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Network-Coded Diversity Protocol for Collision

Recovery in Slotted ALOHA Networks

G. Cocco∗, N. Alagha¶, C. Ibars∗and S. Cioni¶

∗Centre Tecnol`

ogic de Telecomunicacions de Catalunya – CTTC

Parc Mediterrani de la Tecnologia, Av. Carl Friedrich Gauss 7 08860,

Castelldefels – Spain

¶European Space Agency - ESTEC, Noordwijk – The Netherlands

giuseppe.cocco@cttc.es, nader.alagha@esa.int, christian.ibars@cttc.es,

stefano.cioni@esa.int

Abstract

We propose a collision recovery scheme for symbol-synchronous slotted ALOHA (SA) based on

physical layer network coding over extended Galois ﬁelds. Information is extracted from colliding

bursts allowing to achieve higher maximum throughput with respect to previously proposed collision

recovery schemes. An energy efﬁciency analysis is also performed and it is shown that, by adjusting the

transmission probability, high energy efﬁciency can be achieved. A performance evaluation is carried out

using the proposed algorithms, revealing remarkable performance in terms of normalized throughput.

Index Terms

Multiple access, MAC, physical-layer network coding, PNC, slotted ALOHA, collision resolution

I. INTRODU CTION

The throughput of Slotted ALOHA (SA) systems is limited by the collisions that take place

when more than one node accesses the channel in the same time slot. This limitation is par-

ticularly problematic in satellite networks with random access, where the long round-trip time

(RTT) greatly limits the use of feedback from the receiver. Techniques like Diversity Slotted

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ALOHA (DSA) [1], in which each packet is transmitted more than once, have been proposed in

order to increase the probability of successful detection. In [2] a novel scheme called network-

assisted diversity multiple access (NDMA), inspired by signal separation principles borrowed

from signal processing, has been presented. In the scheme the collisions are recovered through

successive retransmissions assuming feedback from the receiver. This interprets the received

signals across successive transmission slots as a matrix which is processed in the analog domain

so that the single bursts are recovered if the matrix is full rank. In Contention Resolution Diversity

Slotted ALOHA (CRDSA) [3] the transmissions are organized in frames. The collided signals

are exploited using a successive interference cancelation (SIC) process. In CRDSA each packet

is transmitted twice and uncollided packets are subtracted from slots in which their replicas

are present. An enhanced version of CRDSA, called CRDSA++, has been presented in [4]. In

CRDSA++ more than two copies of each burst (3to 5) can be transmitted and an iterative

interference cancelation is applied, if needed, within a slot. In [5] a packet-level forward error

correction (FEC) code has been applied to CRDSA, while in [6] a convergence analysis and

optimization of CRDSA has been proposed. Another technique that allows to extract information

from colliding signals is physical layer network coding (PNC). PNC was originally proposed to

increase spectral efﬁciency in two-way relay communication [7] by having the relay decoding

the collision of two signals under the hypothesis of symbol, frequency and phase synchronism. In

[8] a cooperative relaying protocol that leverages on PNC and SIC has been proposed, while in

[9] PNC has been applied in the satellite context for pairwise node communication. In [10] and

[11] it has been proposed to apply PNC to determine the identity of transmitting nodes in case

of ACK collision in multicast networks by using energy detection and ad-hoc coding schemes.

In [12] an overview of the state of the art on PNC has been presented from an information

theoretical point of view. In [13] and [14] PNC has been applied for collision resolution in

multiple access systems with feedback from the receiver, under the assumption of frequency

synchronous transmitters.

In this paper we present a new scheme named Network-Coded Diversity Protocol (NCDP), that

leverages on PNC over extended Galois ﬁelds for recovering collisions in symbol-synchronous

SA systems. Once the PNC is applied to decode the collided bursts, the receiver uses common

matrix manipulation techniques over ﬁnite ﬁelds to recover the original messages, which results

in a high-throughput scheme. The proposed scheme and analysis differ from previous works on

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collision resolutions at both system level and physical level:

•Unlike in [2] and [13] we assume that transmissions are organized in frames. We consider

two different setups, one in which feedback from the receiver is not allowed and another

in which feedback is allowed.

•We take into account the energy consumption in the design of our solution and evaluate

jointly the spectral and the energy efﬁciency of the proposed scheme, comparing it with

other collision resolution schemes previously proposed in the literature.

•We use extended Galois ﬁelds, i.e., GF (2n)with n > 2, instead of GF (2), which is

generally used in PNC. This allows to better exploit the diversity of the system, leading to

an increased spectral efﬁciency and, depending on the system load, to an increased energy

efﬁciency. Unlike in [2], in our scheme most of the processing is done using ﬁnite ﬁeld

arithmetics, which reduces the complexity of the system.

•We present results relative to implementation issues such as decoding in the presence of

frequency offsets, channel estimation and imperfect symbol synchronism for a generic num-

ber of colliding signals. Up to our knowledge such issues have been previously addressed

only for the case of two transmitters [15], [16], [17].

The rest of the paper is organized as follows. In Section II we present the system model. In

Section III the proposed scheme is described, while a theoretical analysis of its performance is

carried out in Section IV. Section V deals with practical issues such as decoding in presence of

frequency offsets, channel estimation and imperfect symbol synchronization. In Section VI we

present the numerical results, while Section VII contains the conclusions.

II. SYSTEM MODE L

Let us consider the return link (i.e, the link from a user terminal to a satellite or a base

station) of a multiple access system with Mtransmitting terminals, T1, ....., TM, and one receiver

R. Packet arrivals at each transmitter are modeled as a Poisson process with rate G

M, which is

independent from one transmitter to the other. Each packet ui= [ui(1), ...., ui(K)] consists of

Kbinary symbols of information ui(ξ)∈ {0,1}, for ξ= 1,...,K. We assume that, upon

receiving a message, each terminal Tiuses the same linear channel code of ﬁxed rate rcc =K

N

to protect its message ui, obtaining the codeword xi= [xi(1), ..., xi(N)], where xi(l)∈ {0,1}

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for l= 1,...,N. Each codeword xiis BPSK modulated (using the mapping 0→ −1,1→+1),

thus obtaining the transmitted signal

si(t) =

N

X

l=1

bi(l)g(t−lTs),(1)

where Tsis the symbol period, bi(l)is the BPSK mapping of xi(l)and g(t)is the square root

raised cosine (SRRC) pulse. The signal si(t)is called burst. Higher order modulations such as

QPSK may also be used, but some preliminary results we obtained showed a consistent increase

of the FER with respect to BPSK in case random phase and frequency offsets are present. In

particular we observed that with QPSK modulation some relative phase rotations introduce an

ambiguity in the received signal that impedes correct decoding even in the absence of noise.

Further investigation is needed to study this issue and the potential countermeasures, such as

the adoption of different quaternary modulations. In the present paper we focus on BPSK which

is more robust to random phase and frequency offsets. As we will show in Section VI, even

with a real-valued modulation our method outperforms other collision resolution schemes that

use complex-valued modulations for certain code rates and SNR values.

In the following we will refer to a time division multiple access (TDMA) scheme. However,

the techniques proposed hereafter can be also applied to other access schemes, such as multi-

frequency-TDMA (MF-TDMA), in which a frame may include several carriers, or code division

multiple access (CDMA), where NCDP can be used to recover collisions in each of the code

sub-channels. The proposed technique still relies on single carrier transmission by each user

terminal. From the user terminal perspective no signiﬁcant change is required.

Transmissions are organized in frames. Each frame is divided into Stime slots. The number

Sof time slots that compose a frame is ﬁxed, i.e., it does not change from one frame to the

other. The duration of each slot is equal to about Nburst symbols. When more than one terminal

transmits its burst in the same slot a collision occurs at the receiver. A collision involving k

transmitters is said to have size k. We assume symbol-synchronous transmissions, i.e., in case

of a collision, the signals from the transmitters add up with symbol synchronism at the receiver

R. The received signal before matched ﬁltering and sampling at R, in case of a collision of size

k(assuming, without loss of generality, the ﬁrst kterminals collide), is:

y(t) = h1(t)s1(t) + ... +hk(t)sk(t) + w(t),(2)

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where si(t)is the burst transmitted by user i,w(t)is a complex additive white Gaussian noise

(AWGN) process while hi(t)takes into account the channel from terminal ito the receiver. hi(t)

can be expressed as:

hi(t) = Aiej(2π∆νit+ϕi),(3)

where (Ai)2=|hi(t)|2is a lognormally distributed random variable modeling the channel power

of transmitter i, while ∆νiand ϕiare the frequency and phase offsets with respect to the

local oscillator in R, respectively. The log-normality of the satellite channel power has been

assumed in several previous works such as [3], [4] and [18] although, as indicated in [4], this

is a pessimistic assumption. As we will refer to the schemes presented in these papers, we

keep the assumption of log-normal power distribution in the following. In general a distribution

characterized by large ﬂuctuations of the channel power are likely to affect the proposed scheme

because it is conditioned by the channel with less power. We will come back on this issue in

the following sections. We assume that the amplitude Aiand the frequency offset ∆νiremain

constant within one frame while ϕiis a random variable uniformly distributed in [−π, +π]that

changes independently from one slot to the other due to the phase noise at the transmitting

terminals as assumed in [3]. The assumption of constant phase within a burst is more accurate

for shorter burst lengths and assuming high class transmitting terminals with stable oscillators

or a high symbol rate (typically above 2Mbaud). We assume this model for ease of exposition.

Further studies are needed to characterize the sensitivity of PNC to phase noise for a generic

number of colliding signals and especially for a large burst size, but this is out of the scope of

the present paper. Assuming that the frequency offset is small compared to the symbol rate 1/Ts

(i.e., ∆νTs≪1), the sample taken at time tlafter matched ﬁltering signal y(t)is:

r(tl) = h1(tl)q1(tl) + ... +hk(tl)qk(tl) + n(tl),(4)

where q(t) = s(t)⊗g(−t),⊗being the convolution operator, while n(tl)’s are i.i.d. zero mean

complex Gaussian random variables with variance N0in each component. Note that even in case

a BPSK modulation is used, as we are assuming in this paper, both the I and Q components of

the received signal are considered by the receiver. This is because the phases of the users have

random relative offsets and thus both components carry information relative to the useful signal.

The frequency and phase relative offsets must be taken into account by the decoder, as they can

not be eliminated by the demodulator. We consider this more in detail in Section III.

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We assume that the receiver has knowledge of the nodes that are transmitting, as well as the full

channel state information at each time slot. As we are considering a random access scheme, the

knowledge about nodes identity cannot be available a priori at the receiver. Thus, nodes identity

must be determined by Rstarting from the received signal, even in case of collision. This can

be achieved by having the transmitting nodes add a pseudonoise preamble in each transmitted

burst. For a length-Npre preamble there are 2Npr e different sequences. Using maximal length

pseudonoise signature sequences the crosscorrelation between any two different sequences is

−1/Npre, which translates in a 10 dB gain for preambles that are just 10 symbols long. We

discuss the issue of node identiﬁcation and channel estimation more in detail in Section V.

III. NETWOR K CODED DIVERSITY PROTO COL

In this section we present our network-coded diversity protocol (NCDP) which aims at in-

creasing the throughput and reducing packet losses in Slotted ALOHA multiple access systems.

In the ﬁrst part of the section we describe the way the received signal is processed by the receiver

in case of collision, while in the rest of the section we describe the NCDP at the transmitter

and at the receiver sides.

A. Multi-User Physical Layer Network Coding

When a collision of size koccurs, i.e., kbursts collide in the same slot, the receiver tries

to decode the bit-wise XOR of the ktransmitted messages. This can be done by feeding the

decoder with the appropriate log-likelihood ratios (LLR). The calculation of the LLRs for a

collision of generic size kin case of BPSK modulation was presented in [13]. In the following

we include the effect of frequency offset in the calculation of the LLR’s, which was not taken

into account in [13].

When signals from ktransmitters collide, the received signal at Ris given by (2). Each

codeword xiis calculated from uias xi=C(ui), where C(.)is the channel encoder operator.

All nodes use the same linear code C(.). Starting from r(t), the receiver Rwants to decode

codeword xs,x1⊕x2⊕...⊕xk, where ⊕denotes the bit-wise XOR. In order to do this the

decoder of Ris fed with vector L⊕= [L⊕(1), ..., L⊕(N)] of LLRs for xs, where:

L⊕(l) = ln

P⌊k+1

2⌋

i=1 P(k

2i−1)

m=1 e−|r(tl)−do(2i−1,m)Th(tl)|2

2N0

P⌊k+1

2⌋

i=1 P(k

2i)

m=1 e−|r(tl)−de(2i,m)Th(tl)|2

2N0

,(5)

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h(tl)being a column vector containing the channel coefﬁcients of the ktransmitters at time

tl(which change at each sample due to frequency offsets), while do(2i−1, m)and de(2i, m)

are column vectors containing one (the m-th) of the k

2i−1or k

2ipossible permutations over k

symbols (without repetitions) of an odd or even number of symbols with value “+1”, respectively.

Equation (5) is derived considering that an even or an odd number of symbols with value +1

adding up at Rmust be interpreted by the decoder as a 0 or a 1, respectively (see [19] and [20]

for an extension to higher order modulations). If the decoding process is successful, Robtains

the message us,u1⊕...⊕uk. In Section V the frame error rate (FER) curves for different

collision sizes obtained using these LLR values are shown.

B. NCDP: Transmitter Side

We call active terminals the nodes that have packets to transmit in a given frame. Each

message is transmitted more than once within a frame, i.e., several replicas of the same message

are transmitted. We will give details about the number of replicas transmitted within a frame in

the next section. Assume that node ihas a message uito deliver to Rduring a given frame, i.e.,

node Tiis an active terminal. Before each transmission, node ipre-encodes uias depicted in

Fig. 1. The pre-coding process works as follows. uiis divided into L=K

nblocks of nbits each.

n bits n bits

Channel

Coding

Modulation

n bits

Fig. 1. NCDP pre-encoding, channel coding and modulation scheme at the transmitter side.

At each slot a different coefﬁcient αij ,j∈ {1,...,S}, is drawn randomly according to a uniform

distribution in GF (2n). If αij = 0, terminal Tidoes not transmit in slot j. Each of the Lblocks

ur

i,r∈ {1,...,L}, is interpreted as an element in GF (2n)and multiplied by αij . We call u′

ij the

message uiafter the multiplication by αij .u′

ij is then channel encoded, generating the codeword

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xij =C(u′

ij ). After channel coding, a header piis added to xij . Such header is generated using

a pseudonoise sequence generator such as the ones used in CDMA. The increase in complexity

at the transmitter side with respect to the case in which all nodes use the same preamble is not

large if maximal length pseudonoise sequences are used, as each node just needs to choose a

random seed and feed it to a shift register which is the same for all nodes and known at the

receiver. On the other hand, at the receiver side the complexity associated to the detection phase

increases in a quasi-linear fashion with the number of correlators used. However, such increase

in complexity may not be an issue if the receiver is located at the gateway station, as it is likely

to have good computational capabilities. The same header piis used for all transmissions of node

Tiwithin frame f. Once the header is attached, xij is BPSK modulated and transmitted. Note

that the multiplication of uiby αij is needed to introduce randomness in the MAC mechanism

and does not modify the number of information bits transmitted.

The choice of the coefﬁcients and of the header is done as follows. Node Tidraws a random

number µ.µis fed to a pseudo-random number generator in GF (2n), which is the same for all

terminals and is known at R. The ﬁrst Soutputs of the generator are used as coefﬁcients. The

header is uniquely determined by µ, i.e, there is a one-to-one correspondence between the set

of values that can be assumed by µand the set of available pseudo-noise sequences. The cross-

correlation properties of the preambles allow the receiver to know which of the active terminals

in frame fis transmitting in each time slot. Moreover, as the header univocally determines µ

and thus the set of coefﬁcients used by each node, Ris able to know which coefﬁcient is used

by each transmitter in a given slot. As we will see in Section III-C, this is of fundamental

importance for the decoding process. As said before, the set of headers is a set of pseudo-noise

sequences, such as those usually adopted in CDMA. The fundamental difference with respect

to a CDMA system is that in the latter the (quasi-)orthogonality of the codes is used to (quasi-

)orthogonalize the channels and expand the spectrum, while in NCDP the low cross-correlation

of the preambles is used only for determining the identity of the transmitting node, which is

obtained without any spectral expansion, as the symbol rate 1/Tsis equal to the chip rate (i.e.,

the rate at which the modulated symbols are transmitted over the channel).

The discrimination capability of pseudo-noise preambles may suffer a consistent degradation

in the presence of strong Doppler shifts which are typical of Low Earth Orbit (LEO) and Medium

Earth Orbit (MEO) satellite systems. For this reason such node identiﬁcation method is mostly

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suited for geostationary satellite networks as well as terrestrial multiple access systems.

C. NCDP: Receiver Side

The decoding scheme at the receiver side is illustrated with an example in Fig. 2(a) and Fig.

2(b). In the example, a frame with S= 4 slots and Ntx = 3 active terminals is considered.

In the ﬁgures only bursts with non-zero coefﬁcients are shown. In each slot the receiver uses

the pseudo-noise preamble of each burst to determine which node is transmitting and which

coefﬁcient has been used for that burst. As described in Section III-B, the coefﬁcients used by a

node in each burst are univocally determined by the preamble. The preamble can be determined

at the receiver Rusing a correlator which calculates the correlation of the received signal with

the maximal length sequence for each possible shift. In order to increase the number of available

sequences of coefﬁcients and to avoid the problems due to the eventual unsuccessful decoding

of some of the slots, the system can be designed so that for each preamble a different coefﬁcient

is associated to each slot. In this way the sequence of coefﬁcients associated to the preamble

changes depending on the slots where the burst that uses that preamble is transmitted. The total

number of different sequences associated to a given preamble is, thus, equal to the number of

possible dispositions of the drepetitions over the Sslots of the frame, that is, S

d. Note that

the use of a preamble is not a peculiarity of NCDP, as usually practical systems make use of

a preamble to perform channel estimation. The preamble is also used by Rto estimate the

channel for each of the transmitters. More details about the channel estimation are given in [21]

and are recalled in Section V. Once the channel has been estimated, the receiver applies PNC

decoding to calculate the bitwise XOR of the transmitted messages, as detailed in Section III-A.

According to arithmetics in Galois ﬁelds and to what is stated in Section III-B, the bitwise

XOR is interpreted as a sum in GF (2n). Thus the slots that have been correctly decoded are

interpreted as a system of linear equations in GF (2n)with coefﬁcients αij, which are known to

the receiver through the headers (see Fig. 2(a)). In order to simplify the notation, in the ﬁgure

we indicated the vector u′

ij =αij u1

i,...,αijuL

i, representing the network coded packet, as

αij ui. At this point, if the coefﬁcient matrix Ahas full rank, Rcan recover all the original

messages using common matrix manipulation techniques in GF (2n)(see Fig. 2(b)). If Ais

not full rank, not all the transmitted packets can be recovered. However, a part of them can

still be retrieved using Gaussian elimination. The decoding process in case of rank deﬁcient

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Decoder

Linear equation in

Received

frame

(a) Step 1: decoding at slot level (PNC). (b) Step 2: decoding at frame level (network coding).

coefﬁcient matrix is analyzed in Section IV. Note that, while in [2] the coefﬁcient matrix A

(called mixing matrix) is a complex matrix whose elements are the terminals’ channel gains,

in NCDP Ais a matrix in a Galois Field. In NCDP each slot is processed only once in the

complex domain (PNC decoding), while all matrix manipulations are done in GF (2n). In [2],

instead, the matrix Ais processed entirely in the complex domain. Operating in GF (2n)has

an important advantage in terms of complexity, as all the processing can be done in the digital

domain and avoids numerical problems that may derive from using a complex matrix, especially

in case of small channel gains. If, on the one hand, using a complex coefﬁcient matrix leads

to a high probability of having full rank (which, however, also depends on the precision of the

quantization in the sampling process), on the other hand in NCDP a relatively small ﬁeld size

(e.g., GF (28)) already achieves almost the same performance in terms of throughput as in the

case of a complex matrix, as we show in Section IV.

IV. THROUGHPUT AND ENE RGY EFFICIENCY ANALYSIS

A. Throughput

During each frame the terminals buffer the packets to be transmitted in the following frame.

Each terminal transmits its packet more than once within a frame, randomly choosing a new

coefﬁcient in GF (2n)independently at each transmission. As described in the previous section,

the coefﬁcients can be generated using a pseudo-random number generator fed with a seed

which is univocally determined by the chosen pseudo-noise preamble. Using the preamble the

receiver can build up a coefﬁcient matrix A∈[GF (2n)]S×Ntx for each frame, with Aj,i =αij,

αij ∈ {1,...,2n−1}. The rows of Arepresent the time slots while the columns represent the

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active terminals, i.e., the terminals that transmit in the present frame. If αij = 0, terminal Ti

does not transmit in slot j. During slot j,Rreceives the sum of the bursts with αij 6= 0. From

the received signal, Rtries to obtain the bit-wise XOR of the encoded messages as described

in Section II. The XOR is interpreted by Ras a linear equation in GF (2n), the coefﬁcients of

which are derived through the pseudo-noise preamble as described in Section III. If Ntx is the

number of active terminals in a frame and assuming that all the received signals are decoded

correctly, a linear system of equations in GF (2n)is obtained with Sequations and Ntx variables.

Each variable corresponds to a different source message. If Ahas rank equal to Ntx , then all the

messages can be obtained by R. A necessary condition for Ato be full rank is Ntx ≤S, i.e.,

the number of active terminals in a frame must be lower than the number of slots in a frame.

Assuming Poisson arrivals with aggregate intensity G, the probability of such event is:

P r{Ntx ≤S}=

S

X

n=0

(GS)ne−GS

n!,(6)

that includes also the case in which there are no active terminals during a frame. For instance, in

case of S= 100 slots and G= 0.8the probability expressed by Eqn. (6) is on the order of 0.99.

Even if Ntx < S, however, it can still happen that Ais not full rank, i.e., not all the messages

can be recovered. The probability that Ais full rank for a given Ntx < S depends on the

MAC policy, and particularly on the probability distribution used to choose the coefﬁcients. One

possibility is to use a uniform distribution for the coefﬁcients (i.e., each coefﬁcient can assume

any value in {0,...,2n−1}with probability 2−n). In this case the number dof transmitted

replicas is a random variable, and the probability that Ais full rank is [22]:

P(S, Ntx) =

Ntx−1

Y

k=0 1−1

2n(S−k).(7)

Using (6) and (7) we ﬁnd the expression for the normalized throughput:

Φ = 1

S

S

X

m=1

m(GS)me−GS

m!P(S, m) = 1

S

S

X

m=1

(GS)me−GS

(m−1)!

m−1

Y

k=0 1−1

2n(S−k)

=1

S

S−1

X

m=0

(GS)m+1 e−GS

(m)!

m

Y

k=0 1−1

2n(S−k)=G

S−1

X

m=0

(GS)me−GS

m!

m

Y

k=0 1−1

2n(S−k).

(8)

From Eqn. (8) we can see that Φgrows with n, which means that the system throughput increases

with the size of the considered ﬁnite ﬁeld. The throughput achievable in case of an asymptotically

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large ﬁeld size nis:

lim

n→∞ Φ = lim

n→∞ "G

S−1

X

m=0

(GS)me−GS

m!

m

Y

k=0 1−1

2n(S−k)#=G

S−1

X

m=0

(GS)me−GS

m!.(9)

Thus, the normalized throughput Φtends to the probability of having less than Stransmitters in

a frame as n→ ∞. Note that this is the same performance that would be achieved by a scheme

that uses coefﬁcient matrix in the complex domain, as in [2]. Further in this section we show

that almost the same performance can be achieved by NCDP using a ﬁnite and relatively small

ﬁeld size.

B. Energy

The MAC scheme we just analyzed presents one main drawback in terms of energy efﬁciency.

As a matter of fact, given the frame length S, a node transmits each message on average E[d] =

S×ptimes, p= (1−2−n)being the probability to choose a non-zero coefﬁcient, i.e., the average

number of transmissions grows linearly with S. In order to decrease the energy consumption,

the probability of choosing the zero coefﬁcient may be increased. However, a reduction in

the transmission probability pmay affect the system throughput. In order to understand the

relationship between the probability pand the throughput Φ, we refer to some results in random

matrix theory. The problem can be formulated as follows: consider an S×Ntx ,S≥Ntx, random

matrix Aover GF (2n)with i.i.d. entries, each of which assumes value 0with probability 1−p

while with probability pit assumes values in {1,...,2n−1}. We are interested in the relationship

between pand the probability that Ais full rank. In [23] the authors show that, in order to achieve

a rank Ntx −O(1) with high probability, then, for Ntx large, pcannot be lower than the threshold

probability log(Ntx )

Ntx . At high loads (i.e., G≃1), on average Ntx ≃S, which means that, setting

p=log(S)

S, the average number of transmissions (and so the energy consumption) for each node

is E[d] = log(S), i.e., it grows logarithmically with the number of slots in a frame. On the

other side, Smust be kept large enough, as this increases the decoding probability (see Eqn.

(9)). With reference to the example considered earlier in this section the average number of

transmissions corresponding to the minimum required pfor S= 100 is equal to about 4.6. We

evaluated numerically the effect a reduction of phas on Φfor the case S= 100 and q= 28.

We considered three cases. In the ﬁrst one the transmission probability in each slot has been

set to p= 1 −2−n= 0.9961, which corresponds to the case studied in the ﬁrst part of this

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

G

Φ

p=0.9961 analytic

analytic asymptotic

p=0.9961 Monte Carlo

p=0.0625

p=log(S)

S= 0.0461

Fig. 2. Normalized throughput plotted against the normalized offered load for different values of the transmission probability

p. We set S= 100 slots per frame while the coefﬁcients were chosen in GF (28). The asymptotic analytical curve (Eqn. (9))

is also plotted.

section and for which the throughput is given by Eqn. (8). In the second case we set pjust

above the threshold, i.e., p= 0.0625 >log(S)

S= 0.0461, while in the last case phas been set

exactly equal to the threshold probability. Fig. 2 shows the results together with the numerical

validation of Eqn. (8). It is interesting to note how passing from p= 0.9961 to p= 0.0628,

with a reduction in transmission probability (or, equivalently, in average energy per message) of

about 93.7%, leaves the throughput unchanged, while a further decrease of pof just another 1.5%

leads to a 10% reduction in the maximum throughput with respect to the case p= 0.9961. The

asymptotic analytical curve described by Eqn. (9) is also plotted in Fig. 2. Such curve represents

the throughput of a system where coefﬁcients are chosen in a ﬁnite ﬁeld with asymptotically

large ﬁeld size. It also represents the throughput of a system derived from the NDMA scheme

proposed in [2], i.e., the coefﬁcient matrix is complex and is processed in the complex domain.

It can be seen that the performance in terms of throughput is almost the same for NDMA and for

NCDP with p= 0.0625, i.e., NCDP does not lose signiﬁcantly with respect to NDMA, while

saves in complexity by performing all the processing in a ﬁnite ﬁeld instead of the complex

March 20, 2018 DRAFT

14

domain.

To further lower the energy consumption and control the number of repetitions d(which,

being a Bernoulli random variable, can theoretically assume values as large as S), an alternative

is to ﬁx the number of transmitted replicas a priori. Although this solution may decrease the

probability of decoding all the transmitted messages (because the resulting Amatrix would be

a subset of all possible matrix of the same size), it may still be possible to recover part of them

by using Gaussian elimination.

V. IMPLEMENTATIO N ASPECT S

For each frame the receiver Rneeds to know which of the active terminals are transmitting

in each slot and must have CSI for each of the users. Both needs are addressed including a

pseudonoise preamble, such as the spreading codes used in CDMA, at the beginning of each

transmitter’s burst. In [3] frequency offset and channel amplitude are derived from the clean

bursts (i.e., bursts that did not experience collision) and assumed to remain constant over the

whole frame. The need for an orthogonal preamble has been removed in CRDSA++ [4]. Unlike

in [3], the method we propose can not rely only on clean bursts. Thus the frequency offset and

the amplitude of each transmitter must be estimated using the collided bursts.

Channel estimation can be performed using the Estimate Maximize (EM) algorithm as we

have shown in [21]. We have adopted the approach described in [24], where the EM algorithm

is used to estimate parameters from superimposed signals. We have applied the same approach

to estimate amplitudes, phases and frequency offsets from the baseband samples of the received

signal in case of a collision of size k. The details on how the EM algorithm is applied can be

found in [21].

In Fig. 3 the FER curves for different collision sizes obtained using the LLR values calculated

in Section III-A are shown. The FER curve for the case of estimated channels using the EM

algorithm are also shown. These results have the purpose of showing the feasibility of channel

estimation from the collided messages. We are currently working on the enhancement of channel

estimation in order to further improve the performance of PNC in terms of FER. In the simulation

all channel amplitudes were assumed to be equal. We performed some preliminary simulations

with unbalanced channels and observed a certain degradation in terms of FER. More speciﬁcally,

the performance of PNC decoding in the case of a collision with different channel gains is

March 20, 2018 DRAFT

15

0 5 10 15 20 25

10−3

10−2

10−1

100

Eb/No in dB

FER

k=5

k=5 (estimated channels)

k=1

Fig. 3. FER curves for the XOR of transmitted messages for a collision of size 5.Ebis the energy per information bit for each

node. A tail-biting duo-binary turbo code with rate rcc = 1/2and codeword length 256 symbols is used by each node. Phase

offsets are uniformly distributed in (−π, +π), frequency offsets are uniformly distributed in (0,∆νmax )with ∆νmax equal

to 1% of the symbol rate on the channel. Amplitudes are constant and equal to 1. The FER curves for the case of estimated

channels using the EM algorithm are also shown [21].

slightly better than that of the weakest channel. This may require the adoption of appropriate

countermeasure depending on the deployment setup. Further analysis on the effect of channel

unbalance is out of the scope of the present paper and will be tackled as future work.

In order to detect eventual decoding errors, a cyclic redundancy check (CRC) can be used.

The CRC operations are done in GF (2) and, by the linearity of the channel encoder, the CRC

ﬁeld in the message obtained by decoding a collision of size kis a good CRC for us, which

is the bitwise XOR of the messages encoded in the kcollided signals. This allows to detect

decoding errors, within the limits of the CRC capabilities, also in collided bursts.

Another important issue is the imperfect symbol synchronism. In [21] we have proposed

several techniques, based on oversampling, aiming at reducing the impairments brought in by

the lack of perfect synchronism. The different methods we proposed in [21] are all based on

oversampling nd show a loss of about 1dB at F E R = 10−2with respect to the case with ideal

synchronism. We do not report here the results for a matter of space.

The lack of timing and phase synchronization determines a certain degradation in the sequence-

detection performance of the code. However, such degradation can be usually tolerated in

practical systems such as CDMA ones. We refer to [25] for further details on this issue.

March 20, 2018 DRAFT

16

VI. NUM ERICA L RESULTS

Our performance metrics are the throughput Φ, deﬁned as the average number of bits per

second per Hertz (bit/s/Hz) that can be correctly decoded, and the packet loss rate (PLR) Υ, i.e.,

the ratio between the number of lost packets and the total number of packets that arrive at the

transmitters. The relationship between these metrics is given by Φ = rdecG(1 −Υ),rdec being

the rate in bits per second per Hz (bit/s/Hz) used by the transmitter and Gbeing the average

rate at which the new messages (i.e., messages that are being transmitted for the ﬁrst time) are

injected in the network. Note that Gis independent from the number of times a message is

repeated within a slot. Also note that rdec is the number of bits per second per Hertz. Thus, for

instance, if a QPSK modulation is adopted, rdec corresponds to twice the rate of the channel

encoder indicated with rcc in Section II.

We consider two benchmarks. The ﬁrst one is the CRDSA++ scheme with three repetitions.

In CRDSA++ a node transmits three copies of a burst (twin bursts) in different slots randomly

chosen within a frame. Each of the twin bursts contains information about the position of the

other twin bursts in the frame. If one of the twin bursts does not experience a collision (i.e., it

is clean) and can be correctly decoded, the positions of the other twin bursts are known. These

bursts may or may not experience a collision with other bursts. If a collision occurs, these are

removed through interference cancelation using the knowledge of the decoded bursts. In order

to do this Rmemorizes the whole frame, decodes the clean bursts, reconstructs the modulated

signals including the effect of each user’s channel, and subtracts them from the slots in which

their replicas are located. The SIC process is iterated for a number Niter of times, at each time

decoding the bursts that appear to be “clean” after the previous SIC iteration. If at the end of

the SIC process not all the bursts have been decoded, the receiver tries to decode each of the

collided bursts considering the interfering bursts as noise. If a burst is successfully decoded all

its replicas are subtracted from the frame and the SIC process starts back.

The second benchmark we consider is a slotted ALOHA system. In a SA system each burst

is transmitted only once. Reception is successful if only one burst is transmitted within a given

slot, while, in case two or more bursts collide, they are discarded. The capture effect is not

considered in the SA scheme.

In both NCDP and CRDSA++ the performance at the physical layer plays an important role

March 20, 2018 DRAFT

17

in both system throughput and packet loss rate. In order to take this into account in the most

general way we adopt an information theoretical approach assuming Gaussian codebooks for

CRDSA++ and Lattice codes for NCDP. We also assume channels are symmetric. We evaluate

the performance of CRDSA++ as in [26]. In case of complex-valued channel symbols and

collision of size k, a burst can be correctly decoded if [27]

rdec ≤log21 + SN R

1 + (k−1) ·SNR,(10)

where rdec is the rate in bits per second per Hertz while SNR is the signal to noise ratio of the

channel (i.e., Es/N0).

As for NCDP, we refer to a result in [28] (Theorem 4) according to which the bitwise XOR

of kcolliding signals using the same rate rdec and real-valued channel symbols can be correctly

decoded if

rdec ≤1

2log21

k+SN R.(11)

We consider two different simulation setups. In the ﬁrst one the nodes do not receive any

feedback from the receiver, while in the second setup Rgives some feedback to the active

terminals. For this last case we consider an automatic repeat request (ARQ) scheme, in which

a node receives an acknowledgement (ACK) or a negative acknowledgement (NACK) from the

receiver in case a message is or is not correctly received, respectively. The amount of feedback

is limited to one ACK/NACK message per node and per frame1. A node that receives a NACK

enters a backlog state. Backlogged nodes retransmit the message for which they received the

NACK in another frame, uniformly chosen at random among the next Bframes. We call Bthe

maximum backlog time. The process goes on until the message is acknowledged [29]. In this

setup we also consider the average energy consumption per received message ηas performance

metric, deﬁned as the average number of transmissions needed for a message to be correctly

received by R.

In both setups we assume a very large population of users and a frame size of S= 100.

1An alternative to the NACK is to have the transmitters using a counter for each transmitted packet, indicating the time

elapsed since it has been transmitted. If the timer exceeds a threshold value (which depends on the system’s RTT), the message

is declared to be lost.

March 20, 2018 DRAFT

18

0 0.5 1 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Φ(bits/s/Hz)

G

NCDP, rdec = 0.94

NCDPp=0.0 625,rdec = 0.94

CRDSA++, rde c = 0.94

Slotted ALOHA, rdec = 0.94

NCDP, rdec = 0.5714

NCDPp=0.0 625,rdec = 0.5714

CRDSA++, rde c = 0.5714

Slotted ALOHA, rdec = 0.57 14

Fig. 4. Normalized throughput Φvs normalized trafﬁc load G. The ﬁeld size for the coefﬁcients of NCDP is 28. No feedback

is assumed from the receiver.

In the ﬁrst setup, in which no feedback is provided by the receiver, the average amount of

energy spent by a node for each message which is correctly received does not change with

the system load G, and is equal to the average number of times a message is repeated within

a frame. In Fig. 4 the normalized throughput Φis plotted against the normalized trafﬁc load

G. In the ﬁgure the throughput curves of NCDP and CRDSA++ schemes for d= 3 replicas

and rates rdec = 0.94 and rdec = 0.5714 (4/7) are shown. The throughput curve for NCDP in

case of a constant retransmission probability p= 0.0625 is also shown (NCDPp= 0.0625).

Note that this probability is above the threshold value we mentioned in Section IV, as for

S= 100 we have log(S)/S = 0.0461. The scheme with p= 0.0625 outperforms all the others

in terms of throughput, achieving a peak value of more than 0.7bit/s/Hz for a rate rdec = 0.94.

The precoding coefﬁcients of NCDP (indicated as αij in Section III-B) are drawn uniformly in

GF (28). The normalized throughput for NCDP with d= 3 and GF (2) (not plotted in the ﬁgure)

has a peak value of about 0.4888, with a loss of about 15% with respect to the case in which

GF (28)is used. In the ﬁgure we see how, depending on the rate and for the same number of

repetitions, either NCDP or CRDSA++ achieve the highest throughput peak. This is due to the

fact that the packet loss rate of CRDSA++ increases when passing from rdec = 0.5714 bpcu

March 20, 2018 DRAFT

19

0 0.5 1 1.5

10−5

10−4

10−3

10−2

10−1

100

Υ(PLR)

G

NCDP, rdec = 0.94

NCDPp=0.0 625,rdec = 0.94

CRDSA++, rde c = 0.94

NCDP, rdec = 0.5714

NCDPp=0.0 625,rdec = 0.5714

CRDSA++, rde c = 0.5714

Slotted ALOHA

Fig. 5. Packet loss rate Υvs normalized trafﬁc load G. The ﬁeld size for the coefﬁcients of NCDP is 28. No feedback is

assumed from the receiver. The PLR curve for SA is the same for both the considered code rates.

to rdec = 0.94 bit/s/Hz, as conﬁrmed by Fig. 5, where the packet loss rates of the considered

schemes are shown. From the ﬁgure we also see that the PLR of NCDP is the same for both

the considered rates and the difference in the peak throughput is only due to the difference in

the rate at the physical level. For both the considered rates NCDP achieves a PLR as low as

10−3for a network load of 0.655, while, for the same PLR, CRDSA++ achieves a throughput

of 1.14 and 0.46 for rdec = 0.5714 and rdec = 0.94, respectively. Interestingly NCDPp= 0.0625,

although achieving the highest peak throughput, never gets to a PLR lower than 10−3, which is

due partly because of the probability that a node chooses not to transmit in a given frame (that

happens with probability (1 −p)S).

It is interesting to note how, in NCDP, increasing the number of transmissions per message

(and so the energy consumption) leads to an increase in the peak throughput of the system.

In the second setup, in which feedback is allowed, we evaluate jointly the throughput Φ

and the energy consumption ηof the schemes under study. In Fig. 6, Φis plotted against G

for a maximum backlog time B= 50 frames. The precoding coefﬁcients of NCDP are drawn

uniformly in GF (28). The ﬁgure shows how Φincreases linearly with Gup to a threshold load

value. Such threshold increases with the (average) number of repetitions and corresponds to the

March 20, 2018 DRAFT

20

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

G

Φ(bits/s/Hz)

NCDP, rdec = 0.94

NCDPp=0.0 625,rdec = 0.94

CRDSA++, rde c = 0.94

Slotted ALOHA, rdec = 0.94

NCDP, rdec = 0.5714

NCDPp=0.0 625,rdec = 0.5714

CRDSA++, rde c = 0.5714

Slotted ALOHA, rdec = 0.57 14

Fig. 6. Normalized throughput Φvs normalized trafﬁc load Gin a system with retransmission.

maximum network load for which the throughput in the setup without feedback (Fig. 4) has an

almost linear behavior, i.e., the PLR is low. This indicates that, if the load is such that a non

negligible fraction of the messages are not decoded at the ﬁrst attempt, the retransmissions

saturate the channel, blocking both the iterative cancelation process of CRDSA++ and the

Gaussian elimination decoding in NCDP. Note that this does not happen in the SA system,

coherently to what shown in [29] for the case of large backlog time. In order to compare

jointly the spectral and the energy efﬁciency of the different schemes, we plot the curves for

the normalized throughput vs the average energy consumption per received message η, which

is shown in Fig. 7. The increase in the number of repetitions corresponds to an increase in

throughput but also to a higher energy consumption for a given transmitter in a given frame.

However, as shown in Fig. 7, this does not necessarily imply a loss in energy efﬁciency. From

the ﬁgure it can be seen that that there is not a scheme that outperforms the others in terms of

both energy and spectral efﬁciency, but which scheme is best depends on the maximum target

throughput. SA achieves a higher throughput with a lower energy consumption with respect

to the other schemes in the region Φ<0.35, while in the region Φ>0.35 both NCDP and

CRDSA++ achieve a higher throughput with lower energy consumption with respect to SA.

March 20, 2018 DRAFT

21

100101102103104105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

η

Φ(bits/s/Hz)

NCDP, rdec = 0.94

NCDPp=0.0625,rdec = 0.94

CRDSA++, rdec = 0.94

Slotted ALOHA, rdec = 0.94

NCDP, rdec = 0.5714

NCDPp=0.0625,rdec = 0.5714

CRDSA++, rdec = 0.5714

Slotted ALOHA, rdec = 0.5714

Fig. 7. Normalized throughput vs average energy consumption per decoded message.

NCDP achieves a maximum Φof 0.653 for rdec = 0.94, slightly higher than CRDSA++, for

which the peak value is 0.628, for rdec = 0.5714. In the NCDP scheme with a retransmission

probability of p= 0.0625 a peak throughput of 0.655 bit/s/Hz is achieved in correspondence of

an average energy consumption of η= 6.25. The maximum average throughput that is achieved

in correspondence to a packet loss rate of 10−3is similar in the two schemes, with CRDSA++

achieving a slightly higher throughput (0.64) than NCDP (0.61).

The simulations show that, for the same number of repetitions, which scheme between NCDP

and CRDSA++ performs better (in terms of both throughput and packet loss rate) depends on

the rate rdec. It is not straightforward to ﬁnd which scheme performs better than the other for

any given (rdec, SNR)pair. However, in the following we derive a subset of the region in the

(rdec, SNR)plane where NCDP outperforms CRDSA++. We start by deriving an upper bound

on the minimum SNR (SNRmin ) required by NCDP in order to decode correctly (at the physical

level, i.e., applying physical layer network coding) a collision of any size. From Eqn. (11) follows

that for a collision of size k, the decoding at physical level of PNC with real-valued modulation

is successful if:

SN R ≥4rdec −1

k.(12)

March 20, 2018 DRAFT

22

Thus, an upper bound on the minimum SNR is:

SN Rub

min = 4rdec ,(13)

which does not depend on the collision size. Let us now consider CRDSA++. If the iterative

decoding stops, in CRDSA++ the receiver tries to decode each burst considering the others as

noise in each slot. In case of symmetric channels and complex-valued modulation the maximum

rate rdec in bit/s/Hz such that the decoding is still possible must satisfy

rdec ≤log21 + SN R

1 + SN R ,(14)

that corresponds to the case in which there is only one interferer. The SNR below which the

decoding (i.e., the collision resolution at physical level within a slot) stops is, thus:

SN R ≥2rdec −1

2−2rdec .(15)

From Eqn. (13) and Eqn. (15) we ﬁnd that NCDP can perform fully while CRDSA++ is limited

to decoding and canceling clean bursts (as in CRDSA) if

4rdec ≤SN R ≤2rdec −1

2−2rdec .(16)

Now it is sufﬁcient to note that, if PNC decoding is successful in each slot, NCDP can perform

in the digital domain the equivalent of the SIC of CRDSA in case the coefﬁcient matrix A

is not full rank. This implies that, when Eqn. (16) holds, NCDP performs at least as well as

CRDSA++. We plot the region deﬁned by Eqn. (13) and Eqn. (15) in Fig. (8). In the region at

the right of the picture where the required SNR of CRDSA++ is higher than that of NCDP, the

latter scheme outperforms the former. This conﬁrms the results of the simulations shown before,

in which, for the same number of repetitions, when (rdec = 0.5714, SNR = 10dB)CRDSA++

outperforms NCDP, while if (rdec = 0.94, SNR = 10dB)NCDP performs better. Interestingly

NCDP outperforms CRDSA++ at higher rates. We must also note that NCDP is limited to BPSK

modulations, while CRDSA++ can be applied also with complex modulations and in principle

may achieve higher rates in bit/s/Hz. However, from Fig. 8 we see that in order to work well at

rates approaching 1bit per channel symbol, CRDSA++ needs an asymptotically large SNR.

Further studies are needed to address a fair comparison in case of asymmetric channels.

An important issue in the collision recovery mechanisms considered in this paper is their

complexity. NCDP needs a more strict synchronization with respect to CRDSA++ (symbol level

March 20, 2018 DRAFT

23

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10

−5

0

5

10

15

20

25

30

rdec (bits/s/Hz)

SNR (dB)

SN Rmin NCDP (real-valued modulation)

SN Rmin CRDSA++ (complex-valued modulation)

Fig. 8. Curves in the (rdec , SN R)plane describing the upper bound on the minimum SNR of NCDP and the minimum SNR

for CRDSA++ required to solve collisions at the physical layer. In the region on the right of the picture where the required

SNR of CRDSA is higher than that of NCDP, the latter scheme outperforms the former.

versus slot level). However, NCDP may be an argument in favor of symbol synchronous multiple

access systems which are currently debated in the satellite communications arena. As for the

decoding complexity, although NCDP has a higher complexity in the physical layer decoder

with respect to a typical point to point system (that has been studied in our previous work [21]),

our scheme has the advantage that, after the PNC decoding, all the operations at the receiver

are done in a ﬁnite ﬁeld, which is particularly suited for implementation in the digital domain.

CRDSA++, instead, applies a SIC, which requires to store and process the whole frame in the

analog domain, requiring by far more memory than in case of a digital frame processing. A fair

complexity comparison between the two schemes is not straightforward and can not be addressed

exhaustively in the present paper for a matter of space.

VII. CO NCLUS IONS

We have proposed a new collision recovery scheme for symbol-synchronous slotted ALOHA

systems based on PNC over extended Galois ﬁelds. The adoption of an EGF allows to better

exploit the diversity of the system, leading to an increased spectral efﬁciency and, depending

March 20, 2018 DRAFT

24

on the system load, to an increased energy efﬁciency. We have compared the proposed scheme

with two benchmarks in two different setups, one without feedback and the other with feedback

from the receiver. In the second setup we have evaluated jointly the spectral efﬁciency and the

energy consumption of the proposed scheme. Once the PNC is applied to decode the collided

bursts, the receiver applies common matrix manipulation techniques over ﬁnite ﬁelds, which

results in a high-throughput scheme. We showed that NCDP achieves a higher or comparable

spectral efﬁciency with respect to the considered benchmarks, while there is not a single scheme

that outperforms the others in terms of both energy and spectral efﬁciency, but the best scheme

depends on the maximum achievable throughput. For completeness, we also reported our previous

results related to several physical layer issues related to multi-user PNC, namely decoding in the

presence of offsets and channel estimation. As a ﬁnal remark, we underline that there is room for

signiﬁcant improvement in the performance of the PNC decoder by using lattice codes, according

to recent result in information theory [30].

ACKNOWLEDGEME NT S

This work was partially supported by the European Commission under project ICT-FP7-

258512 (EXALTED), by the Spanish Government under project TEC2010-17816 (JUNTOS)

and project TEC2010-21100 (SOFOCLES), and by Generalitat de Catalunya under grant 2009-

SGR-940. Giuseppe Cocco was partially supported by the European Space Agency under the

Networking/Partnering Initiative.

Part of the content of this paper has been presented in [31].

This is the pre-peer reviewed version of the following article: G. Cocco, N. Alagha, C. Ibars,

S. Cioni, Network-Coded Diversity Protocol for Collision Recovery in Slotted ALOHA Networks,

in Wiley’s International Journal of Satellite Communications and Networking, which has been

published in ﬁnal form at https://onlinelibrary.wiley.com/doi/abs/10.1002/sat.1056.

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