantaneous SNRs. Denoting by Pstrong the error probability of the strong link, by Pmid the error probability of the middle link, and by Pweak the error probability of the weak link, the pairwise error probability of user i can be written as follows: (i)

P2

s → sˆ = αi Pstrong + βi Pmid + (1 − αi − βi ) Pweak ,





i = 1, 2,

(25)

J. Manssour et al. / Physical Communication 6 (2013) 48–61

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where αi is the probability that link i is the strong link and βi is the probability that link i is the middle link. For U1 , we have

α1 = Φ (k = 2, l = 1) = Pr {Γ1 > Γ2 > Γ3 } + Pr {Γ1 > Γ3 > Γ2 } β1 = Φ (k = 1, l = 2) = Pr {Γ2 > Γ1 > Γ3 } + Pr {Γ3 > Γ1 > Γ2 } ,

(26) (27)

where

γ1k γ2k (γ1 + γ3 ) + γ1k γ3l (γ1 + γ2 ) . (γ1 γ2 + γ2 γ3 + γ3 γ1 ) (γ1 + γ2 ) (γ1 + γ3 )

Φ (k, l) =

A simple replacement of subscripts from 1 to 2 and vice versa in (26) and (27) will give α2 and β2 for U2 . Let us denote by ss the symbol corresponding to the strong link, by sm the symbol corresponding to the middle link, and by sw the symbol corresponding to the weak link. Depending on the order statistics of the SNR of link i = 1, 2, 3, the error probability of a given link can be written as +∞

 Pstrong =

  2  γ ss − sˆs   fstrong(γ ) dγ , Q

(28)

2

0

and

  2  γ sm − sˆm   fmid(γ ) dγ . Q

+∞

 Pmid =

(29)

2

0

When the link is the weakest, an error will occur if and only if an error has occurred in the previous stages (strong or middle). Hence, the error probability of the weak link can be written as follows: Pweak = 1 − 1 − Pstrong (1 − Pmid ) −





 = Pstrong + Pmid − 1 +

1

1

(M − 1)2 

(M − 1)2

Pstrong × Pmid

Pstrong × Pmid ,

(30)

where M is the modulation level, and the third term in the first line represents the case when both the strong and the middle links are both wrong but a correct decision results after network decoding. With the expression of Pweak , the pairwise error probability of user i can be rewritten as (i)

P2



s → sˆ = αi Pstrong + βi Pmid + (1 − αi − βi )



 ×

 Pstrong + Pmid −

1+

1

(M − 1)2



 Pstrong × Pmid



= (1 − βi ) Pstrong + (1 − αi ) Pmid − (1 − αi − βi ) 1 +

1



(M − 1)2

Pstrong × Pmid .

(31)

To derive the two expressions of Pstrong and Pmid , the PDFs of the ith statistic (max or mid) of the received SNRs, fstrong (γ ) and fmid (γ ), are needed, and they are given by [18] fstrong (γ ) = f1 (γ )F2 (γ )F3 (γ ) + f2 (γ )F1 (γ )F3 (γ ) + f3 (γ )F1 (γ )F2 (γ ),

γ ≥0

fmid (γ ) = f1 (γ )[F2 (γ ) (1 − F3 (γ )) + F3 (γ ) (1 − F2 (γ ))] + f2 (γ )[F1 (γ ) (1 − F3 (γ )) + F3 (γ ) (1 − F1 (γ ))] + f3 (γ )[F1 (γ ) (1 − F2 (γ )) + F2 (γ ) (1 − F1 (γ ))], γ ≥ 0. Using the expressions (32) and (33) in (28) and (29), and simplifying, we get

     γk γ12 γ13 γ23 γ123 Pstrong = 1− + + + − 2 γk + 1/δs2 γ12 + 1/δs2 γ13 + 1/δs2 γ23 + 1/δs2 γ123 + 1/δs2 k=1       1 γ12 γ13 γ23 γ123 Pmid = 1− − − +2 , 2 2 2 2 γ12 + 1/δm γ13 + 1/δm γ23 + 1/δm γ123 + 1/δm2 1

where

γik =



γi γk γi + γk

3  

and γijk =

γi γj γk . γi γj + γi γk + γj γk

4. Results and discussion In this section, we evaluate the performance of the three schemes studied: JD, SSC, and MVD.

(32) (33)

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MVD

SSC 10–1

10–2

10–2

10–2

10–3

10–4

BEP

10–1

BEP

BEP

JD 10–1

10–3

10–4

0

10

20

10–3

10–4

0

10

20

0

10

20

Fig. 5. Tightness of the derived upper bounds for JD, SSC, and MVD when γ1 = γ2 = γ3 .

4.1. Tightness of the bounds We first start by comparing the results obtained from simulations with the derived analytical upper bounds, where these results are shown for the case where γ1 = γ2 = γ3 . As can be seen in Fig. 5, the derived upper bounds are in very good accordance with the simulation results for all three network decoding schemes. As a consequence, we use the derived upper bounds for the performance evaluation in the rest of this work. 4.2. Comparison of JD, SSC, and MVD In Fig. 6, we compare the performance of the three schemes by plotting the BEP as a function of SNR, assuming that both users have equal average SNR. Two different cases corresponding to an average relay SNR γ3 of 10 dB and 20 dB are shown in Fig. 6. The trend of the BEP curves is similar for all network decoding schemes at different values of γ3 , with JD slightly outperforming the two proposed schemes (i.e. SSC and MVD). At low γ3 , the performances of JD and SSC coincide, especially at higher SNR values, whereas JD has an improved performance at high γ3 . This behavior is due to the fact that JD exploits better the information provided from the relay signal as it is used jointly for both users, while SSC exploits the network coding operation through the relay’s signal to enhance the performance of the weak user only. Thus, when the relay’s signal has a lower SNR, JD and SSC would have a similar performance, which is slightly better than that of MVD. It is also noteworthy that, when both users have better SNR than the relay link, a slight degradation in performance of MVD is incurred. This is due to the fact that MVD does not consider the weakest link when performing detection or network decoding, which would cause losses when even the weakest link has a relatively high SNR. Similar performance results have been observed while using higher-order modulations (e.g., 8 PSK) meaning that the proposed schemes have a very similar performance to JD. In order to get further insight about the characteristics of the different network decoding schemes, we examine their performance for asymmetric users links. The results are shown in Fig. 7 for the case where γ2 = γ1 + 10 [dB] and γ3 = 20 [dB]. It can be seen that, at low SNR, the BEP of U2 is not significantly better than that of U1 , as U1 would benefit more from the network coding transmission. Examining the BEP of U2 , one can see that it has a similar SSC and MVD performance as long as U1 has an SNR that is lower than the relay node’s. That is because U1 would be seen as the weak user and would be the only transmission exploiting the RN transmission, meaning that U2 is detected based on its direct transmission to the destination. However, the choice of network decoding strategy would be more crucial for U1 in this case as SSC would result in soft combining gains of up to 2 dB when γ1 > 10 dB. These results indicate that the SNR of the users’ links towards the destination, coupled with the relay node’s link quality, have an influence on the individual user’s performance. This is studied in more detail later. 4.3. The effects of user pairing The NC-based MARC scenario makes users cooperate with each other through the network coding operation. This makes user pairing an important factor in determining the system’s performance. As it is not obvious how users should be paired for the NC-based MARC, and since very few works have so far touched upon this issue [19], although without thorough

J. Manssour et al. / Physical Communication 6 (2013) 48–61

10–1

57

10–1

10–2 10–2

BEP

BEP

10–3 10–3

10–4

10–4 10–5

10–5

0

10

20

30

10–6 0

10

20

30

Fig. 6. BEP performance of JD, SSC, and MVD for γ3 = 10 dB and γ3 = 20 dB.

γ3 = 20dB 10–1 10–2

BEP

10–3 user 1

10–4 10–5 user 2

10–6 10–7

0

5

10

15

20

25

30

Fig. 7. BEP performance of JD, SSC, and MVD for γ3 = 20 dB and γ2 = γ1 + 10 dB.

analysis, we investigate the effects of user pairing on the BEP performance of different users and on the network decoding operation. The user pairing is controlled through the parameter γsum , which is defined as

γsum = γ1 + γ2 , (34) where γ1 and γ2 are the SNRs of U1 and U2 in dB. For a specific γsum , one can explore the different pairing possibilities for both users by varying the values of γ1 and γ2 while still satisfying (34). For instance, for the case of γsum = 10 dB, and assuming a step size of 2.5 dB, the following user pairings are possible: (γ1 , γ2 ) = (10, 0); (7.5, 2.5); (5, 5); (2.5, 7.5); (0, 10). Further pairing possibilities are be studied by varying γsum and γ3 . In all subsequent (two-dimensional) graphs, the x-axis will contain values of γ1 = γ2 increasing from 0 dB to γsum dB (where γ2 = γsum − γ1 ), thus having γ2 decreasing from γsum dB to 0 dB. 4.3.1. Effects of user pairing on the system performance (1)

(2)

As a first step, we look at the effects of user pairing on the system performance, which is defined as P = (Pb + Pb )/2. The result is shown in Fig. 8, where γsum = 20 dB and γ3 varies between 0 dB and 40 dB. The assumed network decoding method is JD. One can see that, at low values of γ3 , the best pairing strategy is to pair users with equal SNR, as this results in the lowest average BEP. However, as γ3 increases, the average BEP starts to become more flat, indicating that the significance of user pairing becomes slightly smaller. Another interesting observation is that, at very high γ3 , pairing users with equal SNR results in the worst average BEP. In this region, pairing users with complementary channel conditions would result in the best error performance. These behaviors are explained next.

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J. Manssour et al. / Physical Communication 6 (2013) 48–61

10–1

P 10–2

10–3 20 10 0

30

40

0

10

20

Fig. 8. Average BEP performance using joint network decoding for γsum = γ1 + γ2 = 20 dB.

10–1.7

BEP

10–1.8

10–1.9

0

1

2

3

4

5

6

7

8

9

10

Fig. 9. BEP of the different detection schemes for γsum = 10 dB and γ3 = 20 dB.

4.3.2. Effects of user pairing on individual users and the network decoding operation quality In order to gain more insight about the different phenomena observed above, we define the following two generic cases based on the relation between γsum and γ3 , and study the effect of user pairing on the performance of individual users and on the quality of the network decoding operation for all three network decoding methods.

• γsum < γ3 , where we study the case when γsum = 10 dB and γ3 = 20 dB. • γsum > γ3 , where we study the case when γsum = 20 dB and γ3 = 10 dB. The case when γsum = γ3 has a similar performance to that when γsum > γ3 . The legends used in subsequent figures are JD(Ui ), SSC (Ui ), and MVD(Ui ), being the BEP of user i when using JD, SSC, and MVD, respectively, where i = 1, 2. We first start by examining the case when γsum < γ3 , as shown in Fig. 9. The most interesting observation from this figure is that, when the SNR of Ui increases, Pi also increases until it peaks when both users have equal SNR (note the symmetry in the figure due to the fact that we look at a constant γsum ). In other words, this means that pairing users with similar SNR would result in the worst performance whereas the best performance is achieved when these users have complementary SNR. Although γ3 is high, since γsum is low, the network decoding operation does not always result in a good estimate. In particular, when both users have the same SNR (i.e. γ1 = γ2 = 5 dB) this would result in the worst network decoding operation, as neither of the users has a particularly high SNR. Furthermore, none of the users will have a direct transmission with good enough quality. In contrast, when pairing users with complementary channel conditions (i.e. γ1 = 10 dB and γ2 = 0 dB or vice versa), the signal with the higher SNR will lead to a better quality of the network decoding operation. The above holds regardless of which network decoding scheme is utilized. In other words, as we increase γ1 , γ2 would decrease (to keep a constant γsum ). As long as γ1 < γ2 , γ2 would represent the link with the higher SNR and would thus be used to perform network decoding with the relayed signal. The decrease in γ2 would then lead to a worse network decoding operation, while the consequent increase in γ1 would contribute to a better direct transmission for the weak link. Due to the operating SNR region (i.e. γsum < γ3 ), the loss of information from a worse network decoding operation is larger than the gain of information from a better direct transmission. We then examine the case when γsum > γ3 , and specifically when γsum = 20 dB and γ3 = 10 dB. The BEP performance is shown in Fig. 10. For this case, the BEP curves are similar to the conventional BEP curves: as the SNR of a certain user increases, its BEP decreases, and vice versa. By virtue of having a high γsum , sharing the SNR equally among two users will still result in an acceptable network decoding operation and in a good direct transmission performance. As a result, the losses from a worse network decoding operation are accounted for by a better direct transmission.

J. Manssour et al. / Physical Communication 6 (2013) 48–61

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BEP

10–2

10–3 0

2

4

6

8

10

12

14

16

18

20

Fig. 10. BEP of the different detection schemes for γsum = 20 dB and γ3 = 10 dB.

To summarize the above, the user pairing strategy is independent of the network decoding scheme utilized, as a given pairing strategy will result in a similar trend with slightly differing values when different network decoding schemes are applied. On the other hand, the user pairing strategy is significantly influenced by the SNR of the different links. It has been shown that, for users in good conditions, pairing users with as similar channels as possible results in the best average performance while simultaneously providing good performance for both users. On the other hand, if users with poor channel conditions towards the destination exist (which is a typical case for the utilization of relays), then these users have to be paired with users having complementary (i.e. good) channel conditions towards the destination in order to ensure a good quality of the network decoding operation, hence increasing the likelihood of correct reception for both users. 4.4. Network decoding complexity analysis In this section, we derive the complexity measures of the two proposed network decoding schemes and compare them with the case of joint network decoding. Towards this end, we define one complexity unit as the computation of one metric, where this metric is the Euclidean distance. The complexity calculation is parameterized in terms of k = log2 (M ), where M is the modulation order and k is the number of bits per modulation symbol. For simplicity, we assume that all the links use the same modulation order. However, it is straightforward to derive complexity figures for different special cases with different modulation orders on different links. 4.4.1. Joint network decoding In this case, the receiver performs network decoding through jointly using the three different received signals (r1 , r2 , and r3 ) as done in (2). Although the three received signals represent different information, it should not be forgotten that the relayed signal is a function of the information transmitted by the two users. This means that the receiver needs to try all possibilities over the two users’ symbols only and as a result obtain an estimate of the corresponding transmitted relay signal by a simple bitwise XOR operation. For example, in the case of a BPSK modulation, the possible symbol pairs of {m1 , m2 } are {00; 01; 10; 11}, resulting in {m3 } = {0; 1; 1; 0}, as opposed to iterating over {m1 , m2 , m3 } = {000; 001; 010; 011; 100; 101; 110; 111}. This would lead to a lower complexity for JD as compared to the case of jointly detecting three completely independent signals. This means that, for the case of JD, as the receiver needs to try 2k different possible alternatives for each received signal (keeping in mind that this is needed only for the users’ signals), a total of 22k possible alternatives need to be evaluated, where each alternative results in three Euclidean distance computations (one for each received signal estimate). As a result, searching over all possible combinations would result in a complexity of 3 × 22k . 4.4.2. Selection and Soft Combining In this method, the receiver first detects the strong user, as done in (12). This operation has a complexity of 2k . The detected signal is then network decoded from the relayed signal to form an additional copy of the weak received signal. The complexity of this operation (a bitwise XOR) is assumed to be minimal, and is therefore not included in the complexity computation. The two copies are then combined through, for example, soft combining, and then a detection is performed on the combined estimate, as shown in (13). This operation would result in a complexity of 2 × 2k . Consequently, the total complexity of that scheme is given by 3 × 2k . 4.4.3. Majority vote network decoding In this method, only two links out of three are detected, as done in (24) and (23), and each link has a detection complexity of 2k . The total detection complexity would then be 2k+1 . 4.4.4. Numerical illustration A summary of the derived complexity measures is presented in Table 1. Furthermore, an illustration of the complexity measures using QPSK, 16QAM, and 64QAM modulations as examples is provided. By comparing the above results, one can

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J. Manssour et al. / Physical Communication 6 (2013) 48–61 Table 1 Complexity analysis. Method

Complexity

QPSK

16QAM

64QAM

JD SSC MVD

3 × 22k 3 × 2k 2k+1

48 12 8

768 48 32

12 288 192 128

see that SSC has a lower complexity by 2k as compared to JD, whereas MVD has a further decreased complexity by 50% as compared to SSC. 5. Conclusion In this work, two novel low-complexity network decoding strategies for cooperative network coding, namely Selection and Soft Combining and Majority Vote Network Decoding, were introduced. For these two strategies, and for the optimal joint network decoding, we derived expressions for the bit error probability (BEP) performance as a function of the signal-tonoise ratio (SNR) of the different Rayleigh fading links, and proved the tightness of the derived bounds through simulation results. The two proposed schemes provided a similar BEP performance compared to joint network decoding, while having significantly lower complexity. For instance, assuming a 16QAM modulation, Selection and Soft Combining and Majority Vote Network Decoding require respectively 16 and 20 times fewer operations compared to joint network decoding. A further increase in the modulation order would result in exponential gains in terms of simplicity for the proposed schemes. It was further shown that the user pairing strategy has a similar performance regardless of the network decoding scheme utilized; however, it is the prevailing SNRs of the different links that dictate which pairing strategy to utilize. For users in good conditions, pairing users with as similar channels as possible results in the best average performance, while simultaneously providing good performance for both users. On the other hand, if users with poor channel conditions towards the destination exist, these users have to be paired with users having complementary (i.e. good) channel conditions towards the destination in order to ensure a good quality of the network decoding operation, hence increasing the likelihood of correct reception for both users. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

T. Cover, A.E. Gamal, Capacity theorems for the relay channel, IEEE Trans. Inform. Theory (25) (1979) 572–584. R. Pabst, et al., Relay-based deployment concepts for wireless and mobile broadband radio, IEEE Commun. Mag. (2004). G. Kramer, M. Gastpar, P. Gupta, Cooperative strategies and capacity theorems for relay networks, IEEE Trans. Inform. Theory (2005). G. Kramer, A.J. van Wijngaarden, On the white Gaussian multiple-access relay channel, in: Proc. IEEE ISIT, Sorrento, Italy, 2000, p. 40. R. Ahlswede, N. Cai, S.-Y. Li, R. Yeung, Network information flow, IEEE Trans. Inform. Theory (2000). S.-Y. Li, R.W. Yeung, N. Cai, Linear network coding, IEEE Trans. Inform. Theory IT-49 (2) (2003) 371–381. Y. Wu, P.A. Chou, S.Y. Kung, Information exchange in wireless networks with network coding and physical-layer broadcast, in: 39th CISS, March 2005. S. Katti, D. Kattabi, W. Hu, H.S. Rahul, M. Médard, The importance of being opportunistic: practical network coding for wireless environments, in: SIGCOMM’06, Pisa, Italy, September 2006. P. Popovski, H. Yomo, Bi-directional amplification of throughput in a wireless multi-hop network, in: IEEE 63rd VTC, May 2006. P. Popovski, H. Yomo, The anti-packets can increase the achievable throughput of a wireless multi-hop network, in: Proc. IEEE International Conference on Communications, ICC2006, Istanbul, Turkey, June 2006. C. Hausl, F. Schreckenbach, I. Oikonomidis, Iterative network and channel decoding on a tanner graph, in: 43rd Annual Allerton Conference on Communication, Control, and Computing, Monticello, USA, 2005. C. Hausl, P. Dupraz, Joint network-channel coding for the multiple-access relay channel, in: SECON’06, vol. 3, pp. 817–822, September 2006. Y. Chen, S. Kishore, J.T. Li, Wireless diversity through network coding, in: IEEE WCNC Proceedings, 2006. S. Yang, R. Koetter, Network coding over a noisy relay: a belief propagation approach, in: IEEE ISIT, Nice, France, 2007, pp. 801–804. D. Woldegebreal, H. Karl, Multiple-access relay channel with network coding and non-ideal source–relay channels, in: 2007. 4th International Symposium on Wireless Communication Systems, October 2007, pp. 732–736. X. Bao, J. Li, Matching code-on-graph with network-on-graph: adaptive network coding for wireless relay networks, in: 43rd Allerton Conf. on Communication, Control and Computing, 2005. L. Xiao, T. Fuja, J. Kliewer, D. Costello, A network coding approach to cooperative diversity, IEEE Trans. Inform. Theory (2007). A. Papoulis, Probability, Random Variables and Stochastic Process, second ed., McGraw-Hill, 1984. J. Manssour, A. Osseiran, S.B. Slimane, Wireless network coding in multi-cell networks: analysis and performance, in: ICSPCS, 2008. T.R. Ahsin, S.B. Slimane, Detection strategies in cooperative relaying with network coding, in: Proc. of PIMRC, Istanbul, Turkey, September 2010.

Jawad Manssour received his B.E. degree in Computer and Communications Engineering from Notre Dame University, Lebanon, in 2006. He received his M.Sc. in Wireless Systems from the Royal Institute of Technology (KTH), Stockholm, Sweden, in 2008. Currently, he is working as a system engineer at Ericsson AB, Stockholm, in radio resource management techniques for LTE. He is simultaneously pursuing his Ph.D. degree at the Royal Institute of Technology (KTH), Stockholm. His current research is focused on low-complexity and novel techniques in network coding.

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Tafzeel ur Rehman Ahsin received his B.S degree in Electrical & Electronics Engineering from UET Peshawar, Pakistan, in 2002. He received his Licentiate degree in Telecommunication Engineering from The Royal Institute of Technology (KTH), Stockholm, Sweden, in 2010 and his Ph.D. degree in 2012 at KTH. His research interests include cooperative relaying using network coding in cellular systems.

Slimane Ben Slimane received his B.Sc. degree in electrical engineering from the University of Quebec at Trois-Rivieres, Canada, in 1985, and his M.Sc. and the Ph.D. degrees, both from Concordia University, Montreal, Canada, in 1988 and 1993, respectively. During the period 1993–1995, he worked as a research associate and part-time instructor at Concordia University. He is currently an associate professor in the Radio Communication Systems group, department of Communication Systems (COS), the Royal Institute of Technology (KTH), Stockholm, Sweden. His research interest is in the area of wireless communications, with special emphasis on digital communication techniques for fading channels, error control coding, cooperative communications, spread spectrum communications, and multi-carrier transmission techniques.

Afif Osseiran received his B.Sc. in Electrical and Electronics from Université de Rennes I, France, in 1995, his DEA (B.Sc.E.E) degree in Electrical Engineering from Université de Rennes I and INSA Rennes in 1997, and his M.A.Sc. degree in Electrical and Communication Engineering from École Polytechnique de Montreal, Canada, in 1999. In 2006, he successfully defended his Ph.D. Thesis at the Royal Institute of Technology (KTH), Stockholm, Sweden. Since 1999, he has been with Ericsson, Sweden. In 2004, he joined the European project WINNER as one of Ericsson’s representatives. During 2006 and 2007 he led the spatial temporal processing (i.e. MIMO) task in WINNER. From April 2008 to June 2010, he was the technical manager of the Eureka Celtic project WINNER+. His research interests include many aspects of wireless communications, with a special emphasis on advanced antenna systems, on relaying, on radio resource management, network coding, and cooperative communications. Dr Osseiran is listed in the Who’s Who in the World, and the Who’s Who in Science & Engineering. Dr Osseiran has published over 50 technical papers in international journals and conferences. He has co-authored two books on IMT-Advanced: ‘‘Radio Technologies and Concepts for IMT-Advanced’’ and ‘‘Mobile and Wireless Communications for IMT-Advanced and Beyond’’, published in 2009 and 2011, respectively, by John Wiley & Sons. He is a senior member of IEEE.